Related papers: On the structure of measures constrained by linear…
Neural networks are versatile tools for computation, having the ability to approximate a broad range of functions. An important problem in the theory of deep neural networks is expressivity; that is, we want to understand the functions that…
PDE-constrained optimal control problems require regularisation to ensure well-posedness, introducing small perturbations that make the solutions challenging to approximate accurately. We propose a finite element approach that couples both…
Truss structures composed of members that work exclusively in tension or in compression appear in several problems of science and engineering, e.g., in the study of the resisting mechanisms of masonry structures, as well as in the design of…
We present recent advances in the analysis of constrained optimization problems with constraints given by singular mappings obtained within the framework of the $p$-regularity theory developed over the last twenty years. In particular, we…
These lecture notes grew out of a series of lectures given by the second named author in short courses in Toulouse, Matsumoto, and Darmstadt. The main aim is to explain some aspects of the theory of "Regularity structures" developed…
A generic construction of linear codes over finite fields has recently received a lot of attention, and many one-weight, two-weight and three-weight codes with good error correcting capability have been produced with this generic approach.…
A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory solutions can be defined in algebraically closed p-adic fields.…
Detectability of failures of linear programming (LP) decoding and the potential for improvement by adding new constraints motivate the use of an adaptive approach in selecting the constraints for the underlying LP problem. In this paper, we…
This primary purpose of this paper is to succinctly state a number of verifiable and tractable sufficient conditions under which a particular class of conservative signal processing structures may be readily used to solve a companion class…
This note complements the paper "The quest for optimal sampling: Computationally efficient, structure-exploiting measurements for compressed sensing" [2]. Its purpose is to present a proof of a result stated therein concerning the recovery…
We study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) which include, in particular, the stochastic porous medium equations and the stochastic fast diffusion equation. We propose a fully…
Under very mild assumptions, we give formulas for the correlation and local dimensions of measures on the limit set of a Moran construction by means of the data used to construct the set.
Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural network, and the neural network parameters are calibrated to…
The aim of this article is to study the asymptotic behaviour for large times of solutions to a certain class of stochastic partial differential equations of parabolic type. In particular, we will prove the backward uniqueness result and the…
Optimization problems constrained by partial differential equations (PDEs) naturally arise in scientific computing, as those constraints often model physical systems or the simulation thereof. In an implicitly constrained approach, the…
The purpose of this article is to survey some of the context, achievements, challenges and mysteries of the field of metric dimension reduction, including new perspectives on major older results as well as recent advances.
Nonlinear filtering is a pivotal problem that has attracted significant attention from mathematicians, statisticians, engineers, and various other scientific disciplines. The solution to this problem is governed by the so-called filtering…
We establish existence and uniqueness of global, bounded weak solutions to quasilinear PDEs with bounded, uniformly continuous initial data and investigate their properties. Moreover, we establish existence of bounded weak solutions when…
We investigate a PDE-constrained optimization problem, with an intuitive interpretation in terms of the design of robust membranes made out of an arbitrary number of different materials. We prove existence and uniqueness of solutions for…
We survey a variety of results about partially isometric matrices. We focus primarily on results that are distinctly finite-dimensional. For example, we cover a recent solution to the similarity problem for partial isometries. We also…