Authors: Heinz W. Engl, Gerhard Landl, Thomas Langthaler, and Harald F. Kauffmann
Abstract: In time-resolved fluorescence spectroscopy, a distribution of fluorescence lifetimes resulting from static and dynamic disorder of a polychromophoric ensemble is to be determined form the molecular fluorescence response to the optical probe pulse. To do this, one has to solve a convolution integral equation of the first kind and then invert a Laplace transform. Both problems are ill-posed in the sense of Hadamard. We describe in detail an algorithm that combines coarse discretization for inverting the Laplace transform with a nonlinear least-squares approach based on Newton and quasi-Newton techniques for solving the convolution equation. While this algorithm works well in many cases, it does not completely remove the instabilities due to the ill-posedness. Thus, we also propose an algorithm that combines the approach described above with Tikhonov regularization. Several examples, both with synthetic and with real data, show the performance of our algorithms.