相关论文: Adaptive non-intrusive reconstruction of solutions…
We examine nonlinear dynamical systems of ordinary differential equations or differential algebraic equations. In an uncertainty quantification, physical parameters are replaced by random variables. The inner variables as well as a quantity…
We propose convenient inferential methods for potentially nonstationary multivariate unobserved components models with fractional integration and cointegration. Based on finite-order ARMA approximations in the state space representation,…
We study approximation of the embedding $\ell_p^m \hookrightarrow \ell_q^m$, $1 \leq p < q \leq \infty$, based on randomized algorithms that use up to $n$ arbitrary linear functionals as information on a problem instance where $n \ll m$. By…
We propose and analyze deterministic multilevel approximations for Bayesian inversion of operator equations with uncertain distributed parameters, subject to additive Gaussian measurement data. The algorithms use a multilevel (ML) approach…
To solve high-dimensional parameter-dependent partial differential equations (pPDEs), a neural network architecture is presented. It is constructed to map parameters of the model data to corresponding finite element solutions. To improve…
In this work we investigate the numerical identification of the diffusion coefficient in elliptic and parabolic problems using neural networks. The numerical scheme is based on the standard output least-squares formulation where the…
In this research, we propose an online basis enrichment strategy within the framework of a recently developed constraint energy minimizing generalized multiscale discontinuous Galerkin method (CEM-GMsDGM). Combining the technique of…
In this paper we present an adaptive discretization technique for solving elliptic partial differential equations via a collocation radial basis function partition of unity method. In particular, we propose a new adaptive scheme based on…
We introduce the proximal Galerkin (PG) method for non-symmetric variational inequalities. The proposed approach is asymptotically mesh-independent and yields constraint-preserving approximations. We present both a conforming PG formulation…
Motivated by considering partial differential equations arising from conservation laws posed on evolving surfaces, a new numerical method for an advection problem is developed and simple numerical tests are performed. The method is based on…
We propose a method for the approximation of solutions of PDEs with stochastic coefficients based on the direct, i.e., non-adapted, sampling of solutions. This sampling can be done by using any legacy code for the deterministic problem as a…
We study the time-dependent Navier-Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion, and…
This paper develops and analyzes a class of semi-discrete and fully discrete weak Galerkin finite element methods for unsteady incompressible convective Brinkman-Forchheimer equations. For the spatial discretization, the methods adopt the…
This paper presents an $hp$ a posteriori error analysis for the 2D Helmholtz equation that is robust in the polynomial degree $p$ and the wave number $k$. For the discretization, we consider a discontinuous Galerkin formulation that is…
We propose a high-order adaptive numerical solver for the semilinear elliptic boundary value problem modelling magnetic plasma equilibrium in axisymmetric confinement devices. In the fixed boundary case, the equation is posed on curved…
The state-of-the art proof of a global inf-sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite, second-order linear elliptic PDEs. This paper, therefore, first analyses a nonconforming finite…
This study presents an aposteriori error analysis of adaptive finite element approximations of parabolic boundary control problems with bilateral box constraints that act on a Neumann boundary. The control problem is discretized using the…
We extend the applicability of the popular interior-penalty discontinuous Galerkin (dG) method discretizing advection-diffusion-reaction problems to meshes comprising extremely general, essentially arbitrarily-shaped element shapes. In…
Machine learning methods have been lately used to solve partial differential equations (PDEs) and dynamical systems. These approaches have been developed into a novel research field known as scientific machine learning in which techniques…
As a first step towards a mathematically rigorous understanding of adaptive spectral/$hp$ discretizations of elliptic boundary-value problems, we study the performance of adaptive Legendre-Galerkin methods in one space dimension. These…