Scholars@Duke
Jianfeng Lu

Jianfeng Lu

James B. Duke Distinguished Professor of Mathematics
Mathematics
jianfeng@math.duke.edu
Mathematics Department, Duke University, Box 90320, Durham, NC 27708
242 Physics Bldg, 120 Science Drive, Durham, NC 27708
Office hours By email appointments  

Selected Publications

Algebraic Localization of Wannier Functions Implies Chern Triviality in Non-periodic Insulators

Journal Article Annales Henri Poincare · August 1, 2024 For gapped periodic systems (insulators), it has been established that the insulator is topologically trivial (i.e., its Chern number is equal to 0) if and only if its Fermi projector admits an orthogonal basis with finite second moment (i.e., all basis el ... Full text Cite

Score-based transport modeling for mean-field Fokker-Planck equations

Journal Article Journal of Computational Physics · April 15, 2024 We use the score-based transport modeling method to solve the mean-field Fokker-Planck equations, which we call MSBTM. We establish an upper bound on the time derivative of the Kullback-Leibler (KL) divergence to MSBTM numerical estimation from the exact s ... Full text Cite

Qubit Count Reduction by Orthogonally Constrained Orbital Optimization for Variational Quantum Excited-State Solvers.

Journal Article Journal of chemical theory and computation · April 2024 We propose a state-averaged orbital optimization scheme for improving the accuracy of excited states of the electronic structure Hamiltonian for use on near-term quantum computers. Instead of parameterizing the orbital rotation operator in the conventional ... Full text Cite

Locality of the windowed local density of states

Journal Article Numerische Mathematik · April 1, 2024 We consider a generalization of local density of states which is “windowed” with respect to position and energy, called the windowed local density of states (wLDOS). This definition generalizes the usual LDOS in the sense that the usual LDOS is recovered i ... Full text Cite

DYNATE: Localizing rare-variant association regions via multiple testing embedded in an aggregation tree.

Journal Article Genet Epidemiol · February 2024 Rare-variants (RVs) genetic association studies enable researchers to uncover the variation in phenotypic traits left unexplained by common variation. Traditional single-variant analysis lacks power; thus, researchers have developed various methods to aggr ... Full text Link to item Cite

A Machine Learning Framework for Geodesics Under Spherical Wasserstein–Fisher–Rao Metric and Its Application for Weighted Sample Generation

Journal Article Journal of Scientific Computing · January 1, 2024 Wasserstein–Fisher–Rao (WFR) distance is a family of metrics to gauge the discrepancy of two Radon measures, which takes into account both transportation and weight change. Spherical WFR distance is a projected version of WFR distance for probability measu ... Full text Cite

Deep Network Approximation: Beyond ReLU to Diverse Activation Functions

Journal Article JOURNAL OF MACHINE LEARNING RESEARCH · 2024 Cite

ON THE CONVERGENCE OF SOBOLEV GRADIENT FLOW FOR THE GROSS-PITAEVSKII EIGENVALUE PROBLEM

Journal Article SIAM Journal on Numerical Analysis · January 1, 2024 We study the convergences of three projected Sobolev gradient flows to the ground state of the Gross-Pitaevskii eigenvalue problem. They are constructed as the gradient flows of the Gross-Pitaevskii energy functional with respect to the H1 0 -metric and tw ... Full text Cite

Convergence of flow-based generative models via proximal gradient descent in Wasserstein space

Journal Article IEEE Transactions on Information Theory · January 1, 2024 Flow-based generative models enjoy certain advantages in computing the data generation and the likelihood, and have recently shown competitive empirical performance. Compared to the accumulating theoretical studies on related score-based diffusion models, ... Full text Cite

Regularized Stein Variational Gradient Flow

Journal Article Foundations of Computational Mathematics · January 1, 2024 The stein variational gradient descent (SVGD) algorithm is a deterministic particle method for sampling. However, a mean-field analysis reveals that the gradient flow corresponding to the SVGD algorithm (i.e., the Stein Variational Gradient Flow) only prov ... Full text Cite

Optimal artificial boundary conditions based on second-order correctors for three dimensional random elliptic media

Journal Article Communications in Partial Differential Equations · January 1, 2024 We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale (Formula presented.) in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter ... Full text Cite

ONE-DIMENSIONAL TENSOR NETWORK RECOVERY

Journal Article SIAM Journal on Matrix Analysis and Applications · January 1, 2024 We study the recovery of the underlying graphs or permutations for tensors in the tensor ring or tensor train format. Our proposed algorithms compare the matricization ranks after down-sampling, whose complexity is O(dlogd) for dth-order tensors. We prove ... Full text Cite

REPRESENTATION THEOREM FOR MULTIVARIABLE TOTALLY SYMMETRIC FUNCTIONS*

Journal Article Communications in Mathematical Sciences · January 1, 2024 In this work, we establish a representation theorem for multivariable totally symmetric functions: a multisymmetric continuous function must be the composition of a continuous function and a set of generators of the multisymmetric polynomials. We then stud ... Full text Cite

Coordinate Descent Full Configuration Interaction for Excited States.

Journal Article Journal of chemical theory and computation · November 2023 An efficient excited state method, named xCDFCI, in the configuration interaction framework is proposed. xCDFCI extends the unconstrained nonconvex optimization problem in coordinate descent full configuration interaction (CDFCI) to a multicolumn version f ... Full text Cite

On Explicit L2 -Convergence Rate Estimate for Underdamped Langevin Dynamics

Journal Article Archive for Rational Mechanics and Analysis · October 1, 2023 We provide a refined explicit estimate of the exponential decay rate of underdamped Langevin dynamics in the L2 distance, based on a framework developed in Albritton et al. (Variational methods for the kinetic Fokker–Planck equation, arXiv arXiv:1902.04037 ... Full text Cite

Neural Network-Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions

Journal Article Communications in Mathematics and Statistics · March 1, 2023 In this paper, we propose and study neural network-based methods for solutions of high-dimensional quadratic porous medium equation (QPME). Three variational formulations of this nonlinear PDE are presented: a strong formulation and two weak formulations. ... Full text Cite

Improving the Accuracy of Variational Quantum Eigensolvers with Fewer Qubits Using Orbital Optimization.

Journal Article Journal of chemical theory and computation · February 2023 Near-term quantum computers will be limited in the number of qubits on which they can process information as well as the depth of the circuits that they can coherently carry out. To date, experimental demonstrations of algorithms such as the Variational Qu ... Full text Cite

NUMERICAL ANALYSIS FOR INCHWORM MONTE CARLO METHOD: SIGN PROBLEM AND ERROR GROWTH

Journal Article Mathematics of Computation · January 1, 2023 We consider the numerical analysis of the inchworm Monte Carlo method, which is proposed recently to tackle the numerical sign problem for open quantum systems. We focus on the growth of the numerical error with respect to the simulation time, for which th ... Full text Cite

A REGULARITY THEORY FOR STATIC SCHRÖDINGER EQUATIONS ON R d IN SPECTRAL BARRON SPACES

Journal Article SIAM Journal on Mathematical Analysis · January 1, 2023 Spectral Barron spaces have received considerable interest recently, as it is the natural function space for approximation theory of two-layer neural networks with a dimension-free convergence rate. In this paper, we study the regularity of solutions to th ... Full text Cite

HeteRSGD: Tackling Heterogeneous Sampling Costs via Optimal Reweighted Stochastic Gradient Descent

Conference Proceedings of Machine Learning Research · January 1, 2023 One implicit assumption in current stochastic gradient descent (SGD) algorithms is the identical cost for sampling each component function of the finite-sum objective. However, there are applications where the costs differ substantially, for which SGD sche ... Cite

Convergence of score-based generative modeling for general data distributions

Conference Proceedings of Machine Learning Research · January 1, 2023 Score-based generative modeling (SGM) has grown to be a hugely successful method for learning to generate samples from complex data distributions such as that of images and audio. It is based on evolving an SDE that transforms white noise into a sample fro ... Cite

ON THE GLOBAL CONVERGENCE OF RANDOMIZED COORDINATE GRADIENT DESCENT FOR NONCONVEX OPTIMIZATION*

Journal Article SIAM Journal on Optimization · January 1, 2023 In this work, we analyze the global convergence property of a coordinate gradient descent with random choice of coordinates and stepsizes for nonconvex optimization problems. Under generic assumptions, we prove that the algorithm iterate will almost surely ... Full text Cite

Posterior Computation with the Gibbs Zig-Zag Sampler

Journal Article Bayesian Analysis · January 1, 2023 An intriguing new class of piecewise deterministic Markov processes (PDMPs) has recently been proposed as an alternative to Markov chain Monte Carlo (MCMC). We propose a new class of PDMPs termed Gibbs zig-zag samplers, which allow parameters to be updated ... Full text Cite

GEOMETRY OF BACKFLOW TRANSFORMATION ANSATZE FOR QUANTUM MANY-BODY FERMIONIC WAVEFUNCTIONS

Journal Article Communications in Mathematical Sciences · January 1, 2023 Wave function ansatze based on the backflow transformation are widely used to parametrize anti-symmetric multivariable functions for many-body quantum problems. We study the geometric aspects of such ansatze, in particular we show that in general totally a ... Full text Cite

EDGE STATE DYNAMICS ALONG CURVED INTERFACES

Journal Article SIAM Journal on Mathematical Analysis · January 1, 2023 We study the propagation of wavepackets along weakly curved interfaces between topologically distinct media. Our Hamiltonian is an adiabatic modulation of Dirac operators omnipresent in the topological insulators literature. Using explicit formulas for str ... Full text Cite

On Enhancing Expressive Power via Compositions of Single Fixed-Size ReLU Network

Conference Proceedings of Machine Learning Research · January 1, 2023 This paper explores the expressive power of deep neural networks through the framework of function compositions. We demonstrate that the repeated compositions of a single fixed-size ReLU network exhibit surprising expressive power, despite the limited expr ... Cite

Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions

Conference Proceedings of Machine Learning Research · January 1, 2023 We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small L2 error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either e ... Cite

Global optimality of Elman-type RNNs in the mean-field regime

Conference Proceedings of Machine Learning Research · January 1, 2023 We analyze Elman-type Recurrent Reural Networks (RNNs) and their training in the mean-field regime. Specifically, we show convergence of gradient descent training dynamics of the RNN to the corresponding mean-field formulation in the large width limit. We ... Cite

Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective

Conference Proceedings of Machine Learning Research · January 1, 2023 A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of dimensionality. However, ... Cite

The probability flow ODE is provably fast

Conference Advances in Neural Information Processing Systems · January 1, 2023 We provide the first polynomial-time convergence guarantees for the probability flow ODE implementation (together with a corrector step) of score-based generative modeling with an OU forward process. Our analysis is carried out in the wake of recent result ... Cite

Deep Equilibrium Based Neural Operators for Steady-State PDEs

Conference Advances in Neural Information Processing Systems · January 1, 2023 Data-driven machine learning approaches are being increasingly used to solve partial differential equations (PDEs). They have shown particularly striking successes when training an operator, which takes as input a PDE in some family, and outputs its soluti ... Cite

ON REPRESENTING LINEAR PROGRAMS BY GRAPH NEURAL NETWORKS

Conference 11th International Conference on Learning Representations, ICLR 2023 · January 1, 2023 Learning to optimize is a rapidly growing area that aims to solve optimization problems or improve existing optimization algorithms using machine learning (ML).In particular, the graph neural network (GNN) is considered a suitable ML model for optimization ... Cite

ON REPRESENTING MIXED-INTEGER LINEAR PROGRAMS BY GRAPH NEURAL NETWORKS

Conference 11th International Conference on Learning Representations, ICLR 2023 · January 1, 2023 While Mixed-integer linear programming (MILP) is NP-hard in general, practical MILP has received roughly 100-fold speedup in the past twenty years.Still, many classes of MILPs quickly become unsolvable as their sizes increase, motivating researchers to see ... Cite

Symmetry Breaking and the Generation of Spin Ordered Magnetic States in Density Functional Theory Due to Dirac Exchange for a Hydrogen Molecule

Journal Article Journal of Nonlinear Science · December 1, 2022 We study symmetry breaking in the mean field solutions to the electronic structure problem for the 2 electron hydrogen molecule within the Kohn Sham (KS) local spin density functional theory with Dirac exchange (the XLDA model). This simplified model shows ... Full text Cite

A proximal-gradient algorithm for crystal surface evolution

Journal Article Numerische Mathematik · November 1, 2022 As a counterpoint to recent numerical methods for crystal surface evolution, which agree well with microscopic dynamics but suffer from significant stiffness that prevents simulation on fine spatial grids, we develop a new numerical method based on the mac ... Full text Cite

Fast algorithms of bath calculations in simulations of quantum system-bath dynamics

Journal Article Computer Physics Communications · September 1, 2022 We present fast algorithms for the summation of Dyson series and the inchworm Monte Carlo method for quantum systems that are coupled with harmonic baths. The algorithms are based on evolving the integro-differential equations where the most expensive part ... Full text Cite

On the closedness and geometry of tensor network state sets

Journal Article Letters in Mathematical Physics · August 1, 2022 Tensor network states (TNS) are a powerful approach for the study of strongly correlated quantum matter. The curse of dimensionality is addressed by parametrizing the many-body state in terms of a network of partially contracted tensors. These tensors form ... Full text Cite

Quantum Orbital Minimization Method for Excited States Calculation on a Quantum Computer.

Journal Article Journal of chemical theory and computation · August 2022 We propose a quantum-classical hybrid variational algorithm, the quantum orbital minimization method (qOMM), for obtaining the ground state and low-lying excited states of a Hermitian operator. Given parametrized ansatz circuits representing eigenstates, q ... Full text Cite

Neural collapse under cross-entropy loss

Journal Article Applied and Computational Harmonic Analysis · July 1, 2022 We consider the variational problem of cross-entropy loss with n feature vectors on a unit hypersphere in Rd. We prove that when d≥n−1, the global minimum is given by the simplex equiangular tight frame, which justifies the neural collapse behavior. We als ... Full text Cite

Complexity of zigzag sampling algorithm for strongly log-concave distributions

Journal Article Statistics and Computing · June 1, 2022 We study the computational complexity of zigzag sampling algorithm for strongly log-concave distributions. The zigzag process has the advantage of not requiring time discretization for implementation, and that each proposed bouncing event requires only one ... Full text Cite

Neural-network quantum states for periodic systems in continuous space

Journal Article Physical Review Research · June 1, 2022 We introduce a family of neural quantum states for the simulation of strongly interacting systems in the presence of spatial periodicity. Our variational state is parametrized in terms of a permutationally invariant part described by the Deep Sets neural-n ... Full text Cite

Low-Rank Approximation for Multiscale PDEs

Journal Article Notices of the American Mathematical Society · June 1, 2022 Full text Cite

ON EXPLICIT L2-CONVERGENCE RATE ESTIMATE FOR PIECEWISE DETERMINISTIC MARKOV PROCESSES IN MCMC ALGORITHMS

Journal Article Annals of Applied Probability · April 1, 2022 We establish L2-exponential convergence rate for three popular piecewise deterministic Markov processes for sampling: the randomized Hamiltonian Monte Carlo method, the zigzag process and the bouncy particle sampler. Our analysis is based on a variational ... Full text Cite

Existence and Computation of Generalized Wannier Functions for Non-Periodic Systems in Two Dimensions and Higher

Journal Article Archive for Rational Mechanics and Analysis · March 1, 2022 Exponentially-localized Wannier functions (ELWFs) are an orthonormal basis of the Fermi projection of a material consisting of functions which decay exponentially fast away from their maxima. When the material is insulating and crystalline, conditions whic ... Full text Cite

Fast Localization of Eigenfunctions via Smoothed Potentials

Journal Article Journal of Scientific Computing · January 1, 2022 We study the problem of predicting highly localized low-lying eigenfunctions (- Δ + V) ϕ= λϕ in bounded domains Ω ⊂ Rd for rapidly varying potentials V. Filoche and Mayboroda introduced the function 1/u, where (- Δ + V) u= 1 , as a suitable regularization ... Full text Cite

DEFECT RESONANCES OF TRUNCATED CRYSTAL STRUCTURES

Journal Article SIAM Journal on Applied Mathematics · January 1, 2022 Defects in the atomic structure of crystalline materials may spawn electronic bound states, known as defect states, which decay rapidly away from the defect. Simplified models of defect states typically assume the defect is surrounded on all sides by an in ... Full text Cite

Bloch dynamics with second order Berry phase correction

Journal Article Asymptotic Analysis · January 1, 2022 We derive the semiclassical Bloch dynamics with second-order Berry phase correction in the presence of the slow-varying scalar potential as perturbation. Our mathematical derivation is based on a two-scale WKB asymptotic analysis. For a uniform external el ... Full text Cite

UNIVERSAL APPROXIMATION OF SYMMETRIC AND ANTI-SYMMETRIC FUNCTIONS

Journal Article Communications in Mathematical Sciences · January 1, 2022 We consider universal approximations of symmetric and anti-symmetric functions, which are important for applications in quantum physics, as well as other scientific and engineering computations. We give constructive approximations with explicit bounds on t ... Full text Cite

MANIFOLD LEARNING AND NONLINEAR HOMOGENIZATION

Journal Article Multiscale Modeling and Simulation · January 1, 2022 We describe an efficient domain decomposition-based framework for nonlinear multiscale PDE problems. The framework is inspired by manifold learning techniques and exploits the tangent spaces spanned by the nearest neighbors to compress local solution manif ... Full text Cite

MACHINE LEARNING FOR ELLIPTIC PDES: FAST RATE GENERALIZATION BOUND, NEURAL SCALING LAW AND MINIMAX OPTIMALITY

Conference ICLR 2022 - 10th International Conference on Learning Representations · January 1, 2022 In this paper, we study the statistical limits of deep learning techniques for solving elliptic partial differential equations (PDEs) from random samples using the Deep Ritz Method (DRM) and Physics-Informed Neural Networks (PINNs). To simplify the problem ... Cite

Convergence for score-based generative modeling with polynomial complexity

Conference Advances in Neural Information Processing Systems · January 1, 2022 Score-based generative modeling (SGM) is a highly successful approach for learning a probability distribution from data and generating further samples. We prove the first polynomial convergence guarantees for the core mechanic behind SGM: drawing samples f ... Cite

Optimal Artificial Boundary Condition for Random Elliptic Media

Journal Article Foundations of Computational Mathematics · December 1, 2021 We are given a uniformly elliptic coefficient field that we regard as a realization of a stationary and finite-range ensemble of coefficient fields. Given a right-hand side supported in a ball of size ℓ≫ 1 and of vanishing average, we are interested in an ... Full text Cite

Low-rank approximation for multiscale PDEs

Journal Article · November 24, 2021 Historically, analysis for multiscale PDEs is largely unified while numerical schemes tend to be equation-specific. In this paper, we propose a unified framework for computing multiscale problems through random sampling. This is achieved by incorporating r ... Link to item Cite

Geometry of backflow transformation ansatz for quantum many-body fermionic wavefunctions

Journal Article · November 19, 2021 Wave function ansatz based on the backflow transformation are widely used to parametrize anti-symmetric multivariable functions for many-body quantum problems. We study the geometric aspects of such ansatz, in particular we show that in general totally ant ... Link to item Cite

Stable phase retrieval from locally stable and conditionally connected measurements

Journal Article Applied and Computational Harmonic Analysis · November 1, 2021 In this paper, we study the stability of phase retrieval problems via a family of locally stable phase retrieval frame measurements in Banach spaces, which we call “locally stable and conditionally connected” (LSCC) measurement schemes. For any signal f in ... Full text Open Access Cite

Optimal artificial boundary conditions based on second-order correctors for three dimensional random elliptic media

Journal Article Communications in Partial Differential Equations (2024): 1-62 · September 3, 2021 We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale $\ell$ in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter $L\gg\ell$ aro ... Link to item Cite

ENSEMBLE KALMAN INVERSION FOR NONLINEAR PROBLEMS: WEIGHTS, CONSISTENCY, AND VARIANCE BOUNDS

Journal Article Foundations of Data Science · September 1, 2021 Ensemble Kalman Inversion (EnKI) [23] and Ensemble Square Root Filter (EnSRF) [36] are popular sampling methods for obtaining a target posterior distribution. They can be seem as one step (the analysis step) in the data assimilation method Ensemble Kalman ... Full text Cite

Convergence of stochastic-extended Lagrangian molecular dynamics method for polarizable force field simulation

Journal Article Journal of Computational Physics · August 1, 2021 Extended Lagrangian molecular dynamics (XLMD) is a general method for performing molecular dynamics simulations using quantum and classical many-body potentials. Recently several new XLMD schemes have been proposed and tested on several classes of many-bod ... Full text Cite

Microscopic origins of the crystallographically preferred growth in evaporation-induced colloidal crystals.

Journal Article Proceedings of the National Academy of Sciences of the United States of America · August 2021 Unlike crystalline atomic and ionic solids, texture development due to crystallographically preferred growth in colloidal crystals is less studied. Here we investigate the underlying mechanisms of the texture evolution in an evaporation-induced colloidal a ... Full text Cite

Algebraic localization of Wannier functions implies Chern triviality in non-periodic insulators

Journal Article · July 22, 2021 For gapped periodic systems (insulators), it has been established that the insulator is topologically trivial (i.e., its Chern number is equal to $0$) if and only if its Fermi projector admits an orthogonal basis with finite second moment (i.e., all basis ... Link to item Cite

On the Representation of Solutions to Elliptic PDEs in Barron Spaces

Journal Article · June 14, 2021 Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of $d$-dimensional second-order elliptic PDEs in the Barron spa ... Link to item Cite

Solving parametric PDE problems with artificial neural networks

Journal Article European Journal of Applied Mathematics · June 1, 2021 The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modelled into the equations as random coefficients. However, very often the variability of physical quantities d ... Full text Open Access Cite

Inclusion-exclusion principle for open quantum systems with bosonic bath

Journal Article New Journal of Physics · June 1, 2021 We present two fast algorithms which apply inclusion-exclusion principle to sum over the bosonic diagrams in bare diagrammatic quantum Monte Carlo and inchworm Monte Carlo method, respectively. In the case of inchworm Monte Carlo, the proposed fast algorit ... Full text Cite

Edge state dynamics along curved interfaces

Journal Article · June 1, 2021 We study the propagation of wavepackets along weakly curved interfaces between topologically distinct media. Our Hamiltonian is an adiabatic modulation of Dirac operators omnipresent in the topological insulators literature. Using explicit formulas for str ... Link to item Cite

A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schrödinger Eigenvalue Problems

Journal Article · May 3, 2021 This paper analyzes the generalization error of two-layer neural networks for computing the ground state of the Schr\"odinger operator on a $d$-dimensional hypercube. We prove that the convergence rate of the generalization error is independent of the dime ... Link to item Cite

Optimal Trapping for Brownian Motion: a Nonlinear Analogue of the Torsion Function

Journal Article Potential Analysis · April 1, 2021 We study the problem of maximizing the expected lifetime of drift diffusion in a bounded domain. More formally, we consider the PDE−Δu+b(x)⋅∇u=1inΩ subject to Dirichlet boundary conditions for ∥b∥L∞ fixed. We show that, in any given C2 −domain Ω, the vecto ... Full text Cite

A grid-free approach for simulating sweep and cyclic voltammetry.

Journal Article The Journal of chemical physics · April 2021 We present a computational approach to simulate linear sweep and cyclic voltammetry experiments that does not require a discretized grid in space to quantify diffusion. By using a Green's function solution coupled to a standard implicit ordinary differenti ... Full text Cite

Computing edge states without hard truncation

Journal Article SIAM Journal on Scientific Computing · March 11, 2021 We present a numerical method which accurately computes the discrete spectrum and associated bound states of semi-infinite Hamiltonians which model electronic “edge” states localized at boundaries of one- and two-dimensional crystalline materials. The prob ... Full text Cite

Structure-preserving numerical schemes for Lindblad equations

Journal Article · March 1, 2021 We study a family of structure-preserving deterministic numerical schemes for Lindblad equations. This family of schemes has a simple form and can systemically achieve arbitrary high-order accuracy in theory. Moreover, these schemes can also overcome the n ... Link to item Cite

Iterated projected position algorithm for constructing exponentially localized generalized Wannier functions for periodic and nonperiodic insulators in two dimensions and higher

Journal Article Physical Review B · February 15, 2021 Localized bases play an important role in understanding electronic structure. In periodic insulators, a natural choice of localized basis is given by the Wannier functions which depend on a choice of unitary transform known as a gauge transformation. Over ... Full text Cite

Algebraic localization implies exponential localization in non-periodic insulators

Journal Article · January 7, 2021 Exponentially-localized Wannier functions are a basis of the Fermi projection of a Hamiltonian consisting of functions which decay exponentially fast in space. In two and three spatial dimensions, it is well understood for periodic insulators that exponent ... Link to item Cite

A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations

Journal Article · January 5, 2021 This paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional partial differential equations. We derive the generalization error bounds of t ... Open Access Link to item Cite

On the global convergence of randomized coordinate gradient descent for non-convex optimization

Journal Article · January 4, 2021 In this work, we analyze the global convergence property of coordinate gradient descent with random choice of coordinates and stepsizes for non-convex optimization problems. Under generic assumptions, we prove that the algorithm iterate will almost surely ... Link to item Cite

Efficient construction of tensor ring representations from sampling

Journal Article Multiscale Modeling and Simulation · January 1, 2021 In this paper we propose an efficient method to compress a high dimensional function into a tensor ring format, based on alternating least squares (ALS). Since the function has size exponential in d, where d is the number of dimensions, we propose an effic ... Full text Open Access Cite

Numerical Methods For Stochastic Differential Equations Based On Gaussian Mixture

Journal Article Communications in Mathematical Sciences · January 1, 2021 We develop in this work a numerical method for stochastic differential equations (SDEs) with weak second-order accuracy based on Gaussian mixture. Unlike conventional higher order schemes for SDEs based on Itô-Taylor expansion and iterated Itô integrals, t ... Full text Open Access Cite

A low-rank schwarz method for radiative transfer equation with heterogeneous scattering coefficient

Journal Article Multiscale Modeling and Simulation · January 1, 2021 Random sampling has been used to find low-rank structure and to build fast direct solvers for multiscale partial differential equations of various types. In this work, we design an accelerated Schwarz method for radiative transfer equations that makes use ... Full text Cite

DEEP NETWORK APPROXIMATION FOR SMOOTH FUNCTIONS

Journal Article SIAM Journal on Mathematical Analysis · January 1, 2021 \bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . This paper establishes the optimal approximation error characterization of deep rectified linear unit (ReLU) networks for smooth functions in terms of both width and depth simultaneously. To that end, we first prov ... Full text Cite

COMPLEXITY OF RANDOMIZED ALGORITHMS FOR UNDERDAMPED LANGEVIN DYNAMICS*

Journal Article Communications in Mathematical Sciences · January 1, 2021 We establish an information complexity lower bound of randomized algorithms for simulating underdamped Langevin dynamics. More specifically, we prove that the worst L2 strong error is of order (Formula Presented), for solving a family of d-dimensional unde ... Full text Open Access Cite

Random Coordinate Underdamped Langevin Monte Carlo

Journal Article 24TH INTERNATIONAL CONFERENCE ON ARTIFICIAL INTELLIGENCE AND STATISTICS (AISTATS) · 2021 Cite

Locality of the windowed local density of states

Journal Article Discussion Contributions 10th Vienna Conference on Mathematical Modelling, volume 17. ARGESIM, 2022 · January 1, 2021 We introduce a generalization of local density of states which is "windowed" with respect to position and energy, called the windowed local density of states (wLDOS). This definition generalizes the usual LDOS in the sense that the usual LDOS is recovered ... Link to item Cite

ACTOR-CRITIC METHOD FOR HIGH DIMENSIONAL STATIC HAMILTON-JACOBI-BELLMAN PARTIAL DIFFERENTIAL EQUATIONS BASED ON NEURAL NETWORKS

Journal Article SIAM Journal on Scientific Computing · January 1, 2021 We propose a novel numerical method for high dimensional Hamilton-Jacobi-Bellman (HJB) type elliptic partial differential equations (PDEs). The HJB PDEs, reformulated as optimal control problems, are tackled by the actor-critic framework inspired by reinfo ... Full text Cite

Microscopic origins of the crystallographically preferred growth in evaporation-induced colloidal crystals

Journal Article PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA · 2021 Full text Cite

ANALYSIS OF A FOURTH-ORDER EXPONENTIAL PDE ARISING FROM A CRYSTAL SURFACE JUMP PROCESS WITH METROPOLIS-TYPE TRANSITION RATES

Journal Article Pure and Applied Analysis · January 1, 2021 We analytically and numerically study a fourth-order PDE modeling rough crystal surface diffusion on the macroscopic level. We discuss existence of solutions globally in time and long-time dynamics for the PDE model. The PDE, originally derived by Katsevic ... Full text Cite

On the Representation of Solutions to Elliptic PDEs in Barron Spaces

Conference Advances in Neural Information Processing Systems · January 1, 2021 Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of d-dimensional second-order elliptic PDEs in the Barron space ... Cite

Random Coordinate Underdamped Langevin Monte Carlo

Conference Proceedings of Machine Learning Research · January 1, 2021 The Underdamped Langevin Monte Carlo (ULMC) is a popular Markov chain Monte Carlo sampling method. It requires the computation of the full gradient of the log-density at each iteration, an expensive operation if the dimension of the problem is high. We pro ... Cite

Efficient sampling from the Bingham distribution

Conference Proceedings of Machine Learning Research · January 1, 2021 We give a algorithm for exact sampling from the Bingham distribution p(x) ∝ exp(x⊺Ax) on the sphere Sd-1 with expected runtime of poly(d, λmax(A) - λmin(A)). The algorithm is based on rejection sampling, where the proposal distribution is a polynomial appr ... Cite

A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Partial Differential Equations

Conference Proceedings of Machine Learning Research · January 1, 2021 This paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional partial differential equations. We derive the generalization error bounds of t ... Cite

Temporal-difference learning with nonlinear function approximation: lazy training and mean field regimes

Conference Proceedings of Machine Learning Research · January 1, 2021 We discuss the approximation of the value function for infinite-horizon discounted Markov Reward Processes (MRP) with wide neural networks trained with the Temporal-Difference (TD) learning algorithm. We first consider this problem under a certain scaling ... Cite

Complexity of zigzag sampling algorithm for strongly log-concave distributions

Journal Article Stat Comput · December 20, 2020 We study the computational complexity of zigzag sampling algorithm for strongly log-concave distributions. The zigzag process has the advantage of not requiring time discretization for implementation, and that each proposed bouncing event requires only one ... Open Access Link to item Cite

Continuum limit and preconditioned Langevin sampling of the path integral molecular dynamics

Journal Article Journal of Computational Physics · December 15, 2020 We investigate the continuum limit that the number of beads goes to infinity in the ring polymer representation of thermal averages. Studying the continuum limit of the trajectory sampling equation sheds light on possible preconditioning techniques for sam ... Full text Cite

Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach

Journal Article Journal of Computational Physics · December 15, 2020 We propose a new method to solve eigenvalue problems for linear and semilinear second order differential operators in high dimensions based on deep neural networks. The eigenvalue problem is reformulated as a fixed point problem of the semigroup flow induc ... Full text Open Access Cite

Neural Collapse with Cross-Entropy Loss

Journal Article · December 15, 2020 We consider the variational problem of cross-entropy loss with $n$ feature vectors on a unit hypersphere in $\mathbb{R}^d$. We prove that when $d \geq n - 1$, the global minimum is given by the simplex equiangular tight frame, which justifies the neural co ... Link to item Cite

Efficient posterior sampling for high-dimensional imbalanced logistic regression.

Journal Article Biometrika · December 2020 Classification with high-dimensional data is of widespread interest and often involves dealing with imbalanced data. Bayesian classification approaches are hampered by the fact that current Markov chain Monte Carlo algorithms for posterior computation beco ... Full text Cite

Inchworm Monte Carlo Method for Open Quantum Systems

Journal Article Communications on Pure and Applied Mathematics · November 1, 2020 We investigate in this work a recently proposed diagrammatic quantum Monte Carlo method—the inchworm Monte Carlo method—for open quantum systems. We establish its validity rigorously based on resummation of Dyson series. Moreover, we introduce an integro-d ... Full text Cite

Butterfly-net: Optimal function representation based on convolutional neural networks

Journal Article Communications in Computational Physics · November 1, 2020 Deep networks, especially convolutional neural networks (CNNs), have been successfully applied in various areas of machine learning as well as to challenging problems in other scientific and engineering fields. This paper introduces Butterfly-net, a low-co ... Full text Open Access Cite

Synchronization of Kuramoto oscillators in dense networks

Journal Article Nonlinearity · November 1, 2020 We study synchronization properties of systems of Kuramoto oscillators. The problem can also be understood as a question about the properties of an energy landscape created by a graph. More formally, let G = (V, E) be a connected graph and (ai j)ni, j=1 de ... Full text Cite

ELSI — An open infrastructure for electronic structure solvers

Journal Article Computer Physics Communications · November 1, 2020 Routine applications of electronic structure theory to molecules and periodic systems need to compute the electron density from given Hamiltonian and, in case of non-orthogonal basis sets, overlap matrices. System sizes can range from few to thousands or, ... Full text Cite

Manifold Learning and Nonlinear Homogenization

Journal Article · November 1, 2020 We describe an efficient domain decomposition-based framework for nonlinear multiscale PDE problems. The framework is inspired by manifold learning techniques and exploits the tangent spaces spanned by the nearest neighbors to compress local solution manif ... Link to item Cite

Global optimality of softmax policy gradient with single hidden layer neural networks in the mean-field regime

Journal Article · October 22, 2020 We study the problem of policy optimization for infinite-horizon discounted Markov Decision Processes with softmax policy and nonlinear function approximation trained with policy gradient algorithms. We concentrate on the training dynamics in the mean-fiel ... Open Access Link to item Cite

Random Coordinate Langevin Monte Carlo

Journal Article · October 3, 2020 Langevin Monte Carlo (LMC) is a popular Markov chain Monte Carlo sampling method. One drawback is that it requires the computation of the full gradient at each iteration, an expensive operation if the dimension of the problem is high. We propose a new samp ... Link to item Cite

Optimal Orbital Selection for Full Configuration Interaction (OptOrbFCI): Pursuing the Basis Set Limit under a Budget.

Journal Article Journal of chemical theory and computation · October 2020 Full configuration interaction (FCI) solvers are limited to small basis sets due to their expensive computational costs. An optimal orbital selection for FCI (OptOrbFCI) is proposed to boost the power of existing FCI solvers to pursue the basis set limit u ... Full text Cite

Efficient sampling from the Bingham distribution

Journal Article Algorithmic Learning Theory. PMLR, 2021 · September 30, 2020 We give a algorithm for exact sampling from the Bingham distribution $p(x)\propto \exp(x^\top A x)$ on the sphere $\mathcal S^{d-1}$ with expected runtime of $\operatorname{poly}(d, \lambda_{\max}(A)-\lambda_{\min}(A))$. The algorithm is based on rejection ... Link to item Cite

Fisher information regularization schemes for Wasserstein gradient flows

Journal Article Journal of Computational Physics · September 1, 2020 We propose a variational scheme for computing Wasserstein gradient flows. The scheme builds upon the Jordan–Kinderlehrer–Otto framework with the Benamou-Brenier's dynamic formulation of the quadratic Wasserstein metric and adds a regularization by the Fish ... Full text Cite

Analysis of a continuum theory for broken bond crystal surface models with evaporation and deposition effects

Journal Article Nonlinearity · August 1, 2020 We study a 4th order degenerate parabolic PDE model in one-dimension with a 2nd order correction modeling the evolution of a crystal surface under the influence of both thermal fluctuations and evaporation/deposition effects. First, we provide a non-rigoro ... Full text Cite

On explicit $L^2$-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms

Journal Article Ann. Appl. Probab. 32(2): 1333-1361 (April 2022) · July 29, 2020 We establish $L^2$-exponential convergence rate for three popular piecewise deterministic Markov processes for sampling: the randomized Hamiltonian Monte Carlo method, the zigzag process, and the bouncy particle sampler. Our analysis is based on a variatio ... Open Access Link to item Cite

A Proximal-Gradient Algorithm for Crystal Surface Evolution

Journal Article · June 22, 2020 As a counterpoint to recent numerical methods for crystal surface evolution, which agree well with microscopic dynamics but suffer from significant stiffness that prevents simulation on fine spatial grids, we develop a new numerical method based on the mac ... Link to item Cite

Defect resonances of truncated crystal structures

Journal Article SIAM J. Appl. Math 82 · June 13, 2020 Defects in the atomic structure of crystalline materials may spawn electronic bound states, known as \emph{defect states}, which decay rapidly away from the defect. Simplified models of defect states typically assume the defect is surrounded on all sides b ... Open Access Link to item Cite

Numerical analysis for inchworm Monte Carlo method: Sign problem and error growth

Journal Article · June 13, 2020 We consider the numerical analysis of the inchworm Monte Carlo method, which is proposed recently to tackle the numerical sign problem for open quantum systems. We focus on the growth of the numerical error with respect to the simulation time, for which th ... Link to item Cite

Estimating normalizing constants for log-concave distributions: Algorithms and lower bounds

Journal Article Proceedings of the Annual ACM Symposium on Theory of Computing · June 8, 2020 Estimating the normalizing constant of an unnormalized probability distribution has important applications in computer science, statistical physics, machine learning, and statistics. In this work, we consider the problem of estimating the normalizing const ... Full text Cite

Discontinuous Hamiltonian Monte Carlo for discrete parameters and discontinuous likelihoods

Journal Article Biometrika · June 1, 2020 Hamiltonian Monte Carlo has emerged as a standard tool for posterior computation. In this article we present an extension that can efficiently explore target distributions with discontinuous densities. Our extension in particular enables efficient sampling ... Full text Cite

Variational training of neural network approximations of solution maps for physical models

Journal Article Journal of Computational Physics · May 15, 2020 A novel solve-training framework is proposed to train neural network in representing low dimensional solution maps of physical models. Solve-training framework uses the neural network as the ansatz of the solution map and trains the network variationally v ... Full text Cite

Posterior computation with the Gibbs zig-zag sampler

Journal Article · April 8, 2020 Markov chain Monte Carlo (MCMC) sampling algorithms have dominated the literature on posterior computation. However, MCMC faces substantial hurdles in performing efficient posterior sampling for challenging Bayesian models, particularly in high-dimensional ... Link to item Cite

Analysis of a fourth order exponential PDE arising from a crystal surface jump process with Metropolis-type transition rates

Journal Article Pure Appl. Analysis · March 16, 2020 We analytically and numerically study a fourth order PDE modeling rough crystal surface diffusion on the macroscopic level. We discuss existence of solutions globally in time and long time dynamics for the PDE model. The PDE, originally derived by the seco ... Link to item Cite

Existence and computation of generalized Wannier functions for non-periodic systems in two dimensions and higher

Journal Article Arch. Rational Mech. Anal. 243 · March 14, 2020 Exponentially-localized Wannier functions (ELWFs) are an orthonormal basis of the Fermi projection of a material consisting of functions which decay exponentially fast away from their maxima. When the material is insulating and crystalline, conditions whic ... Open Access Link to item Cite

Ensemble Kalman Inversion for nonlinear problems: weights, consistency, and variance bounds

Journal Article · March 4, 2020 Ensemble Kalman Inversion (EnKI) and Ensemble Square Root Filter (EnSRF) are popular sampling methods for obtaining a target posterior distribution. They can be seem as one step (the analysis step) in the data assimilation method Ensemble Kalman Filter. De ... Link to item Cite

Quadrature Points via Heat Kernel Repulsion

Journal Article Constructive Approximation · February 1, 2020 We discuss the classical problem of how to pick N weighted points on a d-dimensional manifold so as to obtain a reasonable quadrature rule 1|M|∫Mf(x)dx≃∑n=1Naif(xi).This problem, naturally, has a long history; the purpose of our paper is to propose selecti ... Full text Cite

Non-Convex Planar Harmonic Maps

Journal Article · January 5, 2020 We formulate a novel characterization of a family of invertible maps between two-dimensional domains. Our work follows two classic results: The Rad\'o-Kneser-Choquet (RKC) theorem, which establishes the invertibility of harmonic maps into a convex planer d ... Open Access Link to item Cite

The full configuration interaction quantum monte carlo method through the lens of inexact power iteration

Journal Article SIAM Journal on Scientific Computing · January 1, 2020 In this paper, we propose a general analysis framework for inexact power iteration, which can be used to efficiently solve high-dimensional eigenvalue problems arising from quantum many-body problems. Under this framework, we establish the convergence theo ... Full text Open Access Cite

Randomized sampling for basis function construction in generalized finite element methods

Journal Article Multiscale Modeling and Simulation · January 1, 2020 In the framework of generalized finite element methods for elliptic equations with rough coefficients, efficiency and accuracy of the numerical method depend critically on the use of appropriate basis functions. This work explores several random sampling s ... Full text Open Access Cite

Random sampling and efficient algorithms for multiscale pdes

Journal Article SIAM Journal on Scientific Computing · January 1, 2020 We describe a numerical framework that uses random sampling to efficiently capture low-rank local solution spaces of multiscale PDE problems arising in domain decomposition. In contrast to existing techniques, our method does not rely on detailed analytica ... Full text Cite

Dirac operators and domain walls

Journal Article SIAM Journal on Mathematical Analysis · January 1, 2020 We study the eigenvalue problem for a one-dimensional Dirac operator with a spatially varying "mass" term. It is well-known that when the mass function has the form of a kink, or domain wall, transitioning between strictly positive and strictly negative as ... Full text Cite

A stochastic version of stein variational gradient descent for efficient sampling

Journal Article Communications in Applied Mathematics and Computational Science · January 1, 2020 We propose in this work RBM-SVGD, a stochastic version of the Stein variational gradient descent (SVGD) method for efficiently sampling from a given probability measure, which is thus useful for Bayesian inference. The method is to apply the random batch m ... Full text Open Access Cite

A dimension-free hermite-hadamard inequality via gradient estimates for the torsion function

Journal Article Proceedings of the American Mathematical Society · January 1, 2020 Let Ω ⊂ Rn be a convex domain, and let f : Ω → R be a subharmonic function, Δf ≥ 0, which satisfies f ≥ 0 on the boundary ∂Ω. Then (Formula Presented) Our proof is based on a new gradient estimate for the torsion function, Δu = -1 with Dirichlet boundary c ... Full text Cite

Tensor ring decomposition: Optimization landscape and one-loop convergence of alternating least squares

Journal Article SIAM Journal on Matrix Analysis and Applications · January 1, 2020 In this work, we study the tensor ring decomposition and its associated numerical algorithms. We establish a sharp transition of algorithmic difficulty of the optimization problem as the bond dimension increases: On one hand, we show the existence of spuri ... Full text Cite

A mean-field analysis of deep resnet and beyond: Towards provable optimization via overparameterization from depth

Journal Article 37th International Conference on Machine Learning, ICML 2020 · January 1, 2020 Training deep neural networks with stochastic gradient descent (SGD) can often achieve zero training loss on real-world tasks although the optimization landscape is known to be highly non-convex. To understand the success of SGD for training deep neural ne ... Cite

A universal approximation theorem of deep neural networks for expressing probability distributions

Journal Article Advances in Neural Information Processing Systems · January 1, 2020 This paper studies the universal approximation property of deep neural networks for representing probability distributions. Given a target distribution p and a source distribution pz both defined on Rd, we prove under some assumptions that there exists a d ... Cite

Stochastic modified equations for the asynchronous stochastic gradient descent

Journal Article Information and Inference · January 1, 2020 We propose stochastic modified equations (SMEs) for modelling the asynchronous stochastic gradient descent (ASGD) algorithms. The resulting SME of Langevin type extracts more information about the ASGD dynamics and elucidates the relationship between diffe ... Full text Cite

Universal approximation of symmetric and anti-symmetric functions

Journal Article · December 3, 2019 We consider universal approximations of symmetric and anti-symmetric functions, which are important for applications in quantum physics, as well as other scientific and engineering computations. We give constructive approximations with explicit bounds on t ... Link to item Cite

A numerical method for coupling the BGK model and Euler equations through the linearized Knudsen layer

Journal Article Journal of Computational Physics · December 1, 2019 The Bhatnagar-Gross-Krook (BGK) model, a simplification of the Boltzmann equation, in the absence of boundary effect, converges to the Euler equations when the Knudsen number is small. In practice, however, Knudsen layers emerge at the physical boundary, o ... Full text Cite

Approximating pointwise products of Laplacian eigenfunctions

Journal Article Journal of Functional Analysis · November 1, 2019 We consider Laplacian eigenfunctions on a d-dimensional bounded domain M (or a d-dimensional compact manifold M) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions (eℓ)ℓ∈N. We study the subspace of all pointwise products A ... Full text Cite

Exponential Decay of Rényi Divergence Under Fokker–Planck Equations

Journal Article Journal of Statistical Physics · September 1, 2019 We prove the exponential convergence to the equilibrium, quantified by Rényi divergence, of the solution of the Fokker–Planck equation with drift given by the gradient of a strictly convex potential. This extends the classical exponential decay result on t ... Full text Cite

On explicit $L^2$-convergence rate estimate for underdamped Langevin dynamics

Journal Article Arch Rational Mech Anal · August 13, 2019 We provide a refined explicit estimate of exponential decay rate of underdamped Langevin dynamics in $L^2$ distance, based on a framework developed in [1]. To achieve this, we first prove a Poincar\'{e}-type inequality with Gibbs measure in space and Gauss ... Open Access Link to item Cite

Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model

Journal Article Physica D: Nonlinear Phenomena · June 1, 2019 In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study analytically an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface corrugation in 1+1 di ... Full text Open Access Cite

Coordinate Descent Full Configuration Interaction.

Journal Article Journal of chemical theory and computation · June 2019 We develop an efficient algorithm, coordinate descent FCI (CDFCI), for the electronic structure ground-state calculation in the configuration interaction framework. CDFCI solves an unconstrained nonconvex optimization problem, which is a reformulation of t ... Full text Cite

Temporal-difference learning with nonlinear function approximation: lazy training and mean field regimes

Journal Article PMLR · May 26, 2019 We discuss the approximation of the value function for infinite-horizon discounted Markov Reward Processes (MRP) with nonlinear functions trained with the Temporal-Difference (TD) learning algorithm. We first consider this problem under a certain scaling o ... Link to item Cite

Accelerating Langevin Sampling with Birth-death

Journal Article · May 23, 2019 A fundamental problem in Bayesian inference and statistical machine learning is to efficiently sample from multimodal distributions. Due to metastability, multimodal distributions are difficult to sample using standard Markov chain Monte Carlo methods. We ... Link to item Cite

Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance

Journal Article Journal of Mathematical Physics · May 1, 2019 We study the entropy production of the sandwiched Rényi divergence under the primitive Lindblad equation with Gel'fand-Naimark-Segal-detailed balance. We prove that the Lindblad equation can be identified as the gradient flow of the sandwiched Rényi diverg ... Full text Cite

Numerical methods for Kohn-Sham density functional theory

Journal Article Acta Numerica · May 1, 2019 Kohn-Sham density functional theory (DFT) is the most widely used electronic structure theory. Despite significant progress in the past few decades, the numerical solution of Kohn-Sham DFT problems remains challenging, especially for large-scale systems. I ... Full text Cite

Large-scale benchmark of electronic structure solvers with the ELSI infrastructure

Conference ABSTRACTS OF PAPERS OF THE AMERICAN CHEMICAL SOCIETY · March 31, 2019 Link to item Cite

Methodological and Computational Aspects of Parallel Tempering Methods in the Infinite Swapping Limit

Journal Article Journal of Statistical Physics · February 15, 2019 A variant of the parallel tempering method is proposed in terms of a stochastic switching process for the coupled dynamics of replica configuration and temperature permutation. This formulation is shown to facilitate the analysis of the convergence propert ... Full text Open Access Cite

Symmetry Breaking in Density Functional Theory due to Dirac Exchange for a Hydrogen Molecule

Journal Article · February 9, 2019 We study symmetry breaking in the mean field solutions to the 2 electron hydrogen molecule within Kohn Sham (KS) local spin density function theory with Dirac exchange (the XLDA model). This simplified model shows behavior related to that of the (KS) spin ... Link to item Cite

Bold diagrammatic Monte Carlo in the lens of stochastic iterative methods

Journal Article Transactions of Mathematics and Its Applications · February 1, 2019 AbstractThis work aims at understanding of bold diagrammatic Monte Carlo (BDMC) methods for stochastic summation of Feynman diagrams from the angle of stochastic iterative methods. The convergence enhancemen ... Full text Open Access Cite

The simulated tempering method in the infinite switch limit with adaptive weight learning

Journal Article Journal of Statistical Mechanics: Theory and Experiment · January 16, 2019 We investigate the theoretical foundations of the simulated tempering (ST) method and use our findings to design an efficient accelerated sampling algorithm. Employing a large deviation argument first used for replica exchange molecular dynamics (Plattner ... Full text Cite

Learning interacting particle systems: Diffusion parameter estimation for aggregation equations

Journal Article Mathematical Models and Methods in Applied Sciences · January 1, 2019 In this paper, we study the parameter estimation of interacting particle systems subject to the Newtonian aggregation and Brownian diffusion. Specifically, we construct an estimator with partial observed data to approximate the diffusion parameter , and th ... Full text Open Access Cite

Solving for high-dimensional committor functions using artificial neural networks

Journal Article Research in Mathematical Sciences · January 1, 2019 In this note we propose a method based on artificial neural network to study the transition between states governed by stochastic processes. In particular, we aim for numerical schemes for the committor function, the central object of transition path theor ... Full text Cite

Scaling limit of the Stein variational gradient descent: The mean field regime

Journal Article SIAM Journal on Mathematical Analysis · January 1, 2019 We study an interacting particle system in Rd motivated by Stein variational gradient descent [Q. Liu and D. Wang, Proceedings of NIPS, 2016], a deterministic algorithm for approximating a given probability density with unknown normalization based on parti ... Full text Cite

Coordinatewise descent methods for leading eigenvalue problem

Journal Article SIAM Journal on Scientific Computing · January 1, 2019 Leading eigenvalue problems for large scale matrices arise in many applications. Coordinatewise descent methods are considered in this work for such problems based on a reformulation of the leading eigenvalue problem as a nonconvex optimization problem. Th ... Full text Cite

Scaling limit of the Stein variational gradient descent: the mean field regime

Journal Article SIAM Journal on Mathematical Analysis · 2019 Link to item Cite

Trigonometric integrators for quasilinear wave equations

Journal Article Mathematics of Computation · January 1, 2019 Trigonometric time integrators are introduced as a class of explicit numerical methods for quasilinear wave equations. Second-order convergence for the semidiscretization in time with these integrators is shown for a sufficiently regular exact solution. Th ... Full text Cite

Tensorization of the strong data processing inequality for quantum chi-square divergences

Journal Article Quantum · January 1, 2019 It is well-known that any quantum channel E satisfies the data processing inequality (DPI), with respect to various divergences, e.g., quantum χ2κ divergences and quantum relative entropy. More specifically, the data processing inequality states that the d ... Full text Cite

Stop Memorizing: A Data-Dependent Regularization Framework for Intrinsic Pattern Learning

Journal Article SIAM Journal on Mathematics of Data Science · January 2019 Full text Cite

Thermodynamic Limit of Crystal Defects with Finite Temperature Tight Binding

Journal Article Archive for Rational Mechanics and Analysis · November 1, 2018 We consider a tight binding model for localised crystalline defects with electrons in the canonical ensemble (finite Fermi temperature) and nuclei positions relaxed according to the Born–Oppenheimer approximation. We prove that the limit model as the compu ... Full text Open Access Cite

Moderate deviation for random elliptic PDE with small noise

Journal Article Annals of Applied Probability · October 1, 2018 Partial differential equations with random inputs have become popular models to characterize physical systems with uncertainty coming from imprecise measurement and intrinsic randomness. In this paper, we perform asymptotic rare-event analysis for such ell ... Full text Open Access Cite

On Pointwise Products of Elliptic Eigenfunctions

Journal Article · October 1, 2018 We consider eigenfunctions of Schr\"odinger operators on a $d-$dimensional bounded domain $\Omega$ (or a $d-$dimensional compact manifold $\Omega$) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions $(\phi_n)_{n \in \mathb ... Link to item Cite

Detecting localized eigenstates of linear operators

Journal Article Research in Mathematical Sciences · September 1, 2018 We describe a way of detecting the location of localized eigenvectors of the eigenvalue problem Ax = λx for eigenvalues λ with |λ| comparatively large. We define the family of functions fα: {1, 2, …,n} → R fα (k) = log(‖Aα ek ‖ℓ2), where α ≥ 0 is a paramet ... Full text Open Access Cite

Fundamental Limitations for Measurements in Quantum Many-Body Systems

Journal Article Physical Review Letters · August 24, 2018 Dynamical measurement schemes are an important tool for the investigation of quantum many-body systems, especially in the age of quantum simulation. Here, we address the question whether generic measurements can be implemented efficiently if we have access ... Full text Cite

A Concurrent Global–Local Numerical Method for Multiscale PDEs

Journal Article Journal of Scientific Computing · August 1, 2018 We present a new hybrid numerical method for multiscale partial differential equations, which simultaneously captures the global macroscopic information and resolves the local microscopic events over regions of relatively small size. The method couples con ... Full text Cite

Integrated tempering enhanced sampling method as the infinite switching limit of simulated tempering.

Journal Article The Journal of chemical physics · August 2018 A fast and accurate sampling method is in high demand, in order to bridge the large gaps between molecular dynamic simulations and experimental observations. Recently, an integrated tempering enhanced sampling (ITS) method has been proposed and successfull ... Full text Cite

A Mathematical Theory of Optimal Milestoning (with a Detour via Exact Milestoning)

Journal Article Communications on Pure and Applied Mathematics · June 1, 2018 Milestoning is a computational procedure that reduces the dynamics of complex systems to memoryless jumps between intermediates, or milestones, and only retains some information about the probability of these jumps and the time lags between them. Here we a ... Full text Open Access Cite

Stop memorizing: A data-dependent regularization framework for intrinsic pattern learning

Journal Article · May 18, 2018 Deep neural networks (DNNs) typically have enough capacity to fit random data by brute force even when conventional data-dependent regularizations focusing on the geometry of the features are imposed. We find out that the reason for this is the inconsisten ... Open Access Link to item Cite

ELSI: A unified software interface for Kohn-Sham electronic structure solvers

Conference ABSTRACTS OF PAPERS OF THE AMERICAN CHEMICAL SOCIETY · March 18, 2018 Link to item Cite

On discrete Wigner transforms

Journal Article · February 15, 2018 In this work, we derive a discrete analog of the Wigner transform over the space $(\mathbb{C}^p)^{\otimes N}$ for any prime $p$ and any positive integer $N$. We show that the Wigner transform over this space can be constructed as the inverse Fourier transf ... Open Access Link to item Cite

Accelerated sampling by infinite swapping of path integral molecular dynamics with surface hopping.

Journal Article The Journal of chemical physics · February 2018 To accelerate the thermal equilibrium sampling of multi-level quantum systems, the infinite swapping limit of a recently proposed multi-level ring polymer representation is investigated. In the infinite swapping limit, the ring polymer evolves according to ... Full text Open Access Cite

Analysis of multiscale integrators for multiple attractors and irreversible langevin samplers

Journal Article Multiscale Modeling and Simulation · January 1, 2018 We study multiscale integrator numerical schemes for a class of stiff stochastic differential equations (SDEs). We consider multiscale SDEs with potentially multiple attractors that behave as diffusions on graphs as the stiffness parameter goes to its limi ... Full text Cite

Convergence of Phase-Field Free Energy and Boundary Force for Molecular Solvation

Journal Article Archive for Rational Mechanics and Analysis · January 1, 2018 We study a phase-field variational model for the solvation of charged molecules with an implicit solvent. The solvation free-energy functional of all phase fields consists of the surface energy, solute excluded volume and solute-solvent van der Waals dispe ... Full text Open Access Cite

Frozen gaussian approximation with surface hopping for mixed quantum-classical dynamics: A mathematical justification of fewest switches surface hopping algorithms

Journal Article Mathematics of Computation · January 1, 2018 We develop a surface hopping algorithm based on frozen Gaussian approximation for semiclassical matrix Schrödinger equations, in the spirit of Tully's fewest switches surface hopping method. The algorithm is asymptotically derived from the Schrödinger equa ... Full text Open Access Cite

A surface hopping Gaussian beam method for high-dimensional transport systems

Journal Article SIAM Journal on Scientific Computing · January 1, 2018 We consider a set of linear hyperbolic equations coupled by a linear source term and introduce a surface hopping Gaussian beam method as its numerical solver. The Gaussian beam part is basically classic, while the surface hopping part is derived from the e ... Full text Open Access Cite

Point cloud discretization of Fokker-planck operators for committor functions

Journal Article Multiscale Modeling and Simulation · January 1, 2018 The committor functions provide useful information to the understanding of transitions of a stochastic system between disjoint regions in phase space. In this work, we develop a point cloud discretization for Fokker-Planck operators to numerically calculat ... Full text Open Access Cite

Frozen Gaussian approximation for high frequency wave propagation in periodic media

Journal Article Asymptotic Analysis · January 1, 2018 Propagation of high-frequency wave in periodic media is a challenging problem due to the existence of multiscale characterized by short wavelength, small lattice constant and large physical domain size. Conventional computational methods lead to extremely ... Full text Cite

A quasi-nonlocal coupling method for nonlocal and local diffusion models

Journal Article SIAM Journal on Numerical Analysis · January 1, 2018 In this paper, we extend the idea of “geometric reconstruction” to couple a nonlocal diffusion model directly with the classical local diffusion in one dimensional space. This new coupling framework removes interfacial inconsistency, ensures the flux balan ... Full text Open Access Cite

ELSI: A unified software interface for Kohn–Sham electronic structure solvers

Journal Article Computer Physics Communications · January 1, 2018 Solving the electronic structure from a generalized or standard eigenproblem is often the bottleneck in large scale calculations based on Kohn–Sham density-functional theory. This problem must be addressed by essentially all current electronic structure co ... Full text Open Access Cite

A diabatic surface hopping algorithm based on time dependent perturbation theory and semiclassical analysis

Journal Article Multiscale Modeling and Simulation · January 1, 2018 Surface hopping algorithms are popular tools to study dynamics of the quantumclassical mixed systems. In this paper, we propose a surface hopping algorithm in diabatic representations, based on time dependent perturbation theory and semiclassical analysis. ... Full text Open Access Cite

A quantum kinetic monte carlo method for quantum many-body spin dynamics

Journal Article SIAM Journal on Scientific Computing · January 1, 2018 We propose a general framework of a quantum kinetic Monte Carlo algorithm, based on a stochastic representation of a series expansion of the quantum evolution. Two approaches have been developed in the context of quantum many-body spin dynamics, using diff ... Full text Open Access Cite

Stochastic dynamical low-rank approximation method

Journal Article Journal of Computational Physics · 2018 Full text Cite

PEXSI-$\Sigma$: a Green’s function embedding method for Kohn–Sham density functional theory

Journal Article Annals of Mathematical Sciences and Applications · 2018 Full text Cite

Cubic scaling algorithms for RPA correlation using interpolative separable density fitting

Journal Article Journal of Computational Physics · December 15, 2017 We present a new cubic scaling algorithm for the calculation of the RPA correlation energy. Our scheme splits up the dependence between the occupied and virtual orbitals in χ0 by use of Cauchy's integral formula. This introduces an additional integral to b ... Full text Open Access Cite

Lindblad equation and its semiclassical limit of the Anderson-Holstein model

Journal Article Journal of Mathematical Physics · December 1, 2017 For multi-level open quantum systems, the interaction between different levels could pose a challenge to understand the quantum system both analytically and numerically. In this work, we study the approximation of the dynamics of the Anderson-Holstein mode ... Full text Open Access Cite

Riesz Energy on the Torus: Regularity of Minimizers

Journal Article · October 22, 2017 We study sets of $N$ points on the $d-$dimensional torus $\mathbb{T}^d$ minimizing interaction functionals of the type \[ \sum_{i, j =1 \atop i \neq j}^{N}{ f(x_i - x_j)}. \] The main result states that for a class of functions $f$ that behave like Riesz e ... Open Access Link to item Cite

Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem

Journal Article Journal of Statistical Physics · October 1, 2017 We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the ‘fluctuation-dissipation theorem’, the differential equations driven by ... Full text Open Access Cite

A convergent method for linear half-space kinetic equations

Journal Article ESAIM: Mathematical Modelling and Numerical Analysis · September 1, 2017 We give a unified proof for the well-posedness of a class of linear half-space equations with general incoming data and construct a Galerkin method to numerically resolve this type of equations in a systematic way. Our main strategy in both analysis and nu ... Full text Open Access Cite

A variation on the Donsker-Varadhan inequality for the principal eigenvalue.

Journal Article Proceedings. Mathematical, physical, and engineering sciences · August 2017 The purpose of this short paper is to give a variation on the classical Donsker-Varadhan inequality, which bounds the first eigenvalue of a second-order elliptic operator on a bounded domain Ω by the largest mean first exit time of the associated dr ... Full text Open Access Cite

Continuum Limit of a Mesoscopic Model with Elasticity of Step Motion on Vicinal Surfaces

Journal Article Journal of Nonlinear Science · June 1, 2017 This work considers the rigorous derivation of continuum models of step motion starting from a mesoscopic Burton–Cabrera–Frank-type model following the Xiang’s work (Xiang in SIAM J Appl Math 63(1):241–258, 2002). We prove that as the lattice parameter goe ... Full text Open Access Cite

On extending Kohn-Sham density functionals to systems with fractional number of electrons.

Journal Article The Journal of chemical physics · June 2017 We analyze four ways of formulating the Kohn-Sham (KS) density functionals with a fractional number of electrons, through extending the constrained search space from the Kohn-Sham and the generalized Kohn-Sham (GKS) non-interacting v-representable density ... Full text Cite

Orbital minimization method with ℓ1 regularization

Journal Article Journal of Computational Physics · May 1, 2017 We consider a modification of the orbital minimization method (OMM) energy functional which contains an ℓ1 penalty term in order to find a sparse representation of the low-lying eigenspace of self-adjoint operators. We analyze the local minima of the modif ... Full text Open Access Cite

Fractional stochastic differential equations satisfying fluctuation-dissipation theorem

Journal Article · April 23, 2017 We consider in this work stochastic differential equation (SDE) model for particles in contact with a heat bath when the memory effects are non-negligible. As a result of the fluctuation-dissipation theorem, the differential equations driven by fractional ... Open Access Cite

A Hybrid Global-local Numerical Method for Multiscale PDEs

Journal Article · April 23, 2017 We present a new hybrid numerical method for multiscale partial differential equations, which simultaneously captures both the global macroscopic information and resolves the local microscopic events. The convergence of the proposed method is proved for pr ... Open Access Cite

Path integral molecular dynamics with surface hopping for thermal equilibrium sampling of nonadiabatic systems.

Journal Article The Journal of chemical physics · April 2017 In this work, a novel ring polymer representation for a multi-level quantum system is proposed for thermal average calculations. The proposed representation keeps the discreteness of the electronic states: besides position and momentum, each bead in the ri ... Full text Open Access Cite

Dislocation climb models from atomistic scheme to dislocation dynamics

Journal Article Journal of the Mechanics and Physics of Solids · February 1, 2017 We develop a mesoscopic dislocation dynamics model for vacancy-assisted dislocation climb by upscalings from a stochastic model on the atomistic scale. Our models incorporate microscopic mechanisms of (i) bulk diffusion of vacancies, (ii) vacancy exchange ... Full text Open Access Cite

Wavepackets in inhomogeneous periodic media: Effective particle-field dynamics and Berry curvature

Journal Article Journal of Mathematical Physics · February 1, 2017 We consider a model of an electron in a crystal moving under the influence of an external electric field: Schrödinger's equation with a potential which is the sum of a periodic function and a general smooth function. We identify two dimensionless parameter ... Full text Open Access Cite

Validity and Regularization of Classical Half-Space Equations

Journal Article Journal of Statistical Physics · January 1, 2017 Recent result (Wu and Guo in Commun Math Phys 336(3):1473–1553, 2015) has shown that over the 2D unit disk, the classical half-space equation (CHS) for the neutron transport does not capture the correct boundary layer behaviour as long believed. In this pa ... Full text Open Access Cite

Quasi-nonlocal coupling of nonlocal diffusions

Journal Article SIAM Journal on Numerical Analysis · January 1, 2017 We developed a new self-adjoint, consistent, and stable coupling strategy for nonlocal diffusion models, inspired by the quasi-nonlocal atomistic-to-continuum method for crystalline solids. The proposed coupling model is coercive with respect to the energy ... Full text Open Access Cite

Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime

Journal Article SIAM Journal on Mathematical Analysis · January 1, 2017 We study in this work a continuum model derived from a one-dimensional attachmentdetachment-limited type step flow on a vicinal surface, ut = -u2(u3)hhhh, where u, considered as a function of step height h, is the step slope of the surface. We formulate a ... Full text Open Access Cite

An asymptotic preserving method for transport equations with oscillatory scattering coefficients

Journal Article Multiscale Modeling and Simulation · January 1, 2017 We design a numerical scheme for transport equations with oscillatory periodic scattering coefficients. The scheme is asymptotic preserving in the diffusion limit as the Knudsen number goes to zero. It also captures the homogenization limit as the length s ... Full text Open Access Cite

Half-space kinetic equations with general boundary conditions

Journal Article Mathematics of Computation · January 1, 2017 We study half-space linear kinetic equations with general boundary conditions that consist of both given incoming data and various types of reflections, extending our previous work on half-space equations with incoming boundary conditions. As in our previo ... Full text Open Access Cite

Thermalization of oscillator chains with onsite anharmonicity and comparison with kinetic theory.

Journal Article Physical review. E · December 2016 We perform microscopic molecular dynamics simulations of particle chains with an onsite anharmonicity to study relaxation of spatially homogeneous states to equilibrium, and directly compare the simulations with the corresponding Boltzmann-Peierls kinetic ... Full text Open Access Cite

Multiscale implementation of infinite-swap replica exchange molecular dynamics.

Journal Article Proceedings of the National Academy of Sciences of the United States of America · October 2016 Replica exchange molecular dynamics (REMD) is a popular method to accelerate conformational sampling of complex molecular systems. The idea is to run several replicas of the system in parallel at different temperatures that are swapped periodically. These ... Full text Cite

Improved sampling and validation of frozen Gaussian approximation with surface hopping algorithm for nonadiabatic dynamics.

Journal Article The Journal of chemical physics · September 2016 In the spirit of the fewest switches surface hopping, the frozen Gaussian approximation with surface hopping (FGA-SH) method samples a path integral representation of the non-adiabatic dynamics in the semiclassical regime. An improved sampling scheme is de ... Full text Open Access Cite

Traction boundary conditions for molecular static simulations

Journal Article Computer Methods in Applied Mechanics and Engineering · August 15, 2016 This paper presents a consistent approach to prescribe traction boundary conditions in atomistic models. Due to the typical multiple-neighbor interactions, finding an appropriate boundary condition that models a desired traction is a non-trivial task. We f ... Full text Open Access Cite

Decay estimates of discretized Green’s functions for Schrödinger type operators

Journal Article Science China Mathematics · August 1, 2016 For a sparse non-singular matrix A, generally A−1 is a dense matrix. However, for a class of matrices, A−1 can be a matrix with off-diagonal decay properties, i.e., |Aij−1| decays fast to 0 with respect to the increase of a properly defined distance betwee ... Full text Open Access Cite

Localized density matrix minimization and linear-scaling algorithms

Journal Article Journal of Computational Physics · June 15, 2016 We propose a convex variational approach to compute localized density matrices for both zero temperature and finite temperature cases, by adding an entry-wise ℓ1 regularization to the free energy of the quantum system. Based on the fact that the density ma ... Full text Open Access Cite

Sparsifying preconditioner for soliton calculations

Journal Article Journal of Computational Physics · June 15, 2016 We develop a robust and efficient method for soliton calculations for nonlinear Schrödinger equations. The method is based on the recently developed sparsifying preconditioner combined with Newton's iterative method. The performance of the method is demons ... Full text Open Access Cite

PEXSI-$Σ$: A Green's function embedding method for Kohn-Sham density functional theory

Journal Article · June 1, 2016 In this paper, we propose a new Green's function embedding method called PEXSI-$\Sigma$ for describing complex systems within the Kohn-Sham density functional theory (KSDFT) framework, after revisiting the physics literature of Green's function embedding m ... Open Access Link to item Cite

Combining 2D synchrosqueezed wave packet transform with optimization for crystal image analysis

Journal Article Journal of the Mechanics and Physics of Solids · April 1, 2016 We develop a variational optimization method for crystal analysis in atomic resolution images, which uses information from a 2D synchrosqueezed transform (SST) as input. The synchrosqueezed transform is applied to extract initial information from atomic cr ... Full text Open Access Cite

Analysis 0f the divide-and-conquer method for electronic structure calculations

Journal Article Mathematics of Computation · January 1, 2016 We study the accuracy of the divide-and-conquer method for electronic structure calculations. The analysis is conducted for a prototypical subdomain problem in the method. We prove that the pointwise difference between electron densities of the global syst ... Full text Open Access Cite

Gauge-invariant frozen Gaussian approximation method for the schrödinger equation with periodic potentials

Journal Article SIAM Journal on Scientific Computing · January 1, 2016 We develop a gauge-invariant frozen Gaussian approximation (GIFGA) method for the Schrödinger equation (LSE) with periodic potentials in the semiclassical regime. The method generalizes the Herman-Kluk propagator for LSE to the case with periodic media. It ... Full text Open Access Cite

Bloch dynamics with second order Berry phase correction

Journal Article · December 23, 2015 We derive the semiclassical Bloch dynamics with the second-order Berry phase correction in the presence of the slow-varying scalar potential as perturbation. Our mathematical derivation is based on a two-scale WKB asymptotic analysis. For a uniform externa ... Open Access Link to item Cite

Orbital-Free Density Functional Theory of Out-of-Plane Charge Screening in Graphene

Journal Article Journal of Nonlinear Science · December 1, 2015 We propose a density functional theory of Thomas–Fermi–Dirac–von Weizsäcker type to describe the response of a single layer of graphene resting on a dielectric substrate to a point charge or a collection of charges some distance away from the layer. We for ... Full text Open Access Cite

Gentlest ascent dynamics for calculating first excited state and exploring energy landscape of Kohn-Sham density functionals.

Journal Article The Journal of chemical physics · December 2015 We develop the gentlest ascent dynamics for Kohn-Sham density functional theory to search for the index-1 saddle points on the energy landscape of the Kohn-Sham density functionals. These stationary solutions correspond to excited states in the ground stat ... Full text Cite

Fast algorithm for periodic density fitting for Bloch waves

Journal Article · December 1, 2015 We propose an efficient algorithm for density fitting of Bloch waves for Hamiltonian operators with periodic potential. The algorithm is based on column selection and random Fourier projection of the orbital functions. The computational cost of the algorit ... Open Access Link to item Cite

An isoperimetric problem with Coulomb repulsion and attraction to a background nucleus

Journal Article · August 28, 2015 We study an isoperimetric problem the energy of which contains the perimeter of a set, Coulomb repulsion of the set with itself, and attraction of the set to a background nucleus as a point charge with charge $Z$. For the variational problem with constrain ... Open Access Link to item Cite

Diffusion approximations and domain decomposition method of linear transport equations: Asymptotics and numerics

Journal Article Journal of Computational Physics · July 1, 2015 In this paper we construct numerical schemes to approximate linear transport equations with slab geometry by diffusion equations. We treat both the case of pure diffusive scaling and the case where kinetic and diffusive scalings coexist. The diffusion equa ... Full text Open Access Cite

Quantitative Canvas Weave Analysis Using 2-D Synchrosqueezed Transforms: Application of time-frequency analysis to art investigation

Journal Article Signal Processing Magazine, IEEE · July 2015 Quantitative canvas weave analysis has many applications in art investigations of paintings, including dating, forensics, and canvas rollmate identification. Traditionally, canvas analysis is based on X-radiographs. Prior to serving as a painting canvas, a ... Full text Open Access Cite

Numerical scheme for a spatially inhomogeneous matrix-valued quantum Boltzmann equation

Journal Article Journal of Computational Physics · June 5, 2015 We develop an efficient algorithm for a spatially inhomogeneous matrix-valued quantum Boltzmann equation derived from the Hubbard model. The distribution functions are 2 × 2 matrix-valued to accommodate the spin degree of freedom, and the scalar quantum Bo ... Full text Open Access Cite

Emergence of step flow from an atomistic scheme of epitaxial growth in 1+1 dimensions

Journal Article Physical Review E - Statistical, Nonlinear, and Soft Matter Physics · March 4, 2015 The Burton-Cabrera-Frank (BCF) model for the flow of line defects (steps) on crystal surfaces has offered useful insights into nanostructure evolution. This model has rested on phenomenological grounds. Our goal is to show via scaling arguments the emergen ... Full text Open Access Cite

Reactive trajectories and the transition path process

Journal Article Probability Theory and Related Fields · February 2015 Full text Open Access Cite

Efficient rare event simulation for failure problems in random media

Journal Article SIAM Journal on Scientific Computing · January 1, 2015 In this paper we study rare events associated to the solutions of an elliptic partial differential equation with a spatially varying random coefficient. The random coefficient follows the lognormal distribution, which is determined by a Gaussian process. T ... Full text Open Access Cite

Strang splitting methods for a quasilinear Schrödinger equation: Convergence, instability, and dynamics

Journal Article Communications in Mathematical Sciences · January 1, 2015 We study the Strang splitting scheme for quasilinear Schrödinger equations. We establish the convergence of the scheme for solutions with small initial data. We analyze the linear instability of the numerical scheme, which explains the numerical blow-up of ... Full text Open Access Cite

Density matrix minimization with ℓ1 regularization

Journal Article Communications in Mathematical Sciences · January 1, 2015 We propose a convex variational principle to find sparse representation of low-lying eigenspace of symmetric matrices. In the context of electronic structure calculation, this corresponds to a sparse density matrix minimization algorithm with ℓ1 regulariza ... Full text Open Access Cite

Compression of the electron repulsion integral tensor in tensor hypercontraction format with cubic scaling cost

Journal Article Journal of Computational Physics · 2015 © 2015 Elsevier Inc.Electron repulsion integral tensor has ubiquitous applications in electronic structure computations. In this work, we propose an algorithm which compresses the electron repulsion tensor into the tensor hypercontraction format with O(nN2 ... Full text Open Access Cite

Crystal image analysis using 2D synchrosqueezed transforms

Journal Article Multiscale Modeling and Simulation · January 1, 2015 We propose efficient algorithms based on a band-limited version of 2D synchrosqueezed transforms to extract mesoscopic and microscopic information from atomic crystal images. The methods analyze atomic crystal images as an assemblage of nonoverlapping segm ... Full text Open Access Cite

CLASSIFICATION OF WHALE VOCALIZATIONS USING THE WEYL TRANSFORM

Conference 2015 IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING (ICASSP) · 2015 Cite

Nonexistence of a Minimizer for Thomas–Fermi–Dirac–von Weizsäcker Model

Journal Article Communications on Pure and Applied Mathematics · October 2014 Full text Cite

Excitation energies from particle-particle random phase approximation: Davidson algorithm and benchmark studies

Journal Article The Journal of Chemical Physics · September 28, 2014 The particle-particle random phase approximation (pp-RPA) has been used to investigate excitation problems in our recent paper [Y. Yang, H. van Aggelen, and W. Yang, J. Chem. Phys. 139, 224105 (2013)]. It has been shown to be capable of describing ... Full text Cite

Exact dynamical coarse-graining without time-scale separation

Journal Article The Journal of Chemical Physics · July 28, 2014 A family of collective variables is proposed to perform exact dynamical coarse-graining even in systems without time scale separation. More precisely, it is shown that these variables are not slow in general, yet satisfy an overdamped Langevin equa ... Full text Cite

Mathematical theory of solids: From quantum mechanics to continuum models

Journal Article Discrete and Continuous Dynamical Systems · June 2014 Full text Cite

A variational perspective on cloaking by anomalous localized resonance

Journal Article Communications in Mathematical Physics · March 14, 2014 A body of literature has developed concerning “cloaking by anomalous localized resonance.” The mathematical heart of the matter involves the behavior of a divergence-form elliptic equation in the plane, div (a(x) grad u(x)) = f (x). The complex-valued coef ... Full text Open Access Cite

Analysis of time reversible born-oppenheimer molecular dynamics

Journal Article Entropy · January 1, 2014 We analyze the time reversible Born-Oppenheimer molecular dynamics (TRBOMD) scheme, which preserves the time reversibility of the Born-Oppenheimer molecular dynamics even with non-convergent self-consistent field iteration. In the linear response regime, w ... Full text Open Access Cite

Stability of a force-based hybrid method with planar sharp interface

Journal Article SIAM Journal on Numerical Analysis · January 1, 2014 We study a force-based hybrid method that couples an atomistic model with a Cauchy-Born elasticity model with sharp transition interface. We identify stability conditions that guarantee the convergence of the hybrid scheme to the solution of the atomistic ... Full text Open Access Cite

Asymptotic analysis of quantum dynamics in crystals: the Bloch-Wigner transform, Bloch dynamics and Berry phase

Journal Article Acta Mathematicae Applicatae Sinica, English Series · July 2013 Full text Cite

Infinite swapping replica exchange molecular dynamics leads to a simple simulation patch using mixture potentials.

Journal Article The Journal of chemical physics · February 2013 Replica exchange molecular dynamics (REMD) becomes more efficient as the frequency of swap between the temperatures is increased. Recently Plattner et al. [J. Chem. Phys. 135, 134111 (2011)] proposed a method to implement infinite swapping REMD in practice ... Full text Open Access Cite

The landscape of complex networks: Critical nodes and a hierarchical decomposition

Journal Article Methods and Applications of Analysis · 2013 Full text Cite

Seismic modeling using the frozen Gaussian approximation

Conference SEG Technical Program Expanded Abstracts 2013 · 2013 We adopt the frozen Gaussian approximation (FGA) for modeling seismic waves. The FGA method belongs to the category of ray-based beam methods. It decomposes the seismic wavefield into a set of Gaussian functions and propagates these functions along appropr ... Link to item Cite

Convergence of a Force‐Based Hybrid Method in Three Dimensions

Journal Article Communications on Pure and Applied Mathematics · January 2013 AbstractWe study a force‐based hybrid method that couples an atomistic model with the Cauchy‐Born elasticity model. We show that the proposed scheme converges to the solution of the atomistic model with second‐order accurac ... Full text Open Access Cite

Stability and the continuum limit of the spin-polarized Thomas-Fermi-Dirac-von Weizsäcker model

Journal Article Journal of Mathematical Physics · November 1, 2012 The continuum limit of the spin-polarized Thomas-Fermi-Dirac-von Weizsäcker model in an external magnetic field is studied. An extension of the classical Cauchy-Born rule for crystal lattices is established for the electronic structure under sharp ... Full text Open Access Cite

Frozen gaussian approximation for general linear strictly hyperbolic systems: Formulation and eulerian methods

Journal Article Multiscale Modeling and Simulation · September 7, 2012 The frozen Gaussian approximation, proposed in [J. Lu and X. Yang, Commun. Math. Sci., 9 (2011), pp. 663-683], is an efficient computational tool for high frequency wave propagation. We continue in this paper the development of frozen Gaussian approximatio ... Full text Open Access Cite

Convergence of frozen Gaussian approximation for high-frequency wave propagation

Journal Article Communications on Pure and Applied Mathematics · June 1, 2012 The frozen Gaussian approximation provides a highly efficient computational method for high-frequency wave propagation. The derivation of the method is based on asymptotic analysis. In this paper, for general linear strictly hyperbolic systems, we establis ... Full text Open Access Cite

Optimized local basis set for Kohn–Sham density functional theory

Journal Article Journal of Computational Physics · May 2012 Full text Cite

The Kohn-Sham equation for deformed crystals

Journal Article Memoirs of the American Mathematical Society · 2012 Full text Cite

Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool

Journal Article Applied and Computational Harmonic Analysis · March 2011 Full text Cite

The Electronic Structure of Smoothly Deformed Crystals: Wannier Functions and the Cauchy–Born Rule

Journal Article Archive for Rational Mechanics and Analysis · February 2011 Full text Cite

SelInv---An Algorithm for Selected Inversion of a Sparse Symmetric Matrix

Journal Article ACM Transactions on Mathematical Software · February 2011 We describe an efficient implementation of an algorithm for computing selected elements of a general sparse symmetric matrix A that can be decomposed as A Full text Cite

Multiscale modeling

Journal Article Scholarpedia · 2011 Link to item Cite

Markov state models based on milestoning

Journal Article J. Chem. Phys. · 2011 Cite

Fast construction of hierarchical matrix representation from matrix-vector multiplication

Journal Article Journal of Computational Physics · January 1, 2011 We develop a hierarchical matrix construction algorithm using matrix-vector multiplications, based on the randomized singular value decomposition of low-rank matrices. The algorithm uses O(logn) applications of the matrix on structured random test vectors ... Full text Open Access Cite

Frozen Gaussian approximation for high frequency wave propagation

Journal Article Communications in Mathematical Sciences · 2011 We propose the frozen Gaussian approximation for computation of high frequency wave propagation. This method approximates the solution to the wave equation by an integral representation. It provides a highly efficient computational tool based on the asympt ... Open Access Cite

Electronic structure of smoothly deformed crystals: Cauchy‐born rule for the nonlinear tight‐binding model

Journal Article Communications on Pure and Applied Mathematics · November 2010 AbstractThe electronic structure of a smoothly deformed crystal is analyzed using a minimalist model in quantum many‐body theory, the nonlinear tight‐binding model. An extension of the classical Cauchy‐Born rule for crystal ... Full text Cite

Localized bases of eigensubspaces and operator compression

Journal Article Proceedings of the National Academy of Sciences · January 26, 2010 Given a complex local operator, such as the generator of a Markov chain on a large network, a differential operator, or a large sparse matrix that comes from the discretization of a differential operator, we would like to find its best finite dimen ... Full text Cite

Pole-based approximation of the Fermi-Dirac function

Journal Article Chin. Ann. Math. Ser. B · 2009 Open Access Cite

Sequential multiscale modelling using sparse representation

Journal Article Commun. Comput. Phys. · 2008 Cite

The Elastic Continuum Limit of the Tight Binding Model*

Journal Article Chinese Annals of Mathematics, Series B · December 2007 Full text Cite

A sub-linear scaling algorithm for computing the electronic structure of materials

Journal Article Communications in Mathematical Sciences · 2007 Full text Cite

Seamless multiscale modeling via dynamics on fiber bundles

Journal Article Communications in Mathematical Sciences · 2007 Full text Cite

Uniform accuracy of the quasicontinuum method

Journal Article Physical Review B · December 28, 2006 Full text Cite

Removal of Canvas Patterns in Digital Acquisitions of Paintings

Journal Article IEEE Transactions on Image Processing Open Access Cite