@inproceedings{Kaehler2003,
  author    = {K{\"a}hler, Uwe},
  title     = {Multiresolution analysis (MRA) on Qp-spaces},
  doi       = {10.25643/bauhaus-universitaet.317},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20111215-3170},
  year      = {2003},
  abstract  = {In this paper we want to outline the possibility of using methods of Wavelet analysis to study Qp-spaces},
  subject      = {Wavelet},
  language  = {en}
}
@inproceedings{Markwardt2003,
  author    = {Markwardt, Klaus},
  title     = {Biorthogonale Waveletsysteme in der Parameteridentifikation},
  doi       = {10.25643/bauhaus-universitaet.330},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20111215-3308},
  year      = {2003},
  abstract  = {In der vorliegenden Arbeit geht es um die Anwendung von biorthogonalen Waveletsystemen in der Parameteridentifikation. Es sollen Grundlagen geschaffen werden, um bei der Auswertung dynamischer Experimente derartige Wavelets und damit die schnelle Wavelet-Transformation (FWT) systematisch und effektiv zu nutzen. Zu diesem Zweck wird von den Waveletfiltern ein System von Verbindungskoeffizienten abgeleitet. Mit deren Hilfe erfolgen die Projektionen von Operatoren, insbesondere die von Differentiations- und Integrationsoperatoren, in die entsprechenden Wavelet-R{\"a}ume. S{\"a}mtliche Verbindungskoeffizienten k{\"o}nnen rekursiv und in endlich vielen Schritten exakt berechnet werden. Ausgehend von den dynamischen Krafteinwirkungen und den gemessenen Reaktionsbeschleunigungen oder Reaktionsgeschwindigkeiten bez{\"u}glich der einzelnen Freiheitsgrade k{\"o}nnen dann unbekannte Steifigkeiten und D{\"a}mpfungen identifiziert werden. Dazu erfolgt nach entsprechenden Wavelet-Zerlegungen aller relevanten Zeitsignale ein Abgleich auf den einzelnen Frequenzb{\"a}ndern. Dieser f{\"u}hrt insbesondere zu einem System von linearen Matrizengleichungen zur Bestimmung der unbekannten Parameter. Vorgeschlagen wird im Falle einer gr{\"o}ßeren Zahl von Freiheitsgraden und Parametern, ein mehrstufiges Optimierungsverfahren anzuwenden. Gegen{\"u}ber Identifikationsverfahren im Zeitbereich werden aufwendige numerische Quadraturverfahren und die daraus resultierenden Fehlerquellen und Stabilit{\"a}tsprobleme vermieden. Gegen{\"u}ber Verfahren im Frequenzbereich, die ausschließlich mit Hilfe der FFT formuliert werden, sind St{\"o}rungen in den Randspektren besser beherrschbar und eliminierbar. Außerdem werden mit einem FWT-Verfahren einfachere Denoising-Algorithmen anwendbar. Letztendlich wird im Vergleich zu einem FFT-Verfahren ein sp{\"a}terer {\"U}bergang zur Identifikation nichtlinearer MDOF-Systeme methodisch erleichtert.},
  subject      = {Parameteridentifikation},
  language  = {de}
}
@inproceedings{SoaresDahlkeLindemann2003,
  author    = {Soares, Maria Joana and Dahlke, S. and Lindemann, M.},
  title     = {A Wavelet Based Numerical Method for Nonlinear Partial Differential Equations},
  doi       = {10.25643/bauhaus-universitaet.363},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20111215-3631},
  year      = {2003},
  abstract  = {This paper is concerned with the numerical treatment of quasilinear elliptic partial differential equations. In order to solve the given equation we propose to use a Galerkin approach, but, in contrast to conventional finite element discretizations, we work with trial spaces that, not only exhibit the usual approximation and good localization properties, but, in addition, lead to expansions of any element in the underlying Hilbert spaces in terms in multiscale or wavelet bases with certain stability properties. Specifically, we select as trial spaces a nested sequence of spaces from an appropriate biorthogonal multiscale analysis. This gives rise to a nonlinear discretized system. To overcome the problems of nonlinearity, we make use of the machinery of interpolating wavelets to obtain knot oriented quadrature rules. Finally, Newton's method is applied to approximate the solution in the given ansatz space. The results of some numerical experiments with different biorthogonal systems, confirming the applicability of our scheme, are presented.},
  subject      = {Nichtlineare partielle Differentialgleichung},
  language  = {en}
}
@phdthesis{Yousefi,
  author    = {Yousefi, Hassan},
  title     = {Discontinuous propagating fronts: linear and nonlinear systems},
  doi       = {10.25643/bauhaus-universitaet.4717},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20220922-47178},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  pages     = {356},
  abstract  = {The aim of this study is controlling of spurious oscillations developing around discontinuous solutions of both linear and non-linear wave equations or hyperbolic partial differential equations (PDEs). The equations include both first-order and second-order (wave) hyperbolic systems. In these systems even smooth initial conditions, or smoothly varying source (load) terms could lead to discontinuous propagating solutions (fronts). For the first order hyperbolic PDEs, the concept of central high resolution schemes is integrated with the multiresolution-based adaptation to capture properly both discontinuous propagating fronts and effects of fine-scale responses on those of larger scales in the multiscale manner. This integration leads to using central high resolution schemes on non-uniform grids; however, such simulation is unstable, as the central schemes are originally developed to work properly on uniform cells/grids. Hence, the main concern is stable collaboration of central schemes and multiresoltion-based cell adapters. Regarding central schemes, the considered approaches are: 1) Second order central and central-upwind schemes; 2) Third order central schemes; 3) Third and fourth order central weighted non-oscillatory schemes (central-WENO or CWENO); 4) Piece-wise parabolic methods (PPMs) obtained with two different local stencils. For these methods, corresponding (nonlinear) stability conditions are studied and modified, as well. Based on these stability conditions several limiters are modified/developed as follows: 1) Several second-order limiters with total variation diminishing (TVD) feature, 2) Second-order uniformly high order accurate non-oscillatory (UNO) limiters, 3) Two third-order nonlinear scaling limiters, 4) Two new limiters for PPMs. Numerical results show that adaptive solvers lead to cost-effective computations (e.g., in some 1-D problems, number of adapted grid points are less than 200 points during simulations, while in the uniform-grid case, to have the same accuracy, using of 2049 points is essential). Also, in some cases, it is confirmed that fine scale responses have considerable effects on higher scales. In numerical simulation of nonlinear first order hyperbolic systems, the two main concerns are: convergence and uniqueness. The former is important due to developing of the spurious oscillations, the numerical dispersion and the numerical dissipation. Convergence in a numerical solution does not guarantee that it is the physical/real one (the uniqueness feature). Indeed, a nonlinear systems can converge to several numerical results (which mathematically all of them are true). In this work, the convergence and uniqueness are directly studied on non-uniform grids/cells by the concepts of local numerical truncation error and numerical entropy production, respectively. Also, both of these concepts have been used for cell/grid adaptations. So, the performance of these concepts is also compared by the multiresolution-based method. Several 1-D and 2-D numerical examples are examined to confirm the efficiency of the adaptive solver. Examples involve problems with convex and non-convex fluxes. In the latter case, due to developing of complex waves, proper capturing of real answers needs more attention. For this purpose, using of method-adaptation seems to be essential (in parallel to the cell/grid adaptation). This new type of adaptation is also performed in the framework of the multiresolution analysis. Regarding second order hyperbolic PDEs (mechanical waves), the regularization concept is used to cure artificial (numerical) oscillation effects, especially for high-gradient or discontinuous solutions. There, oscillations are removed by the regularization concept acting as a post-processor. Simulations will be performed directly on the second-order form of wave equations. It should be mentioned that it is possible to rewrite second order wave equations as a system of first-order waves, and then simulated the new system by high resolution schemes. However, this approach ends to increasing of variable numbers (especially for 3D problems). The numerical discretization is performed by the compact finite difference (FD) formulation with desire feature; e.g., methods with spectral-like or optimized-error properties. These FD methods are developed to handle high frequency waves (such as waves near earthquake sources). The performance of several regularization approaches is studied (both theoretically and numerically); at last, a proper regularization approach controlling the Gibbs phenomenon is recommended. At the end, some numerical results are provided to confirm efficiency of numerical solvers enhanced by the regularization concept. In this part, shock-like responses due to local and abrupt changing of physical properties, and also stress wave propagation in stochastic-like domains are studied.},
  subject      = {Partielle Differentialgleichung},
  language  = {en}
}
@phdthesis{Zabel2003,
  author    = {Zabel, Volkmar},
  title     = {Anwendungen der Wavelet-Transformation in der Systemidentifikation},
  doi       = {10.25643/bauhaus-universitaet.5},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20040202-66},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  year      = {2003},
  abstract  = {Die meisten traditionellen Methoden der Systemidentifikation beruhen auf der Abbildung der Meßwerte entweder im Zeit- oder im Frequenzbereich. In j{\"u}ngerer Zeit wurden im Zusammenhang mit der Systemidentifikation Verfahren entwicklet, die auf der Anwendung der Wavelet-Transformation beruhen. Das Ziel dieser Arbeit war, einen Algorithmus zu entwickeln, der die Identifikation von Parametern eines Finite-Elemente-Modells, das ein experimentell untersuchtes mechanisches System beschreibt, erm{\"o}glicht. Es wurde eine Methode erarbeitet, mit deren Hilfe die gesuchten Parameter durch L{\"o}sen eines Systems von Bewegungsgleichungen im Zeit-Skalen-Bereich ermittelt werden. Durch die Anwendung dieser Darstellung k{\"o}nnen Probleme, die durch Rauschanteile in den Meßdaten entstehen, reduziert werden. Die Ergebnisse numerischer Simulation und einer experimentellen Studie best{\"a}tigen die Vorteile einer Anwendung der Wavelet-Transformation in der vorgeschlagenen Weise. ...},
  subject      = {Wavelet},
  language  = {de}
}