Existence, Uniqueness and Stability of Solutions of Generalized Tikhonov-Phillips Functionals

Abstract

The Tikhonov-Phillips method is widely used for regularizing ill-posed inverse problems mainly due to the simplicity of its formulation as an optimization problem. The use of different penalizers in the functionals associated to the corresponding optimization problems has originated a variety other methods which can be considered as "variants" of the traditional Tikhonov-Phillips method of order zero. Such is the case for instance of the Tikhonov-Phillips method of order one, the total variation regularization method, etc. In this article we find sufficient conditions on the penalizers in generalized Tikhonov-Phillips functionals which guarantee existence and uniqueness and stability of the minimizers. The particular cases in which the penalizers are given by the bounded variation norm, by powers of seminorms and by linear combinations of powers of seminorms associated to closed operators, are studied. Several examples are presented and a few results on image restoration are shown.

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Document Details

Document Type
Technical Report
Publication Date
Aug 01, 2011
Accession Number
ADA555323

Entities

People

  • G. L. Mazzieri
  • K. G. Temperini
  • RubĂ©n D. Spies

Organizations

  • University of Minnesota

Tags

Communities of Interest

  • Advanced Electronics
  • Energy and Power Technologies

DTIC Thesaurus Topics

  • Atmospheric Motion
  • Banach Space
  • Coercivity
  • Consistency
  • Continuity
  • Convex Sets
  • Data Acquisition
  • Guarantees
  • Hilbert Space
  • Hypotheses
  • Image Restoration
  • Integral Equations
  • Inverse Problems
  • Mathematics
  • Sequences
  • Theorems
  • Vector Spaces

Fields of Study

  • Mathematics

Readers

  • Calculus or Mathematical Analysis
  • Operations Research
  • Wave Propagation and Nonlinear Chaotic Dynamics.