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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. Numerical studies of the Steklov eigenvalue problem via conformal mappings

   https://doi.org/j.amc.2018.11.048  

   Alhejaili, Weaam; Kao, Chiu-Yen (April 2019, Applied mathematics and computation)

   In this paper, spectral methods based on conformal mappings are proposed to solve the Steklov eigenvalue problem and its related shape optimization problems in two dimensions. To apply spectral methods, we first reformulate the Steklov eigenvalue problem in the complex domain via conformal mappings. The eigenfunctions are expanded in Fourier series so the discretization leads to an eigenvalue problem for coefficients of Fourier series. For shape optimization problem, we use a gradient ascent approach to find the optimal domain which maximizes k-th Steklov eigenvalue with a fixed area for a given k. The coefficients of Fourier series of mapping functions from a unit circle to optimal domains are obtained for several different k. 

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3. 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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4. Predicting Protein Folding Pathways with Quadratic Constraints on Rates of Entropy Change: A Nonlinear Optimization-Based Control Approach

   https://doi.org/10.1115/1.4068504  

   Mohammadi, Alireza; Spong, Mark W (April 2025, Journal of Dynamic Systems, Measurement, and Control)

   Abstract This paper addresses the problem of algorithmic prediction of protein folding pathways, namely, the transient three-dimensional conformations of protein molecules during folding, under constrained rates of entropy change. We formulate the physics-based prediction of folding pathways as a control synthesis problem, where the control inputs guide the protein folding simulations. These folding control inputs are obtained from largescale trust-region subproblems (TRS) utilizing a computationally efficient algorithm with no need for outer iterations. The proposed control synthesis approach, which leverages the solutions obtained from a special generalized eigenvalue problem, avoids potentially cumbersome and unpredictable iterative computations at each protein conformation. Moreover, the TRS-based control inputs align the closed-loop dynamics closely with the kinetostatic compliance method (KCM) reference vector field while satisfying ellipsoidal constraints on the folding control inputs. Finally, we provide conditions for existence and uniqueness of the resulting closed-loop solutions, which are the protein folding pathways under constraints on the rate of entropy change. Numerical simulations utilizing the KCM approach on protein backbones confirm the effectiveness of the proposed framework. 

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5. A Spectral Approach to Nondestructive Testing via Electromagnetic Waves

   https://doi.org/10.1109/TAP.2021.3090810  

   Cakoni, Fioralba; Cogar, Samuel; Monk, Peter (December 2021, IEEE transactions on antennas and propagation)

   In recent years, a new approach has been proposed in the study of the inverse scattering problem for electromagnetic waves. In particular, a study is made of the analytic properties of the scattering operator, and the results of this study are used to design target signatures that respond to changes in the electromagnetic parameters of the scattering medium. These target signatures are characterized by novel eigenvalue problems such that the eigenvalues can be determined from measured scattering data. Changes in the structural properties of the material or the presence of flaws cause changes in the measured eigenvalues. In this article, we provide a general framework for developing target signatures and numerical evidence of the efficacy of new target signatures based on recently introduced eigenvalue problems arising in electromagnetic scattering theory for anisotropic media. 

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