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In Riemannian optimization, it is well known that the condition number of the Riemannian Hessian at an optimum strongly influences the asymptotic convergence behavior of optimization algorithms. On the manifold of symmetric positive…
A new approach to solving eigenvalue optimization problems for large structured matrices is proposed and studied. The class of optimization problems considered is related to computing structured pseudospectra and their extremal points, and…
We examine the adjacency matrices of three-regular graphs representing one-face maps. Numerical studies reveal that the limiting eigenvalue statistics of these matrices are the same as those of much larger, and more widely studied classes…
This paper addresses a multi-scale finite element method for second order linear elliptic equations with arbitrarily rough coefficient. We propose a local oversampling method to construct basis functions that have optimal local…
We obtain results for the spectral optimisation of Neumann eigenvalues on rectangles in $\mathbb{R}^2$ with a measure or perimeter constraint. We show that the rectangle with measure $1$ which maximises the $k$'th Neumann eigenvalue…
In spatial statistics and machine learning, the kernel matrix plays a pivotal role in prediction, classification, and maximum likelihood estimation. A thorough examination reveals that for large sample sizes, the kernel matrix becomes…
We study the variation of the Neumann eigenvalues of the $p$-Laplace operator under quasiconformal perturbations of space domains. This study allows to obtain lower estimates of the Neumann eigenvalues in fractal type domains. The suggested…
Survey about extremum of eigenvalues of geometric operators within a conformal class of a compact riemannian manifold.
This paper is concerned with spectral estimates for the first Dirichlet eigenvalue of the degenerate $p$-Laplace operator in bounded simply connected domains $\Omega \subset \mathbb C$. The proposed approach relies on the conformal analysis…
In this paper we extend to non-compact Riemannian manifolds with boundary the use of two important tools in the geometric analysis of compact spaces, namely, the weak maximum principle for subharmonic functions and the integration by parts.…
In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible…
The Laplace-Beltrami operator on (the surface of) a triaxial ellipsoid admits a sequence of real eigenvalues diverging to plus infinity. By introducing ellipsoidal coordinates, this eigenvalue problem for a partial differential operator is…
Sparse polynomial approximation has become indispensable for approximating smooth, high- or infinite-dimensional functions from limited samples. This is a key task in computational science and engineering, e.g., surrogate modelling in…
In this paper we study the eigenvalues of the laplacian matrices of the cyclic graphs with one edge of weight $\alpha$ and the others of weight $1$. We denote by $n$ the order of the graph and suppose that $n$ tends to infinity. We notice…
We study the statistical properties of the entropic optimal (self) transport problem for smooth probability measures. We provide an accurate description of the limit distribution for entropic (self-)potentials and plans as the…
Modern sample points in many applications no longer comprise real vectors in a real vector space but sample points of much more complex structures, which may be represented as points in a space with a certain underlying geometric structure,…
In this work, we investigate the convergence of numerical approximations to coercivity constants of variational problems. These constants are essential components of rigorous error bounds for reduced-order modeling; extension of these…
Some extremely low-dimensional yet crucial geometric eigen-lengths often determine the success of some geometric tasks. For example, the height of an object is important to measure to check if it can fit between the shelves of a cabinet,…
Consider the Laplacian in a bounded domain in R^d with general (mixed) homogeneous boundary conditions. We prove that its eigenfunctions are `quasi-orthogonal' on the boundary with respect to a certain norm. Boundary orthogonality is proved…
Manifold learning and dimensionality reduction techniques are ubiquitous in science and engineering, but can be computationally expensive procedures when applied to large data sets or when similarities are expensive to compute. To date,…