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 manifolds. Our framework is applied to a semilinear
elliptic equation with oscillatory media and a nonlinear radiative transfer
equation; in both cases, significant improvements in efficacy are observed.
This new method does not rely on detailed analytical understanding of the
multiscale PDEs, such as their asymptotic limits, and thus is more versatile
for general multiscale problems.