DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

GLOBAL CONVERGENT ALGORITHM FOR PARABOLIC
COEFFICIENT INVERSE PROBLEMS
QUAN-FANG WANG
The globally convergent algorithm, that is, convexification approach will be applied to
coefficient inverse problems of parabolic differential equations when spatial dimensions
are two. Based on the unified framework of convexification approach, a developed global
iteration scheme for solving numerical solution will be implemented to verify the effectiveness
of convexification approach for 2D parabolic case.
Copyright © 2006 Quan-Fang Wang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Global convergent algorithm is to solve the multidimensional coefficient inverse problems
both in theoretical and computation issues using a unified framework (cf. [6]). Its
application to a class of inverse problems has been reported in literatures (cf. [1–7]). The
main thoughts of convexification approach focus on several aspects. Comparing the local
convergent iteration to exact solution, convexification approach is a global convergent
algorithm, which avoids leading to false solution under incomplete boundary data and inconsistency
of a mathematical model with the reality. A couple of points clearly show its
advantages. The sequential minimization algorithm (i) provides stable approximate solution
via minimization of a finite sequence of strictly convex objective functions, which
is constructed by applying the nonlinear weighted least-squares method with Carleman’s
weight function; (ii) provides the convergence to the “exact” solution independent of
starting vector, which is directly determined from the data eliminating for the descent
methods.
The purpose is to use convexification approach for solving the coefficient inverse problem
arising in parabolic partial differential equations. Numerical study will be implemented
for two dimensions to show the effectiveness.
The contents of this paper are as follows. In Section 2, the formulation is given in the
unified framework of convexification approach. In Section 3, the global convergent algorithm
is applied to parabolic coefficient inverse problems. In Section 4, experiments
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1109–1119