Bunse-Gerstner, Angelika
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Bunse-Gerstner, Angelika
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Bunse-Gerstner, Angelika
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Item-typ:Veröffentlichung, Interpolation Based Parametric Model Order ReductionIn this thesis, we consider model order reduction of parameter-dependent large-scale dynamical systems. The objective is to develop a methodology to reduce the order of the model and simultaneously preserve the dependence of the model on parameters. We use the balanced truncation method together with spline interpolation to solve the problem. The core of this method is to interpolate the reduced transfer function, based on the pre-computed transfer function at a sample in the parameter domain. Linear splines and cubic splines are employed here. The use of the latter, as expected, reduces the error of the method. The combination is proven to inherit the advantages of balanced truncation such as stability preservation and, based on a novel bound for the infinity norm of the matrix inverse, the derivation of error bounds. Model order reduction can be formulated in the projection framework. In the case of a parameter-dependent system, the projection subspace also depends on parameters. One cannot compute this parameter-dependent projection subspace, but has to approximate it by interpolation based on a set of pre-computed subspaces. It turns out that this is the problem of interpolation on Grassmann manifolds. The interpolation process is actually performed on tangent spaces to the underlying manifold. To do that, one has to invoke the exponential and logarithmic mappings which involve some singular value decompositions. The whole procedure is then divided into the offline and online stage. The computation time in the online stage is a crucial point. By investigating the formulation of exponential and logarithmic mappings and analyzing the structure of sums of singular value decompositions, we succeed to reduce the computational complexity of the online stage and therefore enable the use of this algorithm in real time.Dissertation510 319 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Optimal interpolation-based model reductionThis dissertation is devoted to the development and study of new techniques for model reduction of large-scale linear time-invariant dynamical systems. The behavior of processes in electrical networks, mechanics, weather prediction and many others can be described by high-dimensional systems of linear ordinary differential or difference equations. Model reduction methods can then be helpful as they provide automatic processes which construct a reduced-order system whose input-output behavior approximates the behavior of the original system. Most of the current methods are designed for approximating asymptotically stable systems. However, some processes such as weather development are unstable. In this thesis new interpolation-based methods are proposed that aim to compute an optimal reduced model for stable as well as for unstable systems. For these optimization problems tangential interpolation-based first order necessary optimality conditions are derived. On the basis of the established theory an iterative algorithm is proposed which, if it converges, provides a reduced system that satisfies the aforementioned first order conditions. In numerical experiments, the accuracy of the new method is illustrated and compared with other existing techniques. The benefit of the new approach to model reduction of unstable systems is also demonstrated on example of shallow water models. In case of systems with large number of unstable poles, the method is shown to be significantly better than other approaches.Dissertation292 122 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, A Hierarchically Semiseparable Preconditioner for the Numerical Solution of 3D Electromagnetic Scattering ProblemsWe consider the numerical solution of linear systems arising from the discretization of the Electric Field Integral Equation (EFIE) in scattering problems for arbitrarily shaped targets. For some geometries the associated matrix can be poorly conditioned making the use of a preconditioner mandatory to obtain convergence. In this thesis the linear system resulting from the discretized electromagnetic scattering problem is solved by means of a preconditioned GMRES in the context of the multilevel fast multipole method (MLFMM) . For this purpose an approximate hierarchically semiseparable (HSS) representation of the near-field matrix, the part of the matrix capturing interactions among nearby groups in the MLFMM, is constructed and used as preconditioner for the GMRES iterations. As experience shows, the efficiency of an ILU preconditioning for such systems essentially depends on a sufficient fill-in, which apparently sacrifices the sparsity of the near-field matrix and thus leads, for electrically large structures, to storage problems. In the light of this experience we propose to approximate the near-field matrix by means of a HSS approximant so as to build a structured preconditioner to the impedance matrix. In this manner we impose a low-rank structure on the arising fill-in, thus alleviating the storage requirements of the preconditioner. To this end the group distribution at the finest level of the MLFMM is exploited so as to obtain the HSS approximant to the near-field matrix. As a result, the storage problems stated in a previous work by the author are mitigated substantially. The numerical results presented in the thesis show that this kind of algebraic preconditioning can substantially reduce the number of iterations in the solution of the resulting system of equations.Dissertation225 151 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Using model reduction techniques within the incremental 4D-Var methodThis thesis is devoted to the development and study of new numerical methods for data assimilation of large dimensional problems. Incremental four-dimensional variational data assimilation is the method of choice in many operational atmosphere and ocean data assimilation systems. It allows the four dimensional variational technique (4D-Var) to be implemented in a computationally efficient way by replacing the minimization of the full nonlinear 4D-Var cost function with the minimization of a series of simplified linear cost functions. In practice these simplified functions are usually derived from a spatial or spectral truncation of the full system being approximated. This thesis proposes a new method for deriving the simplified problems in incremental 4D-Var based on model reduction techniques developed in the field of control theory. Such a procedure ensures that no essential information is lost. It supplies a better accuracy of the forecast compared to commonly used approximation techniques. A main contribution of this thesis is the derivation of a new interpretation of the data assimilation problem as a control theoretic problem incorporating all statistical information. This new approach makes the use of model reduction techniques within data assimilation methods possible. Moreover, it is important to take into account that numerical weather prediction systems usually lead to unstable control systems. It is shown how a proper choice of the model reduction method is able to cope with this additional difficulty.Various numerical experiments using shallow water test models underline that the combination of model reduction techniques with incremental 4D-Var gives an assimilation method that retains more of the dynamical information of the full system. Numerical tests with varying observing networks illustrate the superior performance of the model reduction approach compared to standard truncation techniques. Additionally, the numerical experiments confirm how the incorporation of all statistical information in the model reduction procedure improve the accuracy of the approximation.Dissertation314 215 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Covariance update in data assimilation for state and parameter estimationThe goal of Data Assimilation (DA) is to estimate, with the aid of numerical methods, the true state of a dynamical system, taking into account observations (measurements), a forecast model and state, as well as statistical information of observation, model and forecast errors. DA is also crucial for improving mathematical models, which are defective or inaccurate and contain unproven data and unknown parameters respectively, for simulation and control. This research field has an extraordinary number of application areas, being the simulation and computation of climate and ocean models among the most important and studied. In the context of data assimilation problems it is common that models depend on poorly known parameters. A well-known approach is to solve both problems, data assimilation and parameter estimation, at the same time following the so-called augmented state approach. The idea is to solve a modified DA problem, where the parameters are considered state variables and included in an augmented state vector, and the parameter evolution dynamics are incorporated into a new augmented forecast model. Typically, parameters are not directly observed; therefore, their estimation depends on the ability of the assimilation scheme to infer parameter updates using the information obtained from the observational data. The interrelation between state variables and parameters is given by their correlations. In this work we investigate the influence of the augmented state covariance matrix on the joint state-parameter data assimilation problem. Moreover, using the augmented approach we propose a novel method based on a low-cost update of the augmented state covariance matrix. Furthermore, we find necessary and sufficient conditions for the convergence of 3D-Var methods, and in particular of our proposed strategy, when applied to linear state-parameter DA problems. The suitability of our proposed method is tested using several benchmark problems.Dissertation494 162