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The dual approach to optimal control in the coefficients of nonlocal nonlinear diffusion

Marcus Schytt, Anton Evgrafov

TL;DR

This work develops a rigorous framework for nonlocal diffusion topology optimization governed by a nonlinear $p$-Laplacian. It introduces a dual variational principle for the nonlocal problem, proves well-posedness, strong duality, and antisymmetry of optimal fluxes, and then analyzes a nonlocal coefficient-control problem with existence of optimal designs. A central contribution is the localization analysis: constructing a flux-recovery operator and a nonlinear recovery operator $\ddot{F}_{\delta}$ to connect nonlocal and local problems, and establishing $\Gamma$-convergence of the nonlocal design problem to the local one as the nonlocal horizon $\delta$ vanishes. The results culminate in consistency analyses for $p>2$ and $1<p<2$, supplemented by numerical verification, yielding a rigorous bridge from nonlocal peridynamic diffusion to classical local models with guaranteed convergence of optimal designs.

Abstract

We derive the dual variational principle (principle of minimal complementary energy) for the nonlocal nonlinear scalar diffusion problem, which may be viewed as the nonlocal version of the $p$-Laplacian operator. We establish existence and uniqueness of solutions (two-point fluxes) as well as their quantitative stability, which holds uniformly with respect to the small parameter (nonlocal horizon) characterizing the nonlocality of the problem. We then focus on the nonlocal analogue of the classical optimal control in the coefficient problem associated with the dual variational principle, which may be interpreted as that of optimally distributing a limited amount of conductivity in order to minimize the complementary energy. We show that this nonlocal optimal control problem $Γ$-converges to its local counterpart, when the nonlocal horizon vanishes.

The dual approach to optimal control in the coefficients of nonlocal nonlinear diffusion

TL;DR

This work develops a rigorous framework for nonlocal diffusion topology optimization governed by a nonlinear -Laplacian. It introduces a dual variational principle for the nonlocal problem, proves well-posedness, strong duality, and antisymmetry of optimal fluxes, and then analyzes a nonlocal coefficient-control problem with existence of optimal designs. A central contribution is the localization analysis: constructing a flux-recovery operator and a nonlinear recovery operator to connect nonlocal and local problems, and establishing -convergence of the nonlocal design problem to the local one as the nonlocal horizon vanishes. The results culminate in consistency analyses for and , supplemented by numerical verification, yielding a rigorous bridge from nonlocal peridynamic diffusion to classical local models with guaranteed convergence of optimal designs.

Abstract

We derive the dual variational principle (principle of minimal complementary energy) for the nonlocal nonlinear scalar diffusion problem, which may be viewed as the nonlocal version of the -Laplacian operator. We establish existence and uniqueness of solutions (two-point fluxes) as well as their quantitative stability, which holds uniformly with respect to the small parameter (nonlocal horizon) characterizing the nonlocality of the problem. We then focus on the nonlocal analogue of the classical optimal control in the coefficient problem associated with the dual variational principle, which may be interpreted as that of optimally distributing a limited amount of conductivity in order to minimize the complementary energy. We show that this nonlocal optimal control problem -converges to its local counterpart, when the nonlocal horizon vanishes.
Paper Structure (17 sections, 30 theorems, 148 equations)

This paper contains 17 sections, 30 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Dual formulation of the state problem
  3. Nonlocal physics
  4. The contribution of this work
  5. Outline
  6. Nonlocal divergence and the nonlocal dual variational principle for $p$-Laplacian
  7. Control in the coefficients of the nonlocal $p$-Lapacian
  8. Localization
  9. An upper estimate for the local problem
  10. A lower estimate for the local problem
  11. An informal derivation of $\ddot{F}_\delta$
  12. Rigorous analysis of $\ddot{F}_\delta$
  13. Application to the lower estimate for the local problem and $\Gamma$-convergence
  14. Consistency of $\ddot{F}_\delta$
  15. Analysis of $J_{p,\delta,\epsilon}[\sigma](\cdot)$ for $p>2$
  16. ...and 2 more sections

Key Result

Proposition 2.1

The following statements hold.

Theorems & Definitions (58)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3: Existence of solutions and optimality conditions
  • proof
  • Lemma 2.4: Symmetric and antisymmetric functions
  • proof
  • Corollary 2.5: Antisymmetry of optimal fluxes
  • proof
  • ...and 48 more