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

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

Variational data assimilation using targetted random walks

The variational approach to data assimilation is a widely used methodology for both online prediction and for reanalysis (offline hindcasting). In either of these scenarios it can be important to assess uncertainties in the assimilated state. Ideally it would be desirable to have complete information concerning the Bayesian posterior distribution for unknown state, given data. The purpose of this paper is to show that complete computational probing of this posterior distribution is now within reach in the offline situation. In this paper we will introduce an MCMC method which enables us to directly sample from the Bayesian\ud posterior distribution on the unknown functions of interest, given observations. Since we are aware that these\ud methods are currently too computationally expensive to consider using in an online filtering scenario, we frame this in the context of offline reanalysis. Using a simple random walk-type MCMC method, we are able to characterize the posterior distribution using only evaluations of the forward model of the problem, and of the model and data mismatch. No adjoint model is required for the method we use; however more sophisticated MCMC methods are available\ud which do exploit derivative information. For simplicity of exposition we consider the problem of assimilating data, either Eulerian or Lagrangian, into a low Reynolds number (Stokes flow) scenario in a two dimensional periodic geometry. We will show that in many cases it is possible to recover the initial condition and model error (which we describe as unknown forcing to the model) from data, and that with increasing amounts of informative data, the uncertainty in our estimations reduces

MCMC methods for functions modifying old algorithms to make\ud them faster

Many problems arising in applications result in the need\ud to probe a probability distribution for functions. Examples include Bayesian nonparametric statistics and conditioned diffusion processes. Standard MCMC algorithms typically become arbitrarily slow under the mesh refinement dictated by nonparametric description of the unknown function. We describe an approach to modifying a whole range of MCMC methods which ensures that their speed of convergence is robust under mesh refinement. In the applications of interest the data is often sparse and the prior specification is an essential part of the overall modeling strategy. The algorithmic approach that we describe is applicable whenever the desired probability measure has density with respect to a Gaussian process or Gaussian random field prior, and to some useful non-Gaussian priors constructed through random truncation. Applications are shown in density estimation, data assimilation in fluid mechanics, subsurface geophysics and image registration. The key design principle is to formulate the MCMC method for functions. This leads to algorithms which can be implemented via minor modification of existing algorithms, yet which show enormous speed-up on a wide range of applied problems

A stochastic model for early placental development

In the human, placental structure is closely related to placental function and consequent pregnancy outcome. Studies have noted abnormal placental shape in small-for-gestational age infants which extends to increased lifetime risk of cardiovascular disease. The origins and determinants of placental shape are incompletely under-stood and are difficult to study in vivo. In this paper we model the early development of the placenta in the human, based on the hypothesis that this is driven by dynamics dominated by a chemo-attractant effect emanating from proximal spiral arteries in the decidua. We derive and explore a two-dimensional stochastic model for these events, and investigate the effects of loss of spiral arteries in regions near to the cord insertion on the shape of the placenta. This model demonstrates that placental shape is highly variable and disruption of spiral arteries can exert profound effects on placental shape, particularly if this disruption is close to the cord insertion. Thus, placental shape reflects the underlying maternal vascular bed. Abnormal placental shape may reflect an abnormal uterine environment, which predisposes to pregnancy complications

Adaptive finite element method assisted by stochastic simulation of chemical systems

Stochastic models of chemical systems are often analysed by solving the corresponding\ud Fokker-Planck equation which is a drift-diffusion partial differential equation for the probability\ud distribution function. Efficient numerical solution of the Fokker-Planck equation requires adaptive mesh refinements. In this paper, we present a mesh refinement approach which makes use of a stochastic simulation of the underlying chemical system. By observing the stochastic trajectory for a relatively short amount of time, the areas of the state space with non-negligible probability density are identified. By refining the finite element mesh in these areas, and coarsening elsewhere, a suitable mesh is constructed and used for the computation of the probability density

Recent Developments in Minnesota Corporate Law

This article summarizes a number of select cases and statutory amendments involving various topics in Minnesota corporate law. Sections I and II address 2003 statutory changes and cases decided in 2003, respectively. Sections III and IV address 2004 statutory changes and cases decided in the first half of 2004, respectively. The purpose of this article is to provide the practitioner with an overview of recent developments in Minnesota corporate law