On the recovery and resolution of exponential relaxation rates from experimental data II. The optimum choice of experimental sampling points for Laplace transform inversion

Abstract

In a previous paper (Bertero, Boccacci & Pike, Proc. R. Soc. Lond . A 383, 15 (1982)) we investigated the benefits of a priori knowledge of finite support of the solution in the inversion of the Laplace transform and determined the number of exponential components that could be recovered from given data in the presence of quantified amounts of noise. The support of the data was assumed to be unrestricted. In the present paper these calculations are extended to cover the case of a necessarily finite number of experimental data points. We consider two cases: uniform distribution and geometric distribution of data points. In both cases the condition number of singular value inversions is minimized with respect to the placing of the data points for fixed numbers of data samples. The results allow optimum placing of the experimental data points as a function of the ratio, γ , of given upper and lower bounds of the support of the solution. The number of exponentials recoverable is less than in the unrestricted case but for γ ≼ 8 we find that with 32 linearly spaced data points, optimally placed, this reduction is rather small. Geometrical sampling of experimental data reduces further the number of sample points required for inversion. For γ ≼ 8 we show that with 5 geometrically spaced data points, points, optimally placed, the ill-conditioning in the restoration of 2 to 4 exponential components is smaller than with 32 linearly spaced data samples. This has important implications for the design of a number of scientific experiments and instruments.