Sparse Dynamics for Partial Differential Equations

TL;DR

This paper introduces a method that enforces sparsity in the basis representation of PDE solutions using soft thresholding, effectively capturing essential dynamics while reducing computational complexity.

Contribution

The authors propose a novel sparse dynamics approach for PDEs that applies soft thresholding to basis coefficients, promoting sparsity and efficiency in simulating complex differential equations.

Findings

Successfully reduces dynamics of convection and diffusion equations

Effectively captures weak shocks and vorticity with high frequency sources

Promotes sparsity using natural bases derived from PDEs

Abstract

We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high frequency source terms.