Hermite spectral method with hyperbolic cross approximations to high-dimensional parabolic PDEs

TL;DR

This paper analyzes the Hermite spectral method combined with hyperbolic cross approximations to efficiently solve high-dimensional parabolic PDEs, demonstrating exponential convergence and error estimates.

Contribution

It provides the first error estimates for Hermite spectral methods with hyperbolic cross approximations in high-dimensional PDEs, showing exponential convergence.

Findings

Exponential convergence of hyperbolic cross approximations with Hermite functions.

Error estimates established in weighted Korobov spaces.

Numerical results confirm theoretical convergence rates.

Abstract

It is well-known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the "curse of dimensionality" in some degree. In this note, we first give the error estimate of hyperbolic cross (HC) approximations with generalized Hermite functions. The exponential convergence in both regular and optimized hyperbolic cross approximations has been shown. Moreover, the error estimate of Hermite spectral method to high-dimensional linear parabolic PDEs with HC approximations has been investigated in the properly weighted Korobov spaces. The numerical result verifies the exponential convergence of this approach.