Asymptotic compatibility of parametrized optimal design problems

TL;DR

This work develops an abstract asymptotic-compatibility framework for parametrized optimal design problems where the design enters the principal part and the governing state equation depends on a modeling parameter that may alter the problem type in the limit. The authors establish well-posedness, Gamma-convergence, and discretization-compatibility results that ensure unconditional convergence to the limiting problem as modeling and discretization parameters vanish, and apply the framework to two nonlocal settings: a scalar fractional nonlocal conductivity problem and a vector-valued bond-based peridynamics problem. They provide discrete schemes, convergence proofs, and an AC theory, followed by numerical illustrations using projected gradient descent that demonstrate convergence to the local limit as the nonlocality parameter tends to the local regime. The study extends AC analysis to parametrized optimal design, offering a robust pathway for reliable approximation across nonlocal-to-local transitions in design problems with nonconvex structure and energy-based objectives.

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.

Figures (3)

• Figure 1: Commutative diagram depicting the notion of asymptotic compatibility for optimal design problems; see Definition \ref{['def:ACAbs']}. The symbol $\overset{\rightharpoonup}{\to}$ denotes that, along the corresponding sequence $\tau$, we have convergence of $\phi(\overline{{\mathfrak{a}}}_\tau)$ and $\overline{u}_\tau$ in $X_1$.
• Figure 2: Optimal design and state for $f \equiv 1$ and different values of $s$ and $R$. The value of $s$ is given in each plot, where $R$ is denoted as delta. Notice that the last row corresponds to the local problem.
• Figure 3: Optimal design and state for $f = 3\chi_D$ with $D = B_{\tfrac{1}{4}}(-0.2,0.1)$. The top row depicts the optimal design and state for $s=0.5$ and $R=0.2$. The bottom row corresponds to the local problem.

Theorems & Definitions (54)

• lemma 1: uniform well posedness
• proof
• remark 1: notation
• theorem 1: existence
• proof
• remark 2: lack of uniqueness
• theorem 2: $\Gamma$-convergence
• proof
• corollary 1: convergence of minimizers
• proof