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We consider the problem of recovering off-the-grid spikes from linear measurements. The state of the art Over-Parametrized Continuous Orthogonal Matching Pursuit (OP-COMP) with Projected Gradient Descent (PGD) successfully recovers those…
The Gradient Scheme framework provides a unified analysis setting for many different families of numerical methods for diffusion equations. We show in this paper that the Gradient Scheme framework can be adapted to elasticity equations, and…
We present a dimension-incremental method for function approximation in bounded orthonormal product bases to learn the solutions of various differential equations. Therefore, we decompose the source function of the differential equation…
We develop both first and second order numerical optimization methods to solve non-smooth optimization problems featuring a shared sparsity penalty, constrained by differential equations with uncertainty. To alleviate the curse of…
Sparse optimization is a central problem in machine learning and computer vision. However, this problem is inherently NP-hard and thus difficult to solve in general. Combinatorial search methods find the global optimal solution but are…
We present a comparison of different multigrid approaches for the solution of systems arising from high-order continuous finite element discretizations of elliptic partial differential equations on complex geometries. We consider the…
In this paper, we consider a finite difference grid-based semi-Lagrangian approach in solving the Vlasov-Poisson (VP) system. Many of existing methods are based on dimensional splitting, which decouples the problem into solving linear…
This paper presents a novel projection-based adaptive algorithm for sparse signal and system identification. The sequentially observed data are used to generate an equivalent sequence of closed convex sets, namely hyperslabs. Each hyperslab…
We present a numerical method for solving the Poisson equation on a nested grid. The nested grid consists of uniform grids having different grid spacing and is designed to cover the space closer to the center with a finer grid. Thus our…
In this paper, we investigate effective sketching schemes via sparsification for high dimensional multilinear arrays or tensors. More specifically, we propose a novel tensor sparsification algorithm that retains a subset of the entries of a…
This article presents two novel adaptive-sparse polynomial dimensional decomposition (PDD) methods for solving high-dimensional uncertainty quantification problems in computational science and engineering. The methods entail global…
In this paper, we present a sparse grid-based Monte Carlo method for solving high-dimensional semi-linear nonlocal diffusion equations with volume constraints. The nonlocal model is governed by a class of semi-linear partial…
A new algorithm for solving large-scale convex optimization problems with a separable objective function is proposed. The basic idea is to combine three techniques: Lagrangian dual decomposition, excessive gap and smoothing. The main…
Spatial frequency estimation from a mixture of noisy sinusoids finds applications in various fields. While subspace-based methods offer cost-effective super-resolution parameter estimation, they demand precise array calibration, posing…
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…
Finite difference schemes are the method of choice for solving nonlinear, degenerate elliptic PDEs, because the Barles-Sougandis convergence framework [Barles and Sougandidis, Asymptotic Analysis, 4(3):271-283, 1991] provides sufficient…
We propose a new solver for the sparse spikes deconvolution problem over the space of Radon measures. A common approach to off-the-grid deconvolution considers semidefinite (SDP) relaxations of the total variation (the total mass of the…
We study a new class of codes for lossy compression with the squared-error distortion criterion, designed using the statistical framework of high-dimensional linear regression. Codewords are linear combinations of subsets of columns of a…
Popular machine learning estimators involve regularization parameters that can be challenging to tune, and standard strategies rely on grid search for this task. In this paper, we revisit the techniques of approximating the regularization…
Discrepancy theory provides powerful tools for producing higher-quality objects which "beat the union bound" in fundamental settings throughout combinatorics and computer science. However, this quality has often come at the price of more…