On the implementation of multilevel Monte Carlo simulation of the stochastic volatility and interest rate model using multi-GPU clusters

Lay, Harold Aldren; Colgin, Zane; Reshniak, Viktor

doi:10.1515/mcma-2018-2025

Cited by 5 publications

(3 citation statements)

References 23 publications

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“…The computation complexity of Monte Carlo Simulation is O(h) where h is the number of steps in each iteration. However, based on [70], in an optimized implementation, it can be reduced to O(h −2 ). It is assumed that numerical discretization of the problem follows weak convergence, and quality obeyed by Euler-Maruyama and Milstein schemes.…”

Section: Tablementioning

confidence: 99%

A Hybrid Modular Approach for Dynamic Fault Tree Analysis

et al. 2020

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Over the years, several approaches have been developed for the quantitative analysis of dynamic fault trees (DFTs). These approaches have strong theoretical and mathematical foundations; however, they appear to suffer from the state-space explosion and high computational requirements, compromising their efficacy. Modularisation techniques have been developed to address these issues by identifying and quantifying static and dynamic modules of the fault tree separately by using binary decision diagrams and Markov models. Although these approaches appear effective in reducing computational effort and avoiding state-space explosion, the reliance of the Markov chain on exponentially distributed data of system components can limit their widespread industrial applications. In this paper, we propose a hybrid modularisation scheme where independent sub-trees of a DFT are identified and quantified in a hierarchical order. A hybrid framework with the combination of algebraic solution, Petri Nets, and Monte Carlo simulation is used to increase the efficiency of the solution. The proposed approach uses the advantages of each existing approach in the right place (independent module). We have experimented the proposed approach on five independent hypothetical and industrial examples in which the experiments show the capabilities of the proposed approach facing repeated basic events and non-exponential failure distributions. The proposed approach could provide an approximate solution to DFTs without unacceptable loss of accuracy. Moreover, the use of modularised or hierarchical Petri nets makes this approach more generally applicable by allowing quantitative evaluation of DFTs with a wide range of failure rate distributions for basic events of the tree.INDEX TERMS Reliability analysis, fault tree analysis, dynamic fault trees, modularisation, petri nets.

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Section: Tablementioning

confidence: 99%

A Hybrid Modular Approach for Dynamic Fault Tree Analysis

et al. 2020

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“…Probabilistic methods (usually implemented with Monte Carlo methods, say, for example, [24,27]) for partial differential equations with/without fractional Laplacian are based on the probabilistic representation of the Laplacian/fractional Laplacian, see e.g. [5].…”

Section: Introductionmentioning

confidence: 99%

A Modified Walk-on-sphere Method for High Dimensional Fractional Poisson Equation

Caiyu¹,

Li²,

Wang³

et al. 2022

Preprint

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“…Probabilistic methods (usually implemented with Monte Carlo methods, say, e.g., [24,25]) for partial differential equations with/without fractional Laplacian are based on the probabilistic representation of the Laplacian/fractional Laplacian, see, for example, [26]. These methods do not require any discretization in space.…”

Section: Introductionmentioning

confidence: 99%

A modified walk‐on‐sphere method for high dimensional fractional Poisson equation

Caiyu

Wang³

et al. 2022

Numerical Methods Partial

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We develop walk-on-sphere method for fractional Poisson equations with Dirichilet boundary conditions in high dimensions. The walk-on-sphere method is based on probabilistic representation of the fractional Poisson equation. We propose efficient quadrature rules to evaluate integral representation in the ball and apply rejection sampling method to drawing from the computed probabilities in general domains. Moreover, we provide an estimate of the number of walks in the mean value for the method when the domain is a ball. We show that the number of walks is increasing in the fractional order and the distance of the starting point to the origin. We also give the relationship between the Green function of fractional Laplace equation and that of the classical Laplace equation. Numerical results for problems in 2-10 dimensions verify our theory and the efficiency of the modified walk-on-sphere method.

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On the implementation of multilevel Monte Carlo simulation of the stochastic volatility and interest rate model using multi-GPU clusters

Cited by 5 publications

References 23 publications

A Hybrid Modular Approach for Dynamic Fault Tree Analysis

A Hybrid Modular Approach for Dynamic Fault Tree Analysis

A Modified Walk-on-sphere Method for High Dimensional Fractional Poisson Equation

A modified walk‐on‐sphere method for high dimensional fractional Poisson equation

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