Wavelet-based finite element methods for approximations of optimal controls for distributed parameter systems /
This dissertation focuses on the application of wavelet analysis in engineering fields, specifically computational mechanics and control problems for distributed parameter systems. In the first part of the dissertation, a wavelet based finite element method is developed for a class of domains. For...
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| Format: | Thesis Book |
| Language: | English |
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[Place of publication not identified] :
[publisher not identified] ;
1996.
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| Online Access: | http://proxy.library.tamu.edu/login?url=http://proquest.umi.com/pqdweb?did=743273741&sid=1&Fmt=2&clientId=2945&RQT=309&VName=PQD |
| Summary: | This dissertation focuses on the application of wavelet analysis in engineering fields, specifically computational mechanics and control problems for distributed parameter systems. In the first part of the dissertation, a wavelet based finite element method is developed for a class of domains. For the construction of tensor product elements and triangular elements, generalized connection coefficients are defined. These coefficients facilitate the wavelet Calerkin formulation over a class of "self-similar" domains. A stable numerical algorithm is adapted to obtain these coefficients without the numerical evaluation of the wavelet functions. Numerical examples are considered for the validation of the methodology.In the second part of the dissertation, the derivation of a multilevel preconditioning method for the wavelet based FEM is presented. These multilevel formulations form the kernel of approximate methods for solving optimal control problems in the last section of the dissertation. It is shown that when the domain of interest is well-aligned with the underlying lattice induced by a multiresolution analysis defined on Rd, we can construct a multilevel preconditioner with optimal complexity. The derived preconditioner is applicable to many wavelets currently available, without modifications to the basis functions. Examples are presented to demonstrate the optimal complexity of the preconditioned system. To enforce prescribed boundary conditions, a Lagrange multiplier method is employed in this dissertation. For an iterative solution of the resulting saddle point problem a modified Uzawa's algorithm is adopted, and multilevel preconditioners are derived that can be employed for a class of distributed control problems. Finally, a formulation of the wavelet based finite element method and the multilevel preconditioning method is presented for optimal control of distributed parameter systems. Specifically, the optimal control problem in which we seek to regulate a temperature distribution along a side of a rectangular domain is formulated. A gradient method is used for the solution of the optimality system and the numerical experiments with the wavelet based FEM equipped with the multilevel preconditioners are given. It is shown that the wavelet based formulation is advantageous in that the fast convergence of approximate solution is achieved independent of the mesh resolution. |
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| Item Description: | Vita. "Major Subject: Aerospace Engineering". |
| Physical Description: | xiv, 151 leaves : illustrations ; 28 cm. Issued also on microfiche from University Microfilms Inc. |
| Bibliography: | Includes bibliographical references. |