A sparse FFT approach for ODE with random coefficients

TL;DR

This paper introduces a sparse FFT-based method for solving ODEs with coefficients depending on spatial and random variables, enabling efficient high-dimensional approximation without restrictive ansatz functions.

Contribution

It develops a novel automatic sparse FFT approach for high-dimensional ODEs with random coefficients, addressing variable influence and coupling without prior restrictions.

Findings

Efficient approximation of ODE solutions with random coefficients.

Automatic variable influence and coupling detection.

Reduced computational costs compared to existing methods.

Abstract

The paper presents a general strategy to solve ordinary differential equations (ODE), where some coefficient depend on the spatial variable and on additional random variables. The approach is based on the application of a recently developed dimension-incremental sparse fast Fourier transform. Since such algorithms require periodic signals, we discuss periodization strategies and associated necessary deperiodization modifications within the occuring solution steps. The computed approximate solutions of the ODE depend on the spatial variable and on the random variables as well. Certainly, one of the crucial challenges of the high dimensional approximation process is to rate the influence of each variable on the solution as well as the determination of the relations and couplings within the set of variables. The suggested approach meets these challenges in a full automatic manner with reasonable computational costs, i.e., in contrast to already existing approaches, one does not need to seriously restrict the used set of ansatz functions in advance.