Methods and Applications of Analysis

Volume 18 (2011)

Number 1

Multi-parameter Tikhonov regularization

Pages: 31 – 46

DOI: http://dx.doi.org/10.4310/MAA.2011.v18.n1.a2

Authors

Kazufumi Ito (Center for Research in Scientific Computation, North Carolina State University, Raleigh)

Bangti Jin (Department of Mathematics, Texas A&M University, College Station, Texas, U.S.A.)

Tomoya Takeuchi (Center for Research in Scientific Computation, North Carolina State University, Raleigh)

Abstract

We study multi-parameter Tikhonov regularization, i.e., with multiple penalties. Such models are useful when the sought-for solution exhibits several distinct features simultaneously. Two choice rules, i.e., discrepancy principle and balancing principle, are studied for choosing an appropriate (vector-valued) regularization parameter, and some theoretical results are presented. In particular, the consistency of the discrepancy principle as well as convergence rate are established, and an a posteriori error estimate for the balancing principle is established. Also two fixed point algorithms are proposed for computing the regularization parameter by the latter rule. Numerical results for several nonsmooth multi-parameter models are presented, which show clearly their superior performance over their single-parameter counterparts.

Keywords

multi-parameter regularization, value function, balancing principle, parameter choic

2010 Mathematics Subject Classification

65J20, 65J22

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