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Author: Trunschke, Philipp × Subject: tensor representation ×

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    Convergence bounds for empirical nonlinear least-squares
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Eigel, Martin; Trunschke, Philipp; Schneider, Reinhold
    We consider best approximation problems in a nonlinear subset of a Banach space of functions. The norm is assumed to be a generalization of the L2 norm for which only a weighted Monte Carlo estimate can be computed. The objective is to obtain an approximation of an unknown target function by minimizing the empirical norm. In the case of linear subspaces it is well-known that such least squares approximations can become inaccurate and unstable when the number of samples is too close to the number of parameters. We review this statement for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and we show sufficient conditions for the RIP to be satisfied with high probability. Several model classes are examined where analytical statements can be made about the RIP. Numerical experiments illustrate some of the obtained stability bounds.
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    Variational Monte Carlo - Bridging concepts of machine learning and high dimensional partial differential equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Eigel, Martin; Trunschke, Philipp; Schneider, Reinhold; Wolf, Sebastian
    A statistical learning approach for parametric PDEs related to Uncertainty Quantification is derived. The method is based on the minimization of an empirical risk on a selected model class and it is shown to be applicable to a broad range of problems. A general unified convergence analysis is derived, which takes into account the approximation and the statistical errors. By this, a combination of theoretical results from numerical analysis and statistics is obtained. Numerical experiments illustrate the performance of the method with the model class of hierarchical tensors.