MLCSPG: An efficient and (provably!) accurate solver for high-dimensional parametric PDEs
I present here a method combining weighted compressed sensing and multi-level approximation for solving high-dimensional parametric PDEs.
Exploiting an analytic dependence of the solution map on the parameters of the PDE, we can show that the solution maps admits a generalized polynomial chaos expansion (with respect to the parameters) which is compressible. A weighted version of the traditional compressed sensing allows to recover the coefficient in a Chebychef basis by means of weighted l1 minimizations. The recovery is based on a number of snapshots of the solution obtained via Petrov-Galerkin approximations.
To further reduce computational cost, a multi-level approach is developed in which a coarser estimate is further refined whilst reducing the number of high-quality pointwise solutions. The overall framework yields an approximation within a targer accuracy uniformly for all parameters in a computing time comparable to that of a single estimation at the finer approximation grid.
Dr. Jean-Luc Bouchot Beijing Institute of Technology