Prof. Dr. Harry Yserentant

yserentant@math.tu-berlin.de


Projects as a project leader

  • SE-AP11

    Multiscale tensor decomposition methods for partial differential equations

    Prof. Dr. Rupert Klein / Prof. Dr. Reinhold Schneider / Prof. Dr. Harry Yserentant

    Project heads: Prof. Dr. Rupert Klein / Prof. Dr. Reinhold Schneider / Prof. Dr. Harry Yserentant
    Project members: -
    Duration: 01.10.2014 - 30.09.2018
    Status: running
    Located at: Freie Universität Berlin / Technische Universität Berlin

    Description

    Novel hierarchical tensor product methods currently emerge as an important tool in numerical analysis and scienti.c computing. One reason is that these methods often enable one to attack high-dimensional problems successfully, another that they allow very compact representations of large data sets. These representations are in some sense optimal and by construction at least as good as approximations by classical function systems like polynomials, trigonometric polynomials, or wavelets. Moreover, the new tensor-product methods are by construction able to detect and to take advantage of self-similarities in the data sets. They should therefore be ideally suited to represent solutions of partial differential equations that exhibit certain types of multiscale behavior.
    The aim of this project is both to develop methods and algorithms that utilize these properties and to check their applicability to concrete problems as they arise in the collobarative research centre. We plan to attack this task from two sides. On the one hand we will try to decompose solutions that are known from experiments, e.g., on Earthquake fault behavior, or large scale computations, such as turbulent flow fields. The question here is whether the new tensor product methods can support the devel­opment of improved understanding of the multiscale behavior and whether they are an improved starting point in the development of compact storage schemes for solutions of such problems relative to linear ansatz spaces.
    On the other hand, we plan to apply such tensor product approximations in the frame­work of Galerkin methods, aiming at the reinterpretation of existing schemes and at the development of new approaches to the ef.cient approximation of partial differential equations involving multiple spatial scales. The basis functions in this setting are not taken from a given library, but are themselves generated and adapted in the course of the solution process.
    One mid-to long-term goal of the project that combines the results from the two tracks of research described above is the construction of a self-consistent closure for Large Eddy Simulations (LES) of turbulent flows that explicitly exploits the tensorproduct approach’s capability of capturing self-similar structures. If this proves successful, we plan to transfer the developed concepts also to Earthquake modelling in joint work with partner project B01.

    http://sfb1114.imp.fu-berlin.de/research/index.php?option=com_projectlog&view=project&id=8
  • CH10

    Analysis and numerics of the chemical master equation

    Prof. Dr. Harry Yserentant

    Project heads: Prof. Dr. Harry Yserentant
    Project members: -
    Duration: 01.06.2014 - 31.05.2017
    Status: completed
    Located at: Technische Universität Berlin

    Description

    The chemical master equation is a fundamental equation in chemical kinetics. It underlies the classical reaction-rate equations and takes the stochastic effects into account that cannot be neglected in the case of small population numbers.

    There is an ongoing effort to tackle the chemical master equation numerically. The major challenge is its high dimensionality: for a system of d interacting species the chemical master equation is a differential equation with state space N_0^d, N_0 the set of nonnegative integers.

    The main goal of project A-CH10 is build a sound mathematical basis for the numerical approximation of the chemical master equation and to put numerical methods for this equation on a firm mathematical ground.

    http://www.tu-berlin.de/?id=168383
  • CH-AP23

    Regularity, complexity, and approximability of electronic wavefunctions

    Prof. Dr. Harry Yserentant

    Project heads: Prof. Dr. Harry Yserentant
    Project members: -
    Duration: 01.10.2013 - 30.09.2016
    Status: completed
    Located at: Technische Universität Berlin

    Description

    The project considers the electronic Schrödinger equation of quantum chemistry that describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. Solutions of this equation depend on 3N variables, three spatial dimensions for each electron. Approximating the solutions is thus inordinately challenging. It is conventionally believed that the accuracy cannot be systematically improved without the effort truly exploding for larger numbers of electrons and that a reduction to simplified models, such as those of the Hartree-Fock method or density functional theory, is the only tenable approach for the approximation of the solutions. Results of the applicant indicate that this conventional wisdom need not be ironclad: the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that of a system of two electrons or even only one electron. Goal of the project is to extend and refine these results and to identify structural properties of the wavefunctions that could ideally enable breaking the curse of dimensionality and to develop the present approximation methods further to true discretications of the Schrödinger equation.

    http://www.dfg-spp1324.de/abstracts.php?lang=de#20