ICOSAHOM 2020
International Conference on Spectral and High Order Methods
12th  16th July 2021  Vienna, Austria
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).
Please note that all times are shown in the time zone of the conference. The current conference time is: 30th Nov 2022, 10:04:19pm CET

Session Overview 
Session  
MS 16a: high order, tensorstructured methods and low rank approximation
 
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 higherorder discretizations. In stead, tensor compression methods realize a form of high order adaptivity by means of singular value decomposition at runtime in tensorformatted 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 lowrank 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, spacetime tensor decomposition and dynamic low rank approximation, tensor formatted solution of partial differential equations.  
Presentations  
12:00pm  12:30pm
Tensorized Adaptive Biasing Force method for molecular dynamics simulations (joint work with Tony Lelièvre and Pierre Monmarché) 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 highdimensional space, is to use MonteCarlo methods. More precisely, some timedependent stochastic dynamics, which are ergodic with respect to the probability measure of interest, are simulated and the desired averages can be obatined as longtime limits of time averages along one trajectory of the stochastic process. For instance, the socalled overdamped Langevin dynmaics is on example of MonteCarlo process that is known to be regodic with respect to the Gibbs measure. While this approach should enable in principle to overcome the socalled 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 longtime limit of these trajectorial averages. Among them, a very popular one is the socalled "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 HartreeFock energies to optical spectra of compact molecules MaxPlanck 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 lowrank representation of multivariate functions and operators. Based on representation of $d$variate functions and operators in the rankstructured tensor formats, they provide $O(dn)$ complexity of numerical calculations instead of $O(n^d)$ by conventional methods [7]. A starting point was the HartreeFock (HF) solver based on the tensor calculation of all involved operators in a Gausssiantype basis, discretized on $ n\times n\times n$ 3D Cartesian grids. For all operators in the nonlinear 3D integrodifferential HF equation the usual 3D analytical integration is substituted by the gridbased tensor algorithms in $O(n\log n)$ complexity [1,3]. The 4th order tensor of the twoelectron 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 BetheSalpeter 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 blockdiagonal plus lowrank 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 ShermanMorrisonWoodbury 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 HartreeFock equation in multilevel tensorstructured format. SIAM J. Sci. Comp., 33 (1), 4565, 2011. [2] V. Khoromskaia, B.N. Khoromskij and R. Schneider. Tensorstructured calculation of the twoelectron integrals in a general basis. SIAM J. Sci. Comp., {\bf 35} (2), A987A1010, 2013. [3] V. Khoromskaia. Blackbox HartreeFock solver by tensor numer. methods. CMAM, 14 (1), 89111, 2014. [4] P. Benner, V. Khoromskaia and B. N. Khoromskij. A Reduced basis approach for calculation of the BetheSalpeter excitation energies using lowrank tensor factorizations. Mol. Physics, 114 (78) 2016. [5] P. Benner, S. Dolgov, V. Khoromskaia and B. N. Khoromskij. Fast iterative solution of the BetheSalpeter eigenvalue problem using lowrank and QTT tensor approximation. J. Comp. Phys., 334, 2017 pp. 221239. [6] P. Benner, V. Khoromskaia, B. N. Khoromskij and C. Yang. Computing the density of states for optical spectra of molecules by lowrank and QTT tensor approximation. J. Comp. Phys., 382, 2019 pp. 221239.. [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 Skoltech, Russian Federation Tensor decompositions (CP, Tucker, TensorTrain, Hierarchical Tucker) have proven to be a very effective tool to approximate multivariate functions in the recent 1015 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 highdimensional FokkerPlanck equation C) Optimization of multivariate functions 1:30pm  2:00pm
Range separated tensor formats in numerical modeling of large multiparticle systems MaxPlanck 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 manyparticle 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 biomolecular systems via the PoissonBoltzmann 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 lowrank representation of the long range part in the multiparticle 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 rankstructured 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. Rangeseparated tensor formats for manyparticle modeling. SIAM J. Sci. Comp., (2): A1034A1062, 2018. [4] Boris N. Khoromskij. Rangeseparated 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 PoissonBoltzmann Type Equations with Singular Source Terms Using the RangeSeparated Tensor Format. SIAM Journal on Scientific Computing, 43 (1), A415A445, 2021; EPreprint arXiv:1901.09864, 2019. http://personalhomepages.mis.mpg.de/bokh 
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