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We address composite optimization problems, which consist in minimizing the sum of a smooth and a merely lower semicontinuous function, without any convexity assumptions. Numerical solutions of these problems can be obtained by proximal…
In this work we study a DPG method for an ultra-weak variational formulation of a reaction-diffusion problem. We improve existing a priori convergence results by sharpening an approximation result for the numerical flux. By duality…
Estimators derived from a divergence criterion such as $\varphi-$divergences are generally more robust than the maximum likelihood ones. We are interested in particular in the so-called MD$\varphi$DE, an estimator built using a dual…
Linear system solving is one of the main workhorses in applied mathematics. Recently, theoretical computer scientists have contributed sophisticated algorithms for solving linear systems with symmetric diagonally dominant matrices (a class…
The main purpose of this article is to establish new uniqueness results for Calder\'on type inverse problems related to damped nonlocal wave equations. To achieve this goal we extend the theory of very weak solutions to our setting, which…
We compare the convergence behavior of best polynomial approximations and Legendre and Chebyshev projections and derive optimal rates of convergence of Legendre projections for analytic and differentiable functions in the maximum norm. For…
We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton-Jacobi-Bellman (HJB) equations. We consider diffusion corrected difference-quadrature schemes from the literature and new…
Convergence is proven for Schwarz-like methods applied to degenerate elliptic-parabolic equations with a $p$-structure. This family of PDEs, e.g., arises when modelling nonlinear diffusion processes. The Schwarz-like approximation methods…
This paper concerns with numerical approximations of solutions of second order fully nonlinear partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for second order fully nonlinear…
We consider additive Schwarz methods for boundary value problems involving the $p$-Laplacian. While existing theoretical estimates suggest a sublinear convergence rate for these methods, empirical evidence from numerical experiments…
We propose an asympotically optimal choice of shared concentrated real poles of a family of rational approximants of time-dependent exponential functions $\exp(-tz)$ for $z \geq 0$ and $t$ in a positive time interval $T$. Our result extends…
We establish efficient approximate counting algorithms for several natural problems in local lemma regimes. In particular, we consider the probability of intersection of events and the dimension of intersection of subspaces. Our approach is…
A numerical method for an inverse problem for an elliptic equation with the running source at multiple positions is presented. This algorithm does not rely on a good first guess for the solution. The so-called "approximate global…
Error estimates are rigorously derived for a semi-discrete version of a conservative spectral method for approximating the space-homogeneous Fokker-Planck-Landau (FPL) equation associated to hard potentials. The analysis included shows that…
Consider the problem of minimizing a convex differentiable function on the probability simplex, spectrahedron, or set of quantum density matrices. We prove that the exponentiated gradient method with Armjo line search always converges to…
Inverse problems involving systems of partial differential equations (PDEs) with many measurements or experiments can be very expensive to solve numerically. In a recent paper we examined dimensionality reduction methods, both stochastic…
New numerical algorithms based on rational functions are introduced that can solve certain Laplace and Helmholtz problems on two-dimensional domains with corners faster and more accurately than the standard methods of finite elements and…
A reduced-order model algorithm, based on approximations of Lax pairs, is proposed to solve nonlinear evolution partial differential equations. Contrary to other reduced-order methods, like Proper Orthogonal Decomposition, the space where…
During the past three decades, the advantageous concept of the Green's function has been extended from linear systems to nonlinear ones. At that, there exist a rigorous and an approximate extensions. The rigorous extension introduces the…
In distributed optimization problems, a technique called gradient coding, which involves replicating data points, has been used to mitigate the effect of straggling machines. Recent work has studied approximate gradient coding, which…