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# Research

- [1]
- © TU Berlin

I'm working on an effcient approximation of
integral operators, as well as high-dimensioan functions and
coresponding operator equations. Mathematical tools are mostly based
on an adptive hierarchical framwework like wavelet compression, sparse
grid approximation, and (hierarchical) tensor product
approximation.

Traditionally we considered mainly boundary
integral equations. Recently we are focusing our effort on electronic
structure calculations. There the major issue is the numerical
solution of the electronic Schrödinger equation (quantum chemistry
and solid state physics) together with ab inito models like Density
Funtional Theory. BigDFT [2]

The solution procees leads
after an appropriate discretization to large-scale non-linear
problems of eignvlaue type and constrained optimization
problem.

## DFG priority program SPP 1145

*"Adaptive solution of coupled cluster
equation and tensor product approximation of two-electron
integrals" *

Homepage

**Abstract:** In
order to achieve optimal computational complexity in electronic
structure calculations, i.e. linear scaling concerning system size and
more general with respect to the number of degrees of freedom, it is
crucial to utilize adaptivity in various places. Within the first part
of our project we want to apply recent ideas from multiscale analysis
to develop adaptive algorithms for coupled cluster and configuration
interaction methods. Here, adaptivity means to select those excitation
amplitudes from the full configuration space which contribute most to
the energy. Such kind of schemes have been already discussed in the
literature, however, we intend to incorporate some new mathematical
insights by selecting amplitudes according to appropriate norms, and
to apply a posteriori error estimators which enable a quantitative
estimate of the remaining approximation error. The second part of our
proposal concerns the efficient computation of two-electron integrals
which constitutes a major bottleneck of standard quantum chemistry
methods. We want to incorporate adaptivity into our previously
developed density-fitting scheme based on optimal tensor product
approximations. This can be accomplished by means of hierarchical
tensor product decompositions as well as adaptive algorithms for
convolutions with the Coulomb potential.

## DFG priority program SPP 1324

- [3]
- © TU-Berlin

*"Tensor methods in multi-dimensional
spectral problems with particular application in electronic structure
calculations"*

Homepage [4]**Abstract:**

Many challenging problems of
numerical computations arise from problems involving a high spatial
dimension. For a fine grid resolution even 3 dimensions cause a
problem, but 6 or even much higher dimensions require quite new
methods, since the standard approaches have a computational complexity
growing exponentially in the dimension ( curse of dimensionality). A
remedy is the use of data-sparse matrices or corresponding
constructions exploiting tensor product representations. Here, we
focus on eigenvalue problems in this field. While the design of the
algorithms is rather general, the main application are problems from
electronic structure calculations.

Many of the developed
methods may be applied to general problems stemming from elliptic
differential or integral operators. In particular, the basic
electronic Schrödinger equation is an eigenvalue problem for an
elliptic 2nd order partial differential equation in high dimensions.
Alternative to a direct treatment of this original problem we would
like to exploit successful developments in quantum chemistry, mainly
putting newly developed methods on top of well established electronic
structure programs. A major focus will be on eigenvalue problems in
Density Functional Methods. Perhaps there are further instances where
the development of the project would contribute to numerical methods
in electronic structure calculation, e.g. adaptive configuration
interaction (CI) and coupled cluster (CC) methods and Jastrow factor
calculation.

## Project A7 from the DFG Resaerch Center Matheon

- [5]
- © TU-Berlin

*"Numerical Discretization Methods in
Quantum Chemistry"*Homepage

**Abstract:**

Computer simulation plays an ever expanding role in modern scientific research, and the fields of chemistry, biochemistry, and pharmaceutical research are no exceptions. The model on which our physical understanding of chemistry rests is the Schrödinger equation, the basic equation of quantum mechanics. Approximation techniques for its solutions is an active area of research spanning the fields of chemistry, physics, and applied mathematics. The main problem is that this equation is an equation in 3N space dimensions for a system consisting of N electrons and nuclei. The so-called "curse of dimensionality" prohibits direct approximation techniques for even reasonably small systems, and a host of methods have been discovered over the past decades which attack the problem from other approaches. However recent developments indicate that the curse of dimensionality might be broken---or at least brought into the realm of numerical tractability. Among these developments is an improved understanding of the "regularity" of the solutions, together with advances in sparse grid techniques from numerical analysis. The goal of this project is to further refine these ideas and to implement them in efficient numerical algorithms.

## Address

**Technische Universität Berlin**

Institute of Mathematics

Faculty II - Mathematics and Sciences

sec. MA 5-3

Str. des 17. Juni 136

10623 Berlin

Tel.: +49 30 314-28579

Fax.: +49 30 314-28967

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