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TAMIDS SciML Lab Seminar Series: Simo Särkkä: Probabilistic Solvers for ODEs and PDEs

November 14, 2022

2:00 pm - 3:00 pm

Online via Zoom

Meeting ID: 998 4499 3279

Passcode: 724615

Speaker: Simo Särkkä

Host: Ulisses Braga-Neto

Abstract: We discuss probabilistic solvers for ordinary and partial differential equations (ODEs/PDEs) and their implementation using Bayesian filters and smoothers. Probabilistic numerical solving of ODEs can be formulated as Gaussian process (GP) regression, where the observations are derivatives of the vector field (i.e., the right hand side) of the ODE. When the GP has a state-space representation, the problem can be reduced to a non-linear Bayesian filtering and smoothing problem. In particular, the iterated extended Kalman smoother (IEKS) can be used to compute maximum a posteriori (MAP) estimate of the solution along with its uncertainty. We also discuss the extension of the IEKS solution to PDEs of non-linear Cauchy type. In these models, the PDE can be approximated in the spatial direction using finite differences or basis function expansions, which then reduces the PDE to an ODE, where the IEKS solution can be applied. As opposed to the ODE case, in the PDE case, prior information on both the temporal and spatial smoothness of the PDE solution can be encoded into the spatio-temporal GP prior, which in the ODE case is only possible in the temporal direction.

Biography: Simo Särkkä received his Master of Science (Tech.) degree in engineering physics and mathematics, and Doctor of Science (Tech.) degree in electrical and communications engineering from Helsinki University of Technology, Espoo, Finland, in 2000 and 2006, respectively. Currently, he is an Associate Professor at Aalto University and an Adjunct Professor (= Docent) at Tampere University and LUT University. He is also a Fellow of European Laboratory for Learning and Intelligent Systems (ELLIS) and the Leader of AI Across Fields (AIX) program of Finnish Center for Artificial Intelligence (FCAI). His and his group’s research interests are in multi-sensor data processing systems with applications in location sensing, health and medical technology, machine learning, inverse problems, and brain imaging. He has authored or coauthored more than 150 peer-reviewed scientific articles and his books “Bayesian Filtering and Smoothing” and “Applied Stochastic Differential Equations” along with the Chinese translation of the former were recently published via the Cambridge University Press. He is a Senior Member of IEEE and serving as a Senior Area Editor of IEEE Signal Processing Letters.