Sparse grids for the Schrödinger equation

Griebel, Michael; Hamaekers, Jan

doi:10.1051/m2an:2007015

Cited by 71 publications

(75 citation statements)

References 58 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recent approaches, like the wavelet multiscale methods [12] and the hyperbolic cross (sparse grids) approximation [8,88,95,101,28,29,30] allow to relax the curse of dimensionality, and already enable to handle the moderate dimensional problems, e.g., with d ≤ 10. The sparse Schur complement methods circumvent the curse of dimension by reduction to the interface or to the wire basket of the boundary [51,50].…”

Section: Methods Of Separation Of Variablesmentioning

confidence: 99%

Tensors-structured numerical methods in scientific computing: Survey on recent advances

Khoromskij¹

2012

Chemometrics and Intelligent Laboratory Systems

160

153

View full text Add to dashboard Cite

In the present paper, we give a survey of the recent results and outline future prospects of the tensor-structured numerical methods in applications to multidimensional problems in scientific computing. The guiding principle of the tensor methods is an approximation of multivariate functions and operators relying on certain separation of variables. Along with the traditional canonical and Tucker models, we focus on the recent quantics-TT tensor approximation method that allows to represent N -d tensors with log-volume complexity, O(d log N ). We outline how these methods can be applied in the framework of tensor truncated iteration for the solution of the high-dimensional elliptic/parabolic equations and parametric PDEs. Numerical examples demonstrate that the tensor-structured methods have proved their value in application to various computational problems arising in quantum chemistry and in the multi-dimensional/parametric FEM/BEM modeling-the tool apparently works and gives the promise for future use in challenging high-dimensional applications.

show abstract

Section: Methods Of Separation Of Variablesmentioning

confidence: 99%

Tensors-structured numerical methods in scientific computing: Survey on recent advances

Khoromskij¹

2012

Chemometrics and Intelligent Laboratory Systems

160

153

View full text Add to dashboard Cite

show abstract

“…Furthermore, sparse grid approaches using Fourier basis functions and Meyer basis functions were implemented and studied in [3] and [4], respectively. Here, it turned out that, in principle, the sparse grid approach indeed possesses favourable approximation rates and cost complexities for the solution of Schrödinger's equation.…”

Section: Hψ = Eψmentioning

confidence: 99%

“…Note that the regularity of a function is directly related to the decay properties of its Fourier transform. In particular, the standard isotropic Sobolev spaces as well as the standard Sobolev spaces of dominating mixed smoothness [14], both generalized to the N -particle case [3,4], are included in the definition of H t,r mix . They can be written as…”

Section: Hψ = Eψmentioning

confidence: 99%

Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation

Griebel¹,

Hamaekers²

2010

Zeitschrift Für Physikalische Chemie

Self Cite

View full text Add to dashboard Cite

In this article we combine the favorable properties of efficient Gaussian type orbitals basis sets, which are applied with good success in conventional electronic structure methods, and tensor product multiscale bases, which provide guaranteed convergence rates and allow for adaptive resolution. To this end, we develop and study a new approach for the treatment of the electronic Schrödinger equation based on a modified adaptive sparse grid technique and a certain particle-wise decomposition with respect to oneparticle functions obtained by a nonlinear rank-1 approximation. Here, we employ a multiscale Gaussian frame for the sparse grid spaces and we use Gaussian type orbitals to represent the rank-1 approximation. With this approach we are able to treat small atoms and molecules with up to six electrons.2

show abstract

“…An approach for improving the efficiency of the Fourier method is the so-called mapped Fourier method where the coordinates are transformed according to the properties of the Hamiltonian [21,35]. Also, a sparse grid algorithm for the Fourier method was devised by Hallatschek [28] and later employed for the Schrödinger equation [24,25]. Still these improvements do not enable solution-or residual-based mesh adaption.…”

Section: Introductionmentioning

confidence: 99%

A Time-Space Adaptive Method for the Schrödinger Equation

Kormann

2016

Commun. Comput. Phys.

View full text Add to dashboard Cite

Abstract. In this paper, we present a discretization of the time-dependent Schrödinger equation based on a Magnus-Lanczos time integrator and high-order Gauss-Lobatto finite elements in space. A truncated Galerkin orthogonality is used to obtain dualitybased a posteriori error estimates that address the temporal and the spatial error separately. Based on this theory, a space-time adaptive solver for the Schrödinger equation is devised. An efficient matrix-free implementation of the differential operator, suited for spectral elements, is used to enable computations for realistic configurations. We demonstrate the performance of the algorithm for the example of matter-field interaction.

show abstract

Sparse grids for the Schrödinger equation

Cited by 71 publications

References 58 publications

Tensors-structured numerical methods in scientific computing: Survey on recent advances

Tensors-structured numerical methods in scientific computing: Survey on recent advances

Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation

A Time-Space Adaptive Method for the Schrödinger Equation

Contact Info

Product

Resources

About