Goals of Research Area J

Expanding our research we will give special emphasis to multiscale algorithms for high-dimensional problems and aim at the derivation of effective models with a particular focus on meanfield and limit theories describing the macroscopic behaviour of large systems with a huge number of degrees of freedom. The Trimester Program “Analysis and Numerics for High Dimensional Problems” (May–August 2011) at HIM co-organised by Griebel, Hackbusch, Hegland, and Schwab provides a platform for exchange and cooperation on the range from sparse grids methods over manifold learning and tensor product techniques to stochastic PDEs. Furthermore, Bonn hosts a Workshop on High-Dimensional Approximation in 2011 and a Summerschool on H-Matrices.

Derivation of effective models.

In physics and chemistry, systems of interest are typically characterised by a huge number of degrees of freedom. Here, one of the main goals of statistical mechanics is the derivation of simpler effective theories which describe the macroscopic behaviour of these systems. In particular, we plan to continue the analysis of many body quantum dynamics. Here, several important questions remain open, such as the effective dynamics of unstable many body systems with a typically exhibited blow-up in finite time and the effective evolution equations for the dynamics of fermionic systems. In this case, the mean-field regime is naturally associated with a semiclassical limit, which renders the analysis more involved. A rigorous understanding of the dynamics of superconductors represents a further important challenge, where one, for example, asks how the time-dependent Ginzburg-Landau equation arises from the microscopic many-body theory.
Simultaneously, we will further advance our algorithms for the direct numerical simulation of the electronic Schrödinger equation. Here, the configuration interaction approach (CI), which is not size consistent, resembles just an ANOVA-type decomposition, whereas the coupled cluster approach (CC), which is size-consistent, relates to the exponential of an ANOVA decomposition.
A generalisation by means of Fulde’s cumulant expansion allows us to obtain a new, size consistent approach based on the ANOVA decomposition and sparse grids. Thereby, for molecular systems with localised electrons, a fast convergence of this expansion can be observed and a truncation after few terms results in an numerical method with linear scaling properties. This way, a practically computable, accurate, solely numerical and convergent approximation is within reach, which circumvents the curse of dimension and indeed renders  many particle systems computationally tractable.

Multiscale methods for high dimensional problems.

The dimension-adaptive multiscale representation of lower dimensional manifolds in higher dimensional spaces will be addressed by a vector-valued sparse grid approach. We will develop and analyse algorithms for the approximate treatment of functions and differential operators on such manifolds. Thereby the aim is to detect the relevant intrinsic low-dimensional structures in nominally high-dimensional spaces and thus circumvent the curse of dimensionality. This enables algorithms with a cost complexity where only the effective, hopefully low intrinsic dimension enters exponentially. Specific applications of this methodology will be studied in close collaboration with the division on virtual material design of the Fraunhofer Institute SCAI. The efficient numerical treatment of problems arising from stochastic market models and aggregation in Research Area H might benefit from this approach.
In addition, we will employ recently developed compressed sensing approaches for the numerical solution of stochastic partial differential equations and we will extend our results on matrix completion and recovery of low rank matrices from incomplete measurements to low rank tensor recovery. Here, the development of efficient algorithms will require more than a mere generalisation due to fundamental computational problems that arise when passing from matrices to tensors. Furthermore, we will work on the use of compressive sensing to overcome the curse of dimension in high-dimensional function recovery. In particular, by using sublinear Fourier transform algorithms, we expect to achieve a significantly lower computational complexity than with classical methods. These aspects are closely intertwined with our aims to generalise the adaptive cross approximation (ACA) technique to tensors of high order. On the practical side, dimension-trees will be employed to reduce the curse of dimension. From the theoretical point of view, significantly sharper stability estimates will be derived for the convergence analysis of ACA. The principle of concentration of measure will potentially be used to prove efficiency in high dimensions.
Another major direction of research during the next years will be on the efficient multiscale treatment of rate-independent evolution equations with nonconvex energy functionals via their relaxation with polyconvex and rank-one convex envelopes. The relaxation of the energy functionals requires the solution of high-dimensional problems that result from a discretisation of certain associated phase spaces. Moreover, for the evolution problems, an accurate approximation is needed. We aim at realising parallel implementations of already developed approximation schemes, which combine multilevel and adaptive active set methods. These tools will then enable reliable simulations with applications in plasticity and in phase transitions in crystalline solids. There is a close relation of the numerical multiscale treatment of evolution equations and the corresponding analytical approaches pursued by S. Müller and Conti in Research Area B.
Furthermore, we will advance the efficient solution of boundary value problems of partial differential operators with high-contrast coefficients. Here, we aim at establishing weighted Poincaré inequalities with weight-independent constants for microstructured, porous media type domains, and we will investigate a new, promising approach based on so-called flux norm estimates.