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Hongkai Zhao

Ruth F. DeVarney Distinguished Professor of Mathematics
Mathematics
Office hours Wednesday 3-5pm  

Selected Publications


A data-driven and model-based accelerated Hamiltonian Monte Carlo method for Bayesian elliptic inverse problems

Journal Article Statistics and Computing · August 1, 2023 In this paper, we consider a Bayesian inverse problem modeled by elliptic partial differential equations (PDEs). Specifically, we propose a data-driven and model-based approach to accelerate the Hamiltonian Monte Carlo (HMC) method in solving large-scale B ... Full text Cite

How Much Can One Learn a Partial Differential Equation from Its Solution?

Journal Article Foundations of Computational Mathematics · January 1, 2023 In this work, we study the problem of learning a partial differential equation (PDE) from its solution data. PDEs of various types are used to illustrate how much the solution data can reveal the PDE operator depending on the underlying operator and initia ... 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

How much can one learn from a single solution of a PDE?

Journal Article Pure and Applied Functional Analysis · 2023 Cite

Quantitative PAT with simplified P N approximation

Journal Article Inverse Problems · May 1, 2021 The photoacoustic tomography (PAT) is a hybrid modality that combines the optics and acoustics to obtain high resolution and high contrast imaging of heterogeneous media. In this work, our objective is to study the inverse problem in the quantitative step ... Full text Cite

Marchenko-Pastur law with relaxed independence conditions

Journal Article Random Matrices: Theory and Applications · 2021 Full text Cite

Scalable Incremental Nonconvex Optimization Approach for Phase Retrieval

Journal Article Journal of Scientific Computing · 2021 Cite

A Dual Iterative Refinement Method for Non-rigid Shape Matching

Conference Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition · January 1, 2021 In this work, a robust and efficient dual iterative refinement (DIR) method is proposed for dense correspondence between two nearly isometric shapes. The key idea is to use dual information, such as spatial and spectral, or local and global features, in a ... Full text Cite

Robust Inexact Alternating Optimization for Matrix Completion with Outliers

Journal Article Journal of Computational Mathematics · June 2020 Full text Cite

Solving phase retrieval via graph projection splitting

Journal Article Inverse Problems · May 1, 2020 Full text Cite

Efficient and robust shape correspondence via sparsity-enforced quadratic assignment

Journal Article Proceedings / CVPR, IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEE Computer Society Conference on Computer Vision and Pattern Recognition · 2020 Full text Cite

Neural-Response-Based Extreme Learning Machine for Image Classification

Journal Article IEEE Transactions on Neural Networks and Learning Systems · February 2019 Full text Cite

A hybrid adaptive phase space method for reflection traveltime tomography

Journal Article SIAM Journal on Imaging Sciences · January 1, 2019 We present a hybrid imaging method for a challenging traveltime tomography problem which includes both unknown medium and unknown scatterers in a bounded domain. The goal is to recover both the medium and the boundary of the scatterers from the scattering ... Full text Cite

Intrinsic complexity and scaling laws: From random fields to random vectors

Journal Article Multiscale Modeling and Simulation · January 1, 2019 Random fields are commonly used for modeling of spatially (or timely) dependent stochastic processes. In this study, we provide a characterization of the intrinsic complexity of a random field in terms of its second order statistics, e.g., the covariance f ... Full text Open Access Cite

Approximate Separability of the Green's Function of the Helmholtz Equation in the High Frequency Limit

Journal Article Communications on Pure and Applied Mathematics · November 2018 AbstractThe minimum number of terms that are needed in a separable approximation for a Green's function reveals the intrinsic complexity of the solution space of the underlying differential equation. It also has implication ... Full text Cite

A data-driven and model-based accelerated Hamiltonian Monte Carlo method for Bayesian elliptic inverse problems

Journal Article Statistics and Computing · August 1, 2023 In this paper, we consider a Bayesian inverse problem modeled by elliptic partial differential equations (PDEs). Specifically, we propose a data-driven and model-based approach to accelerate the Hamiltonian Monte Carlo (HMC) method in solving large-scale B ... Full text Cite

How Much Can One Learn a Partial Differential Equation from Its Solution?

Journal Article Foundations of Computational Mathematics · January 1, 2023 In this work, we study the problem of learning a partial differential equation (PDE) from its solution data. PDEs of various types are used to illustrate how much the solution data can reveal the PDE operator depending on the underlying operator and initia ... 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

How much can one learn from a single solution of a PDE?

Journal Article Pure and Applied Functional Analysis · 2023 Cite

Quantitative PAT with simplified P N approximation

Journal Article Inverse Problems · May 1, 2021 The photoacoustic tomography (PAT) is a hybrid modality that combines the optics and acoustics to obtain high resolution and high contrast imaging of heterogeneous media. In this work, our objective is to study the inverse problem in the quantitative step ... Full text Cite

Marchenko-Pastur law with relaxed independence conditions

Journal Article Random Matrices: Theory and Applications · 2021 Full text Cite

Scalable Incremental Nonconvex Optimization Approach for Phase Retrieval

Journal Article Journal of Scientific Computing · 2021 Cite

A Dual Iterative Refinement Method for Non-rigid Shape Matching

Conference Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition · January 1, 2021 In this work, a robust and efficient dual iterative refinement (DIR) method is proposed for dense correspondence between two nearly isometric shapes. The key idea is to use dual information, such as spatial and spectral, or local and global features, in a ... Full text Cite

Robust Inexact Alternating Optimization for Matrix Completion with Outliers

Journal Article Journal of Computational Mathematics · June 2020 Full text Cite

Solving phase retrieval via graph projection splitting

Journal Article Inverse Problems · May 1, 2020 Full text Cite

Efficient and robust shape correspondence via sparsity-enforced quadratic assignment

Journal Article Proceedings / CVPR, IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEE Computer Society Conference on Computer Vision and Pattern Recognition · 2020 Full text Cite

Neural-Response-Based Extreme Learning Machine for Image Classification

Journal Article IEEE Transactions on Neural Networks and Learning Systems · February 2019 Full text Cite

A hybrid adaptive phase space method for reflection traveltime tomography

Journal Article SIAM Journal on Imaging Sciences · January 1, 2019 We present a hybrid imaging method for a challenging traveltime tomography problem which includes both unknown medium and unknown scatterers in a bounded domain. The goal is to recover both the medium and the boundary of the scatterers from the scattering ... Full text Cite

Intrinsic complexity and scaling laws: From random fields to random vectors

Journal Article Multiscale Modeling and Simulation · January 1, 2019 Random fields are commonly used for modeling of spatially (or timely) dependent stochastic processes. In this study, we provide a characterization of the intrinsic complexity of a random field in terms of its second order statistics, e.g., the covariance f ... Full text Open Access Cite

Approximate Separability of the Green's Function of the Helmholtz Equation in the High Frequency Limit

Journal Article Communications on Pure and Applied Mathematics · November 2018 AbstractThe minimum number of terms that are needed in a separable approximation for a Green's function reveals the intrinsic complexity of the solution space of the underlying differential equation. It also has implication ... Full text Cite

Variational Hamiltonian Monte Carlo via Score Matching

Journal Article Bayesian Analysis · June 1, 2018 Full text Cite

A weak formulation for the multiphase Stokes flow problem without body fitting grids

Journal Article Pure and Applied Mathematics Quarterly · 2018 Full text Cite

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

Journal Article Research in the Mathematical Sciences · December 2017 Full text Cite

Convergence analysis of the fast sweeping method for static convex Hamilton–Jacobi equations

Journal Article Research in the Mathematical Sciences · December 2016 Full text Cite

Geometric Understanding of Point Clouds Using Laplace-Beltrami Operator

Journal Article Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR) · 2016 Cite

Real-Time Adaptive Video Compression

Journal Article SIAM Journal on Scientific Computing · January 2015 Full text Cite

Computing Surface Uniformization Using Discrete Beltrami Flow

Journal Article SIAM Journal on Scientific Computing · January 2015 Full text Cite

Determining scattering support of anisotropic acoustic mediums and obstacles

Journal Article Communications in Mathematical Sciences · 2015 Full text Cite

A cell based particle method for modeling dynamic interfaces

Journal Article Journal of Computational Physics · September 2014 Full text Cite

A static PDE Approach for MultiDimensional Extrapolation Using Fast Sweeping Methods

Journal Article SIAM Journal on Scientific Computing · January 2014 Full text Cite

Computation of Quasi-Conformal Surface Maps Using Discrete Beltrami Flow

Journal Article SIAM Journal on Imaging Sciences · January 2014 Full text Cite

Partially Blind Deblurring of Barcode from Out-of-Focus Blur

Journal Article SIAM Journal on Imaging Sciences · January 2014 Full text Cite

Exploring accurate Poisson–Boltzmann methods for biomolecular simulations

Journal Article Computational and Theoretical Chemistry · November 2013 Full text Cite

A weak formulation for solving elliptic interface problems without body fitted grid

Journal Article Journal of Computational Physics · September 2013 Full text Cite

An efficient algorithm for the generalized Foldy–Lax formulation

Journal Article Journal of Computational Physics · February 2013 Full text Cite

Quantitative Fluorescence Photoacoustic Tomography

Journal Article SIAM Journal on Imaging Sciences · January 2013 Full text Cite

A local mesh method for solving PDEs on point clouds

Journal Article Inverse Problems & Imaging · 2013 Full text Cite

Solving Partial Differential Equations on Point Clouds

Journal Article SIAM Journal on Scientific Computing · January 2013 Full text Cite

Numerical Poisson–Boltzmann model for continuum membrane systems

Journal Article Chemical Physics Letters · January 2013 Full text Cite

A Hybrid Reconstruction Method for Quantitative PAT

Journal Article SIAM Journal on Imaging Sciences · January 2013 Full text Cite

Analysis of a numerical solver for radiative transport equation

Journal Article Mathematics of Computation · April 11, 2012 Full text Cite

Higher-order schemes for 3D first-arrival traveltimes and amplitudes

Journal Article GEOPHYSICS · March 2012 In the geometrical-optics approximation for the Helmholtz equation with a point source, traveltimes and amplitudes have upwind singularities at the point source. Hence, both first-order and higher-order finite-difference solvers exhibit formally at ... Full text Cite

Numerical Study of Surfactant-Laden Drop-Drop Interactions

Journal Article Communications in Computational Physics · August 2011 AbstractIn this paper, we numerically investigate the effects of surfactant on drop-drop interactions in a 2D shear flow using a coupled level-set and immersed interface approach proposed in (Xu et al., J. Comput. Phys., 21 ... Full text Cite

Robust principal component analysis-based four-dimensional computed tomography

Journal Article Physics in Medicine and Biology · June 7, 2011 Full text Cite

Uniformly Accurate Discontinuous Galerkin Fast Sweeping Methods for Eikonal Equations

Journal Article SIAM Journal on Scientific Computing · January 2011 Full text Cite

Euclidean skeletons using closest points

Journal Article Inverse Problems & Imaging · 2011 Full text Cite

Bioluminescence tomography with Gaussian prior

Journal Article Biomedical Optics Express · December 1, 2010 Full text Cite

Generalized Foldy-Lax formulation

Journal Article Journal of Computational Physics · June 2010 Full text Cite

A Compact Upwind Second Order Scheme for the Eikonal Equation

Journal Article Journal of Computational Mathematics · June 2010 Full text Cite

A Nonparametric Approach for Noisy Point Data Preprocessing

Journal Article International Journal of CAD/CAM · 2010 Full text Cite

A grid based particle method for evolution of open curves and surfaces

Journal Article Journal of Computational Physics · November 2009 Full text Cite

A Fast-Forward Solver of Radiative Transfer Equation

Journal Article Transport Theory and Statistical Physics · September 25, 2009 Full text Cite

Fast sweeping method for the factored eikonal equation

Journal Article Journal of Computational Physics · September 2009 Full text Cite

A grid based particle method for moving interface problems

Journal Article Journal of Computational Physics · May 2009 Full text Cite

On removal of charge singularity in Poisson–Boltzmann equation

Journal Article The Journal of Chemical Physics · April 14, 2009 The Poisson–Boltzmann theory has become widely accepted in modeling electrostatic solvation interactions in biomolecular calculations. However the standard practice of atomic point charges in molecular mechanics force fields introduces singularity ... Full text Cite

A multilevel and multigrid optical tomography based on radiative transfer equation

Journal Article Proceedings of SPIE-OSA Biomedical Optics (Optical Society of America, 2009) · 2009 Full text Cite

Expectation-Maximization Algorithm with Local Adaptivity

Journal Article SIAM Journal on Imaging Sciences · January 2009 Full text Cite

A phase and space coherent direct imaging method

Journal Article The Journal of the Acoustical Society of America · January 1, 2009 A direct imaging algorithm for point and extended targets is presented. The algorithm is based on a physical factorization of the response matrix of a transducer array. The factorization is used to transform a passive target problem to an active so ... Full text Cite

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

Journal Article Journal of Computational Physics · September 2008 Full text Cite

A phase-space formulation for elastic-wave traveltime tomography

Journal Article Journal of Physics: Conference Series · July 1, 2008 Full text Cite

A direct imaging method using far-field data

Journal Article Inverse Problems · August 1, 2007 Full text Cite

A Fast Sweeping Method for Static Convex Hamilton–Jacobi Equations

Journal Article Journal of Scientific Computing · May 2007 Full text Cite

Parallel Implementation of Fast Sweeping Method

Journal Article Journal of Computational Mathematics · 2007 Cite

Fast Sweeping Methods for Eikonal Equations on Triangular Meshes

Journal Article SIAM Journal on Numerical Analysis · January 2007 Full text Cite

High Order Fast Sweeping Methods for Static Hamilton–Jacobi Equations

Journal Article Journal of Scientific Computing · October 24, 2006 Full text Cite

A direct imaging algorithm for extended targets

Journal Article Inverse Problems · August 1, 2006 Full text Cite

Efficient numerical simulation for long range wave propagation

Journal Article Journal of Computational Physics · July 2006 Full text Cite

A level-set method for interfacial flows with surfactant

Journal Article Journal of Computational Physics · March 2006 Full text Cite

Fixed-point Iterative Sweeping Methods for Static Hamilton-Jacobi Equations

Journal Article Methods and Applications of Analysis · 2006 Full text Cite

A Surface Reconstruction Method for Highly Noisy Point Clouds

Journal Article International Workshop on Variational, Geometric, and Level Set Methods in Computer Vision · 2006 Cite

A New Modeling-Based Algorithm for Implicit Surface Polygonization

Journal Article Proceedings of SPIE, Proceedings of Vision Geometry XIII · 2005 Full text Cite

Swept Volumes

Journal Article Computer-Aided Design and Applications · January 2005 Full text Cite

Imaging of location and geometry for extended targets using the response matrix

Journal Article Journal of Computational Physics · September 2004 Full text Cite

A fast sweeping method for Eikonal equations

Journal Article Mathematics of Computation · May 21, 2004 In this paper a fast sweeping method for computing the numerical solution of Eikonal equations on a rectangular grid is presented. The method is an iterative method which uses upwind difference for discretization and uses Gauss-Seidel iterations with al ... Full text Cite

Coupled Parabolic Equations for Wave Propagation

Journal Article Methods and Applications of Analysis · 2004 Full text Cite

Analysis of the Response Matrix for an Extended Target

Journal Article SIAM Journal on Applied Mathematics · January 2004 Full text Cite

Fast Sweeping Algorithms for a Class of Hamilton--Jacobi Equations

Journal Article SIAM Journal on Numerical Analysis · January 2003 Full text Cite

Generalized Snell's Law for Weighted Minimal Surface in Heterogeneous Media

Journal Article Methods and Applications of Analysis · 2003 Full text Cite

Reactive Autophobic Spreading of Drops

Journal Article Journal of Computational Physics · December 2002 Full text Cite

Super-resolution in time-reversal acoustics

Journal Article The Journal of the Acoustical Society of America · January 1, 2002 The phenomenon of super-resolution in time-reversal acoustics is analyzed theoretically and with numerical simulations. A signal that is recorded and then retransmitted by an array of transducers, propagates back though the medium, and refocuses ap ... Full text Cite

Fast Surface Reconstruction and Deformation Using the Level Set Method

Journal Article Proceedings IEEE Workshop on Variational and Level Set Methods in Computer Vision · 2001 Full text Cite

A PDE-Based Fast Local Level Set Method

Journal Article Journal of Computational Physics · November 1999 Full text Cite

Absorbing boundary conditions for domain decomposition

Journal Article Applied Numerical Mathematics · August 1998 Full text Cite

A Variational Level Set Approach to Multiphase Motion

Journal Article Journal of Computational Physics · August 1996 Full text Cite