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 surfa ce temperature and/or heat flux histories q(t) is considered. To analy se 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 ord er to give a convenient regularized solution. We show some advantages of using wavelets (or hierarchical bases) to determine such a decompos ition for ill-posed problems. Wavelets are capable of decomposing the sought function space V into the direct summation of subspaces such th at the sensitivity of the observations with respect to the variation o f the function to be determined q(t) in each subspace provided by the decomposition has quite a different magnitude. Based on the results de rived from sensitivity analysis, we propose a hierarchical method usin g the discretization size h (or scale level j) as a regularization par ameter. When the level of noise is unknown, the hierarchical method al so gives a simple rule to get a suboptimal regularization parameter. N umerical results are presented.