Related papers: Quasi-interpolators with application to postproces…
This work proposes and analyzes a compressed sensing approach to polynomial approximation of complex-valued functions in high dimensions. Of particular interest is the setting where the target function is smooth, characterized by a rapidly…
This paper investigates the numerical approximation of integrals for functions in fractional Gaussian Sobolev spaces $W^s_{p}(\mathbb{R}^d,\gamma)$ with dominating mixed smoothness defined via kernel related to the fractional…
We study a fourth-order div problem and its approximation by the discontinuous Petrov-Galerkin method with optimal test functions. We present two variants, based on first and second-order systems. In both cases we prove well-posedness of…
We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space $\dot W^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of…
We propose a fast and stable method for constructing matrix approximations to fractional integral operators applied to series in the Chebyshev fractional polynomials. This method utilizes a recurrence relation satisfied by the fractional…
We introduce a new class of fractional backward orthogonal functions designed for the spectral approximation of weakly singular adjoint Volterra integral equations. These basis functions generate an approximation space that naturally…
We introduce an interpolation--regression operator for polynomial approximation on the unit sphere $\mathbb{S}^2$ from discrete samples. The approximant is a spherical polynomial of degree $r$ which interpolates the data on a prescribed…
In this paper, a novel pre- and post-processing algorithm is presented that can double the convergence rate of finite element approximations for linear wave problems. In particular, it is shown that a $q$-step pre- and post-processing…
We present novel model reduction methods for rapid solution of parametrized nonlinear partial differential equations (PDEs) in real-time or many-query contexts. Our approach combines reduced basis (RB) space for rapidly convergent…
A numerical analysis for the fully discrete approximation of an operator Lyapunov equation related to linear SPDEs (stochastic partial differential equations) driven by multiplicative noise is considered. The discretization of the Lyapunov…
We introduce the concept of data-driven finite element methods. These are finite-element discretizations of partial differential equations (PDEs) that resolve quantities of interest with striking accuracy, regardless of the underlying mesh…
Certain energy-conservative Galerkin discretizations for nonlinear dispersive wave equations have revealed an unusual convergence behavior: optimal convergence is attained when continuous Lagrange finite element spaces of odd polynomial…
Operator learning has been highly successful for continuous mappings between infinite-dimensional spaces, such as PDE solution operators. However, many operators of interest-including differential operators-are discontinuous or set-valued,…
In this work, we propose a novel formulation for the solution of partial differential equations using finite element methods on unfitted meshes. The proposed formulation relies on the discrete extension operator proposed in the aggregated…
We present two approximate versions of the proximal subgradient method for minimizing the sum of two convex functions (not necessarily differentiable). The algorithms involve, at each iteration, inexact evaluations of the proximal operator…
The challenge to measure exposures regularly forces financial institutions into a choice between an overwhelming computational burden or oversimplification of risk. To resolve this unsettling dilemma, we systematically investigate replacing…
This article considers the extension of two-grid $hp$-version discontinuous Galerkin finite element methods for the numerical approximation of second-order quasilinear elliptic boundary value problems of monotone type to the case when…
We develop and analyze an ultraweak formulation of linear PDEs in nondivergence form where the coefficients satisfy the Cordes condition. Based on the ultraweak formulation we propose discontinuous Petrov--Galerkin (DPG) methods. We…
A new generalization of shifted thin plate splines $$\varphi(x)=(c^{2d}+||x||^{2d})\log\left(c^{2d}+||x||^{2d}\right),\qquad x\in\mathbb{R}^n, d\in \mathbb{N}, c>0$$ is presented to increase the accuracy of quasi-interpolation further. With…
We study approximations of the partition function of dense graphical models. Partition functions of graphical models play a fundamental role is statistical physics, in statistics and in machine learning. Two of the main methods for…