Fast Functional Integrals with Application to Differential Equations Models

A new method is introduced which uses higher-order Laplace approximation to evaluate functional integrals much faster than existing methods. An implementation in MATLAB is called SLAM-FIT (Sparse Laplace Approximation Method for Functional Integration on Time) or simply SLAM. In this paper SLAM is applied to estimate parameters of mixed models that require functional integration. It is compared with two more general packages which can be used to do functional integration. One is Stan, a recent and very general package for integrating and estimating using hybrid Monte Carlo. The other is INLA, a recent R package which uses Laplace approximations for Gaussian Markov random fields. In both cases it is able to get near-identical or equivalent results, in significantly less time, even for moderately sized data sets. The fundamental speed advantage of the algorithm may be greater than it appears, because SLAM is running in pure MATLAB while the other two packages use optimized compiled code.