Scientific, Computing and Applied & Industrial Mathematics: William Carlquist

The process of optimally fitting a differential-equation model to data is usually approached in an iterative manner by solving the equations numerically with some choice of parameters and using some algorithm (e.g. gradient descent) to improve the choice of parameters with successive steps of the iteration. We propose a new method that steps back from an exact numerical method and instead allows the numerical solution to emerge as part of the optimization. We introduce the objective function (1-s) || x - data ||^2 + s || Dx - f(x;p) ||^2 where x is the model values, Dx=f(x;p) is the differential equation in discrete form (i.e. Dx is the discretization of the differential operator), where we must optimize for model values x and parameters p. We use s to implement niches in a genetic optimization algorithm and extract the best fit in the limit as s approaches 1. This method bypasses the need for implicit solution methods and, interestingly, admits conservative quantities, which allow us to gauge the accuracy of our optimization. I will discuss the theory, benefits, and examples of the method.