Numerical methods for the solution of an ill-posed problem associated with interstellar dust models

This thesis focuses on numerical methods for the solution of an ill-posed problem concerning interstellar dust models. After defining ill-posed problems, we show some examples that underline difficulties into the process of finding a reasonable solution to such problems. In particular, we deal with a Fredholm integral equation of the first kind, which is a typical ill-posed problem. Subsequently, we give a taste of a successful way of modifying the problem, i.e. regularizing it, which results in adding some information about the solution. The purpose of what we call regularization is precisely reaching a solution the closest possible to the better one, which is typically smooth. We explore a Matlab package of routines called Regularization Tools, taking into consideration the function tikhonov and tsvd (truncated svd), although efficient and versatile, for our case-study the result was unsatisfactory. After the description of an a priori analysis of the ill-posed problem, which means an analysis before trying to solve it, we illustrate the original approach by Tikhonov, which turns out to be more suitable. Such method is based on some a priori information about the solution we are looking for, such as its nonnegativity or convexity, and on the application of the method of projection of conjugate gradients. To be specific we use the method of conjugate gradients with projection into the set of vectors with nonnegative components, with the constraint of nonnegativity, to save the physical meaning of our solution. We translate in C-language original FORTRAN routines implementing the method, we report some tests, and at the end we describe our case-study regarding interstellar dust models. Our sought solution is the grains size distribution of the dust clouds. All of our framework is taken from Zubko et al. (2004). The results in the article Zubko et al. (2004) have been partially reproduced. This is due to the fact that we focused on data and results excluding very small grains.