358 citations · 807 across the 21 of their papers we have counts for
25 papers
On low-rank tensor train approximability for linear nearest neighbor systems
Patrick Gelß, Sebastian Matera, Reinhold Schneider +1
Low-rank tensor methods are an important tool in the numerical treatment of equations with a high-dimensional state space. Nearest neighbor interaction systems like the Ising model…
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…
Approximating optimal feedback controllers of finite horizon control problems using hierarchical tensor formats
Mathias Oster, Leon Sallandt, Reinhold Schneider
Controlling systems of ordinary differential equations (ODEs) is ubiquitous in science and engineering. For finding an optimal feedback controller, the value function and associate…
Approximative Policy Iteration for Exit Time Feedback Control Problems driven by Stochastic Differential Equations using Tensor Train format
Konstantin Fackeldey, Mathias Oster, Leon Sallandt +1
We consider a stochastic optimal exit time feedback control problem. The Bellman equation is solved approximatively via the Policy Iteration algorithm on a polynomial ansatz space…
Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks
Moritz Geist, Philipp Petersen, Mones Raslan +2
We perform a comprehensive numerical study of the effect of approximation-theoretical results for neural networks on practical learning problems in the context of numerical analysi…
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…
Approximating the Stationary Bellman Equation by Hierarchical Tensor Products
Mathias Oster, Leon Sallandt, Reinhold Schneider
We treat infinite horizon optimal control problems by solving the associated stationary Hamilton-Jacobi-Bellman (HJB) equation numerically to compute the value function and an opti…
A Theoretical Analysis of Deep Neural Networks and Parametric PDEs
Gitta Kutyniok, Philipp Petersen, Mones Raslan +1
We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge…
Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations
Martin Eigel, Manuel Marschall, Max Pfeffer +1
Stochastic Galerkin methods for non-affine coefficient representations are known to cause major difficulties from theoretical and numerical points of view. In this work, an adaptiv…
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…
Numerical and Theoretical Aspects of the DMRG-TCC Method Exemplified by the Nitrogen Dimer
Fabian M. Faulstich, Mihály Máté, Andre Laestadius +8
In this article, we investigate the numerical and theoretical aspects of the coupled-cluster method tailored by matrix-product states. We investigate chemical properties of the use…
Analysis of The Tailored Coupled-Cluster Method in Quantum Chemistry
Fabian M. Faulstich, Andre Laestadius, Örs Legeza +2
In quantum chemistry, one of the most important challenges is the static correlation problem when solving the electronic Schrödinger equation for molecules in the Born--Oppenheimer…
A Randomized Tensor Train Singular Value Decomposition
Benjamin Huber, Reinhold Schneider, Sebastian Wolf
The hierarchical SVD provides a quasi-best low rank approximation of high dimensional data in the hierarchical Tucker framework. Similar to the SVD for matrices, it provides a fund…
Low rank tensor recovery via iterative hard thresholding
Holger Rauhut, Reinhold Schneider, Zeljka Stojanac
We study extensions of compressive sensing and low rank matrix recovery (matrix completion) to the recovery of low rank tensors of higher order from a small number of linear measur…
Iterative Methods Based on Soft Thresholding of Hierarchical Tensors
Markus Bachmayr, Reinhold Schneider
We construct a soft thresholding operation for rank reduction of hierarchical tensors and subsequently consider its use in iterative thresholding methods, in particular for the sol…
Tensor product methods and entanglement optimization for ab initio quantum chemistry
Szilárd Szalay, Max Pfeffer, Valentin Murg +4
The treatment of high-dimensional problems such as the Schrödinger equation can be approached by concepts of tensor product approximation. We present general techniques that can be…
Tensor completion in hierarchical tensor representations
Holger Rauhut, Reinhold Schneider, Zeljka Stojanac
Compressed sensing extends from the recovery of sparse vectors from undersampled measurements via efficient algorithms to the recovery of matrices of low rank from incomplete infor…
Tree tensor network state study of the ionic-neutral curve crossing of LiF
V. Murg, F. Verstraete, R. Schneider +2
We present a tree-tensor-network-state (TTNS) method study of the ionic-neutral curve crossing of LiF. For this ansatz, the long-range correlation deviates from the mean-field valu…
Convergence results for projected line-search methods on varieties of low-rank matrices via Łojasiewicz inequality
Reinhold Schneider, André Uschmajew
The aim of this paper is to derive convergence results for projected line-search methods on the real-algebraic variety of real matrices of rank a…
Tensor Product Approximation (DMRG) and Coupled Cluster method in Quantum Chemistry
Örs Legeza, Thorsten Rohwedder, Reinhold Schneider +1
We present the Copupled Cluster (CC) method and the Density matrix Renormalization Grooup (DMRG) method in a unified way, from the perspective of recent developments in tensor prod…
Explicit Green operators for quantum mechanical Hamiltonians. I. The hydrogen atom
Heinz-Jürgen Flad, Gohar Harutyunyan, Reinhold Schneider +1
We study a new approach to determine the asymptotic behaviour of quantum many-particle systems near coalescence points of particles which interact via singular Coulomb potentials.…
Direct minimization for calculating invariant subspaces in density functional computations of the electronic structure
Reinhold Schneider, Thorsten Rohwedder, Alexej Neelov +1
We analyse three related preconditioned steepest descent algorithms, which are partially popular in Hartree-Fock and Kohn-Sham theory as well as invariant subspace computations, fr…
Daubechies wavelets as a basis set for density functional pseudopotential calculations
Luigi Genovese, Alexey Neelov, Stefan Goedecker +8
Daubechies wavelets are a powerful systematic basis set for electronic structure calculations because they are orthogonal and localized both in real and Fourier space. We describe…
Adaptive Eigenvalue Computation - Complexity Estimates
W. Dahmen, T. Rohwedder, R. Schneider +1
This paper is concerned with the design and analysis of a fully adaptive eigenvalue solver for linear symmetric operators. After transforming the original problem into an equivalen…