Search
Results
-
Multilevel quasi-Monte Carlo uncertainty quantification for advection-diffusion-reaction
(2019)SAM Research ReportWe survey the numerical analysis of a class of deterministic, higher-order QMC integration methods in forward and inverse uncertainty quantification algorithms for advection-reaction-diffusion (ARD) equations in polygonal domains $D \subset \mathbb{R}^2$ with distributed uncertain inputs. We admit spatially heterogeneous material properties. For the parametrization of the uncertainty, we assume at hand systems of functions which are ...Report -
Multi-level Monte Carlo Finite Element method for parabolic stochastic partial differential equations
(2011)SAM Research ReportWe analyze the convergence and complexity of multi-level Monte Carlo (MLMC) discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show, under regularity assumptions on the solution that are minimal under certain criteria, that the judicious combination of piecewise linear, continuous multi-level Finite Element discretizations in space and Euler--Maruyama discretizations in time ...Report -
Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations
(2021)SAM Research ReportReport -
Deep learning in high dimension: ReLU network Expression Rates for Bayesian PDE inversion
(2020)SAM Research ReportWe establish dimension independent expression rates by deep ReLU networks for so-called (b,ε,X)-holomorphic functions. These are mappings from [−1,1]N→X, with X being a Banach space, that admit analytic extensions to certain polyellipses in each of the input variables. The significance of this function class has been established in previous works, where it was shown that functions of this type occur widely in uncertainty quantification ...Report -
Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs
(2022)SAM Research ReportWe establish summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions for countably-parametric solutions of linear elliptic and parabolic divergence-form partial differential equations with Gaussian random field inputs. The novel proof technique developed here is based on analytic continuation of parametric solutions into the complex domain. It differs from previous works that used bootstrap arguments ...Report -
Deep ReLU neural network expression for elliptic multiscale problems
(2020)SAM Research ReportReport -
hp-FEM for reaction-diffusion equations. II: Robust exponential convergence for multiple length scales in corner domains
(2020)SAM Research ReportReport -
ReLU Neural Network Galerkin BEM
(2022)SAM Research ReportWe introduce Neural Network (NN for short) approximation architectures for the numerical solution of Boundary Integral Equations (BIEs for short). We exemplify the proposed NN approach for the boundary reduction of the potential problem in two spatial dimensions. We adopt a Galerkin formulation based approach, in polygonal domains with a finite number of straight sides. Trial spaces used in the Galerkin discretization of the BIEs are built ...Report -
-