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We design and analyze an iterative two-grid algorithm for the finite element discretizations of strongly nonlinear elliptic boundary value problems in this paper. We propose an iterative two-grid algorithm, in which a nonlinear problem is…
In this paper, we study the stabilizer-free weak Galerkin methods on polytopal meshes for a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient. With certain…
We consider a minimal residual discretization of a simultaneous space-time variational formulation of parabolic evolution equations. Under the usual `LBB' stability condition on pairs of trial- and test spaces we show quasi-optimality of…
In this paper, a new iterative two-level algorithm is presented for solving the finite element discretization for nonsymmetric or indefinite elliptic problems. The iterative two-level algorithm uses the same coarse space as the traditional…
Elliptic partial differential equations arise in many fields of science and engineering such as steady state distribution of heat, fluid dynamics, structural/mechanical engineering, aerospace engineering and seismology etc. In three…
Differential privacy has become a cornerstone in the development of privacy-preserving learning algorithms. This work addresses optimizing differentially private kernel learning within the empirical risk minimization (ERM) framework. We…
Developing efficient methods for solving parametric partial differential equations is crucial for addressing inverse problems. This work introduces a Least-Squares-based Neural Network (LS-Net) method for solving linear parametric PDEs. It…
Motivated by many applications in complex domains with boundaries exposed to large topological changes or deformations, fictitious domain methods regard the actual domain of interest as being embedded in a fixed Cartesian background. This…
In this article, we present a method for increasing adaptivity of an existing robust estimation algorithm by learning two parameters to better fit the residual distribution. The analyzed method uses these two parameters to calculate weights…
We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of…
In this contribution, kernel approximations are applied as ansatz functions within the Deep Ritz method. This allows to approximate weak solutions of elliptic partial differential equations with weak enforcement of boundary conditions using…
We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three…
Partial differential equation (PDE) models with multiple temporal/spatial scales are prevalent in several disciplines such as physics, engineering, and many others. These models are of great practical importance but notoriously difficult to…
This paper proposes a new gradient-based optimization approach for designing optimal feedback kernels for parabolic distributed parameter systems with boundary control. Unlike traditional kernel optimization methods for parabolic systems,…
We present a framework for solving partial different equations on evolving surfaces. Based on the grid-based particle method (GBPM) [18], the method can naturally resample the surface even under large deformation from the motion law. We…
This paper introduces a nonconforming virtual element method for general second-order elliptic problems with variable coefficients on domains with curved boundaries and curved internal interfaces. We prove arbitrary order optimal…
The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length and timescales. Often, it is computationally intractable to resolve the finest features…
We construct least squares formulations of PDEs with inhomogeneous essential boundary conditions, where boundary residuals are not measured in unpractical fractional Sobolev norms, but which formulations nevertheless are shown to yield a…
This paper presents a numerical method for variable coefficient elliptic PDEs with mostly smooth solutions on two dimensional domains. The PDE is discretized via a multi-domain spectral collocation method of high local order (order 30 and…
We study the discretization of an elliptic partial differential equation, posed on a two- or three-dimensional domain with smooth boundary, endowed with a generalized Robin boundary condition which involves the Laplace-Beltrami operator on…