Sparse grids for the Schrödinger equation

DOI10.1051/m2an:2007015zbMath1145.65096OpenAlexW2132119675MaRDI QIDQ5447898

Jan Hamaekers, Michael Griebel

Publication date: 20 March 2008

Published in: ESAIM: Mathematical Modelling and Numerical Analysis (Search for Journal in Brave)

Full work available at URL: http://www.numdam.org/item?id=M2AN_2007__41_2_215_0

zbMATH Keywords

convergencescalabilityfermionic systemsmany-particle problemselectronic Schrödinger equationanti-symmetric sparse gridsspace grid method

Mathematics Subject Classification ID

Other elementary particle theory in quantum theory (81V25) Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics (81Q05) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) PDEs in connection with quantum mechanics (35Q40) Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs (65N50) Numerical methods for eigenvalue problems for boundary value problems involving PDEs (65N25)

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Uses Software

• SLEPc
• PETSc

Cites Work

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• On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives
• Symmetric iterative interpolation processes
• Exact non-reflecting boundary conditions
• A general framework of compactly supported splines and wavelets
• Number of lattice points in the hyperbolic cross
• Numerical integration using sparse grids
• On additive Schwarz preconditioners for sparse grid discretizations
• Hyperbolic wavelet approximation
• Efficient solution of symmetric eigenvalue problems using multigrid preconditioners in the locally optimal block conjugate gradient method
• Dimension-adaptive tensor-product quadrature
• Optimized tensor-product approximation spaces
• Sparse grids for boundary integral equations
• The electron density is smooth away from Nuclei
• On the abstract theory of additive and multiplicative Schwarz algorithms
• Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems
• Information complexity of multivariate Fredholm integral equations in Sobolev classes
• A note on the complexity of solving Poisson's equation for spaces of bounded mixed derivatives
• Sharp regularity results for Coulombic many-electron wave functions
• MATHEMATICAL CONCEPTS OF OPEN QUANTUM BOUNDARY CONDITIONS
• Multigrid Methods for Nearly Singular Linear Equations and Eigenvalue Problems
• On the eigenfunctions of many-particle systems in quantum mechanics
• The approximation of classes of periodic functions of many variables
• Computational chemistry from the perspective of numerical analysis
• SLEPc
• APPROXIMATION BY TRIGONOMETRIC POLYNOMIALS OF FUNCTIONS OF SEVERAL VARIABLES ON THE TORUS
• Ten Lectures on Wavelets
• Comparative Analysis of the Cuthill–McKee and the Reverse Cuthill–McKee Ordering Algorithms for Sparse Matrices
• Multilevel Algorithms Considered as Iterative Methods on Semidefinite Systems
• Design of Absorbing Boundary Conditions for Schrödinger Equations in $\mathbbR$^d
• Numerical solution of the Dirac equation by a mapped Fourier grid method
• Absorbing Boundary Conditions for the Schrödinger Equation
• On approximate approximations using Gaussian kernels
• Sparse grids
• Subspace correction multi‐level methods for elliptic eigenvalue problems
• BestN-term approximation in electronic structure calculations I. One-electron reduced density matrix
• On the computation of the eigenproblems of hydrogen and helium in strong magnetic and electric fields with the sparse grid combination technique
• Electron wavefunctions and densities for atoms