Bayesian data assimilation in shape registration

In this paper we apply a Bayesian framework to the problem of geodesic curve matching. Given a template curve, the geodesic equations provide a mapping from initial conditions\ud for the conjugate momentum onto topologically equivalent shapes. Here, we aim to recover the well defined posterior distribution on the initial momentum which gives rise to observed points on the target curve; this is achieved by explicitly including a reparameterisation in the formulation. Appropriate priors are chosen for the functions which together determine this field and the positions of the observation points, the initial momentum p0 and the reparameterisation vector field v, informed by regularity results about the forward model. Having done this, we illustrate how Maximum Likelihood Estimators (MLEs) can be used to find regions of high posterior density, but also how we can apply recently developed MCMC methods on function spaces to characterise the whole of the posterior density. These illustrative examples also include scenarios where the posterior distribution is multimodal and irregular, leading us to the conclusion that knowledge of a state of global maximal posterior density does not always give us the whole picture, and full posterior sampling can give better quantification of likely states and the overall uncertainty inherent in the problem

Approximation of Bayesian inverse problems for PDEs

Inverse problems are often ill posed, with solutions that depend sensitively on data. In any numerical approach to the solution of such problems, regularization of some form is needed to counteract the resulting instability. This paper is based on an approach to regularization, employing a Bayesian formulation of the problem, which leads to a notion of well posedness for inverse problems, at the level of probability measures. The stability which results from this well posedness may be used as the basis for quantifying the approximation, in finite dimensional spaces, of inverse problems for functions. This paper contains a theory which utilizes this stability property to estimate the distance between the true and approximate posterior distributions, in the Hellinger metric, in terms of error estimates for approximation of the underlying forward problem. This is potentially useful as it allows for the transfer of estimates from the numerical analysis of forward problems into estimates for the solution of the related inverse problem. It is noteworthy that, when the prior is a Gaussian random field model, controlling differences in the Hellinger metric leads to control on the differences between expected values of polynomially bounded functions and operators, including the mean and covariance operator. The ideas are applied to some non-Gaussian inverse problems where the goal is determination of the initial condition for the Stokes or Navier–Stokes equation from Lagrangian and Eulerian observations, respectively

Bert Salwen- A Recollection

A recollection of the life and work of Bert Salwen

A Constrained Approach to Multiscale Stochastic Simulation of\ud Chemically Reacting Systems

Stochastic simulation of coupled chemical reactions is often computationally intensive, especially if a chemical system contains reactions occurring on different time scales. In this paper we introduce a multiscale methodology suitable to address this problem. It is based on the Conditional Stochastic Simulation Algorithm (CSSA) which samples from the conditional distribution of the suitably defined fast variables, given values for the slow variables. In the Constrained Multiscale Algorithm (CMA) a single realization of the CSSA is then used for each value of the slow variable to approximate the effective drift and diffusion terms, in a similar manner to the constrained mean-force computations in other applications such as molecular dynamics. We then show how using the ensuing Stochastic Differential Equation (SDE) approximation, we can in turn approximate average switching times in stochastic chemical systems