Concepts of Data-Sparse Tensor-Product Approximation in Many-Particle Modelling

Flad, Heinz-Jürgen; Hackbusch, Wolfgang; Khoromskij, Boris N.; Schneider, Reinhold

doi:10.1142/9789812836021_0020

Cited by 10 publications

(9 citation statements)

References 79 publications

(95 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In connection with wavelet bases, in [10,19,38] an approach for efficient computation of integrals using separable approximations has been developed, focusing on the efficient computation of individual wavelet coefficients and discretization matrix entries and accordingly tailored error estimates. Since for our purposes we want to avoid explicitly assembling matrices as in (5.3) whenever possible, we are rather interested in a different point of view: we replace the potential terms in (5.3) by separable approximations (5.8) and exploit the tensor product structure for applying the operators efficiently.…”

Section: Separable Approximation Of Potentialsmentioning

confidence: 99%

Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation

Bachmayr

2012

ESAIM: M2AN

View full text Add to dashboard Cite

Abstract. In the framework of an explicitly correlated formulation of the electronic Schrödinger equation known as the transcorrelated method, this work addresses some fundamental issues concerning the feasibility of eigenfunction approximation by hyperbolic wavelet bases. Focusing on the two-electron case, the integrability of mixed weak derivatives of eigenfunctions of the modified problem and the improvement compared to the standard formulation are discussed. Elements of a discretization of the eigenvalue problem based on orthogonal wavelets are described, and possible choices of tensor product bases are compared especially from an algorithmic point of view. The use of separable approximations of potential terms for applying operators efficiently is studied in detail, and estimates for the error due to this further approximation are given.Mathematics Subject Classification. 35B65, 35J10, 65T60, 81Q05.

show abstract

Section: Separable Approximation Of Potentialsmentioning

confidence: 99%

Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation

Bachmayr

2012

ESAIM: M2AN

View full text Add to dashboard Cite

show abstract

“…These can be obtained from the familiar representation of the Coulomb potential via a Gaussian transform, we refer to Ref. [61] for further details. In Paper I, we have discussed a wavelet based approach for the canonical format which furthermore provides sparse approximations of the Kronecker factors.…”

Section: Tensor Product Approximationmentioning

confidence: 99%

Canonical Tensor Products as a Generalization of Gaussian-type Orbitals

Chinnamsetty¹,

Espig²,

Flad³

et al. 2010

Zeitschrift Für Physikalische Chemie

Self Cite

View full text Add to dashboard Cite

We propose a possible generalization of Gaussian-type orbital (GTO) bases by means of canonical tensor products. The present work focus on the application of tensor product as an alternative to conventional GTO based density fitting schemes. Tensor product approximation leads to highly nonlinear optimization problems which require sophisticated algorithms. We give a brief description of the optimization problem and algorithm. The present work extends our previous paper [S. R. Chinnamsetty, M. Espig, B. N. Khoromskij, W. Hackbusch and H.-J. Flad, J. Chem. Phys. 127 (2007), 084110], where we discussed tensor product approximations of the electron density and the Hartree potential, to orbital products which are required for the exchange part of Hartree-Fock and in post Hartree-Fock methods. We provide a detailed error analysis for the Coulomb and exchange terms in Hartree-Fock calculations. Furthermore, a comparison is given between all-electron and pseudopotential calculations.

show abstract

“…Moreover, there are interesting results on the tensor-product approximation of the 3D Newton and Yukawa potentials in the framework of the waveletand polynomial-based multiresolution schemes [16,4], as well as using the grid-based approximations [13,8,25,29].…”

Section: Introductionmentioning

confidence: 99%

QTT Representation of the Hartree and Exchange Operators in Electronic Structure Calculations

Khoromskaia

Khoromskij

Schneider

2011

Comput. Methods Appl. Math.

Self Cite

View full text Add to dashboard Cite

In this paper, the tensor-structured numerical evaluation of the Coulomb and exchange operators in the Hartree-Fock equation is supplemented by the usage of recent quantized-TT (QTT) formats. It leads to O(log n) complexity at computationally extensive stages in the rank-structured calculation with the respective 3D Hartree and exchange potentials discretized on large n × n × n Cartesian grids. The numerical examples for some volumetric organic molecules confirm that the QTT ranks of these potentials are nearly independent of the one-dimension grid size n. Thus, paradoxically, the complexity of the grid-based evaluation of the Coulumb and exchange matrices becomes almost independent of the grid size, being regulated only by the structure of a molecular system. As a result, the grid approximation of the Hartree-Fock equation allows to gain the high resolution with a guaranteed accuracy.

show abstract

Concepts of Data-Sparse Tensor-Product Approximation in Many-Particle Modelling

Cited by 10 publications

References 79 publications

Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation

Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation

Canonical Tensor Products as a Generalization of Gaussian-type Orbitals

QTT Representation of the Hartree and Exchange Operators in Electronic Structure Calculations

Contact Info

Product

Resources

About