Related papers: Picard iterations for diffusions on symmetric matr…

We analyze, mainly using bifurcation methods, an elliptic superlinear problem in one-dimension with periodic boundary conditions. One of the main novelties is that we follow for the first time a bifurcation approach, relying on a…

In this article we derive Talagrand's $T_2$ inequality on the path space w.r.t. the maximum norm for various stochastic processes, including solutions of one-dimensional stochastic differential equations with measurable drifts, backward…

Some tools and ideas are interchanged between random matrix theory and multivariate statistics. In the context of the random matrix theory, classes of spherical and generalised Wishart random matrix ensemble, containing as particular cases…

The Banach-Picard iteration is widely used to find fixed points of locally contractive (LC) maps. This paper extends the Banach-Picard iteration to distributed settings; specifically, we assume the map of which the fixed point is sought to…

We show that an idea, originating initially with a fundamental recursive iteration scheme (usually referred as "the" Kaczmarz algorithm), admits important applications in such infinite-dimensional, and non-commutative, settings as are…

Many iterative optimization algorithms involve compositions of special cases of Lipschitz continuous operators, namely firmly nonexpansive, averaged and nonexpansive operators. The structure and properties of the compositions are of…

Random matrix theory provides new insights into multiple scattering in random media. In a recent study, we demonstrated the statistical separation of single- and multiple-scattering components based on a Wishart random matrix. The first-…

Ptychography employs a set of diffraction patterns that capture redundant information about an illuminated specimen as a localized beam is moved over the specimen. The robustness of this method comes from the redundancy of the dataset that…

Several recent works have developed a new, probabilistic interpretation for numerical algorithms solving linear systems in which the solution is inferred in a Bayesian framework, either directly or by inferring the unknown action of the…

We study a class of semilinear diffusion equations on infinite, connected, weighted graphs, focusing on two types of nonlinearities: monotone decreasing and Lipschitz continuous. Under minimal structural assumptions on the graph, we…

This paper aims to develop a theory for linear-quadratic Nash systems and Master equations in possibly infinite-dimensional Hilbert spaces. As a first step and motivated by the recent results in [31], we study a more general model in the…

The computation of the matrix exponential is a ubiquitous operation in numerical mathematics, and for a general, unstructured $n\times n$ matrix it can be computed in $\mathcal{O}(n^3)$ operations. An interesting problem arises if the input…

We present the Deep Picard Iteration (DPI) method, a new deep learning approach for solving high-dimensional partial differential equations (PDEs). The core innovation of DPI lies in its use of Picard iteration to reformulate the typically…

This paper is concerned with stochastic processes that model multiple (or iterated) scattering in classical mechanical systems of billiard type, defined below. From a given (deterministic) system of billiard type, a random process with…

In this paper, we investigated the solvability of G-SVIEs under two cases: time-varying Lipschitz coefficients and integral-Lipschitz coefficients. Using the Picard iteration method, we established the existence and uniqueness of solutions…

In this article we show the existence of limiting spectral distribution of a symmetric random matrix whose entries come from a stationary Gaussian process with covariances satisfying a summability condition. We provide an explicit…

For a sufficiently nice 2 dimensional shape, we define its approximating matrix (or patterned matrix) as a random matrix with iid entries arranged according to a given pattern. For large approximating matrices, we observe that the…

Estimating the diagonal entries of a matrix, that is not directly accessible but only available as a linear operator in the form of a computer routine, is a common necessity in many computational applications, especially in image…

We construct solutions to the randomly-forced Navier--Stokes--Poisson system in periodic three-dimensional domains or in the whole three-dimensional Euclidean space. These solutions are weak in the sense of PDEs and also weak in the sense…

A matrix-compression algorithm is derived from a novel isogenic block decomposition for square matrices. The resulting compression and inflation operations possess strong functorial and spectral-permanence properties. The basic observation…