17 citations · 32 across the 14 of their papers we have counts for
15 papers
Multilevel Sparse Tensor Approximation for High-Dimensional Parametric PDEs
Martin Eigel, Philipp Trunschke, Dana Wrischnig
In this paper the efficiency of multilevel sparse tensor approximation methods for high-dimensional affine parametric diffusion equations is investigated. Methodologically, the rec…
Estimating systematic errors in Bayesian inversion using transport maps
Maren Casfor, Philipp Trunschke, Nando Hegemann +1
In indirect measurements, the sought parameters have to be determined by solving an inverse problem, typically in a Bayesian framework. Often, the accurate numerical simulation of…
Optimal sampling for least squares approximation with general dictionaries
Philipp Trunschke, Anthony Nouy
We consider the problem of approximating an unknown function from point evaluations. This problem is a crucial subproblem in many modern (nonlinear) approximation schemes. When obt…
Sample-based almost-sure quasi-optimal approximation in reproducing kernel Hilbert spaces
Nando Hegemann, Anthony Nouy, Philipp Trunschke
This paper addresses the problem of approximating an unknown function from point evaluations. When obtaining these point evaluations is costly, minimising the required sample size…
Optimal sampling for stochastic and natural gradient descent
Robert Gruhlke, Anthony Nouy, Philipp Trunschke
We consider the problem of optimising the expected value of a loss functional over a nonlinear model class of functions, assuming that we have only access to realisations of the gr…
Weighted sparsity and sparse tensor networks for least squares approximation
Philipp Trunschke, Anthony Nouy, Martin Eigel
Approximation of high-dimensional functions is a problem in many scientific fields that is only feasible if advantageous structural properties, such as sparsity in a given basis, c…
Convergence bounds for local least squares approximation
Philipp Trunschke
We consider the problem of approximating a function in a general nonlinear subset of , when only a weighted Monte Carlo estimate of the -norm can be computed. Of particul…
Adaptive non-intrusive reconstruction of solutions to high-dimensional parametric PDEs
Martin Eigel, Nando Farchmin, Sebastian Heidenreich +1
Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerk…
Convergence bounds for nonlinear least squares and applications to tensor recovery
Philipp Trunschke
We consider the problem of approximating a function in general nonlinear subsets of when only a weighted Monte Carlo estimate of the -norm can be computed. Of particular…
Efficient approximation of high-dimensional exponentials by tensornetworks
Martin Eigel, Nando Farchmin, Sebastian Heidenreich +1
In this work a general approach to compute a compressed representation of the exponential of a high-dimensional function is presented. Such exponential functions play…
A block-sparse Tensor Train Format for sample-efficient high-dimensional Polynomial Regression
Michael Götte, Reinhold Schneider, Philipp Trunschke
Low-rank tensors are an established framework for high-dimensional least-squares problems. We propose to extend this framework by including the concept of block-sparsity. In the co…
Pricing high-dimensional Bermudan options with hierarchical tensor formats
Christian Bayer, Martin Eigel, Leon Sallandt +1
An efficient compression technique based on hierarchical tensors for popular option pricing methods is presented. It is shown that the "curse of dimensionality" can be alleviated f…
Convergence bounds for empirical nonlinear least-squares
Martin Eigel, Reinhold Schneider, Philipp Trunschke
We consider best approximation problems in a nonlinear subset of a Banach space of functions . The norm is assumed to be a generalization o…
The Oracle of DLphi
Dominik Alfke, Weston Baines, Jan Blechschmidt +24
We present a novel technique based on deep learning and set theory which yields exceptional classification and prediction results. Having access to a sufficiently large amount of l…
Variational Monte Carlo - Bridging Concepts of Machine Learning and High Dimensional Partial Differential Equations
Martin Eigel, Reinhold Schneider, Philipp Trunschke +1
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 mod…