Multilevel monte carlo for cortical circuit models

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

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Space-time multilevel Monte Carlo methods and their application to cardiac electrophysiology

Journal of Computational Physics ◽

10.1016/j.jcp.2021.110164 ◽

2021 ◽

Vol 433 ◽

pp. 110164

Author(s):

S. Ben Bader ◽

P. Benedusi ◽

A. Quaglino ◽

P. Zulian ◽

R. Krause

Keyword(s):

Monte Carlo ◽

Monte Carlo Methods ◽

Cardiac Electrophysiology ◽

Space Time ◽

Multilevel Monte Carlo

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Central limit theorem for the multilevel Monte Carlo Euler method

The Annals of Applied Probability ◽

10.1214/13-aap993 ◽

2015 ◽

Vol 25 (1) ◽

pp. 211-234 ◽

Cited By ~ 11

Author(s):

Mohamed Ben Alaya ◽

Ahmed Kebaier

Keyword(s):

Monte Carlo ◽

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Euler Method ◽

Multilevel Monte Carlo

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Multilevel Monte Carlo by using the Halton sequence

Monte Carlo Methods and Applications ◽

10.1515/mcma-2020-2065 ◽

2020 ◽

Vol 26 (3) ◽

pp. 193-203

Author(s):

Shady Ahmed Nagy ◽

Mohamed A. El-Beltagy ◽

Mohamed Wafa

Keyword(s):

Monte Carlo ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Computational Cost ◽

Random Numbers ◽

Multilevel Monte Carlo ◽

Halton Sequence ◽

Random Generator ◽

Mc Simulation ◽

Multilevel Monte Carlo Method

AbstractMonte Carlo (MC) simulation depends on pseudo-random numbers. The generation of these numbers is examined in connection with the Brownian motion. We present the low discrepancy sequence known as Halton sequence that generates different stochastic samples in an equally distributed form. This will increase the convergence and accuracy using the generated different samples in the Multilevel Monte Carlo method (MLMC). We compare algorithms by using a pseudo-random generator and a random generator depending on a Halton sequence. The computational cost for different stochastic differential equations increases in a standard MC technique. It will be highly reduced using a Halton sequence, especially in multiplicative stochastic differential equations.

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Multilevel Monte Carlo method with applications to stochastic partial differential equations

International Journal of Computer Mathematics ◽

10.1080/00207160.2012.701735 ◽

2012 ◽

Vol 89 (18) ◽

pp. 2479-2498 ◽

Cited By ~ 26

Author(s):

Andrea Barth ◽

Annika Lang

Keyword(s):

Monte Carlo ◽

Partial Differential Equations ◽

Monte Carlo Method ◽

Differential Equations ◽

Stochastic Partial Differential Equations ◽

Multilevel Monte Carlo ◽

Partial Differential ◽

Multilevel Monte Carlo Method

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Multilevel Monte Carlo for Smoothing via Transport Methods

SIAM Journal on Scientific Computing ◽

10.1137/17m1156071 ◽

2018 ◽

Vol 40 (4) ◽

pp. A2315-A2335 ◽

Cited By ~ 2

Author(s):

Jeremie Houssineau ◽

Ajay Jasra ◽

Sumeetpal S. Singh

Keyword(s):

Monte Carlo ◽

Multilevel Monte Carlo

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A multilevel Monte Carlo finite difference method for random scalar degenerate convection–diffusion equations

Journal of Hyperbolic Differential Equations ◽

10.1142/s021989161750014x ◽

2017 ◽

Vol 14 (03) ◽

pp. 415-454 ◽

Cited By ~ 5

Author(s):

Ujjwal Koley ◽

Nils Henrik Risebro ◽

Christoph Schwab ◽

Franziska Weber

Keyword(s):

Monte Carlo ◽

Finite Difference ◽

Convergence Rate ◽

Finite Volume ◽

Monte Carlo Algorithm ◽

Diffusion Equations ◽

Entropy Solutions ◽

Finite Volume Scheme ◽

Multilevel Monte Carlo ◽

Convection Diffusion

This paper proposes a finite difference multilevel Monte Carlo algorithm for degenerate parabolic convection–diffusion equations where the convective and diffusive fluxes are allowed to be random. We establish a notion of stochastic entropy solutions to these equations. Our chief goal is to efficiently compute approximations to statistical moments of these stochastic entropy solutions. To this end, we design a multilevel Monte Carlo method based on a finite volume scheme for each sample. We present a novel convergence rate analysis of the combined multilevel Monte Carlo finite volume method, allowing in particular for low [Formula: see text]-integrability of the random solution with [Formula: see text], and low deterministic convergence rates (here, the theoretical rate is [Formula: see text]). We analyze the design and error versus work of the multilevel estimators. We obtain that the maximal rate (based on optimizing possibly the pessimistic upper bounds on the discretization error) is obtained for [Formula: see text], for finite volume convergence rate of [Formula: see text]. We conclude with numerical experiments.

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