Duality between data assimilation/nonlinear filtering and optimal control has a rich history tracing back to Kalman-Bucy’s original 1961 paper. Duality is manifested in many guises, e.g., with the time arrow reversed, the Riccati equation of optimal control is the same as equation for the covariance update equation of the Kalman filter. In recent years, the duality relationship has been used to derive control type algorithms for data assimilation and simulation problems. This has led to several new classes of control-type algorithms such as (i) nonlinear smoothers based on approximate solution of the Bellman’s equation of optimal control; (ii) forward-backward algorithms based on a Schrodinger bridge-type construction; (iii) feedback particle filter based on a diffusion map approximation of the solution of a certain Poisson equation; and (iv) gradient flow type interpretations of linear and nonlinear filters. In numerical evaluations, it is often found that these control algorithms exhibit smaller simulation variance and better scaling properties with problem dimension when compared to the traditional methods based on importance sampling.
This session will serve to provide a snapshot of some of the exciting news developments in this historically significant area.
08:30
A Schroedinger perspective on data assimilaton
Sebastian Reich | University of Potsdam | Germany
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Sebastian Reich | University of Potsdam | Germany
In 1931, Erwin Schroedinger posed a certain boundary value problem in the space of probability measures which has recently found connections and applications to a wide range of problems arising from data science including data assimilation, clustering of data, and learning. In my talk I will summarise this Schroedinger perspective on data assimilation and its link to the classical Kalman filter for stochastic processes; a more detailed exposition of which can be found in my 2019 Acta Numerica paper (arXiv:1807.08351).
09:00
Adaptive simulation of rare events in high dimensions: a stochastic control approach
Carsten Hartmann | BTU Cottbus - Senftenberg | Germany
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Carsten Hartmann | BTU Cottbus - Senftenberg | Germany
We propose a Monte-Carlo scheme for the simulation of rare events that combines an adaptive importance sampling method with a control variates technique. The scheme is based on a Gibbs variational principle that is used to determine the optimal (i.e. minimum variance) change of measure and exploits the fact that the latter can be rephrased as a stochastic optimal control problem. The control problem can be solved by a stochastic approximation algorithm, using the Feynman–Kac representation of the associated dynamic programming equations. We discuss numerical aspects for high-dimensional problems along with simple toy examples.
This is joint work with Omar Kebiri, Lara Neureither and Lorenz Richter.
09:30
Wasserstein gradient flow for filtering and control: theory and algorithms
Abhishek Halder | University of California at Santa Cruz | United States
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Abhishek Halder | University of California at Santa Cruz | United States
This talk will outline a recent development in systems-control theory, where new geometric interpretations for the filtering and optimal control problems are emerging. At the heart of this development, lies the Wasserstein metric and the theory of optimal mass transport, which induces a Riemannian geometric structure on the manifold of joint probability density functions supported over the state space. It turns out that the equations of filtering can be viewed as the gradient flow of certain Lyapunov functional with respect to suitable notion of distance on such infinite dimensional manifold. These ideas lead to infinite dimensional proximal recursions. The well-known exact filters, such as the Kalman-Bucy and the Wonham filter, have been explicitly recovered in this setting. Perhaps more interestingly, the same framework can be used to design gradient descent algorithms numerically implementing the proximal recursions over probability weighted scattered point cloud, avoiding function approximation or spatial discretization, and hence have extremely fast runtime. The same ideas appear naturally in the finite horizon density control (a.k.a. Schrodinger bridge) problems, and there too, the Wasserstein proximal algorithms help solve certain Schrodinger bridge problems with nonlinear prior dynamics. The latter can be seen as the continuum limit of decentralized optimal control, and is of contemporary engineering interest.
This is joint work with Kenneth Caluya.
10:00
Feedback particle filters on manifolds for point process observations
Simone Carlo Surace | University of Bern | Switzerland
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Simone Carlo Surace | University of Bern | Switzerland
The filtering of a Markov diffusion process from counting process observations leads to ‘large’ changes in the conditional distribution upon an observed event, corresponding to a multiplication of the density by the intensity function of the observation process. If that distribution is represented by unweighted samples or particles, they need to be jointly transformed such that they sample from the modified distribution. When the hidden state evolves on a manifold, there are additional considerations that make existing approaches unsuitable. We present some recent efforts to address these problems, and tie it to more general observations about the relations between nonlinear filtering with particles and geometry.