Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

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Session Overview
MS 16a: high order, tensor-structured methods and low rank approximation
Thursday, 15/July/2021:
12:00pm - 2:00pm

Session Chair: Carlo Marcati
Session Chair: Christoph Schwab
Session Chair: Maxim Rakhuba
Virtual location: Zoom 2

Session Abstract

Tensor compression based numerical methods have been, in recent years, successfully em-

ployed for approximation and solution of PDEs in a wide range of scientific domains, provid-

ing at least the high order performance of hp and spectral methods. Unlike the latter methods

though, their deployment does not require explicit coding of higher-order discretizations. In-

stead, tensor compression methods realize a form of high order adaptivity by means of singular

value decomposition at runtime in tensor-formatted numerics.

In addition, tensor formatted methods have proven to be useful both in tackling high dimen-

sional problems —by overcoming the curse of dimensionality— and in reducing the complexity

and providing highly accurate approximations in low dimensional settings. Indeed, tensor com-

pression techniques work by identifying and taking advantage of approximate low-rank struc-

tures in data. Like spectral and high order methods, they adaptively exploit the regularity of the

underlying functions to obtain compact and rapidly converging representations.

The goal of this minisymposium is to bring together specialists in the field of tensor com-

pression methods and discuss recent advances in their mathematical analysis and their compu-

tational implementation, with a particular focus on their application on model order reduction

at runtime for the approximation of the solution of physical problems and of partial differen-

tial equations. The minisymposium covers, but is not restricted to, topics such as: analysis and

implementation of tensor train and hierarchical tensor decompositions, quantized tensor ap-

proximation, space-time tensor decomposition and dynamic low rank approximation, tensor

formatted solution of partial differential equations.

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12:00pm - 12:30pm

Tensorized Adaptive Biasing Force method for molecular dynamics simulations (joint work with Tony Lelièvre and Pierre Monmarché)

Virginie Ehrlacher

Ecole des Ponts ParisTech & INRIA, France

This talk will focus on the presentation some recent developments in molecular dynamics simulations based on tensor methods.

One objective of molecular dynamics simulations is to compute averages of some microscopic quantities depending on the positions of the different atoms or molecules composing the molecular system with respect to probability measures of interest, for instance the Gibbs measure. These averages can be interpreted as macroscopic quantities associated to the molecular system, such as the pressure inside a fluid for instance.

Standard methods for computing such avergages, which amounts to computing integrals of functions defined on a very high-dimensional space, is to use Monte-Carlo methods. More precisely, some time-dependent stochastic dynamics, which are ergodic with respect to the probability measure of interest, are simulated and the desired averages can be obatined as long-time limits of time averages along one trajectory of the stochastic process. For instance, the so-called overdamped Langevin dynmaics is on example of Monte-Carlo process that is known to be regodic with respect to the Gibbs measure.

While this approach should enable in principle to overcome the so-called curse of dimensionality, it faces yet another challenging difficulty which is called the "metastability problem". Indeed, the convergence of the long time limits of the trajectorial averages may be very slow because of the fact that the stochastic process remains "trapped" during very long times into some metastable states. There are several numerical approaches to avoid this problem and accelerate the convergence in the long-time limit of these trajectorial averages. Among them, a very popular one is the so-called "Adaptive Biasing Force Method" (ABF). This method makes use of a certain number of coarse degrees of freedom of the molecular system, like the distance between two particular atoms, or the angle between two particular bonds, in order to bias the original stochastic dynamics and accelerate the convergence of the trajectorial averages.

However, the ABF method cannot be used in practice in situations where the number of relevant coarse degrees of freedom of ths sytem is large.

The aim of this talk is to present recent results obtained in a joint work with Tony Lelièvre and Pierre Monmarché about the theoretical properties of a Tensorized Adaptive Biasing Force (TABF) algorithm, where tensor methods are coupled together with the ABF method in order to allow simulations where the number of coarse degrees of freedom is large. Numerical experiments which illustrate the efficiency of the method and demonstrate the fact that it enables to capture complex correlation effects will also be presented on large molecular systems.

12:30pm - 1:00pm

Tensor numerical methods in quantum chemistry: from Hartree-Fock energies to optical spectra of compact molecules

Venera Khoromskaya

Max-Planck Institute for Mathematics in the Sciences, Germany

The novel tensor numerical methods appeared as bridging of the algebraic tensor ecompositions originating from chemometrics and the nonlinear approximation theory on separable low-rank representation of multivariate functions and operators. Based on representation of $d$-variate functions and operators in the rank-structured tensor formats, they provide $O(dn)$ complexity of numerical calculations instead of $O(n^d)$ by conventional methods [7].

A starting point was the Hartree-Fock (HF) solver based on the tensor calculation of all involved operators in a Gausssian-type basis, discretized on $ n\times n\times n$ 3D Cartesian grids.

For all operators in the non-linear 3D integro-differential HF equation the usual 3D analytical integration is substituted by the grid-based tensor algorithms in $O(n\log n)$ complexity [1,3].

The 4th order tensor of the two-electron integrals (TEI)) is computed only as selected columns of its Cholesky factorization [2]. This construction of TEI is a unique prerequisite for efficient calculation of the excitation energies for molecules by using the Bethe-Salpeter equation (BSE) which relaxes the numerical costs to $O(N^2)$ in a size of atomic orbitals basis set $N$ [5], instead of practically intractable $O(N^6)$ by conventional diagonalization of the BSE matrix.

The block-diagonal plus low-rank approximation to the large BSE matrix is constructed using the Cholesky form of TEI, enabling fast iterative solution for the smallest in modulus eigenvalues (corresponding to optical spectra) by using the Sherman-Morrison-Woodbury scheme [6]. Using this decomposition of the BSE matrix we also developed an economical method [6] for calculating the density of states (DOS) for optical spectra of compact molecules. Numerical examples for small amino acid molecules are presented.

[1] B. N. Khoromskij, V. Khoromskaia and H.-J. Flad. Numerical solution of the Hartree-Fock equation in multilevel tensor-structured format. SIAM J. Sci. Comp., 33 (1), 45-65, 2011.

[2] V. Khoromskaia, B.N. Khoromskij and R. Schneider. Tensor-structured calculation of the two-electron integrals in a general basis. SIAM J. Sci. Comp., {\bf 35} (2), A987-A1010, 2013.

[3] V. Khoromskaia. Black-box Hartree-Fock solver by tensor numer. methods. CMAM, 14 (1), 89-111, 2014.

[4] P. Benner, V. Khoromskaia and B. N. Khoromskij. A Reduced basis approach for calculation of the Bethe-Salpeter excitation energies using low-rank tensor factorizations. Mol. Physics, 114 (7-8) 2016.

[5] P. Benner, S. Dolgov, V. Khoromskaia and B. N. Khoromskij. Fast iterative solution of the Bethe-Salpeter eigenvalue problem using low-rank and QTT tensor approximation. J. Comp. Phys., 334, 2017 pp. 221-239.

[6] P. Benner, V. Khoromskaia, B. N. Khoromskij and C. Yang. Computing the density of states for optical spectra of molecules by low-rank and QTT tensor approximation. J. Comp. Phys., 382, 2019 pp. 221-239..

[7] Venera Khoromskaia and Boris N. Khoromskij. Tensor numerical methods in quantum chemistry. De Gruyter, Berlin, 2018.

1:00pm - 1:30pm

Tensor decompositions for multivariate functions approximation with applications

Ivan Oseledets

Skoltech, Russian Federation

Tensor decompositions (CP, Tucker, Tensor-Train, Hierarchical Tucker) have proven to be a very effective tool to approximate multivariate functions in the recent 10-15 years. A lot of results have been obtained both on algorithmic and theoretical side. In the talk, I will overview some of our recent results for using tensor approximations for:

A) Approximation of probability density from samples

B) Solving high-dimensional Fokker-Planck equation

C) Optimization of multivariate functions

1:30pm - 2:00pm

Range separated tensor formats in numerical modeling of large multi-particle systems

Boris Khoromskij

Max-Planck Institute for Mathematics in the Sciences, Germany

Tensor numerical methods allow to construct high accuracy computational schemes for solving $d$-dimensional PDEs with the linear complexity scaling in dimension, [1,2]. We discuss how the tensor numerical methods apply to calculation of electrostatic potential of many-particle systems by using the novel Range Separated (RS) tensor format [3]. The particular application of the RS tensor representation for numerical modeling of electrostatics in large bio-molecular systems via the Poisson-Boltzmann equation in 3D will be discussed [5]. The approach is based on application of the RS tensor decomposition of the discretized Dirac delta [4] combined with the low-rank representation of the long range part in the multi-particle electrostatic potential. We sketch how the numerical tensor decomposition of large function related tensors can be performed by using the concept of machine learning. The storage and complexity reduction for the rank-structured representations can be based on the QTT tensor approximation. The numerical illustrations will be presented.

[1] B.N. Khoromskij. Tensor Numerical Methods in Scientific Computing.

Research monograph, De Gruyter, Berlin, 2018.

[2] V. Khoromskaia and B.N. Khoromskij. Tensor Numerical Methods in Quantum Chemistry.

Research monograph, De Gruyter, Berlin, 2018.

[3] P. Benner, V. Khoromskaia and B. N. Khoromskij.

Range-separated tensor formats for many-particle modeling.

SIAM J. Sci. Comp., (2): A1034-A1062, 2018.

[4] Boris N. Khoromskij. Range-separated tensor representation of the discretized multidimensional Dirac delta and elliptic operator inverse. Journal of Computational Physics, v. 401, 108998, 2020.

[5] P. Benner, V. Khoromskaia, B. N. Khoromskij, C. Kweyu and M. Stein.

Regularization of Poisson--Boltzmann Type Equations with Singular Source Terms Using the Range-Separated Tensor Format. SIAM Journal on Scientific Computing, 43 (1), A415-A445, 2021; E-Preprint arXiv:1901.09864, 2019.

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