Linear multistep methods, particle filtering and sequential Monte Carlo

Numerical integration is the main bottleneck in particle filter methodologies for dynamic inverse problems to estimate model parameters, initial values, and non-observable components of an ordinary differential equation (ODE) system from partial, noisy observations, because proposals may result in stiff systems which first slow down or paralyze the time integration process, then end up being discarded. The immediate advantage of formulating the problem in a sequential manner is that the integration is carried out on shorter intervals, thus reducing the risk of long integration processes followed by rejections. We propose to solve the ODE systems within a particle filter framework with higher order numerical integrators which can handle stiffness and to base the choice of the variance of the innovation on estimates of the discretization errors. The application of linear multistep methods to particle filters gives a handle on the stability and accuracy of the propagation, and linking the innovation variance to the accuracy estimate helps keep the variance of the estimate as low as possible. The effectiveness of the methodology is demonstrated with a simple ODE system similar to those arising in biochemical applications.