A duality-based optimization approach for model adaptivity in heterogeneous multiscale problems

TL;DR

This paper presents a duality-based optimization framework for adaptive modeling in heterogeneous multiscale problems, enabling goal-oriented model tuning without strict a priori knowledge, validated through numerical experiments on complex diffusion and advection problems.

Contribution

It introduces a novel optimization-based approach for model adaptivity using the Dual Weighted Residual method, allowing systematic tuning of models based on local error indicators.

Findings

Effective in heterogeneous, random coefficient problems

Enables goal-oriented model refinement

Improves accuracy in multiscale simulations

Abstract

This paper introduces a novel framework for model adaptivity in the context of heterogeneous multiscale problems. The framework is based on the idea to interpret model adaptivity as a minimization problem of local error indicators, that are derived in the general context of the Dual Weighted Residual (DWR) method. Based on the optimization approach a post-processing strategy is formulated that lifts the requirement of strict a priori knowledge about applicability and quality of effective models. This allows for the systematic, "goal-oriented" tuning of effective models with respect to a quantity of interest. The framework is tested numerically on elliptic diffusion problems with different types of heterogeneous, random coefficients, as well as an advection-diffusion problem with strong microscopic, random advection field.