Optimal experimental designs for linear inverse problems

Optimal experimental designs for inverse and ill-posed problems are investigated. Given discrete points s(i), for i = 1, ..., m, a Fredholm integral equation of the first kind can be discretized into a semi-discrete form g(s(i)) = integral (1)(0)k(s(i), t)f(t)dt by using a collocation method, and into a fully discrete form g(s(i)) = integral (1)(0)k(s(i) , t)f(n)(t)dt by restricting the unknown function f(n)(t) within an n-dimensional subspace. Our optimal design problem is to determine a distribution of the discrete points s(i), i = 1, ...,m maximizing the nth singular value lambda (n) of the semi-discrete or the fully discrete operators, where the number n plays the role of the regularization parameter (it could be fixed a priori or be selected a posteriori). The optimal experimental designs for various inverse problems - numerical differentiation, inversion of Laplace transformation, small particles sizing by light extinction and two-dimensional steady-state inverse heat conduction problem, are studied numerically in detail. Several new and interesting features for the considered problems are found.