Prof. Dr. Helmut Harbrecht

Imported from Grants Tool 4703028

The present project aims at studying the mathematical theory of tensor approximation schemes for representing high-dimensional functions. Starting point are high-dimensional functions, which are defined on an m-fold product of domains of arbitrary dimensions, providing smoothness either in standard Sobolev spaces or in Sobolev spaces of dominating mixed smoothness. Of practical interest is also the situation that such spaces are equipped with dimension weights. We will provide estimates on the cost to approximate such functions in terms of the required ranks and also in terms of storage requirements for the eigenfunctions.

Shape optimization is indispensable for designing and constructing industrial components. Many problems that arise in application, particularly in structural mechanics and in the optimal control of distributed parameter systems, can be formulated as the minimization of functionals which are defined over a class of admissible domains. Shape optimization problems can be solved by means of gradient based minimization algorithms, which involve the shape functionals' derivative with respect to the domain under consideration. The computation of the shape gradient and the implementation of appropriate numerical optimization algorithms is meanwhile well understood, provided that the state equation's input data are given exactly. In practice, however, input data for numerical simulations in engineering are often not exactly known. One must thus address how to account for uncertain input data in the state equation. This project is concered with shape optimization under uncertainty. The uncertainty might be caused by different sources like uncertain geometric entities, uncertain loads, or uncertain material parameters.

The Basler Edition der Bernoulli-Briefwechsel (BEBB) has the aim of publishing a full online edition of the letters exchanged by the mathematicians and physicists Daniel Bernoulli (1700-1782), Jacob I Bernoulli (1654-1705), Jacob II Bernoulli (1759-1789), Johann I Bernoulli (1667-1748), Johann II Bernoulli (1710-1790), Nicolaus I Bernoulli (1687-1759), Nicolaus II Bernoulli (1695-1726), and Jacob Hermann (1678-1733) with over 400 correspondents, many of them of world renown. The Bernoulli correspondence represents a central part of the scientific network of the 17th and 18th century and is an essential text source for the study of western intellectual history, particularly of the history of mathematics and physics.

This project combines shape optimization and homogenization techniques by looking for the optimal design of the microstructure in composite materials and of scaffolds. The development of materials with specific properties is of huge practical interest, for example, for medical applications or for the development of light weight structures in aeronautics. In particular, the optimal design of microstructures leads to fundamental questions for porous media: what is the sensitivity of homogenized coefficients with respect to the shape of the microstructure? Based on Hadamard's shape gradient, we develop optimization algorithms and provide numerical simulations for several problems arising from the homogenization of elastic materials.

This project aims at developing wavelet methods that adaptively solve boundary integral equations posed in three space dimensions. Since isotropic refinement is not optimal for the approximation of singularities which are of anisotropic nature, we will use (anisotropic) tensor wavelets for the discretization. Tensor wavelets are known to be able to resolve anisotropic singularities in an optimal way. Especially they are able to optimally resolve edge singularities which arise in case of non-smooth geometries and are typically of anisotropic nature. Therefore, the classical, isotropic adaptive wavelet algorithm has to be extended to this more general setting and necessary building blocks such as quadrature and compression routines have to be further developed. We will construct an adaptive algorithm which is optimal in the sense that it computes the approximate solution at an expense that scales proportional to the best $N$-term approximation of the unknown solution.

The scope of the present project is the analysis and efficient solution of shape optimization problems with time-dependent state equation. The main ingredient is the shape sensitivity analysis, i.e., the analytic computation of the shape gradient. The efficient discretization of both, the shape and the state equation, is a further emphasis of this project. In particular, we intend to apply space-time methods for the solution of the state equation.

Viele Probleme in ingenieurswissenschaftlichen Anwendungen führen auf Probleme der Formoptimierung: ein Werkstück (Gebiet) soll hinsichtlich einer gewissen Eigenschaft (Kostenfunktional) unter Beachtung gewisser Eigenschaften (Zustandsgleichung und andere Nebenbedingungen) optimiert werden. Beipiele finden sich in der Baustatik, im Flugzeugdesign, etc. Insbesondere können zahlreiche freie Randprobleme und inverse Probleme als Formoptimierungsprobleme formuliert und behandelt werden. Die Entwicklung von effizienten Formoptimierungsmethoden ist unerlässlich für die computergestützte Simulation in den Ingenieurs- und Naturwissenschaften. Als Modellproblem soll in diesem Projekt Bernoullis freies Randproblem in drei Raumdimensionen betrachtet werden, wobei das gesuchte Gebiet durch eine Levelset-Funktion dargestellt werden soll. Um die zugrundeliegende Zustandsgleichung zu lösen, soll eine schnelle, parametrische Randelementmethode verwendet werden, wie etwa die adaptive Kreuzapproximation, das Multipol-Verfahren oder ein Wavelet-Galerkin- Verfahren. Hierfür muss der Rand des durch die Levelset-Funktion implizit gegebenen Gebiets extrahiert und parametrisiert werden. Speziell muss die Levelset-Funktion durch eine Diskretisierung höherer Ordnung behandelt werden.

The present project is concerned with the efficient solution of large scale Riccati and Lyapunov equations. Such problems typically appear in state-feedback control of systems governed by partial differential equations. We apply a sparse grid discretization in order to arrive at a solution method which scales essentially linear in the number of unknowns.

The Basler Edition der Bernoulli-Briefwechsel (BEBB) has the aim of publishing a full online edition of the letters exchanged by the mathematicians and physicists Daniel Bernoulli (1700-1782), Jacob I Bernoulli (1654-1705), Jacob II Bernoulli (1759-1789), Johann I Bernoulli (1667-1748), Johann II Bernoulli (1710-1790), Nicolaus I Bernoulli (1687-1759), Nicolaus II Bernoulli (1695-1726), and Jacob Hermann (1678-1733) with over 400 correspondents, many of them of world renown. The Bernoulli correspondence represents a central part of the scientific network of the 17th and 18th century and is an essential text source for the study of western intellectual history, particularly of the history of mathematics and physics.