Statistical and computational guarantees for gradient based MCMC in some PDE inverse problems
25 Mar 2024, 16:00 — Room 705 @ DIBRIS/DIMA, Via Dodecaneso 35, Genoa
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Speaker:
Richard Nickl — University of Cambridge
Richard Nickl — University of Cambridge
Abstract:
We discuss recent results, both positive and negative, about the run-time of gradient based Markov chain Monte Carlo (MCMC) algorithms that target posterior distributions arising from high-dimensional non-linear statistical regression models with Gaussian process priors. Prototypical applications include inverse problems with partial differential equations (PDEs). We show that cold-start MCMC may not work (have `exponential in dimension’ runtime) even for target measures that are radially strictly decreasing away from their unique mode, but that warm-start Langevin type MCMC can achieve polynomial runtime for posterior computation under certain `gradient stability’ conditions that can be verified in a large class of relevant PDE models.
We discuss recent results, both positive and negative, about the run-time of gradient based Markov chain Monte Carlo (MCMC) algorithms that target posterior distributions arising from high-dimensional non-linear statistical regression models with Gaussian process priors. Prototypical applications include inverse problems with partial differential equations (PDEs). We show that cold-start MCMC may not work (have `exponential in dimension’ runtime) even for target measures that are radially strictly decreasing away from their unique mode, but that warm-start Langevin type MCMC can achieve polynomial runtime for posterior computation under certain `gradient stability’ conditions that can be verified in a large class of relevant PDE models.
Bio:
Richard Nickl is a Professor of Mathematical Statistics at the University of Cambridge. He is a fellow of Gonville and Caius College. He obtained his academic degrees from the University of Vienna, including a PhD in 2005. He has made contributions to various areas of mathematical statistics; including non-parametric and high-dimensional statistics, empirical process theory, and Bayesian inference for statistical inverse problems and partial differential equations. Jointly with Evarist Giné, he is the author of the book `Mathematical foundations of infinite-dimensional statistical models', published with Cambridge University Press, which won the 2017 PROSE Award for best monograph in the mathematics category. He was an invited speaker at the 2022 International Congress of Mathematicians (ICM) and at the 8th European Congress of Mathematics (ECM).He has been awarded the 2017 Ethel Newbold Prize of the Bernoulli Society as well as a Consolidator Grant and an Advanced Grant by the European Research Council.
Richard Nickl is a Professor of Mathematical Statistics at the University of Cambridge. He is a fellow of Gonville and Caius College. He obtained his academic degrees from the University of Vienna, including a PhD in 2005. He has made contributions to various areas of mathematical statistics; including non-parametric and high-dimensional statistics, empirical process theory, and Bayesian inference for statistical inverse problems and partial differential equations. Jointly with Evarist Giné, he is the author of the book `Mathematical foundations of infinite-dimensional statistical models', published with Cambridge University Press, which won the 2017 PROSE Award for best monograph in the mathematics category. He was an invited speaker at the 2022 International Congress of Mathematicians (ICM) and at the 8th European Congress of Mathematics (ECM).He has been awarded the 2017 Ethel Newbold Prize of the Bernoulli Society as well as a Consolidator Grant and an Advanced Grant by the European Research Council.
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