Special Session 142: Recent developments for PDE constrained shape and topological optimization and their applications

Numerical optimization of Stokes eigenvalues

Pedro Antunes University of Lisbon Portugal Co-Author(s):    Nuno Martins

Abstract: We study a volume-constrained shape optimization problem for the eigenvalues of the Stokes operator in two and three dimensions, aiming to identify domains that minimize individual eigenvalues. While the ball is known to satisfy optimality conditions in two dimensions (Henrot, Mazari-Fouquer and Privat), this is no longer the case in three dimensions. We propose a numerical framework based on the Method of Fundamental Solutions (MFS) to approximate the solutions of the Stokes eigenvalue problem. The validity of this approach is supported by density results established in this work. The optimization is carried out using a gradient-type method. To address the difficulties associated with eigenvalue multiplicities, we adopt a smoothing strategy based on lower-order eigenvalues. Instead of computing the shape gradient of the target eigenvalue directly, we consider the gradients of a set of preceding eigenvalues. This approach allows us to define a smooth objective functional, in contrast with the standard formulation, which is typically non-differentiable in a neighborhood of the optimal shape due to eigenvalue multiplicities. The effectiveness of the method is demonstrated through numerical experiments, which highlight qualitative differences between optimal shapes in two and three dimensions. This is a joint work with Nuno Martins (Universidade Nova de Lisboa)

Numerical Exploration of Blaschke--Santal\`o Diagrams Using Neural Networks

Eloi Martinet Insutite for Mathematics, JMU Wuerzburg Germany Co-Author(s):

Abstract: We are interested in the use of neural networks for shape optimization and their application in the exploration of shape functionals, also called Blaschke--Santal\`o diagrams. In a first part, we show how a certain neural network architecture allows one to represent convex bodies in any dimension without constraints. By leveraging the automatic differentiation capabilities of PyTorch, we demonstrate that this representation can be applied effortlessly to shape optimization problems. In a second part, in collaboration with Ilias Ftouhi, we show how to take advantage of this parametrization to provide a faithful numerical description of various Blaschke--Santal\`o diagrams, by representing each convex body as a repulsive electric charge in the diagram.

Graph and domain partitioning based upon escape time optimization

Jeremy L Marzuola University of North Carolina USA Co-Author(s):    Zach Boyd, Nico Fraiman, Peter Mucha, Braxton Osting, Arunima Bhattacharya, Matthias Kurzke, Dan Weser

Abstract: We provide a rearrangement based algorithm for fast detection of subgraphs of k vertices with long escape times for directed or undirected networks. Complementing other notions of densest subgraphs and graph cuts, our method is based on the mean hitting time required for a random walker to leave a designated set and hit the complement. We provide a new relaxation of this notion of hitting time on a given subgraph and use that relaxation to construct a fast subgraph detection algorithm and a generalization to K-partitioning schemes. Using a modification of the subgraph detector on each component, we propose a graph partitioner that identifies regions where random walks live for comparably large times. Importantly, our method implicitly respects the directed nature of the data for directed graphs while also being applicable to undirected graphs. We apply the partitioning method for community detection to a large class of model and real-world data sets. We also discuss existence and boundary regularity for a related algorithm on domains/manifolds.

Convergence of thresholding schemes

Idriss Mazari-Fouquer CEREMADE, Paris Dauphine University PSL France Co-Author(s):    A. Chambolle, Y. Privat

Abstract: We will give an overview of recent progress in the study of quantitative inequalities for optimal control problems. In particular, we will show how they can be used to obtain convergence results for thresholding schemes, which are of great importance in the simulation of optimal control problems. This is a joint work with A. Chambolle and Y. Privat.

Accelerated Rearrangement Methods for Two-Phase PDE-Constrained Optimisation

Seyyed Abbas Mohammadi University of Dundee; Wits University Scotland Co-Author(s):

Abstract: We study a class of PDE-constrained optimisation problems in which the control is restricted to take two values under a global constraint, leading naturally to a topological optimisation framework. Such problems arise in applications including material design, population dynamics, and spectral optimisation. From an analytical perspective, optimal solutions exhibit a bang--bang structure and can be characterised using rearrangement ttheory and symmetrisation techniques, providing insight into the geometry of optimal configurations, although explicit solutions are typically available only in special cases. On the computational side, rearrangement methods offer an efficient iterative approach based on thresholding of state or adjoint variables. These methods are known to converge and, in certain settings, achieve linear convergence rates; however, their performance may still be limited in practice. Motivated by acceleration techniques in optimisation, we introduce an accelerated rearrangement method (ARM) for two-phase problems. The method incorporates momentum-type updates through extrapolation of Fr\`echet derivatives while preserving admissibility of the control. Numerical results demonstrate significantly improved convergence compared to classical rearrangement schemes.

Steklov-Wentzell Eigenvalues of Nearly Spherical Domains

Chee Han Tan National Sun Yat-Sen University Taiwan Co-Author(s):    Robert Viator

Abstract: We consider the Steklov-Wentzell eigenproblem on nearly spherical domains in $\mathbb{R}^{d + 1}$ with $d + 1 \ge 3$, where we include the Laplace-Beltrami operator in the usual Steklov boundary condition. Treating such domains as perturbations of the unit ball, we derive the first-order asymptotic expansion for the Laplace-Beltrami operator and combine this with our previous results from the Steklov eigenproblem to compute the first-order asymptotic expansion for a family of scaled-invariant Steklov-Wentzell eigenvalues. By decomposing the first-order perturbation matrix using the addition theorem for spherical harmonics, we prove that the ball is substationary for an infinite list of scale-invariant Steklov-Wentzell eigenvalues. We extend the analysis to show that the ball is stationary for certain functionals of Steklov-Wentzell eigenvalues and obtain, as a special case, related local isoperimetric results for functionals of Steklov eigenvalues.

Extremal Steklov-Neumann Eigenvalues

Robert Viator Denison University USA Co-Author(s):    Chiu-Yen Kao, Braxton Osting, Chee Han Tan

Abstract: We consider the Steklov-Neumann eigenproblem on an open bounded planar domain $\Omega$ with smooth connected boundary $\Gamma$. We pose the extremal eigenvlue problems of minimizing/maximizing the $k$-th non-trivial Steklov-Neumann eigenvalue among boundary partitions into Steklov subdomains $\Gamma_S$ and Neumann subdomains $\Gamma_N$ of prescribed measure. We formulate a relaxation of these EEPs in terms of $L^{\infty}(\Gamma)$ densities rather than partitions of $\Gamma$, and establish existence and optimality conditions for these relaxed EEPs. We also establish a homogenization result that allows solutions of the relaxed EEPs to infer properties of the original EEPs. Given time, we will explore numerical evidence of symmetries of minimizing arrangements of $\Gamma_S$ and $\Gamma_N$ for the $k$-th Steklov-Neumann eigenvalue of the disk.

Efficient iterative-convolution thresholding methods for interface related optimization problems

Dong Wang The Chinese University of Hong Kong, Shenzhen & Shenzhen International Center for Industrial and Applied Mathematics Peoples Rep of China Co-Author(s):    Dong Wang

Abstract: Interface related problems have a lot of applications in fluid dynamics, material science, image processing, shape and topology optimization, and so on. In this talk, we propose to use indicator functions to implicitly represent the interface, introduce a concave approximation to the energy/objective functional, and derive an unconditionally stable iterative method for interface related problems. We will introduce a topology optimization problem as one example for optimization cases. Some other applications including image segmentation, configuration of foam bubbles, optimal partition will also be presented.