Enhancing adaptive sparse grid approximations and improving refinement strategies using adjoint-based a posteriori error estimates

TL;DR

This paper introduces an adaptive sparse grid algorithm that leverages adjoint-based a posteriori error estimates to improve approximation accuracy and refine strategies for quantities of interest in PDEs.

Contribution

It presents a novel framework combining adjoint-based error estimates with sparse grid adaptivity, enhancing accuracy and efficiency over traditional methods.

Findings

Enhanced approximation accuracy using adjoint-based error estimates.

Alternative refinement strategies outperform traditional hierarchical surplus methods.

Framework effectively balances physical discretization and stochastic interpolation errors.

Abstract

In this paper we present an algorithm for adaptive sparse grid approximations of quantities of interest computed from discretized partial differential equations. We use adjoint-based a posteriori error estimates of the physical discretization error and the interpolation error in the sparse grid to enhance the sparse grid approximation and to drive adaptivity of the sparse grid. Utilizing these error estimates provides significantly more accurate functional values for random samples of the sparse grid approximation. We also demonstrate that alternative refinement strategies based upon a posteriori error estimates can lead to further increases in accuracy in the approximation over traditional hierarchical surplus based strategies. Throughout this paper we also provide and test a framework for balancing the physical discretization error with the stochastic interpolation error of the enhanced sparse grid approximation.