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1. Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

   https://doi.org/10.1051/cocv/2021033  

   Oudet, Édouard; Kao, Chiu-Yen; Osting, Braxton (January 2021, ESAIM: Control, Optimisation and Calculus of Variations)

   null (Ed.)

   Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e. , a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally (doi: 10.1007/s00222-015-0604-x ). In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus γ and b boundary components, we maximize σ j (Σ, g )  L ( ∂ Σ, g ) over a class of smooth metrics, g , where σ j (Σ, g ) is the j th nonzero Steklov eigenvalue and L ( ∂ Σ, g ) is the length of ∂ Σ. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. For genus γ = 0 and b = 2, …, 9, 12, 15, 20 boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which we display in striking images. For higher eigenvalues, numerical evidence suggests that the maximizers are degenerate, but we compute local maximizers for the second and third eigenvalues with b = 2 boundary components and for the third and fifth eigenvalues with b = 3 boundary components. 

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2. Computational approaches for extremal geometric eigenvalue problems

   C.-Y. Kao; B. Osting; and E ́. Oudet (January 2023, Handbook of numerical analysis)

   In an extremal eigenvalue problem, one considers a family of eigenvalue problems, each with discrete spectra, and extremizes a chosen eigenvalue over the family. In this chapter, we consider eigenvalue problems defined on Riemannian manifolds and extremize over the metric structure. For example, we consider the problem of maximizing the principal Laplace–Beltrami eigenvalue over a family of closed surfaces of fixed volume. Computational approaches to such extremal geometric eigenvalue problems present new computational challenges and require novel numerical tools, such as the parameterization of conformal classes and the development of accurate and efficient methods to solve eigenvalue problems on domains with nontrivial genus and boundary. We highlight recent progress on computational approaches for extremal geometric eigenvalue problems, including (i) maximizing Laplace–Beltrami eigenvalues on closed surfaces and (ii) maximizing Steklov eigenvalues on surfaces with boundary. 

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3. Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials

   https://doi.org/10.1017/S0956792521000127  

   CHERKAEV, ELENA; KIM, MINWOO; LIM, MIKYOUNG (May 2021, European Journal of Applied Mathematics)

   null (Ed.)

   The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain. 

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4. Halpern Iteration for Near-Optimal and Parameter-Free Monotone Inclusion and Strong Solutions to Variational Inequalities

   Diakonikolas, J (January 2020, Proceedings of Thirty Third Conference on Learning Theory)

   null (Ed.)

   We leverage the connections between nonexpansive maps, monotone Lipschitz operators, and proximal mappings to obtain near-optimal (i.e., optimal up to poly-log factors in terms of iteration complexity) and parameter-free methods for solving monotone inclusion problems. These results immediately translate into near-optimal guarantees for approximating strong solutions to variational inequality problems, approximating convex-concave min-max optimization problems, and minimizing the norm of the gradient in min-max optimization problems. Our analysis is based on a novel and simple potential-based proof of convergence of Halpern iteration, a classical iteration for finding fixed points of nonexpansive maps. Additionally, we provide a series of algorithmic reductions that highlight connections between different problem classes and lead to lower bounds that certify near-optimality of the studied methods. 

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5. Target signatures for thin surfaces

   https://doi.org/10.1088/1361-6420/ac4154  

   Cakoni, Fioralba; Monk, Peter; Zhang, Yangwen (January 2022, Inverse Problems)

   Abstract We investigate an inverse scattering problem for a thin inhomogeneous scatterer in R m , m = 2, 3, which we model as an m − 1 dimensional open surface. The scatterer is referred to as a screen. The goal is to design target signatures that are computable from scattering data in order to detect changes in the material properties of the screen. This target signature is characterized by a mixed Steklov eigenvalue problem for a domain whose boundary contains the screen. We show that the corresponding eigenvalues can be determined from appropriately modified scattering data by using the generalized linear sampling method. A weaker justification is provided for the classical linear sampling method. Numerical experiments are presented to support our theoretical results. 

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