Fitting data with linear combinations of real or complex exponentials is pervasive within many disciplines in Sciences and Engineering as diverse as biomedical data processing, signals in speech recognition or atmospheric transfer functions. One obvious reason is that combinations of exponentials are solutions of homogeneous linear ordinary differential equations and as such they naturally model many different physical processes. Thus, if we have measurements of a quantity that can be modeled as the solution of such an equation, fitting this data to a linear combination of exponentials can give valuable information on decay rates or other material properties of the physical system. Also, exponentials have good approximation properties on compact domains and complex exponentials are naturally related to Fourier series.

We will explore first the nuances of the problem that turns out to be a difficult one, and then survey some of the more successful numerical methods used in exponential fitting including a modified Prony method, variable projections and subspace-based methods, and finally describe several important applications.

Using available computer programs the students will compare the performance of the different algorithms when applied to some real-world data.

References:

[1] Separable nonlinear least squares: the Variable Projection method and its application, G. H. Golub and V. Pereyra, Inverse Problems 19:R1-R26 (2003).
[2] Exponential analysis in physical phenomena, A. A. Istratov and O. F. Vyvenko, Rev. Sc. Instruments, 70:1233-1257 (1999).
[3] A modified Prony algorithm for exponential function fitting, M. R. Osborne and G. Smyth, SIAM J. Sci. Comp. 16:119-138 (1995).
[4] Fitting Nature's Basic Functions, Parts I-IV, B. W. Rust, Computing in Sc. and Eng., 2001-2002.

Lecture 1 & 2:
Mathematical background, difficulties associated with exponential fitting, model validation

Lecture 3:
Nonlinear optimization techniques: variable projections

Lecture 4 & 5:
"Linear" methods: Prony-type and matrix-pencil methods