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Spectral algorithms leverage spectral regularization techniques to analyze and process data, providing a flexible framework for addressing supervised learning problems. To deepen our understanding of their performance in real-world…
Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value…
The starting point of this paper is that a spectral method is essentially a combination of an orthonormal basis of the underlying Hilbert space with Galerkin conditions. The choice of an orthonormal basis depends on a number of desirable…
The kernel polynomial method (KPM) is a powerful numerical method for approximating spectral densities. Typical implementations of the KPM require an a prior estimate for an interval containing the support of the target spectral density,…
The properties of the first (largest) eigenvalue and its eigenvector (first eigenvector) are investigated for large sparse random symmetric matrices that are characterized by bimodal degree distributions. In principle, one should be able to…
The goal of this paper is to survey the properties of the eigenvalue relaxation for least squares binary problems. This relaxation is a convex program which is obtained as the Lagrangian dual of the original problem with an implicit compact…
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…
We consider three mathematically equivalent variants of the conjugate gradient (CG) algorithm and how they perform in finite precision arithmetic. It was shown in [{\em Behavior of slightly perturbed Lanczos and conjugate-gradient…
We study generalized eigenvalue problems for meet and join matrices with respect to incidence functions on semilattices. We provide new bounds for generalized eigenvalues of meet matrices with respect to join matrices under very general…
In this paper, we describe a new algorithm that approximates the extreme eigenvalue/eigenvector pairs of a symmetric matrix. The proposed algorithm can be viewed as an extension of the Jacobi eigenvalue method for symmetric matrices…
We present a novel, high-order, efficient, and exponentially convergent shifted Gegenbauer integral pseudospectral method (SGIPSM) to solve numerically Lane-Emden equations provided with some mixed Neumann and Robin boundary conditions. The…
We consider solution of multiply shifted systems of nonsymmetric linear equations, possibly also with multiple right-hand sides. First, for a single right-hand side, the matrix is shifted by several multiples of the identity. Such problems…
We propose a multigrid correction scheme to solve a new Steklov eigenvalue problem in inverse scattering. With this scheme, solving an eigenvalue problem in a fine finite element space is reduced to solve a series of boundary value problems…
We consider the forward problem of uncertainty quantification for the generalised Dirichlet eigenvalue problem for a coercive second order partial differential operator with random coefficients, motivated by problems in structural…
The graph partition problem is the problem of partitioning the vertex set of a graph into a fixed number of sets of given sizes such that the sum of weights of edges joining different sets is optimized. In this paper we simplify a known…
A nonlinear optimization method is proposed for the solution of inverse medium problems with spatially varying properties. To avoid the prohibitively large number of unknown control variables resulting from standard grid-based…
In several geophysical applications, such as full waveform inversion and data modelling, we are facing the solution of inhomogeneous Helmholtz equation. The difficulties of solving the Helmholtz equa- tion are two fold. Firstly, in the case…
This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems…
The recently developed data-driven eigenmatrix method shows very promising reconstruction accuracy in sparse recovery for a wide range of kernel functions and random sample locations. However, its current implementation can lead to…
The dynamic matrix inverse problem is to maintain the inverse of a matrix undergoing element and column updates. It is the main subroutine behind the best algorithms for many dynamic problems whose complexity is not yet well-understood,…