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
Abstract:
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
email: parimah.kazemi@gmail.com
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Robert J. Renka
Department of Computer Science & Engineering
University of North Texas
Denton, TX 76203-1366, USA
email: robert.renka@unt.edu
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