Gibbs flow for approximate transport with applications to Bayesian computation

Let $\pi_{0}$ and $\pi_{1}$ be two distributions on the Borel space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$. Any measurable function $T:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ such that $Y=T(X)\sim\pi_{1}$ if $X\sim\pi_{0}$ is called a transport map from $\pi_{0}$ to $\pi_{1}$. For any $\pi_{0}$ and $\pi_{1}$, if one could obtain an analytical expression for a transport map from $\pi_{0}$ to $\pi_{1}$, then this could be straightforwardly applied to sample from any distribution. One would map draws from an easy-to-sample distribution $\pi_{0}$ to the target distribution $\pi_{1}$ using this transport map. Although it is usually impossible to obtain an explicit transport map for complex target distributions, we show here how to build a tractable approximation of a novel transport map. This is achieved by moving samples from $\pi_{0}$ using an ordinary differential equation with a velocity field that depends on the full conditional distributions of the target. Even when this ordinary differential equation is time-discretized and the full conditional distributions are numerically approximated, the resulting distribution of mapped samples can be efficiently evaluated and used as a proposal within sequential Monte Carlo samplers. We demonstrate significant gains over state-of-the-art sequential Monte Carlo samplers at a fixed computational complexity on a variety of applications.Comment: Significantly revised with new methodology and numerical example

Parameter estimation by implicit sampling

Implicit sampling is a weighted sampling method that is used in data assimilation, where one sequentially updates estimates of the state of a stochastic model based on a stream of noisy or incomplete data. Here we describe how to use implicit sampling in parameter estimation problems, where the goal is to find parameters of a numerical model, e.g.~a partial differential equation (PDE), such that the output of the numerical model is compatible with (noisy) data. We use the Bayesian approach to parameter estimation, in which a posterior probability density describes the probability of the parameter conditioned on data and compute an empirical estimate of this posterior with implicit sampling. Our approach generates independent samples, so that some of the practical difficulties one encounters with Markov Chain Monte Carlo methods, e.g.~burn-in time or correlations among dependent samples, are avoided. We describe a new implementation of implicit sampling for parameter estimation problems that makes use of multiple grids (coarse to fine) and BFGS optimization coupled to adjoint equations for the required gradient calculations. The implementation is "dimension independent", in the sense that a well-defined finite dimensional subspace is sampled as the mesh used for discretization of the PDE is refined. We illustrate the algorithm with an example where we estimate a diffusion coefficient in an elliptic equation from sparse and noisy pressure measurements. In the example, dimension\slash mesh-independence is achieved via Karhunen-Lo\`{e}ve expansions

Data assimilation in the low noise regime with application to the Kuroshio

On-line data assimilation techniques such as ensemble Kalman filters and particle filters lose accuracy dramatically when presented with an unlikely observation. Such an observation may be caused by an unusually large measurement error or reflect a rare fluctuation in the dynamics of the system. Over a long enough span of time it becomes likely that one or several of these events will occur. Often they are signatures of the most interesting features of the underlying system and their prediction becomes the primary focus of the data assimilation procedure. The Kuroshio or Black Current that runs along the eastern coast of Japan is an example of such a system. It undergoes infrequent but dramatic changes of state between a small meander during which the current remains close to the coast of Japan, and a large meander during which it bulges away from the coast. Because of the important role that the Kuroshio plays in distributing heat and salinity in the surrounding region, prediction of these transitions is of acute interest. Here we focus on a regime in which both the stochastic forcing on the system and the observational noise are small. In this setting large deviation theory can be used to understand why standard filtering methods fail and guide the design of the more effective data assimilation techniques. Motivated by our analysis we propose several data assimilation strategies capable of efficiently handling rare events such as the transitions of the Kuroshio. These techniques are tested on a model of the Kuroshio and shown to perform much better than standard filtering methods.Comment: 43 pages, 12 figure