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Neural and gpc operator surrogates: construction and expression rate bounds
(2022)SAM Research ReportApproximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising e.g. as data-to-solution maps of linear and nonlinear partial differential equations. Specifically, we study approximation rates for Deep Neural Operator and Generalized Polynomial Chaos (gpc) Operator surrogates for nonlinear, holomorphic maps between infinite-dimensional, separable Hilbert spaces. Operator in- and outputs ...Report -
Multilevel approximation of Gaussian random fields: Covariance compression, estimation and spatial prediction
(2021)SAM Research ReportCentered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded domains in Euclidean space or smooth, compact and orientable manifolds are determined by their covariance operators. We consider centered GRFs given sample-wise as variational solutions to coloring operator equations driven by spatial white noise, with pseudodifferential coloring operator being elliptic, self-adjoint and positive from the Hörmander class. ...Report -
Constructive Deep ReLU Neural Network Approximation
(2021)SAM Research ReportWe propose an efficient, deterministic algorithm for constructing exponentially convergent deep neural network (DNN) approximations of multivariate, analytic maps f:[−1,1]K→R. We address in particular networks with the rectified linear unit (ReLU) activation function. Similar results and proofs apply for many other popular activation functions. The algorithm is based on collocating f in deterministic families of grid points with small ...Report -
Deep ReLU Neural Network Expression Rates for Data-to-QoI Maps in Bayesian PDE Inversion
(2020)SAM Research ReportFor Bayesian inverse problems with input-to-response maps given by well-posed partial differential equations (PDEs) and subject to uncertain parametric or function space input, we establish (under rather weak conditions on the ``forward'', input-to-response maps) the parametric holomorphy of the data-to-QoI map relating observation data 𝛿� to the Bayesian estimate for an unknown quantity of interest (QoI). We prove exponential expression ...Report -
Quasi-Monte Carlo Bayesian estimation under Besov priors in elliptic inverse problems
(2019)SAM Research ReportReport -
Multilevel approximation of Gaussian random fields: fast simulation
(2019)SAM Research ReportReport -
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 -
Uncertainty Quantification for Spectral Fractional Diffusion: Sparsity Analysis of Parametric Solutions
(2018)SAM Research ReportReport -
Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients
(2017)SAM Research ReportReport -
Multilevel QMC with Product Weights for Affine-Parametric, Elliptic PDEs
(2016)Research reports / Seminar for Applied MathematicsReport