Related papers: Stochastic Galerkin reduced basis methods for para…
In this paper we apply the stochastic variance reduced gradient (SVRG) method, which is a popular variance reduction method in optimization for accelerating the stochastic gradient method, to solve large scale linear ill-posed systems in…
In this work we analyze the inverse problem of recovering the space-dependent potential coefficient in an elliptic / parabolic problem from distributed observation. We establish novel (weighted) conditional stability estimates under very…
In uncertainty quantification, critical parameters of mathematical models are substituted by random variables. We consider dynamical systems composed of ordinary differential equations. The unknown solution is expanded into an orthogonal…
Partial differential equations can be used to model many problems in several fields of application including, e.g., fluid mechanics, heat and mass transfer, and electromagnetism. Accurate discretization methods (e.g., finite element or…
A local weighted discontinuous Galerkin gradient discretization method for solving elliptic equations is introduced. The local scheme is based on a coarse grid and successively improves the solution solving a sequence of local elliptic…
While much research effort has been dedicated to scaling up sparse Gaussian process (GP) models based on inducing variables for big data, little attention is afforded to the other less explored class of low-rank GP approximations that…
The high-order accurate continuous Galerkin finite element method offers attractive computational efficiency for computational fluid dynamics. A challenge is however spurious oscillations which result for convection dominated flows over…
In recent years, there has been an increasing interest in using deep learning and neural networks to tackle scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods, such…
We develop efficient hierarchical preconditioners for optimal control problems governed by partial differential equations with uncertain coefficients. Adopting a discretize-then-optimize framework that integrates finite element…
In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations…
We extend the finite element method introduced by Lakkis and Pryer [2011] to approximate the solution of second order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin (DG) framework. This is done by viewing…
A novel and scalable geometric multi-level algorithm is presented for the numerical solution of elliptic partial differential equations, specially designed to run with high occupancy of streaming processors inside Graphics Processing…
In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems.…
In this study, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the…
We develop and analyze a nonlinear reduced basis (RB) method for parametrized elliptic partial differential equations based on a binary-tree partition of the parameter domain into tensor-product structured subdomains. Each subdomain is…
Critical points of energy functionals, which are of broad interest, for instance, in physics and chemistry, in solid and quantum mechanics, in material science, or in general diffusion-reaction models arise as solutions to the associated…
Under the guidance of the general theory developed for classical partial differential equations (PDEs), we investigate the Riesz bases of wavelets in the spaces where fractional PDEs usually work, and their applications in numerically…
We consider the computation of averaged coefficients for the homogenization of elliptic partial differential equations. In this problem, like in many multiscale problems, a large number of similar computations parametrized by the…
Training nonlinear parametrizations such as deep neural networks to numerically approximate solutions of partial differential equations is often based on minimizing a loss that includes the residual, which is analytically available in…
Neural networks are increasingly recognized as a powerful numerical solution technique for partial differential equations (PDEs) arising in diverse scientific computing domains, including quantum many-body physics. In the context of…