Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems.
Electron. J. Diff. Eqns., Conference 21 (2014), pp. 173--181.

Tikhonov regularization using Sobolev metrics

Parimah Kazemi, Robert J. Renka

Given an ill-posed linear operator equation Au=f in a Hilbert space, we formulate a variational problem using Tikhonov regularization with a Sobolev norm of u, and we treat the variational problem by a Sobolev gradient flow. We show that the gradient system has a unique global solution for which the asymptotic limit exists with convergence in the strong sense using the Sobolev norm, and that the variational problem therefore has a unique global solution. We present results of numerical experiments that demonstrates the benefits of using a Sobolev norm for the regularizing term.

Published February 10, 2014.
Math Subject Classifications: 47A52, 65D25, 65F22.
Key Words: Gradient system; Ill-posed problem; least squares; Sobolev gradient; Tikhonov regularization.

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Parimah Kazemi
Department of Mathematics and Computer Science
Ripon College, P. O. Box 248
Ripon, WI 54971-0248, USA
Robert J. Renka
Department of Computer Science & Engineering
University of North Texas
Denton, TX 76203-1366, USA

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