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

Title: Tikhonov regularization using Sobolev metrics

Authors: Parimah Kazemi (Ripon College, Ripon, WI, USA)
         Robert J. Renka (Univ. of North Texas, Denton, TX, USA)
 
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.