A sensitivity decomposition for the regularized solution of inverse heat conduction problems by wavelets

In this paper, an extremely ill-posed problem of determining the surface temperature and/or heat flux histories q(t) is considered. To analyse precisely its degree of ill-posedness and its resolution limit, we have studied the problem using different analysis tools-singular-value decomposition and multiresolution analysis. The main purpose of this paper is to develop a 'sensitivity decomposition' concept that splits the sought function space V into ill-posed and well-posed parts in order to give a convenient regularized solution. We show some advantages of using wavelets (or hierarchical bases) to determine such a decomposition for ill-posed problems. Wavelets are capable of decomposing the sought function space V into the direct summation of subspaces such that the sensitivity of the observations with respect to the variation of the function to be determined q(t) in each subspace provided by the decomposition has quite a different magnitude. Based on the results derived from sensitivity analysis, we propose a hierarchical method using the discretization size h (or scale level j) as a regularization parameter. When the level of noise is unknown, the hierarchical method also gives a simple rule to get a suboptimal regularization parameter. Numerical results are presented.