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Numerical methods for elliptic inver...

Khan, Akhtar A.

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Numerical methods for elliptic inverse problems with interior data.

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Numerical methods for elliptic inverse problems with interior data./
Author:	Khan, Akhtar A.
Description:	208 p.
Notes:	Adviser: Mark S. Gockenbach.
Contained By:	Dissertation Abstracts International66-07B.
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3182576
ISBN:	9780542225369

Numerical methods for elliptic inverse problems with interior data.
Khan, Akhtar A.

Numerical methods for elliptic inverse problems with interior data. - 208 p.

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.

ISBN: 9780542225369Subjects--Topical Terms:

515831
Mathematics.

Numerical methods for elliptic inverse problems with interior data.
LDR:03207nam 2200289 a 45 001 953809
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020 $a 9780542225369
035 $a (UMI)AAI3182576
035 $a AAI3182576
040 $a UMI $c UMI
100 1 $a Khan, Akhtar A. $3 1066871
245 1 0 $a Numerical methods for elliptic inverse problems with interior data.
300 $a 208 p.
500 $a Adviser: Mark S. Gockenbach.
500 $a Source: Dissertation Abstracts International, Volume: 66-07, Section: B, page: 3743.
502 $a Thesis (Ph.D.)--Michigan Technological University, 2005.
520 $a 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.
520 $a 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.
520 $a 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.
590 $a School code: 0129.
650 4 $a Mathematics. $3 515831
690 $a 0405
710 2 $a Michigan Technological University. $3 1032921
773 0 $t Dissertation Abstracts International $g 66-07B.
790 $a 0129
790 1 0 $a Gockenbach, Mark S., $e advisor
791 $a Ph.D.
792 $a 2005
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3182576