Search
Results
-
Compressive Space-Time Galerkin Discretizations of Parabolic Partial Differential Equations
(2015)Research ReportWe study linear parabolic initial-value problems in a space-time variational formulation based on fractional calculus. This formulation uses “time derivatives of order one half” on the bi-infinite time axis. We show that for linear, parabolic initial-boundary value problems on (0, ∞), the corresponding bilinear form admits an inf-sup condition with sparse tensor product trial and test function spaces. We deduce optimality of compressive, ...Report -
Deep ReLU networks and high-order finite element methods II: Chebyshev emulation
(2023)SAM Research ReportExpression rates and stability in Sobolev norms of deep ReLU neural networks (NNs) in terms of the number of parameters defining the NN for continuous, piecewise polynomial functions, on arbitrary, finite partitions \(\mathcal{T}\) of a bounded interval \((a,b)\) are addressed. Novel constructions of ReLU NN surrogates encoding the approximated functions in terms of Chebyshev polynomial expansion coefficients are developed. Chebyshev ...Report -
A-posteriori QMC-FEM error estimation for Bayesian inversion and optimal control with entropic risk measure
(2023)SAM Research ReportWe propose a novel a-posteriori error estimation technique where the target quantities of interest are ratios of high-dimensional integrals, as occur e.g. in PDE constrained Bayesian inversion and PDE constrained optimal control subject to an entropic risk measure. We consider in particular parametric, elliptic PDEs with affine-parametric diffusion coefficient, on high-dimensional parameter spaces. We combine our recent a-posteriori ...Report -
Multilevel Domain Uncertainty Quantification in Computational Electromagnetics
(2022)SAM Research ReportWe continue our study [Domain Uncertainty Quantification in Computational Electromagnetics, JUQ (2020), {\bf 8}:301--341] of the numerical approximation of time-harmonic electromagnetic fields for the Maxwell lossy cavity problem for uncertain geometries. We adopt the same affine-parametric shape parametrization framework, mapping the physical domains to a nominal polygonal domain with piecewise smooth maps. The regularity of the pullback ...Report -
-
Monte-Carlo Finite-Volume Methods in Uncertainty Quantification for Hyperbolic Conservation Laws
(2017)SAM Research ReportReport -
-
Finite elements with mesh refinement for elastic wave propagation in polygons
(2014)Research ReportError estimates for the space-semidiscrete finite element approximation of solutions to initial boundary value problems for linear, second-order hyperbolic systems in bounded polygons G⊂R² inline image with straight sides are presented. Using recent results on corner asymptotics of solutions of linear wave equations with time-independent coefficients in conical domains, it is shown that continuous, simplicial Lagrangian finite elements ...Report -
Multilevel MCMC Bayesian Inversion of Parabolic PDEs under Gaussian Prior
(2020)SAM Research ReportWe analyze the convergence of a multi-level Markov Chain Monte-Carlo (MLMCMC) algorithm for the Bayesian estimation of solution functionals for linear, parabolic partial differential equations subject to uncertain diffusion coefficient. The multilevel convergence analysis is performed for a time-independent, log-gaussian diffusion coefficient and for observations which are assumed to be corrupted by additive, centered gaussian observation ...Report