Asymptotic compatibility of parametrized optimal design problems

TL;DR

This paper establishes a framework ensuring that solutions to parametrized optimal design problems converge to solutions of their limiting problems, even as the model type changes, with applications to nonlocal and fractional equations.

Contribution

The paper introduces a novel framework for asymptotic compatibility in parametrized optimal design problems, accommodating changing model types and ensuring convergence.

Findings

Proves unconditional convergence for fractional equations.

Demonstrates convergence for nonlocal peridynamics models.

Provides a unified approach for different nonlocal models.

Abstract

We study optimal design problems where the design corresponds to a coefficient in the principal part of the state equation. The state equation, in addition, is parameter dependent, and we allow it to change type in the limit of this (modeling) parameter. We develop a framework that guarantees asymptotic compatibility, that is unconditional convergence with respect to modeling and discretization parameters to the solution of the corresponding limiting problems. This framework is then applied to two distinct classes of problems where the modeling parameter represents the degree of nonlocality. Specifically, we show unconditional convergence of optimal design problems when the state equation is either a scalar-valued fractional equation, or a strongly coupled system of nonlocal equations derived from the bond-based model of peridynamics.