Related papers: Computing spectral measures of self-adjoint operat…
Trigonometric polynomials are widely used for the approximation of a smooth function $f$ from a set of nonuniformly spaced samples $\{f(x_j)\}_{j=0}^{N-1}$. If the samples are perturbed by noise, controlling the smoothness of the…
We present SPEC-RE, a new algorithm to sort complex eigenvalues, generated as the solutions to algebraic equations, whose coefficients are analytic functions of one or many, possibly complex parameters. The fact that the eigenvalues are…
Non-self-adjoint second-order ordinary differential operators on a finite interval with complex weights are studied. Properties of spectral characteristics are established and the inverse problem of recovering operators from their spectral…
This paper establishes convergence rates for learning elliptic pseudo-differential operators, a fundamental operator class in partial differential equations and mathematical physics. In a wavelet-Galerkin framework, we formulate learning…
We address the problem of the exact computation of two joint spectral characteristics of a family of linear operators, the joint spectral radius (in short JSR) and the lower spectral radius (in short LSR), which are well-known different…
We study the spectral theory of operators, generated as direct sums of self-adjoint extensions of quasi-differential minimal operators on a multi-interval set (self-adjoint vector-operators), acting in a Hilbert space. Spectral theorems for…
The problem of variation of spectral subspaces for linear self-adjoint operators under an additive bounded perturbation is considered. The aim is to find the best possible upper bound on the norm of the difference of two spectral…
We develop Spectral-Operator Calculus (SOC), an axiomatic calculus for scalar evaluation of operator-generated spectral observables. This paper (SOC-I) treats the self-adjoint setting, where observables are bounded Borel transforms and…
Recently, sparsity-based algorithms are proposed for super-resolution spectrum estimation. However, to achieve adequately high resolution in real-world signal analysis, the dictionary atoms have to be close to each other in frequency,…
Eigenmaps are important in analysis, geometry, and machine learning, especially in nonlinear dimension reduction. Approximation of the eigenmaps of a Laplace operator depends crucially on the scaling parameter $\epsilon$. If $\epsilon$ is…
A new computational algorithm, the discrete singular convolution (DSC), is introduced for computational electromagnetics. The basic philosophy behind the DSC algorithm for the approximation of functions and their derivatives is studied.…
Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume that operator $L$ generates the analytic semigroup $\{e^{-tL}\}_{t>0}$ whose kernels satisfy the standard Gaussian upper bounds. This paper investigates the sharp weighted…
In a mixed generalized linear model, the goal is to learn multiple signals from unlabeled observations: each sample comes from exactly one signal, but it is not known which one. We consider the prototypical problem of estimating two…
Joint spectra of tuples of operators are subsets in complex projective space. The corresponding tuple of operators can be viewed as an infinite dimensional analog of a determinantal representation of the joint spectrum. We investigate the…
The paper aims to study the spectral properties of elliptic operators with highly inhomogeneous coefficients and related issues concerning wave propagation in high-contrast media. A unified approach to solving problems in bounded domains…
We consider the approximate computation of spectral projectors for symmetric banded matrices. While this problem has received considerable attention, especially in the context of linear scaling electronic structure methods, the presence of…
Proximal algorithms have gained popularity in recent years in large-scale and distributed optimization problems. One such problem is the phase retrieval problem, for which proximal operators have been proposed recently. The phase retrieval…
We consider the numerical approximation of the spectrum of a second-order elliptic eigenvalue problem by the hybridizable discontinuous Galerkin (HDG) method. We show for problems with smooth eigenfunctions that the approximate eigenvalues…
Based on the spectral divide-and-conquer algorithm by Nakatsukasa and Higham [SIAM J. Sci. Comput., 35(3): A1325-A1349, 2013], we propose a new algorithm for computing all the eigenvalues and eigenvectors of a symmetric banded matrix. For…
We approximate the spectral data (eigenvalues and eigenfunctions) of compact Riemannian manifold by the spectral data of a sequence of (computable) discrete Laplace operators associated to some graphs immersed in the manifold. We give an…