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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. Surface Registration with Eigenvalues and Eigenvectors

   https://doi.org/10.1109/TVCG.2019.2915567  

   Hamidian, Hajar; Zhong, Zichun; Fotouhi, Farshad; Hua, Jing (May 2019, IEEE Transactions on Visualization and Computer Graphics)

   This paper presents a novel surface registration technique using the spectrum of the shapes, which can facilitate accurate localization and visualization of non-isometric deformations of the surfaces. In order to register two surfaces, we map both eigenvalues and eigenvectors of the Laplace-Beltrami of the shapes through optimizing an energy function. The function is defined by the integration of a smoothness term to align the eigenvalues and a distance term between the eigenvectors at feature points to align the eigenvectors. The feature points are generated using the static points of certain eigenvectors of the surfaces. By using both the eigenvalues and the eigenvectors on these feature points, the computational efficiency is improved considerably without losing the accuracy in comparison to the approaches that use the eigenvectors for all vertices. After the alignment, the eigenvectors can be employed to calculate the point-to-point correspondence of the surfaces. Therefore, the proposed method can accurately define the displacement of the vertices. We evaluate our method by conducting experiments on synthetic and real data using hippocampus, heart, and hand models. We also compare our method with non-rigid ICP and a similar spectrum-based methods. These experiments demonstrate the advantages and accuracy of our method. 

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3. Surface Registration with Eigenvalues and Eigenvectors

   Hamidian, Hajar; Zhong, Zichun; Fotouhi, Farshad; Hua, Jing (January 2019, IEEE transactions on visualization and computer graphics)

   This paper presents a novel surface registration technique using the spectrum of the shapes, which can facilitate accurate localization and visualization of non-isometric deformations of the surfaces. In order to register two surfaces, we map both eigenvalues and eigenvectors of the Laplace-Beltrami of the shapes through optimizing an energy function. The function is defined by the integration of a smoothness term to align the eigenvalues and a distance term between the eigenvectors at feature points to align the eigenvectors. The feature points are generated using the static points of certain eigenvectors of the surfaces. By using both the eigenvalues and the eigenvectors on these feature points, the computational efficiency is improved considerably without losing the accuracy in comparison to the approaches that use the eigenvectors for all vertices. In our technique, the variation of the shape is expressed using a scale function defined at each vertex. Consequently, the total energy function to align the two given surfaces can be defined using the linear interpolation of the scale function derivatives. Through the optimization of the energy function, the scale function can be solved and the alignment is achieved. After the alignment, the eigenvectors can be employed to calculate the point-to-point correspondence of the surfaces. Therefore, the proposed method can accurately define the displacement of the vertices. We evaluate our method by conducting experiments on synthetic and real data using hippocampus, heart, and hand models. We also compare our method with non-rigid Iterative Closest Point (ICP) and a similar spectrum-based methods. These experiments demonstrate the advantages and accuracy of our method 

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4. Morse inequalities for ordered eigenvalues of generic self-adjoint families

   https://doi.org/10.1007/s00222-024-01284-y  

   Berkolaiko, Gregory; Zelenko, Igor (October 2024, Inventiones mathematicae)

   In many applied problems one seeks to identify and count the critical points of a particular eigenvalue of a smooth parametric family of self-adjoint matrices, with the parameter space often being known and simple, such as a torus. Among particular settings where such a question arises are the Floquet–Bloch decomposition of periodic Schrödinger operators, topology of potential energy surfaces in quantum chemistry, spectral optimization problems such as minimal spectral partitions of manifolds, as well as nodal statistics of graph eigenfunctions. In contrast to the classical Morse theory dealing with smooth functions, the eigenvalues of families of self-adjoint matrices are not smooth at the points corresponding to repeated eigenvalues (called, depending on the application and on the dimension of the parameter space, the diabolical/Dirac/Weyl points or the conical intersections). This work develops a procedure for associating a Morse polynomial to a point of eigenvalue multiplicity; it utilizes the assumptions of smoothness and self-adjointness of the family to provide concrete answers. In particular, we define the notions of non-degenerate topologically critical point and generalized Morse family, establish that generalized Morse families are generic in an appropriate sense, establish a differential first-order conditions for criticality, as well as compute the local contribution of a topologically critical point to the Morse polynomial. Remarkably, the non-smooth contribution to the Morse polynomial turns out to depend only on the size of the eigenvalue multiplicity and the relative position of the eigenvalue of interest and not on the particulars of the operator family; it is expressed in terms of the homologies of Grassmannians. 

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5. Radial Basis Approximation of Tensor Fields on Manifolds: From Operator Estimation to Manifold Learning

   Harlim, John; Jiang, Shixiao W; Peoples, John W (November 2023, Journal of machine learning research)

   Mahoney, Michael (Ed.)

   In this paper, we study the Radial Basis Function (RBF) approximation to differential operators on smooth tensor fields defined on closed Riemannian submanifolds of Euclidean space, identified by randomly sampled point cloud data. The formulation in this paper leverages a fundamental fact that the covariant derivative on a submanifold is the projection of the directional derivative in the ambient Euclidean space onto the tangent space of the submanifold. To differentiate a test function (or vector field) on the submanifold with respect to the Euclidean metric, the RBF interpolation is applied to extend the function (or vector field) in the ambient Euclidean space. When the manifolds are unknown, we develop an improved second-order local SVD technique for estimating local tangent spaces on the manifold. When the classical pointwise non-symmetric RBF formulation is used to solve Laplacian eigenvalue problems, we found that while accurate estimation of the leading spectra can be obtained with large enough data, such an approximation often produces irrelevant complex-valued spectra (or pollution) as the true spectra are real-valued and positive. To avoid such an issue, we introduce a symmetric RBF discrete approximation of the Laplacians induced by a weak formulation on appropriate Hilbert spaces. Unlike the non-symmetric approximation, this formulation guarantees non-negative real-valued spectra and the orthogonality of the eigenvectors. Theoretically, we establish the convergence of the eigenpairs of both the Laplace-Beltrami operator and Bochner Laplacian for the symmetric formulation in the limit of large data with convergence rates. Numerically, we provide supporting examples for approximations of the Laplace-Beltrami operator and various vector Laplacians, including the Bochner, Hodge, and Lichnerowicz Laplacians. 

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