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

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  • ANC/DTC Seminar
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 accept-reject

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/.,

33:102-130, 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):1600-1622, 2014.


[3] A. Abdulle, G. Vilmart, and K.C. Zygalakis, Long time accuracy of

Lie-Trotter splitting methods for Langevin dynamics./SIAM J. Numer.

Anal./, 53(1):1-16, 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):1--48, 2016.


[5] A. Duncan, G. A. Pavliotis and T. Lelievre, Variance Reduction using

Nonreversible Langevin Samplers, /J. Stat. Phys./ 163(3):457-491, 2016.


[6] M.B. Giles, Mutlilevel Monte Carlo methods, /Acta Numerica,/

24:259-328, 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



[8] T. Nagapetyan,  L. Szpruch, S. Vollmer, K. C. Zygalakis and M. B.

Giles, Multilevel Monte Carlo for Scalable Bayesian Computations.

arXiv:1609.06144 <>