Sparse Approximate Solution of Partial Differential Equations

TL;DR

This paper introduces an adaptive finite element method leveraging sparsity and compressed sensing principles to efficiently approximate solutions of PDEs, refining only significant solution components.

Contribution

It presents a novel recursive mesh refinement technique using linear programming to target sparse solutions in PDEs, improving efficiency and accuracy.

Findings

Effective approximation of sparse PDE solutions demonstrated

Refined error estimates support the method's reliability

Procedure efficiently identifies significant solution components

Abstract

A new concept is introduced for the adaptive finite element discretization of partial differential equations that have a sparsely representable solution. Motivated by recent work on compressed sensing, a recursive mesh refinement procedure is presented that uses linear programming to find a good approximation to the sparse solution on a given refinement level. Then only those parts of the mesh are refined that belong to large expansion coefficients. Error estimates for this procedure are refined and the behavior of the procedure is demonstrated via some simple elliptic model problems.