# Wavelet-Fourier CORSING techniques for multi-dimensional advection-diffusion-reaction equations

## TL;DR

This paper introduces a wavelet-Fourier CORSING method for efficiently solving multi-dimensional advection-diffusion-reaction equations, combining compressed sensing with Petrov-Galerkin techniques to achieve accurate coefficient recovery.

## Contribution

It provides the first rigorous error bounds and implementation strategies for multi-dimensional wavelet-Fourier CORSING methods in PDEs.

## Key findings

- Achieves accurate approximation of PDE solutions using sparse recovery.
- Provides stability and robustness demonstrated through numerical experiments.
- Develops new estimates for local a-coherence in multi-dimensional settings.

## Abstract

We present and analyze a novel wavelet-Fourier technique for the numerical treatment of multidimensional advection-diffusion-reaction equations based on the CORSING (COmpRessed SolvING) 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 PDE. In this paper, we provide the first rigorous recovery error bounds and effective recipes for the implementation of the CORSING technique in the multi-dimensional 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 is shown by numerical illustrations in the one-, two-, and three-dimensional case.