Wavelet-based Methods for Numerical Solutions of Differential Equations

TL;DR

This paper explores the advantages of wavelet-based methods in numerically solving differential equations, highlighting their efficiency, sparse matrix properties, and successful application to high-frequency problems.

Contribution

It demonstrates the effectiveness of wavelet methods for differential equations, including high wave number Helmholtz problems, and discusses their advantages over traditional approaches.

Findings

Wavelet methods produce sparse matrices with small condition numbers.

Effective in solving high wave number Helmholtz equations.

Showcases wavelet efficiency in one-dimensional differential equations.

Abstract

Wavelet theory has been well studied in recent decades. Due to their appealing features such as sparse multiscale representation and fast algorithms, wavelets have enjoyed many tremendous successes in the areas of signal/image processing and computational mathematics. This paper primarily intends to shed some light on the advantages of using wavelets in the context of numerical differential equations. We shall identify a few prominent problems in this field and recapitulate some important results along these directions. Wavelet-based methods for numerical differential equations offer the advantages of sparse matrices with uniformly bounded small condition numbers. We shall demonstrate wavelets' ability in solving some one-dimensional differential equations: the biharmonic equation and the Helmholtz equation with high wave numbers (of magnitude $O(10^4)$ or larger).