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CATEGORIES:Statistics
SUMMARY:Explicit stabilised Runge-Kutta methods and their 
 application to Bayesian inverse problems  - Kostas
  Zygalakis\, University of Edinburgh
DTSTART;TZID=Europe/London:20190308T160000
DTEND;TZID=Europe/London:20190308T170000
UID:TALK115927AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/115927
DESCRIPTION:The concept of Bayesian inverse problems provides 
 a coherent mathematical and algorithmic framework 
 that enables researchers to combine mathematical m
 odels with the (often vast) datasets routinely ava
 ilable today in many fields of engineering science
  and technology. The ability to solve such inverse
  problems depends crucially on the efficient calcu
 lation of quantities relating to the posterior dis
 tribution\, giving rise to computationally challen
 ging high dimensional optimization and sampling pr
 oblems. In this talk\, we will connect the corresp
 onding optimization and sampling problems to the l
 arge time behaviour of solutions to (stochastic) d
 ifferential equations. Establishing such a connect
 ion allows utilising existing knowledge from the f
 ield of numerical analysis of differential equatio
 ns. In particular\, numerical stability is key for
  a good performing optimization or sampling algori
 thm since the larger the time-step used while the 
 limiting behaviour of the underlying differential 
 equation is preserved\, the more computationally e
 fficient an algorithm is. With this in mind we wil
 l explore the applicability of explicit stabilised
  Runge-Kutta methods for optimization and sampling
  problems\; These methods are optimal in terms of 
 their stability properties within the class of exp
 licit integrators and we will show that when used 
 as optimization methods they match the optimal con
 vergence rate of the conjugate gradient method for
  quadratic optimization problems. Numerical invest
 igations indicate  that in the general case they a
 re able to outperform state of the art optimizatio
 n methods like   Nesterov's accelerated method. In
  the case of sampling\, we will investigate their 
 applicability to Bayesian inverse problems arising
  in computational imaging. An additional complexit
 y arises there due to the fact that many of them c
 ontain non-differentiable terms\, which when regul
 arised lead to extra stiffness\, hence making expl
 icit stabilised methods even more suitable for the
 se problems as illustrated by a range of numerical
  experiments that show that for the same computati
 onal cost as current state of the arts methods\, e
 xplicit stabilised methods deliver much better MCM
 C samples. \n\nThis is joint work with Armin Eftek
 hari (EPFL)\, Bart Vandereycken (Geneva)\, Gilles 
 Vilmart (Geneva)\, Marcelo Pereyra (Heriot-Watt) a
 nd Luis Vargas (Edinburgh)
LOCATION:MR12
CONTACT:Dr Sergio Bacallado
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