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作者 Khan, Akhtar A
書名 Numerical methods for elliptic inverse problems with interior data
國際標準書號 9780542225369
book jacket
說明 208 p
附註 Source: Dissertation Abstracts International, Volume: 66-07, Section: B, page: 3743
Adviser: Mark S. Gockenbach
Thesis (Ph.D.)--Michigan Technological University, 2005
The coefficients in a variety of linear elliptic partial differential equations can be estimated from interior measurements of the solution. An abstract framework for this inverse problem has the advantage that seemingly different problems can be examined in a unified framework. This thesis is devoted to the study of elliptic inverse problems from an abstract point of view. This work contains a new modified output least-squares approach for elliptic inverse problems where an energy based norm is used. It turns out that when the (coefficient-dependent) energy norm is used, the result is a smooth, convex output least-squares functional. This circumvents a serious deficiency of classical output least squares functional of being, in general, a non-convex functional. Using total variation regularization, it is possible to estimate discontinuous coefficients by the modified output least-squares approach. The minimization problem is guaranteed to have a solution, and this solution can be obtained as the limit of the solution to finite-dimensional discretization of the problem. All of these properties hold in an abstract framework that encompasses several interesting problems: The standard (scalar) elliptic BVP in divergence form, the system of isotropic elasticity, and others
Another major issue examined in this thesis is the study of the augmented Lagrangian method for the modified OLS in an extended and abstract framework. Posing the estimation problem as a constrained optimization problem with the PDE as the constraint allows the use of the augmented Lagrangian method, which is guaranteed to converge. Moreover, the convergence analysis encompasses discretization by finite element methods, so the proposed algorithm can be implemented and will produce a solution to the constrained minimization problem
Based on the proposed modified OLS, new and effective fixed point type iterative schemes are proposed and a complete convergence analysis is given under less stringent conditions. An abstract version of the equation error approach is given which is capable of identifying even the discontinuous coefficients. The principle of iterative regularization is extended to nonlinear elliptic inverse problems. Several numerical examples and comparison of different approaches is given
School code: 0129
Host Item Dissertation Abstracts International 66-07B
主題 Mathematics
Alt Author Michigan Technological University
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