Related papers: Integral Operator Approaches for Scattered Data Fi…
We study the inverse medium scattering problem to reconstruct the unknown inhomogeneous medium from the far-field patterns of scattered waves. The inverse scattering problem is generally ill-posed and nonlinear, and the iterative…
We consider the problem of state estimation in dynamical systems and propose a different mechanism for handling unmodeled system uncertainties. Instead of injecting random process noise, we assign different weights to measurements so that…
Operator learning is a data-driven approximation of mappings between infinite-dimensional function spaces, such as the solution operators of partial differential equations. Kernel-based operator learning can offer accurate, theoretically…
This paper presents a novel {\em Interpolated Factored Green Function} method (IFGF) for the accelerated evaluation of the integral operators in scattering theory and other areas. Like existing acceleration methods in these fields, the IFGF…
The attempt to solve inverse scattering problems often leads to optimization and sampling problems that require handling moderate to large amounts of partial differential equations acting as constraints. We focus here on determining…
For linear ill-posed problems with nontrivial null spaces, Tikhonov regularization and truncated singular value decomposition (TSVD) typically yield solutions that are close to the minimum norm solution. Such a bias is not always desirable,…
This paper is concerned with the introduction of Tikhonov regularization into least squares approximation scheme on $[-1,1]$ by orthonormal polynomials, in order to handle noisy data. This scheme includes interpolation and…
Analyzing the structure of sampled features from an input data distribution is challenging when constrained by limited measurements in both the number of inputs and features. Traditional approaches often rely on the eigenvalue spectrum of…
Distributed estimation in the context of sensor networks is considered, where distributed agents are given a set of sensor measurements, and are tasked with estimating a target variable. A subset of sensors are assumed to be faulty. The…
Regularized kernel methods such as support vector machines (SVM) and support vector regression (SVR) constitute a broad and flexible class of methods which are theoretically well investigated and commonly used in nonparametric…
In this work the numerical solution of acoustic tomography problem based on the iterative and functional-analytical algorithms is considered. The mathematical properties of these algorithms were previously described in works of R.G.Novikov…
Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training…
We consider solving a probably infinite dimensional operator equation, where the operator is not modeled by physical laws but is specified indirectly via training pairs of the input-output relation of the operator. Neural operators have…
We propose a new approach to linear ill-posed inverse problems. Our algorithm alternates between enforcing two constraints: the measurements and the statistical correlation structure in some transformed space. We use a non-linear multiscale…
Neural networks have emerged as effective tools for solving ill-posed inverse problems. In many scientific applications, however, observational training data are insufficient, and learned inverse operators must instead be trained on…
Filtering and smoothing algorithms for linear discrete-time state-space models with skewed and heavy-tailed measurement noise are presented. The algorithms use a variational Bayes approximation of the posterior distribution of models that…
Higher order numerical schemes for stochastic partial differential equations that do not possess commutative noise require the simulation of iterated stochastic integrals. In this work, we extend the algorithms derived by Kloeden, Platen,…
The joint bidiagonalization process of a matrix pair $\{A,L\}$ can be used to develop iterative regularization algorithms for large scale ill-posed problems in general-form Tikhonov regularization…
Functions with discontinuities appear in many applications such as image reconstruction, signal processing, optimal control problems, interface problems, engineering applications and so on. Accurate approximation and interpolation of these…
We present an extension of the linear sampling method for solving the sound-soft inverse scattering problem in two dimensions with data generated by randomly distributed small scatterers. The theoretical justification of our novel sampling…