Related papers: Spectral discretization errors in filtered subspac…
We analyze numerical approximation of the fractional elliptic problem $L^{\beta}u=f$, ${\beta>0}$, where $L$ is a second-order self-adjoint elliptic operator with homogeneous Dirichlet or Neumann boundary conditions. The paper develops a…
Solutions of partial differential equations can often be written as surface integrals having a kernel related to a singular fundamental solution. Special methods are needed to evaluate the integral accurately at points on or near the…
Discrete regularization methods are often applied for obtaining stable approximate solutions for ill-posed operator equations $Tx=y$, where $T: X\to Y$ is a bounded operator between Hilbert spaces with non-closed range $R(T)$ and $y\in…
We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the $\ell_0$ norm. Specifically, the $\ell_0$ model has an objective function that is the sum of a convex fidelity term and a…
Phase retrieval in real or complex Hilbert spaces is the task of recovering a vector, up to an overall unimodular multiplicative constant, from magnitudes of linear measurements. In this paper, we assume that the vector is normalized, but…
In this paper, a new iterative two-level algorithm is presented for solving the finite element discretization for nonsymmetric or indefinite elliptic problems. The iterative two-level algorithm uses the same coarse space as the traditional…
We study the convergence of a random iterative sequence of a family of operators on infinite dimensional Hilbert spaces, inspired by the Stochastic Gradient Descent (SGD) algorithm in the case of the noiseless regression, as studied in [1].…
When modeling propagation and scattering phenomena using integral equations discretized by the boundary element method, it is common practice to approximate the boundary of the scatterer with a mesh comprising elements of size approximately…
The method of second order relative spectra has been shown to reliably approximate the discrete spectrum for a self-adjoint operator. We extend the method to normal operators and find optimal convergence rates for eigenvalues and…
In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method…
We propose a new random sketching approach for embedding high-dimensional Hilbert-Schmidt operators, using random input-output pairs. Such operator can then be approximated in a low-dimensional subspace of operators by solving a small…
This paper deals with the design of discrete-time algorithms for the robust filtering differentiator. Two discrete-time realizations of the filtering differentiator are introduced. The first one, which is based on an exact discretization of…
Low-rank tensor approximation techniques attempt to mitigate the overwhelming complexity of linear algebra tasks arising from high-dimensional applications. In this work, we study the low-rank approximability of solutions to linear systems…
We provide lower error bounds for randomized algorithms that approximate integrals of functions depending on an unrestricted or even infinite number of variables. More precisely, we consider the infinite-dimensional integration problem on…
In this article, we study eigenvalue problems associated to self-adjoint operators and their approximation obtained by subspace projection, as used in the reduced basis method for instance. We provide error bounds between the exact…
We consider a class of statistical inverse problems involving the estimation of a regression operator from a Polish space to a separable Hilbert space, where the target lies in a vector-valued reproducing kernel Hilbert space induced by an…
We establish a spectral duality for certain unbounded operators in Hilbert space. The class of operators includes discrete graph Laplacians arising from infinite weighted graphs. The problem in this context is to establish a practical…
We start by showing how to approximate unitary and bounded self-adjoint operators by operators in finite dimensional spaces. Using ultraproducts we give a precise meaning for the approximation. In this process we see how the spectral…
This article deals with the computation of guaranteed lower bounds of the error in the framework of finite element (FE) and domain decomposition (DD) methods. In addition to a fully parallel computation, the proposed lower bounds separate…
This work concerns the distance in 2-norm from a matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Perturbations are allowed only on the constant coefficient…