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Mathematisch-Naturwissenschaftliche Fakultät - Jahrgang 2005


Titel Fast Optimised Wavelet Methods for Control Problems Constrained by Elliptic PDEs
Autor Carsten Burstedde
Publikationsform Dissertation
Abstract In this thesis, a wavelet method for the numerical solution of an optimal control problem constrained by a linear elliptic partial differential equation is developed. The particular challenge here lies in considering and combining two areas of research, namely the efficient solution of an elliptic partial differential equation (shortly PDE) on the one hand and an optimisation problem specified by an objective functional and PDE constraints on the other.
To cope with the finite amount of computer memory, the problem needs to be discretised. Already for the numerical solution of a single PDE, this gives rise to a large and ill-conditioned sparse linear system of equations, which necessitates the use of iterative solvers combined with suitable preconditioning techniques. The reformulation of the control problem in terms of a Lagrangian functional leads to a coupled system of PDEs. Its iterative solution requires repeated solutions of a single PDE in inner loops, such that the computation time is multiplied accordingly. Moreover, the introduction of control and adjoint variables leads to a significant increase of the memory requirements.
Here we address these difficulties in a unified way by the systematic use of biorthogonal B-spline wavelet bases, which results in optimally preconditioned operators. Therefore, iterative solution schemes such as the method of conjugate gradients need only a constant amount of iterations to reduce the error by a fixed factor. The introduction of specific transformations additionally improves the condition numbers of the wavelet bases and the discretised differential operators, which leads to a significant speedup of the computations. Furthermore, the wavelet framework permits the numerical evaluation of fractional Sobolev norms in the objective functional by means of Riesz matrices, for which we present a novel construction which yields exact results for a wider range of functions and smoothness indices than the currently used approaches.
To construct an algorithm of optimal computational complexity, that is, with a runtime proportional to the number of unknowns, we design a two-layer nested iteration strategy and combine it with an inner-outer conjugate gradient scheme, employing specifically balanced error tolerances and stopping criteria. An adaptive variant of the algorithm is devised by the incorporation of routines which have been recently proposed as part of adaptive wavelet methods for elliptic PDEs and nonlinear variational problems. This ansatz allows for different distributions of active wavelet coefficients for the state, adjoint and control variables.
Extensive parameter studies are presented for both uniform and adaptive discretisations. It is demonstrated that the freedom in modelling introduced by the enhanced construction of Riesz operators allows to influence the character of the state and the control by varying the Sobolev norms in the objective functional. The algorithm is indeed of optimal linear computational complexity. Moreover, we verify that the adaptive scheme leads to a considerable reduction of active wavelet coefficients and a slightly superlinear rate of convergence.
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© Universitäts- und Landesbibliothek Bonn | Veröffentlicht: 2005