Optimal approximation with exponential sums by a maximum likelihood modification of Prony’s method

Abstract

We consider a modification of Prony’s method to solve the problem of best approximation of a given data vector by a vector of equidistant samples of an exponential sum in the 2-norm. We survey the derivation of the corresponding non-convex minimization problem that needs to be solved and give its interpretation as a maximum likelihood method. We investigate numerical iteration schemes to solve this problem and give a summary of different numerical approaches. With the help of an explicitly derived Jacobian matrix, we review the Levenberg-Marquardt algorithm which is a regularized Gauss-Newton method and a new iterated gradient method (IGRA). We compare this approach with the iterative quadratic maximum likelihood (IQML). We propose two further iteration schemes based on simultaneous minimization (SIMI) approach. While being derived from a different model, the scheme SIMI-I appears to be equivalent to the Gradient Condition Reweighted Algorithm (GRA) by Osborne and Smyth. The second scheme SIMI-2 is more stable with regard to the choice of the initial vector. For parameter identification, we recommend a pre-filtering method to reduce the noise variance. We show that all considered iteration methods converge in numerical experiments.