Wavelet–Fourier CORSING techniques for multidimensional advection–diffusion–reaction equations

Abstract

We present and analyze a novel wavelet–Fourier technique for the numerical treatment of multidimensional advection–diffusion–reaction equations based on the COmpRessed SolvING (CORSING) paradigm. Combining the Petrov–Galerkin technique with the compressed sensing approach the proposed method is able to approximate the largest coefficients of the solution with respect to a biorthogonal wavelet basis. Namely, we assemble a compressed discretization based on randomized subsampling of the Fourier test space and we employ sparse recovery techniques to approximate the solution to the partial differential equation (PDE). In this paper we provide the first rigorous recovery error bounds and effective recipes for the implementation of the CORSING technique in the multidimensional setting. Our theoretical analysis relies on new estimates for the local $a$-coherence, which measures interferences between wavelet and Fourier basis functions with respect to the metric induced by the PDE operator. The stability and robustness of the proposed scheme are shown by numerical illustrations in the one-, two- and three-dimensional cases.

Sparse recovery in bounded Riesz systems with applications to numerical methods for PDEs

“…Compared to other nonlinear approximation methods for PDEs, such as adaptive finite elements or adaptive wavelets methods (see, e.g., [49] and references therein), CORSING has the advantages that no a posteriori error indicators are needed and that the assembly of the discretization matrix as well as the sparse recovery step (here performed via Orthogonal Matching Pursuit (OMP)) can be easily parallelized. We refer to [16] for a more detailed discussion and to [12,[14][15][16] for numerical experiments for multi-dimensional advection-diffusion-reaction equations and the Stokes problem.…”

“…Here, we will focus on the restricted isometry analysis and sparse recovery guarantees for CORSING. For numerical experiments for multi-dimensional advection-diffusion-reaction equations and for the Stokes problem, we refer the reader to [15,12,16,14]. We also note in passing that the CORSING paradigm can be adapted to the framework of collocation techniques for PDEs (see [13]).…”

“…However, tensorizing the hierarchical basis of hat functions does not preserve orthogonality. In order to obtain a stable discretization, one can instead resort to biorthogonal spline wavelets to obtain a Riesz basis (see [14] for more details).…”