A Compressive Sampling Approach To Adaptive Multi-Resolution Approximation of Differential Equations With Random Inputs

TL;DR

This paper introduces a compressive sampling method for adaptively approximating solutions to stochastic differential equations using sparse recovery in multi-wavelet bases, demonstrating efficiency with benchmark problems.

Contribution

It presents a novel adaptive approximation technique combining compressive sampling and multi-wavelet bases for stochastic differential equations.

Findings

Accurately approximates mean and variance with fewer samples.

Outperforms traditional Monte Carlo in efficiency.

Validated on three benchmark problems.

Abstract

In this paper, a novel method to adaptively approximate the solution to stochastic differential equations, which is based on compressive sampling and sparse recovery, is introduced. The proposed method consider the problem of sparse recovery with respect to multi-wavelet basis (MWB) from a small number of random samples to approximate the solution to problems. To illustrate the robustness of developed method, three benchmark problems are studied and main statistical features of solutions such as the variance and the mean of solutions obtained by proposed method are compared with the ones obtained from Monte Carlo simulations.