MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Wavelet approximation of correlated wavefunctions. II. Hyperbolic wavelets and adaptive approximation schemes

Hongjun Luo, Dietmar Kolb, Heinz-Jürgen Flad, Wolfgang Hackbusch and Thomas Koprucki


We have studied various aspects concerning the use of hyperbolic wavelets and adaptive approximation schemes for wavelet expansions of correlated wavefunctions. In order to analyze the consequences of reduced regularity of the wavefunction at the electron-electron cusp, we first considered a realistic exactly solvable many-particle model in one dimension. Convergence rates of wavelet expansions, with respect to $L^2$ and $H^1$ norms and the energy, were established for this model. We compare the performance of hyperbolic wavelets and their extensions through adaptive refinement in the cusp region, to a fully adaptive treatment based on the energy contribution of individual wavelets. Although hyperbolic wavelets show an inferior convergence behavior, they can be easily refined in the cusp region yielding an optimal convergence rate for the energy. Preliminary results for the helium atom are presented, which demonstrate the transferability of our observations to more realistic systems. We propose a contraction scheme for wavelets in the cusp region, which reduces the number of degrees of freedom and yields a favorable cost to benefit ratio for the evaluation of matrix elements.


Related publications

2002 Repository Open Access
Hongjun Luo, Dietmar Kolb, Heinz-Jürgen Flad, Wolfgang Hackbusch and Thomas Koprucki

Wavelet approximation of correlated wave functions. II. Hyperbolic wavelets and adaptive approximation schemes

In: The journal of chemical physics, 117 (2002) 8, pp. 3625-3638