AG Numerik

Drittmittelprojekte / External funded projects

DFG-Projects

  • New Smoothness Spaces on Domains and their Discrete Characterizations
    within the D-A-CH framework (DA 360/22-1) (2018 - 2021)

    Since the discovery of wavelets, the past decades have seen an explosion concerning the design of novel representation systems for functions or distributions. The main intent of that work has been to find representation systems which are optimal for the sparse approximation of various signal classes with the main applications lying in the area of signal processing. As an example we mention the efficient detection of directional information that can be performed by shearlet or curvelet or ridgelet systems which possess vastly superior approximation properties as compared to standard discretization methods such as finite elements or wavelets. Having the various spectacular results concerning the approximation properties of these new representation systems in mind, a natural next step would be to employ them also for the numerical treatment of operator equations. However, the development of numerical methods based on these new representation system is currently facing the bottleneck that to date no useful constructions of these representation systems on bounded domains exist. The aim of this project is to remove this bottleneck by constructing and analyzing new discrete representation systems on finite domains which on the one hand enjoy the same optimal approximation properties as, for example, shearlets while still forming a stable discretization of the energy space for a large class of PDEs (for example Sobolev spaces). Our focus will be on the development of a comprehensive theory for the adaption of function spaces to finite domains. We will be especially interested in the discrete characterization of such spaces, the regularity theory of various PDEs on such spaces and the compressibility properties of Galerkin matrices of various PDEs with respect to these discretizations. These studies are expected to lay the ground work for the subsequent development and implementation of a large class of novel discretization methods for operator equations which outperform current wavelet- or finite-element-based methods for a large class of important problems such as reaction-diffusion equations, linear transport equations or elliptic PDEs. We cooperate with Prof. Dr. P. Grohs and Prof. Dr. H.-G. Feichtinger.

  • Regularity Theory of Stochastic Partial Differential Equations in (Quasi-)Banach Spaces
    (2014 - 2015)

    This project is concerned with regularity estimates for stochastic partial differential equations (SPDEs, for short) on bounded Lipschitz domains. We use specific (quasi-)Banach spaces to measure the smoothness of the solution. Our analysis is motivated by some fundamental problems arising in the context of the numerical treatment of SPDEs. We divide our investigations into three parts which are closely related to each other. In the first two parts we use a specific scale of Besov spaces to measure the spatial regularity of the solution process. This scale determines the convergence order that can be achieved by adaptive (wavelet) schemes and other non-linear approximation methods. It consists mostly of Besov spaces with summability parameter less than one, thus, of quasi-Banach spaces. In the first part we want to derive refined regularity results in weighted Sobolev spaces which yield the desired Besov regularity results by embedding strategies. In the second part, we strive for a more direct approach. To this end, the well-known theory of stochastic integration in UMD-Banach spaces has to be generalised as far as possible to quasi-Banach spaces. In the third part of the project, we want to derive regularity estimates in tensor products of weighted Sobolev spaces which would justify the use of anisotropic full space-time adaptive tensor wavelet methods.

  • Optimal Adaptive Finite Element and Wavelet Methods for p-Poisson Equations
    (2013 - 2016)

    This project is concerned with the design of adaptive strategies for certain classes of quasilinear problems, in particular p-Poisson equations, their convergence analysis, and the proof of optimality in terms of the number of degrees of freedom and the algebraic complexity, respectively. Our approach is based on an adaptive regularization of so-called Kacanov iterations, whose regularization parameter is tuned according to a posteriori error estimators which have also the function of guiding adaptive discretizations. We shall focus on both, adaptive finite element and wavelet methods.The motivation is twofold: on the one hand, appropriate reliable error estimators for finite element discretizations have already been defined and studied for the p-Poisson equation, and we expect that we will be able to "port" this knowledge to wavelet methods for which, in this particular problem, reliable error estimators are not yet available. On the other hand, the strong analytical properties of wavelets can usually be exploited to derive more simply and sometimes more rigorously a convergence and optimality analysis for adaptive wavelet schemes compared to finite element approaches; moreover, the understanding of Besov regularity of solutions of any type of known elliptic equations so far considered has been based on the use of wavelets. Let us stress the fact that Besov regularity of solutions is a fundamental issue when it comes to address the rate of convergence or the complexity of both adaptive finite element and wavelet methods. In addition to the analysis of the adaptive methods for p-Poisson equations, we also plan to perform extensive numerical simulations in order to demonstrate the validity of the theoretical results. We cooperate with Prof. Dr. L. Diening and Prof. Dr. M. Fornasier.

  • Adaptive Wavelet and Frame Techniques for Acoustic BEM
    within the D-A-CH framework
    (2013 - 2016)

    In BIOTOP, we will develop new methods to model 3D acoustic wave problems, especially the calculation of head related transfer functions (HRTFs), which play an important role in the localization of sound sources in 3D. Head related transfer functions describe the filtering effect of the head, the torso, and especially the outer ear (pinna) on incoming sounds and can be used to describe and simulate spatial hearing in virtual acoustics. For the simulation of acoustic problems, the boundary element method (BEM) is an important tool. Unfortunately, the matrices generated by the boundary element method are dense and their dimension grows with the wavenumber/frequency. To overcome this obstruction, we will design efficient variants of the boundary element method based on adaptive wavelet methods and the concept of frames. They will be particularly tuned to the requirements in acoustics, thus allowing an efficient computation of sound fields also for high frequencies. We will combine wavelet compression strategies with adaptive techniques and design new frames adapted to the problem at hand, and investigate the mathematical and numerical properties. BIOTOP is a multi-disciplinary project by necessity, involving mathematics, numerics and acoustical modeling. As a D-A-CH project, it combines the expertise of the research groups from Germany (Prof. Dahlke, Dr. M. Weimar: adaptive schemes, regularity theory), Austria (Dr. P. Balasz, Dr. G. Chardon, Dr. W. Kreuzer, H. Ziegelwanger: frames, calculation of HRTFs, acoustic modeling) and Switzerland (Prof. Dr. H. Harbrecht, MSc M. Utzinger: wavelet BEM, adaptivity).

  • Adaptive Wavelet Methods for SPDEs, Part II (Poster)
    within DFG-SPP 1324 "Extraction of quantifiable information from complex systems"
    (2012 - 2015)

    This project is the continuation of the DFG-Project 'Adaptive Wavelet Methods for SPDEs, Part I'. It is concerned with the numerical treatment of complex stochastic dynamical systems which are described by stochastic partial differential equations of parabolic type on Lipschitz domains. The overall goal of this project is to establish fully adaptive numerical schemes based on wavelets. By using a variant of the Rothe method, we intend to use a semi-implicit time discretization scheme involving a suitable step-size control. Then, in each time step, an elliptic subproblem has to be solved. To this end, suitable variants of recently developed adaptive wavelet/frame schemes can be used. To get a theoretical underpinning of our algorithms, our investigations must involve regularity estimates of the solutions in specific scales of Besov spaces. In the first period of the SPP 1324, we have introduced a new, wavelet-based noise model and the adaptive wavelet schemes for the elliptic subproblems have already been designed, analysed, implemented and tested. Moreover, spatial Besov regularity results for the solution of linear equations on Lipschitz domains have been established. In the second period of SPP 1324, these investigations will be systematically continued. The focus will be on the development of efficient variants of the Rothe method and the generalization of the Besov regularity results to nonlinear equations and full space-time regularity. As in the first period, the overall scheme will be implemented, tested and integrated int the Marburg software library. This is a joint project with Prof. Dr. K. Ritter and Prof. Dr. R. L. Schilling.

  • Coordination of SPP 1324 "Mathematical Methods for Extracting Quantifiable Information from Complex Systems", Phase II
    (2011-2014)

  • Adaptive Wavelet Frame Methods for Operator Equations: Sparse Grids, Vector-Valued Spaces and Applications to Nonlinear Inverse Parabolic Problems, Part II (Poster)
    within DFG-SPP 1324 "Extraction of quantifiable information from complex systems"
    (2011 - 2013)

    This project is the continuation of the project 'Adaptive Wavelet Frame Methods for Operator Equations: Sparse Grids, Vector-Valued Spaces and Applications to Nonlinear Inverse Parabolic Problems Part I'. For the forward problems, we use generalized tensor product approximation techniques that realize dimension independent convergence rates. In the first period of SPP 1324, these tensor wavelet bases have already been provided and associated adaptive wavelet algorithms have been designed, implemented, and tested. The tests performed during the first period include the numerical solution of a prototypical inverse parabolic parameter identification problem. In addition, the theoretical prerequesites for applying sparsity constrained Tikhonov regularization to such inverse problems have been proved. These investigations will be systematically continued in the second period. In particular, one central goal will be the generalization of such adaptive wavelet algorithms to nonlinear equations. Furthermore we aim at extending the theoretical investigation of Tikhonov-regularization schemes with sparsity constraints by incorporating positivity constraints. As a model problem we will study the parameter identification problem for a parabolic reaction-diffusion system which describes the gene concentrations in embroys at an early state of development (embryogenesis). Within this project, we cooperate with Prof. Dr. P. Maaß and Prof. Dr. R. Stevenson.

  • Adaptive Wavelet Methods for SPDEs, Part I
    within DFG-SPP 1324 "Extraction of quantifiable information from complex systems"
    (2009 - 2012)

    This project is concerned with the numerical treatment of complex stochastic dynamical systems which are described by stochastic partial differential equations of parabolic type on piecewise smooth domains. These equations are driven by a (cylindrical) Wiener process W and may be interpreted as abstract Cauchy problems in a suitable function space. We study the pathwise approximation of the solution process, and the aim of our joint research project of numerical analysts and probabilists is to derive a fully adaptive numerical scheme in time and space. By using a variant of the Rothe method, we intend to use a semi-implicit time discretization scheme involving a suitable step-size control. Then, in each time step, an elliptic subproblem has to be solved. To this end, suitable variants of recently developed adaptive wavelet/frame schemes will be employed. Moreover, alternative noise representations based on biorthogonal wavelet or bi-frame wavelet expansions will be derived. We intend to establish the optimal convergence of the scheme, and we also want to ensure that adaptivity really pays for these problems. Therefore our investigations will be accompanied by regularity estimates for the exact solutions in specific scales of Besov spaces. Another central goal is the implementation and testing of the resulting algorithms. This part will be based on the Marburg software library. This is a joint project with Prof. Dr. K. Ritter and Prof. Dr. R. L. Schilling.

  • Coordination of SPP 1324 "Mathematical Methods for Extracting Quantifiable Information from Complex Systems", Phase I
    (2008-2011)

  • Adaptive Wavelet Frame Methods for Operator Equations: Sparse Grids, Vector-Valued Spaces and Applications to Nonlinear Inverse Parabolic Problems, Part I
    within DFG-SPP 1324 "Extraction of quantifiable information from complex systems"
    (2008 - 2011)

    The aim of this project is the development of optimal convergent adaptive wavelet schemes for complex systems. Especially, we shall be concerned with (nonlinear) elliptic and parabolic operator equations on nontrivial domains as well as with the related inverse parameter identification problems. As always, the construction of suitable wavelet bases on these domains is a principal problem. In this project, we shall use variants of adaptive frame schemes in combination with domain decomposition ideas. Another bottleneck is the curse of dimensionality. We attack this problem by means of tensor product approximation techniques that realize dimension independent convergence rates. Our findings will be applied to well-posed elliptic and parabolic operator equations. The resulting adaptive forward solvers will be combined with regularization techniques for the related inverse parameter identification problem. We aim at analyzing and developing Tikhonovregularization schemes with sparsity constraints for such nonlinear inverse problems. As a model problem we will study the parameter identification problem for a parabolic reactiondiffusion system which describes the gene concentrations in embryos at an early state of development (embryogenesis). This is a joint project with Prof. Dr. P. Maaß and Prof. Dr. R. Stevenson.

  • Adaptive Wavelet Methods for Inverse Problems and Inverse Parabolic Equations
    (2006 - 2009)

    In this project, two fields of applied mathematics, which were developed almost independently from each other, will be consolidated. On one hand, adaptive wavelet methods for the numerical treatment of operator equations were examined for many years. However the achieved convergence results refer exclusively to continuously invertible operators.On the other hand, the general convergence theory, at least for linear inverse problems, has been worked out for a long time. However, adaptive wavelet methods and the results of non-linear wavelet approximation schemes have hardly been used so far in this context, with the exception of the recent results in [14, 42].In this project we would like to analyze different approaches to adaptive wavelet methods for inverse problems: two-step regularization schemes, where the data are pre-smoothed in a first step; regularization methods based on adaptive procedures for the forward operator combined with classical regularization procedures; regularization by wavelet discretization. These methods will be extended to some non-linear operator equations.As a prototypical application an inverse heat conduction problem will be examined. This is a joint project with Prof. Dr. P. Maaß.

  • Multivariate Wavelet Analysis: Constructions, Specific Applications, and Data Structures, Part III (Poster)
    within DFG-SPP 1114 "Mathematische Methoden der Zeitreihenanalyse und digitalen Bildverarbeitung"
    (2005 - 2007)

    This project is the continuation of the DFG-Project: Multivariate Wavelet Analysis: Constructions, Specific Applications, and Data Structures within SPP 1114. In the second period of SPP 1114, we have mainly investigated the following issues. Firstly, we were concerned with the construction of nonseparable multivariate wavelets/multiwavelets with respect to general scalings. Secondly, by using a variant of the famous Feichtinger/Gröchenig theory, we have started to derive a systematic approach to the construction of frames on domains and manifolds. Thirdly, we have investigated the applicability of multiwavelets and frames in image/signal analysis. In this third project, we intend to continue the research of these topics, again in a parallel and very systematic fashion. We shall focus on the construction of biorthogonal multiwavelets for general scalings, and we shall investigate to what extent additional features such as symmetry and approximation properties can be incorporated. Moreover, in the context of our frame approach, we want to extend our research on generalized coorbit spaces and on the associated frame decomposition, with special amphasis on spaces of mixed smoothness and on localization properties. In addition, we intend to investigate further if our multiwavelet/frame approach is helpful in applications from signal/image analysis, in particular concerning denoising algorithms.

  • Multivariate Wavelet Analysis: Constructions, Specific Applications, and Data Structures, Part II (Poster)
    within DFG-SPP 1114 "Mathematische Methoden der Zeitreihenanalyse und digitalen Bildverarbeitung"
    (2003 - 2005)

    This project is the continuation of the DFG-Project: Multivariate Wavelet Analysis: Constructions, Specific Applications, and Data Structures within SPP 1114. In the first period of SPP 1114, we were concerned with the constructions of nonseparable multivariate wavelets/multiwavelets with respect to general scalings and with their applications in image/signal analysis. In this second project, we intend to continue these researches, but with a slightly different focus. In the meantime, it has turned out that in many applications stable multiscale bases are not flexible enough. One possible way out would be the use of frames. By using a variant of the famous Feichtiner/Gröching theory, we have already been able to derive a quite general approach to the construction of frames on domains and manifolds. Therefore, besides the work on wavelets for general scalings, we intend to generalize these results as far as possible. Especially, generalized coorbit spaces and the associated frame decompositions will be studied in detail. Moreover, we want to investigate if the frames approach is helpful in certain applications from signal/image analysis, especially concerning denoising algorithms.

  • Multivariate Wavelet Analysis: Constructions, Specific Applications, and Data Structures, Part I (Poster)
    within DFG-SPP 1114 "Mathematische Methoden der Zeitreihenanalyse und digitalen Bildverarbeitung"
    (2001 - 2003)

    This project is concerned with the construction and the applications of wavelets. We are especially interested in nonseparable multivariate wavelets with respect to general scaling matrices. The use of general scalings provides some principal advantages. One aim of this project is to investigate if these superiorities really count when it comes to practical applications, or if they are wasted by an overhead of technical difficulties. We therefore want to develop construction principles which yield suitable wavelet/multiwavelet bases in an economical way. We shall focus on wavelets with some additional features such as certain interpolation properties. Furthermore, we intend to investigate if general scaling matrices can be helpful in certain applications from signal/image analysis, especially concerning denoising algorithms for general kinds of noise. The analysis will be accompanied by the development of efficient data structures which are adapted to the specific requirements of wavelet algorithms.

LOEWE-Center 'Synthetic Microbiology'

  • Sensitivitätsanalyse, Parameterbestimmung und Modellvalidierung für komplexe biologische Prozesse
    (2010 - 2014)

    Mathematische Modelle gewinnen bei der Beschreibung biologischer Prozesse mehr und mehr an Bedeutung. Ihr Hauptziel ist die Bereitstellung wissenschaftlicher Erkenntnisse über biologische Prozesse. Auf Simulationsergebnisse ist jedoch nur dann Verlass, wenn das zugrundeliegende Modell den gegebenen Prozess genau beschreibt. Das bedeutet, dass mathematische Modelle ür biologische Prozesse diesen nicht nur qualitativ beschreiben, sondern auch sein quantitatives Verhalten widerspiegeln müssen. Dies verlangt ein Modell, das anhand experimenteller Daten validiert wurde, mit hinreichend guten Schätzungen für die Modellparameter. Die Entwicklung und quantitative Validierung komplexer, nichtlinearer Differentialgleichungsmodelle ist eine schwierige Aufgabe, die der Unterstützung durch numerische Verfahren zur Datenanalyse bedarf, wie z. B. raum-zeitliche Datenextraktion aus Bildfolgen, Sensitivitätsanalyse, Parameterschätzung und optimale Versuchsplanung. Der Schwerpunkt unserer Forschung ist die Entwicklung innovativer numerischer Methoden zur Modellvalidierung und die Übertragung dieser Verfahren auf anspruchsvolle Anwendungen von SYNMIKRO. Es besteht eine Kooperation mit B. Eckhardt und E. Kostina


  • Dynamik regulatorischer Netzwerke für Zellpolarität
    (2010 - 2014)

    Im Rahmen von SYNMIKRO beschäftigen wir uns primär mit der mathematischen Modellierung und der numerischen Simulation von zellbiologischen Systemen. Das generelle Ziel ist es, mittels der Analyse geeigneter mathematischer Modelle ein tieferes Verständnis der zugrunde liegenden Systeme zu gewinnen. Langfristig sollen die gewonnenen Erkenntnisse dazu dienen, verlässliche Vorhersagen für das Design von Experimenten zu machen. Im Zentrum der Forschungen steht derzeit die Entwicklung von mathematischen Modellen zur Beschreibung von Zellpolarität, speziell beim Bakterium Myxococcus xanthus. Die Bewegungsmuster von M. xanthus sind eng mit Proteinoszillationen innerhalb der Zelle gekoppelt. Deshalb wurde zunächst untersucht, ob solche Proteinoszillationen durch äußere Anregung (Trigger) induziert sein müssen oder ob eigenständig oszillierende Systeme mit wenigen Spezies möglich sind. Zu diesem Zwecke wurde ein mathematisches Modell für ein System aus nur zwei Proteinen unter möglichst wenig einschränkenden Annahmen entwickelt. Die prinzipiellen Annahmen dieses Modells sind: eine Wechselwirkung der Proteine findet nur an den Zellpolen statt, der Transport der Proteine durch das Innere der Zelle erfolgt durch Diffusion ohne Wechselwirkung. In beiden Zellpolen gelten identische Gesetze für die Wechselwirkungen, keiner der Pole ist ausgezeichnet, Oszillationen sind eine Folge der Dynamik im Modell. Tatsächlich konnte gezeigt werden, dass im Rahmen eines solchen Modells Oszillationen ohne äußere Anregungen möglich sind! In naher Zukunft wird das Modell weiter verfeinert und verallgemeinert werden, stochastische Einflüsse werden berücksichtigt und andere Wechselwirkungsszenarien systematisch studiert. Ein weiterer Schwerpunkt ist die Entwicklung geeigneter mathematischer Verfahren zur Analyse elektronenmikroskopischer Bilder, die bei der Analyse zellbiologischer Systeme auftreten. Die Stichworte sind hier unter anderem Segmentierung und Tracking. Diese Forschungen dürften wegen der generellen Problematik auch für zahlreiche andere SYNMIKRO-Projekte relevant sein. Es besteht eine Kooperation mit P. Lenz und L. Sogaard-Andersen

EU Research Networks

  • Uncertainty Principles versus Localization Properties, Function Systems for Efficient Coding Schemes - UNLocX
    (2010 - 2013)

    Stephan Dahlke is on the advisory board for this project. Click here to get more Information.

  • Wavelets and Multiscale Methods in Numerical Analysis (concluded)
  • Harmonic Analysis and Statistics for Signal and Image Processing (local HASSIP page)

DAAD-Projects