# Workshop Programme

## for period 14 - 15 June 2010

### Workshop on Filtering

14 - 15 June 2010

Timetable

 Monday 14 June 09:30-09:50 Registration and a word from the organisers 09:50-10:40 Krylov, N (University of Minnesota) Kalman-Bucy filter and SPDEs with growing lower-order coefficients in W1p spaces without weights. Sem 1 We consider divergence form uniformly parabolic SPDEs with VMO bounded leading coefficients, bounded coefficients in the stochastic part, and possibly growing lower-order coefficients in the deterministic part. We look for solutions which are summable to the p-th power, p=2, with respect to the usual Lebesgue measure along with their first-order derivatives with respect to the spatial variable. Our methods allow us to include Zakai's equation for the Kalman-Bucy filter into the general filtering theory. 10:40-11:20 Morning coffee 11:20-12:10 Weis, L (Karlsruhe Institute of Technology (KIT)) Regularity results for parabolic stochastic PDEs . Sem 1 We present a new approach to maximal regularity results for parabolic stochastic partial differential equations, which also applies to systems of higher order ellictic equations on domains and manifolds and Stokes operators. It combines methods from stochastic analysis and spectral theory. Our results are motivated and will be compared to results of Z. Brzezniak and N.V. Krylov. 12:10-13:00 Elworthy, D (Warwick) Geometric approach to filtering some illustrations Sem 1 Suppose we have an SDE on Rn+p which lies over an SDE on Rn for the natural projection of Rn+p to Rn. With some "cohesiveness" assumptions on the SDE on Rn we can decompose the SDE on the big space and so describe the conditional law of its solution given knowledge of its projection. The same holds for suitable SDE's on manifolds, and in some infinite dimensional examples arising from SPDE's and stochastic flows. The method also relates to a canonical decomposition of one diffusion operator lying over another. This approach will be illustrated by considering the conditional law of solutions of a simple evolutionary SPDE given the integral of the solution over the space variables, and by looking at the problem of conditioning a stochastic flow by knowledge of its one-point motion, with a related application to standard gradient estimates. This is joint work taken from a monograph by myself, Yves LeJan, and Xue-Mei Li, The Geometry of Filtering to appear in Birkhauser's "Frontiers in Mathematics" series 13:00-14:00 Lunch 14:00-14:50 Gyongy, I (Edinburgh) On accelerated numerical schemes for nonlinear filtering. Sem 1 Some numerical schemes, in particular, finite difference approximations are considered to calculate nonlinear filters for partially observed diffusion processes. Theorems on Richardson's acceleration of the convergence of numerical schemes are presented. The talk is based on joint result with Nicolai Krylov. 14:50-15:40 Tretyakov, M (University of Leicester) Nonlinear filtering algorithms based on averaging over characteristics and on the innovation approach. Sem 1 It is well known that numerical methods for nonlinear filtering problems, which directly use the Kallianpur-Striebel formula, can exhibit computational instabilities due to the presence of very large or very small exponents in both the numerator and denominator of the formula. We obtain computationally stable schemes by exploiting the innovation approach. We propose Monte Carlo algorithms based on the method of characteristics for linear parabolic stochastic partial differential equations. Convergence and some properties of the considered algorithms are studied. Variance reduction techniques are discussed. Results of some numerical experiments are presented. The talk is based on a joint work with G.N. Milstein.
 Tuesday 15 June 09:00-09:50 Del Moral, P (Bordeaux) A backward particle interpretation of Feynman-Kac formulae with applications to filtering and smoothing problems Sem 1 We design a particle interpretation of Feynman-Kac measures on path spaces based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of their final time marginals. In contrast to traditional genealogical tree based models, these new particle algorithms can be used to compute normalized additive functionals on-the-fly as well as their limiting occupation measures with a given precision degree that does not depend on the final time horizon. We provide uniform convergence results w.r.t. the time horizon parameter as well as functional central limit theorems and exponential concentration estimates, yielding what seems to be the first results of this type for this class of models. We also illustrate these results in the context of filtering and path estimation problems, as well as in computational physics and imaginary time Schroedinger type partial differential equations, with a special interest in the numerical approximation of the invariant measure associated to h-processes. This is joint work with Arnaud Doucet, and Sumeetpal Singh. 09:50-10:40 Rogers, C (Cambridge) Least-action filtering Sem 1 This talk studies the filtering of a partially-observed multidimensional diffusion process using the principle of least action, equivalently, maximum-likelihood estimation. We show how the most likely path of the unobserved part of the diffusion can be determined by solving a shooting ODE, and then we go on to study the (approximate) conditional distribution of the diffusion around the most likely path; this turns out to be a zero-mean Gaussian process which solves a linear SDE whose time-dependent coefficients can be identified by solving a first-order ODE with an initial condition. This calculation of the conditional distribution can be used as a way to guide SMC methods to search relevant parts of the state space, which may be valuable in high-dimensional problems, where SMC struggles; in contrast, ODE solution methods continue to work well even in moderately large dimension. 10:40-11:20 Morning coffee 11:20-12:10 Stannat, W (Technische Universitat Darmstadt) Stability of the optimal filter for nonergodic signals - a variational approach Sem 1 We give an overview on results on stability of the optimal filter for signal processes with state space $\mathbb{R}^d$ observed with independent additive noise, both in discrete and continuous time. Explicit lower bounds on the rate of stability in terms of the coefficients of the signal and the observation are obtained. For the time-continuous case the bounds are uniform w.r.t. appropriate time-discrete approximations. I also discuss a particular extension to signals observed with independent multiplicative noise. 12:10-13:00 Veretennikov, A (Leeds) On filtering equations with non-specified initial data Sem 1 The talk will be mainly devoted to the question of filtering with non-specified data, of how mixing rate for the (ergodic) signal component of the filtering system may be used more or less directly so as to estimate the rate of forgetting initial error in the filtering measure (nonlinear) dynamics. Some other relevant issues may be addressed if time allows. 13:00-14:00 Lunch 14:00-14:50 Friz, P (Technische Universitat Berlin) Rough path stability of SPDEs arising in non-linear filtering and beyond Sem 1 We present a (rough)pathwise view on stochastic partial differential equations. Our results are based on the marriage of rough path analysis with (2nd order) viscosity theory. Joint work with M. Caruana and H. Oberhauser. 14:50-15:40 Lee, W (Warwick) Filtering of wave equation in high dimension Sem 1 The aim of this talk is to study a so-called data-model mismatch problem. It consists of two more or less independent parts. In the first part, we present infinite dimensional Kalman Filter for the advection equation on the torus. We see the velocity difference between the true signal and the model leads to various limit behaviors of the posterior mean. In the second part, Fourier diagonal Filter would be examined in the context of the Majda-McLaughlin-Tabak wave turbulence model. It is demonstrated that nonlinear wave interactions renormalize the dynamics, leading to a possible destruction of scaling structures in the bare wave systems. The Filter performance is improved when this renormalized dispersion relation is considered.