Prof. Dr. Reinhold Schneider

schneidr@math.tu-berlin.de


Research focus

Numerical methods for solving high-dimensional PDE's, tensor product approximation, quantum chemistry

Projects as a project leader

  • D-SE9

    Optimal control of evolution Maxwell equations and low rank approximation

    Prof. Dr. Reinhold Schneider / Prof. Dr. Fredi Tröltzsch

    Project heads: Prof. Dr. Reinhold Schneider / Prof. Dr. Fredi Tröltzsch
    Project members: Benjamin Huber
    Duration: 01.06.2014 - 31.05.2017
    Status: running
    Located at: Technische Universität Berlin

    Description

    The project D-SE9 focuses on the analysis and efficient numerical solution of optimal control problems for nonlinear evolution equations with Maxwell's evolution equations as a challenging benchmark example. In particular, we aim at developing low rank approximation techniques for the solution of forward-backward optimality systems that arise whenever optimal control problems for evolution equations are considered. In this project, we thus merge existing expertise in optimal control and low rank matrix and tensor approximation.

    http://www.d-se9.de
  • D-SE10

    Low rank tensor recovery

    Prof. Dr. Reinhold Schneider

    Project heads: Prof. Dr. Reinhold Schneider
    Project members: Sebastian Wolf
    Duration: 01.06.2014 - 31.05.2017
    Status: running
    Located at: Technische Universität Berlin

    Description

    In the project D-SE10 we aspire to recover higher order tensors from a relatively small number of measurements using low rank assumptions. As straight forward generalizations of the matrix recovery techniques to the problem of tensor recovery are often either infeasible or impossible, the focus of this project is twofold. First, to investigate those generalizations that might still be feasible in a tensor setting in particular Riemannian methods on low rank tensor manifolds, and second, to apply and specialize existing techniques from tensor product approximation like the ALS to the tensor recovery and completion settings.

    http://d-se10.de
  • C-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
  • C-SE13

    Topology optimization of wind turbines under uncertainties

    Dr. Martin Eigel / PD Dr. René Henrion / Prof. Dr. Dietmar Hömberg / Prof. Dr. Reinhold Schneider

    Project heads: Dr. Martin Eigel / PD Dr. René Henrion / Prof. Dr. Dietmar Hömberg / Prof. Dr. Reinhold Schneider
    Project members: Dr. Johannes Neumann / Dr. Thomas Petzold
    Duration: 01.06.2014 - 31.05.2017
    Status: running
    Located at: Technische Universität Berlin / Weierstraß-Institut

    Description

    The application focus of this project is the topology optimization of the main frame of wind turbines. This is the central assembly platform at the tower head accommodating the drive train, the generator carrier, the azimuth bearing and drives and a lot of small components. Topology optimization should not be mistaken for legally mandated structural analysis computations. For the latter, it is standard to solicit a number of single load scenarios based on available time series data. While this approach is questionable already for stress analysis, it is prohibitive for topology optimization. Disregarding the multivariate distribution of the random loads would not provide any probabilistic certificate for bounding stresses. Moreover, the natural way to choose weights is to derive a stochastic load from available time series data. The main frame is made of cast iron which is prone to a number of material impurities like shrink holes, dross, and chunky graphite. This motivates the additional consideration of randomness for the material stiffness. Structures resulting from topology optimization often exhibit unacceptably high stresses necessitating costly subsequent shape design works. To avoid this already during the optimization, state constraints have to be included in the optimization problem. The main novelty of this project is that it combines a phase field relaxed topology optimisation problem not only with uncertain loading and material data but also with chance state constraints. Even in the finite-dimensional case, the derivation of optimality conditions including gradient formulas is completely open. In the long run, including an appropriate damage model as additional state equation will be a further task of great practical importance.

    http://www.wias-berlin.de/projects/ECMath-SE13/
  • A-AP15

    Generalized tensor methods in quantum chemistry

    Prof. Dr. Reinhold Schneider

    Project heads: Prof. Dr. Reinhold Schneider
    Project members: -
    Duration: 01.06.2013 - 31.05.2016
    Status: completed
    Located at: Technische Universität Berlin

    Description

    The computation of the electronic structure is of utmost importance for the task of molecular engineering in modern chemistry and material science. In this context, the accurate computation of the electron correlation is a fundamental and extremely difficult problem. In contrast to the tremendous progress made in calculating weakly correlated systems by Density Functional Theory (DFT) for extended systems or Coupled Cluster Methods for highly accurate calculations, there are two major types of systems for which current quantum chemical methods have deficiencies: (1) Open-shell systems with a large number of unpaired electrons, as they occur in multiple transition metal complexes or in molecular magnets; (2) Extended or periodic systems without a band gap, where the limit of the applicability of the available size consistent methods is reached.The aim of this proposal is to develop a general tensor network state (TNS) based algorithm that can be applied efficiently to these open problems of quantum chemistry. Realization of such an algorithm relies on carrying out a variety of complex tasks. Several new formal methods and methodological concepts of tensor decompositions will have to be designed to comply with the specific, nonlocal nature of the Hamiltonian, and to this end, the applicants will join their rather complementary expertise regarding the powerful DMRG method and similar recent developments from physics, mathematics and information technology. To arrive at an efficient implementation of the quantum chemistry TNS algorithm, our contributions will be implemented and tested based on existing program structures of the QC-DMRG- Budapest [Legeza-2011] and the TTNS-Vienna [Murg-2010c] codes.

    http://gepris.dfg.de/gepris/projekt/234056486?language=en