Tensor product methods and entanglement optimization for ab initio quantum chemistry
The treatment of high‐dimensional problems such as the Schrödinger equation can be
approached by concepts of tensor product approximation. We present general techniques …
approached by concepts of tensor product approximation. We present general techniques …
Low-rank tensor methods for partial differential equations
M Bachmayr - Acta Numerica, 2023 - cambridge.org
Low-rank tensor representations can provide highly compressed approximations of
functions. These concepts, which essentially amount to generalizations of classical …
functions. These concepts, which essentially amount to generalizations of classical …
The alternating linear scheme for tensor optimization in the tensor train format
S Holtz, T Rohwedder, R Schneider - SIAM Journal on Scientific Computing, 2012 - SIAM
Recent achievements in the field of tensor product approximation provide promising new
formats for the representation of tensors in form of tree tensor networks. In contrast to the …
formats for the representation of tensors in form of tree tensor networks. In contrast to the …
Convergence rates for greedy algorithms in reduced basis methods
The reduced basis method was introduced for the accurate online evaluation of solutions to
a parameter dependent family of elliptic PDEs. Abstractly, it can be viewed as determining a …
a parameter dependent family of elliptic PDEs. Abstractly, it can be viewed as determining a …
On manifolds of tensors of fixed TT-rank
S Holtz, T Rohwedder, R Schneider - Numerische Mathematik, 2012 - Springer
Recently, the format of TT tensors (Hackbusch and Kühn in J Fourier Anal Appl 15: 706–722,
2009; Oseledets in SIAM J Sci Comput 2009, submitted; Oseledets and Tyrtyshnikov in SIAM …
2009; Oseledets in SIAM J Sci Comput 2009, submitted; Oseledets and Tyrtyshnikov in SIAM …
On the low-rank approximation by the pivoted Cholesky decomposition
H Harbrecht, M Peters, R Schneider - Applied numerical mathematics, 2012 - Elsevier
The present paper is dedicated to the application of the pivoted Cholesky decomposition to
compute low-rank approximations of dense, positive semi-definite matrices. The resulting …
compute low-rank approximations of dense, positive semi-definite matrices. The resulting …
Constructive quantization: Approximation by empirical measures
S Dereich, M Scheutzow, R Schottstedt - Annales de l'IHP Probabilités …, 2013 - numdam.org
In this article, we study the approximation of a probability measure μ on Rd by its empirical
measure ˆμN interpreted as a random quantization. As error criterion we consider an …
measure ˆμN interpreted as a random quantization. As error criterion we consider an …
A projection method to solve linear systems in tensor format
J Ballani, L Grasedyck - Numerical linear algebra with …, 2013 - Wiley Online Library
In this paper, we propose a method for the numerical solution of linear systems of equations
in low rank tensor format. Such systems may arise from the discretisation of PDEs in high …
in low rank tensor format. Such systems may arise from the discretisation of PDEs in high …
Dynamical approximation by hierarchical Tucker and tensor-train tensors
C Lubich, T Rohwedder, R Schneider… - SIAM Journal on Matrix …, 2013 - SIAM
We extend results on the dynamical low-rank approximation for the treatment of time-
dependent matrices and tensors (Koch and Lubich; see [SIAM J. Matrix Anal. Appl., 29 …
dependent matrices and tensors (Koch and Lubich; see [SIAM J. Matrix Anal. Appl., 29 …
Adaptivity and variational stabilization for convection-diffusion equations∗
In this paper we propose and analyze stable variational formulations for convection diffusion
problems starting from concepts introduced by Sangalli. We derive efficient and reliable a …
problems starting from concepts introduced by Sangalli. We derive efficient and reliable a …