Scientific Computation and Applied & Industrial Mathematics: Tyrone Rees

One of the central problems in computational mathematics is to fit a suitable model to observed data. Mathematically, this can be posed as a nonlinear least-squares problem. Standard methods for solving such problems are based on the Gauss-Newton and Newton approximation, solved either within a trust-region or with an additional regularization term (e.g., the Levenberg-Marquardt method).

I will describe a method that combines Gauss-Newton and Newton approximations, where appropriate, to produce a hybrid method that exhibits better convergence properties. I will then describe a newly proposed algorithm, the tensor-Newton method, that minimizes a tensor model locally. Since this shares the sum-of-squares nature of the problem being solved, it makes better use of second derivative information that has been computed than the traditional Newton approximation. Part of the motivation of this work is improving the fitting capabilities of the widely used data analysis and visualization package Mantid. As well as standard test examples, I present results on real-world examples from ISIS, a pulsed neutron and muon source located at the Rutherford Appleton Laboratory. The algorithms described in this talk are available as part of the open source nonlinear least-squares solver RALFit (https://github.com/ralna/RALFit).