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A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It…
Recently, it was discovered that for a given function class $\mathbf{F}$ the error of best linear recovery in the square norm can be bounded above by the Kolmogorov width of $\mathbf{F}$ in the uniform norm. That analysis is based on deep…
Sparsity finds applications in areas as diverse as statistics, machine learning, and signal processing. Computations over sparse structures are less complex compared to their dense counterparts, and their storage consumes less space. This…
The low-degree polynomial framework has emerged as a powerful tool for providing evidence of statistical-computational gaps in high-dimensional inference. For detection problems, the standard approach bounds the low-degree advantage through…
In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only…
A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of…
We introduce a Prony-like method to recover a continuous domain 2-D piecewise smooth image from few of its Fourier samples. Assuming the discontinuity set of the image is localized to the zero level-set of a trigonometric polynomial, we…
The sparse polynomial approximation of continuous functions has emerged as a prominent area of interest in function approximation theory in recent years. A key challenge within this domain is the accurate estimation of approximation errors.…
In this paper, we propose a novel sparse recovery method based on the generalized error function. The penalty function introduced involves both the shape and the scale parameters, making it very flexible. The theoretical analysis results in…
This paper addresses the Bayesian calibration of dynamic models with parametric and structural uncertainties, in particular where the uncertain parameters are unknown/poorly known spatio-temporally varying subsystem models. Independent…
We introduce a method to recover a continuous domain representation of a piecewise constant two-dimensional image from few low-pass Fourier samples. Assuming the edge set of the image is localized to the zero set of a trigonometric…
A general stochastic algorithm for solving mixed linear and nonlinear problems was introduced in [11]. We show in this paper how it can be used to solve the fault inverse problem, where a planar fault in elastic half-space and a slip on…
Statistical inverse learning aims at recovering an unknown function $f$ from randomly scattered and possibly noisy point evaluations of another function $g$, connected to $f$ via an ill-posed mathematical model. In this paper we blend…
Partial differential equations often contain unknown functions that are difficult or impossible to measure directly, hampering our ability to derive predictions from the model. Workflows for recovering scalar PDE parameters from data are…
The purpose of this work is the study of solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions. These operators can be realized as the…
Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited…
In this paper, we construct an asymptotic-preserving neural networks (APNNs) [21] for the linearized Boltzmann equation in the acoustic scaling and with uncertain parameters. Utilizing the micro-macro decomposition, we design the loss…
We give the first polynomial-time algorithm for performing linear or polynomial regression resilient to adversarial corruptions in both examples and labels. Given a sufficiently large (polynomial-size) training set drawn i.i.d. from…
In this note we study the problem of sampling and reconstructing signals which are assumed to lie on or close to one of several subspaces of a Hilbert space. Importantly, we here consider a very general setting in which we allow infinitely…
Supervised learning is an active research area, with numerous applications in diverse fields such as data analytics, computer vision, speech and audio processing, and image understanding. In most cases, the loss functions used in machine…