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. In the first part of the talk I will look at methods that use the Gauss-Newton and Newton approximation, solved either within a trust-region or with an additional regularization term. While these are well-known methods, we describe a new hybrid method combining these two approaches.
I will then describe a newly proposed algorithm that minimizes a tensor model locally. Since this shares the sum-of-squares nature of the problem being solved, it arguably makes better use of second derivative information that has been computed than the traditional Newton approximation, and I will present evidence in support of this claim.
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.
This is joint work with Nick Gould and Jennifer Scott.