Directional differentiability for shape optimization with variational inequalities as constraints

IF 1.3 3区数学 Q4 AUTOMATION & CONTROL SYSTEMS

Esaim-Control Optimisation and Calculus of Variations Pub Date : 2023-07-21 DOI:10.1051/cocv/2023056

V. A. Kovtunenko, K. Kunisch

引用次数: 0

Abstract

For equilibrium constrained optimization problems subject to nonlinear state equations, the property of directional differentiability with respect to a parameter is studied. An abstract class of parameter dependent shape optimization problems is investigated with penalty constraints linked to variational inequalities. Based on the Lagrange multiplier approach, on smooth penalties due to Lavrentiev regularization, and on adjoint operators, a shape derivative is obtained. The explicit formula provides a descent direction for the gradient algorithm identifying the shape of the breaking-line from a boundary measurement. A numerical example is presented for a nonlinear Poisson problem modeling Barenblatt’s surface energies and non-penetrating cracks.

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以变分不等式为约束的形状优化的方向可微性

对于非线性状态方程的平衡约束优化问题，研究了其对参数的方向可微性。研究了一类抽象的参数依赖形状优化问题，该问题具有与变分不等式相关的惩罚约束。基于拉格朗日乘子方法，在Lavrentiev正则化的光滑惩罚和伴随算子上，得到了一个形状导数。该显式公式为从边界测量中识别断线形状的梯度算法提供了下降方向。给出了一个模拟Barenblatt表面能和非穿透性裂纹的非线性泊松问题的数值例子。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Esaim-Control Optimisation and Calculus of Variations Mathematics-Computational Mathematics

自引率

7.10%

发文量

期刊介绍： ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations. Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines. Targeted topics include: in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory; in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis; in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.