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Best N-term approximation in electronic structure calculations I. One-electron reduced density matrix

Heinz-Jürgen Flad, Wolfgang Hackbusch, Reinhold Schneider (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

We discuss best N-term approximation spaces for one-electron wavefunctions φ i and reduced density matrices ρ emerging from Hartree-Fock and density functional theory. The approximation spaces A q α ( H 1 ) for anisotropic wavelet tensor product bases have been recently characterized by Nitsche in terms of tensor product Besov spaces. We have used the norm equivalence of these spaces to weighted ℓ q spaces of wavelet coefficients to proof that both φ i and ρ are in A q α ( H 1 ) for all α > 0 with α = 1 q - 1 2 . Our proof is based on the...

Best N-term approximation in electronic structure calculations. II. Jastrow factors

Heinz-Jürgen Flad, Wolfgang Hackbusch, Reinhold Schneider (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a novel application of best N-term approximation theory in the framework of electronic structure calculations. The paper focusses on the description of electron correlations within a Jastrow-type ansatz for the wavefunction. As a starting point we discuss certain natural assumptions on the asymptotic behaviour of two-particle correlation functions ℱ ( 2 ) near electron-electron and electron-nuclear cusps. Based on Nitsche's characterization of best N-term approximation spaces A q α ( H 1 ) , we prove...

Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton-Jacobi equations

Nicolas Bacaër (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Using the min-plus version of the spectral radius formula, one proves: 1) that the unique eigenvalue of a min-plus eigenvalue problem depends continuously on parameters involved in the kernel defining the problem; 2) that the numerical method introduced by Chou and Griffiths to compute this eigenvalue converges. A toolbox recently developed at I.n.r.i.a. helps to illustrate these results. Frenkel-Kontorova models serve as example. The analogy with homogenization of Hamilton-Jacobi equations is emphasized....

Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton-Jacobi equations

Nicolas Bacaër (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Using the min-plus version of the spectral radius formula, one proves: 1) that the unique eigenvalue of a min-plus eigenvalue problem depends continuously on parameters involved in the kernel defining the problem; 2) that the numerical method introduced by Chou and Griffiths to compute this eigenvalue converges. A toolbox recently developed at I.n.r.i.a. helps to illustrate these results. Frenkel-Kontorova models serve as example. The analogy with homogenization of Hamilton-Jacobi equations...

Cover pages

(2008)

Programs and Algorithms of Numerical Mathematics

Different boundary conditions for LES solver Palm 6.0 used for ABL in tunnel experiment

Řezníček, Hynek, Geletič, Jan, Bureš, Martin, Krč, Pavel, Resler, Jaroslav, Vrbová, Kateřina, Trush, Arsenii, Michálek, Petr, Beneš, Luděk, Sühring, Matthias (2023)

Programs and Algorithms of Numerical Mathematics

We tried to reproduce results measured in the wind tunnel experiment with a CFD simulation provided by numerical model PALM. A realistic buildings layout from the Prague-Dejvice quarter has been chosen as a testing domain because solid validation campaign for PALM simulation of Atmospheric Boundary Layer (ABL) over this quarter was documented in the past. The question of input data needed for such simulation and capability of the model to capture correctly the inlet profile and its turbulence structure...

Eigenvalues and eigenfunctions of the Laplace operator on an equilateral triangle

Milan Práger (1998)

Applications of Mathematics

A boundary value problem for the Laplace equation with Dirichlet and Neumann boundary conditions on an equilateral triangle is transformed to a problem of the same type on a rectangle. This enables us to use, e.g., the cyclic reduction method for computing the numerical solution of the problem. By the same transformation, explicit formulae for all eigenvalues and all eigenfunctions of the corresponding operator are obtained.

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