Section 2 Inverse Problem

**Chapter 6**

**Abstract**

proposed method.

**1. Introduction**

ent application in mind [1–15].

discussion on some numerical aspects.

**105**

A Numerical Approach to Solving

Levenberg-Marquardt Algorithm

This chapter is intended to provide a numerical algorithm involving the combined use of the Levenberg-Marquardt algorithm and the Galerkin finite element method for estimating the diffusion coefficient in an inverse heat conduction problem (IHCP). In the present study, the functional form of the diffusion coefficient is an unknown priori. The unknown diffusion coefficient is approximated by the polynomial form and the present numerical algorithm is employed to find the solution. Numerical experiments are presented to show the efficiency of the

**Keywords:** parabolic equation, inverse problem, Levenberg-Marquardt

The numerical solution of the inverse heat conduction problem (IHCP) requires

to determine diffusion coefficient from an additional information. Inverse heat conduction problems have many applications in various branches of science and engineering, mechanical and chemical engineers, mathematicians and specialists in many other sciences branches are interested in inverse problems, each with differ-

In this work, we propose an algorithm for numerical solving an inverse heat conduction problem. The algorithm is based on the Galerkin finite element method and Levenberg-Marquardt algorithm [16–17] in conjunction with the least-squares scheme. It is assumed that no prior information is available on the functional form of the unknown diffusion coefficient in the present study, thus, it is classified as the function estimation in inverse calculation. Run the numerical algorithm to solve the unknown diffusion coefficient which is approximated by the polynomial form. The Levenberg-Marquardt optimization is adopted to modify the estimated values.

The plan of this paper is as follows: in Section 2, we formulate a one-dimensional IHCP. In Section 3, the numerical algorithm is derived. Calculation of sensitivity coefficients will be discussed in Section 4. In order to discuss on some numerical aspects, two examples are given in Section 5. Section 6 ends this paper with a brief

an Inverse Heat Conduction

*Tao Min, Xing Chen, Yao Sun and Qiang Huang*

Problem Using the
