Wed, 22 July, 2009, 10:30-11:30, C1
In this talk we will briefly review the development of the theory of local
regularization as an efficient "structure-preserving" method for solving
linear and nonlinear ill-posed problems. The application of this method has
primarily been for problems of Volterra type, and in recent years an
effective discrepancy principle has been added to the theory. In addition,
the theory has been extended to nonlinear problems such as the Hammerstein
problem and the autoconvolution problem.
One appealing feature of the method of local regularization for nonlinear
problems is that localized domains defined simply for the purposes
of regularization can also be used to facilitate the linearization of such
problems, further adding to the efficiency of these regularization methods.
Finally, we will describe a class of "generalized local regularization
methods" which includes both the method of local regularization as well
as the method of Lavrentiev (or "simplified regularization") as special cases.
Not only does the generalized method have the potential for application
to far more general inverse problems, but the theory behind generalized
local regularization also gives insight into why local regularization often
outperforms the method of Lavrentiev in numerical testing.
The work presented here is joint with Cara Brooks and includes collaborative
efforts with Zhewei Dai and Xiaoyue Luo.
Presentation slides (pdf, 2.7 MB)
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