Colin Cotter

Alastair Gregory, Colin Cotter, Sebastian Reich

This paper extends the Multilevel Monte Carlo variance reduction technique to nonlinear filtering. In particular, Multilevel Monte Carlo is applied to a certain variant of the particle filter, the Ensemble Transform Particle Filter. A key aspect is the use of optimal transport methods to re-establish correlation between coarse and fine ensembles after resampling; this controls the variance of the estimator. Numerical examples present a proof of concept of the effectiveness of the proposed method, demonstrating significant computational cost reductions (relative to the single-level ETPF counterpart) in the propagation of ensembles.

Colin Cotter, Simon Cotter, Paul Russell

Markov chain Monte Carlo methods are a powerful and commonly used family of numerical methods for sampling from complex probability distributions. As applications of these methods increase in size and complexity, the need for efficient methods increases. In this paper, we present a particle ensemble algorithm. At each iteration, an importance sampling proposal distribution is formed using an ensemble of particles. A stratified sample is taken from this distribution and weighted under the posterior, a state-of-the-art ensemble transport resampling method is then used to create an evenly weighted sample ready for the next iteration. We demonstrate that this ensemble transport adaptive importance sampling (ETAIS) method outperforms MCMC methods with equivalent proposal distributions for low dimensional problems, and in fact shows better than linear improvements in convergence rates with respect to the number of ensemble members. We also introduce a new resampling strategy, multinomial transformation (MT), which while not as accurate as the ensemble transport resampler, is substantially less costly for large ensemble sizes, and can then be used in conjunction with ETAIS for complex problems. We also focus on how algorithmic parameters regarding the mixture proposal can be quickly tuned to optimise performance. In particular, we demonstrate this methodology's superior sampling for multimodal problems, such as those arising from inference for mixture models, and for problems with expensive likelihoods requiring the solution of a differential equation, for which speed-ups of orders of magnitude are demonstrated. Likelihood evaluations of the ensemble could be computed in a distributed manner, suggesting that this methodology is a good candidate for parallel Bayesian computations.

Andreas Bock, Alexis Arnaudon, Colin Cotter

We present a framework for shape matching in computational anatomy allowing users control of the degree to which the matching is diffeomorphic. This control is given as a function defined over the image and parameterises the template deformation. By modelling localised template deformation we have a mathematical description of growth only in specified parts of an image. The location can either be specified from prior knowledge of the growth location or learned from data. For simplicity, we consider landmark matching and infer the distribution of a finite dimensional parameterisation of the control via Markov chain Monte Carlo. Preliminary numerical results are shown and future paths of investigation are laid out. Well-posedness of this new problem is studied together with an analysis of the associated geodesic equations.

Alastair Gregory, Colin Cotter

This paper presents a seamless algorithm for the application of the multilevel Monte Carlo (MLMC) method to the ensemble transform particle filter (ETPF). The algorithm uses a combination of optimal coupling transformations between coarse and fine ensembles in difference estimators within a multilevel framework, to minimise estimator variance. It differs from that of Gregory et al. (2016) in that strong coupling between the coarse and fine ensembles is seamlessly maintained during all stages of the assimilation algorithm, instead of using independent transformations to equal weights followed by recoupling with an assignment problem. This modification is found to lead to an increased rate in variance decay between coarse and fine ensembles with level in the hierarchy, a key component of MLMC. This offers the potential for greater computational cost reductions. This is shown, alongside evidence of asymptotic consistency, in numerical examples.

Colin Cotter

Given a fluid equation with reduced Lagrangian $l$ which is a functional of velocity $\MM{u}$ and advected density $D$ given in Eulerian coordinates, we give a general method for semidiscretising the equations to give a canonical Hamiltonian system; this system may then be integrated in time using a symplectic integrator. The method is Lagrangian, with the variables being a set of Lagrangian particle positions and their associated momenta. The canonical equations obtained yield a discrete form of Euler-Poincaré equations for $l$ when projected onto the grid, with a new form of discrete calculus to represent the gradient and divergence operators. Practical symplectic time integrators are suggested for a large family of equations which include the shallow-water equations, the EP-Diff equations and the 3D compressible Euler equations, and we illustrate the technique by showing results from a numerical experiment for the EP-Diff equations.