On the local regularization of inverse problems of Volterra type

Patricia K. Lamm
Department of Mathematics
Michigan State University
E. Lansing, MI 48824-1027

In Proc. 1995 ASME Design Engineering Technical Conferences: Parameter Identification Symposium, September 1995, Boston, MA.



Abstract: We consider a local regularization method for the solution of first-kind Volterra integral equations with convolution kernel. The local regularization is based on a splitting of the original Volterra operator into "local" and "global" parts, and a use of Tikhonov regularization to stabilize the inversion of the local operator only. The regularization parameters for the local procedure include the standard Tikhonov parameter, as well as a parameter that represents the length of the local regularization interval.

We present a convergence theory for the infinite-dimensional regularization problem and show that the regularized solutions converge to the true solution as the regularization parameters go to zero (in a prescribed way). In addition, we show how numerical implementation of the ideas of local regularization can lead to the notion of "sequential Tikhonov regularization" for Volterra problems; this approach has been shown in (Lamm and Elden, 1995) to be just as effective as Tikhonov regularization, but to be much more efficient computationally.

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Contact: lamm@math.msu.edu