MANIFOLD LEARNING AND NONLINEAR HOMOGENIZATION

We describe an efficient domain decomposition-based framework for nonlinear mul-tiscale 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 frame-work 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 a detailed analytical understanding of multiscale PDEs, such as their asymptotic limits, and thus is more versatile for general multiscale problems.