ANC Seminar: Kostas Zygalakis (from Maths), Chair: Amos Storkey
What 


When 
Nov 22, 2016 from 11:00 AM to 12:00 PM 
Where  IF 4.31/4.33 
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Ergodic Stochastic Differential Equations and Sampling: A numerical
analysis perspective
Understanding the long time behaviour of solutions to ergodic stochastic
differential equations is an important question with relevance in many
field of applied mathematics and statistics. Hence, designing
appropriate numerical algorithms that are able to capture such behaviour
correctly is extremely important. A recently introduced framework [1,2]
using backward error analysis allows us to characterise the bias with
which one approximates the invariant measure (in the absence of the
accept/reject correction). Using this framework we will analyse
splitting [3] and stochastic gradient algorithms [4] arising in
molecular dynamics and machine learning respectively. These ideas will
also be used to design numerical methods exploiting the variance
reduction of recently introduced nonreversible Langevin samplers [5].
Finally if there is time we will discuss, how things ideas can be
combined with the idea of Multilevel Monte Carlo [6] to produce unbiased
estimates of ergodic averages without the need the of an acceptreject
correction [7] and optimal computational cost, and how can this
exploited in the case of big data [8].
[1] K.C. Zygalakis. On the existence and applications of modified
equations for stochastic differential equations. /SIAM J. Sci. Comput/.,
33:102130, 2011.
[2] A. Abdulle, G. Vilmart, and K. C. Zygalakis. High order numerical
approximation of the invariant measure of ergodic sdes. /SIAM J. Numer.
Anal./, 52(4):16001622, 2014.
[3] A. Abdulle, G. Vilmart, and K.C. Zygalakis, Long time accuracy of
LieTrotter splitting methods for Langevin dynamics./SIAM J. Numer.
Anal./, 53(1):116, 2015.
[4] S. J. Vollmer, K.C. Zygalakis and Y. W. Teh, Exploration of the
(Non)asymptotic Bias and Variance of Stochastic Gradient Langevin
Dynamics./J. Mach. Learn. Res., / 17(159):148, 2016.
[5] A. Duncan, G. A. Pavliotis and T. Lelievre, Variance Reduction using
Nonreversible Langevin Samplers, /J. Stat. Phys./ 163(3):457491, 2016.
[6] M.B. Giles, Mutlilevel Monte Carlo methods, /Acta Numerica,/
24:259328, 2015
[7] L. Szpruch, S. Vollmer, K. C. Zygalakis and M. B. Giles, Multi
Level Monte Carlo methods for a class of ergodic
stochastic di
fferential equations. arXiv:1605.01384
<https://arxiv.org/abs/1605.01384>
[8] T. Nagapetyan, L. Szpruch, S. Vollmer, K. C. Zygalakis and M. B.
Giles, Multilevel Monte Carlo for Scalable Bayesian Computations.
arXiv:1609.06144 <https://arxiv.org/abs/1609.06144>