线性不适定问题中选取Tikhonov正则化参数的线性模型函数方法

June2013

CHINESE JOURNAL

OF ENGINEERING MATHEMATICS

Vol.30No.3

doi:10.3969/j.issn.1005-3085.2013.03.016Article ID:1005-3085(2013)03-0451-16

On the Linear Model Function Method for Choosing Tikhonov Regularization Parameters in Linear Ill-posed Problems?

WANG Ze-wen1,XU Ding-hua2

(1-School of Sciences,East China Institute of Technology,Nanchang330013;

2-School of Sciences,Zhejiang Sci-Tech University,Hangzhou310018)

Abstract:How to choose regularization parameters is an important issue in Tikhonov regular-ization of ill-posed problems.Based on the damped Morozov discrepancy principle,

this paper studies the linear model function method for choosing regularization

parameters.The linear model function is derived from the point of view of the

Hermite interpolation,and two linear model function algorithms(a basic algorithm

and a modi?ed algorithm)with their convergence results are discussed for choosing

regularization parameters.Then,a new relaxation algorithm for the linear model

function is proposed to overcome the local convergence of the basic algorithm.Fur-

thermore,two hybrid algorithms,the linear-to-linear model function algorithm and

the hyperbolic-to-linear model function algorithm,are proposed with global con-

vergence.E?ciency of the proposed algorithms is illustrated through numerical

experiments.

Keywords:ill-posed problem;regularization parameter;linear model function;Morozov dis-crepancy principle

Classi?cation:AMS(2000)47A52;65R30;65R32CLC number:O241

Document code:A

1Introduction

Many linear ill-posed problems[1-4]are formulated as operator equations of the form

Kx=y,(1)

where K:X→Y is a bounded linear operator between Hilbert space X and Y with scalar product ·,· and norm · .In general,our problem is essentially ill-posed: the range R(K)is not closed in Y.The non-closed range yields the discontinuity of

Received:13July2011. Accepted:26Dec2011.Biography:Wang Zewen(Born in1974),Male,Doctor,Professor. Research?eld:inverse problems of mathematical physics.

?Foundation item:The National Natural Science Foundation of China(11161002;11071221);the Young Scientist Training Project of Jiangxi Province(20122BCB23024).