Journal ArticleDOI
On the computation of the eigenproblems of hydrogen helium in strong magnetic and electric fields with the sparse grid combination technique
Jochen Garcke,Michael Griebel +1 more
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TLDR
The combination technique is introduced for the numerical solution of d -dimensional eigenproblems on sparse grids and is applied to solve the three-dimensional Schrodinger equation for hydrogen (one-electron problem) and the six-dimensionalSchrodinger equations for helium (two-electrons problem) in strong magnetic and electric fields.About:
This article is published in Journal of Computational Physics.The article was published on 2000-12-10. It has received 66 citations till now. The article focuses on the topics: Sparse grid & Helium atom.read more
Citations
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Regularity and Approximability of Electronic Wave Functions
TL;DR: In this paper, Fourier analysis and the Electronic Schrodinger Equation (ESE) are used to describe the spectrum and exponential decay of mixed derivatives of the Eigenfunction Expansions.
Journal ArticleDOI
Data mining with sparse grids
TL;DR: It turns out that the new method achieves correctness rates which are competitive to that of the best existing methods, i.e. the amount of data to be classified.
Journal ArticleDOI
Computational chemistry from the perspective of numerical analysis
TL;DR: The results of mathematical analysis are outlined, recent results in numerical analysis are laid on, recent developments of new methods and challenging open issues are focused on.
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A sparse grid space-time discretization scheme for parabolic problems
Michael Griebel,Daniel Oeltz +1 more
TL;DR: The space-time sparse grid approach can be employed together with adaptive refinement in space and time and then leads to similar approximation rates as the non-adaptive method for smooth functions.
Journal ArticleDOI
Sparse grids for the Schrödinger equation
Michael Griebel,Jan Hamaekers +1 more
TL;DR: The antisymmetric sparse grid discretization to the electronic Schrodinger equation is applied and costs, accuracy, convergence rates and scalability are compared with respect to the number of electrons present in the system.
References
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The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Journal ArticleDOI
Error Estimates for Adaptive Finite Element Computations
I. Babuvška,W. C. Rheinboldt +1 more
TL;DR: The main theorem gives an error estimate in terms of localized quantities which can be computed approximately, and the estimate is optimal in the sense that, up to multiplicative constants which are independent of the mesh and solution, the upper and lower error bounds are the same.
Journal ArticleDOI
Numerical integration using sparse grids
Thomas Gerstner,Michael Griebel +1 more
TL;DR: The usage of extended Gauss (Patterson) quadrature formulas as the one‐dimensional basis of the construction is suggested and their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw–Curtis and Gauss rules is shown.
Book
Approximation of functions with bounded mixed derivative
TL;DR: In this age of modern era, the use of internet must be maximized, to get the on-line approximation of functions with bounded mixed derivative book, as the world window, as many people suggest.
Journal ArticleDOI
Radius of convergence and analytic behavior of the 1/Z expansion.
TL;DR: In this article, Neville-Richardson et al. performed a 401-order perturbation calculation to resolve the controversy over the radius of convergence of the 1-Z$ expansion for the ground-state energy of helium-like ions.