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Multilevel Monte Carlo Methods and Applications to Elliptic PDEs with Random Coefficients
"... We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevel Mo ..."
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Cited by 46 (15 self)
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We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevel Monte Carlo method. The main result is that in certain circumstances the asymptotic cost of solving the stochastic problem is a constant (but moderately large) multiple of the cost of solving the deterministic problem. Numerical calculations demonstrating the effectiveness of the method for one and twodimensional model problems arising in groundwater flow are presented. 1
Antithetic multilevel Monte Carlo estimation for multidimensional SDEs without Lévy area simulation. Arxiv preprint arXiv:1202.6283
, 2012
"... Abstract In this paper we develop antithetic multilevel Monte Carlo (MLMC) estimators for multidimensional SDEs driven by Brownian motion. Giles has previously shown that if we combine a numerical approximation with strong order of convergence O( ∆ t) with MLMC we can reduce the computational comp ..."
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Cited by 10 (4 self)
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Abstract In this paper we develop antithetic multilevel Monte Carlo (MLMC) estimators for multidimensional SDEs driven by Brownian motion. Giles has previously shown that if we combine a numerical approximation with strong order of convergence O( ∆ t) with MLMC we can reduce the computational complexity to estimate expected values of Lipschitz functionals of SDE solutions with a rootmeansquare error of ε from O(ε−3) to O(ε−2). However, in general, to obtain a rate of strong convergence higher than O( ∆ t1/2) requires simulation, or approximation, of Lévy areas. Recently, Giles and Szpruch [5] constructed an antithetic multilevel estimator that avoids the simulation of Lévy areas and still achieves an MLMC correction variance which is O( ∆ t2) for smooth payoffs and almost O( ∆ t3/2) for piecewise smooth payoffs, even though there is only O( ∆ t1/2) strong convergence. This results in an O(ε−2) complexity for estimating the value of financial European and Asian put and call options. In this paper, we extend these results to more complex payoffs based on the path minimum. To achieve this, an approximation of the Lévy areas is needed, resulting in O( ∆ t3/4) strong convergence. By modifying the antithetic MLMC estimator we are able to obtain O(ε−2 log(ε)2) complexity for estimating financial barrier and lookback options. 1
Parallel Simulations for Analysing Portfolios of Catastrophic Event Risk
 Workshop of the International Conference for High Performance Computing, Networking, Storage and Analysis (SC
, 2012
"... Abstract—At the heart of the analytical pipeline of a modern quantitative insurance/reinsurance company is a stochastic simulation technique for portfolio risk analysis and pricing process referred to as Aggregate Analysis. Support for the computation of risk measures including Probable Maximum Los ..."
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Cited by 6 (6 self)
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Abstract—At the heart of the analytical pipeline of a modern quantitative insurance/reinsurance company is a stochastic simulation technique for portfolio risk analysis and pricing process referred to as Aggregate Analysis. Support for the computation of risk measures including Probable Maximum Loss (PML) and the Tail Value at Risk (TVAR) for a variety of types of complex property catastrophe insurance contracts including Cat eXcess
Optimization of mesh hierarchies in Multilevel Monte Carlo samplers. ArXiv eprints:1403.2480
, 2014
"... Abstract We perform a general optimization of the parameters in the Multilevel Monte Carlo (MLMC) discretization hierarchy based on uniform discretization methods with general approximation orders and computational costs. Moreover, we discuss extensions to nonuniform discretizations based on a prio ..."
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Cited by 5 (1 self)
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Abstract We perform a general optimization of the parameters in the Multilevel Monte Carlo (MLMC) discretization hierarchy based on uniform discretization methods with general approximation orders and computational costs. Moreover, we discuss extensions to nonuniform discretizations based on a priori refinements and the effect of imposing constraints on the largest and/or smallest mesh sizes. We optimize geometric and nongeometric hierarchies and compare them to each other, concluding that the geometric hierarchies, when optimized, are nearly optimal and have the same asymptotic computational complexity. We discuss how enforcing domain constraints on parameters of MLMC hierarchies affects the optimality of these hierarchies. These domain constraints include an upper and lower bound on the mesh size or enforcing that the number of samples and the number of discretization elements are integers. We also discuss the optimal tolerance splitting between the bias and the statistical error contributions and its asymptotic behavior. To provide numerical grounds for our theoretical results, we apply these optimized hierarchies together with the Continuation MLMC Algorithm [13]
MultiGPU Computing for Achieving Speedup in Realtime Aggregate Risk Analysis. High Performance Computing on Graphics Processing Units (hgpu.org
, 2013
"... Abstract—Stochastic simulation techniques employed for portfolio risk analysis, often referred to as Aggregate Risk Analysis, can benefit from exploiting stateoftheart highperformance computing platforms. In this paper, we propose parallel methods to speedup aggregate risk analysis for supporti ..."
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Cited by 2 (0 self)
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Abstract—Stochastic simulation techniques employed for portfolio risk analysis, often referred to as Aggregate Risk Analysis, can benefit from exploiting stateoftheart highperformance computing platforms. In this paper, we propose parallel methods to speedup aggregate risk analysis for supporting realtime pricing. To achieve this an algorithm for analysing aggregate risk is proposed and implemented in C and OpenMP for multicore CPUs and in C and CUDA for manycore GPUs. An evaluation of the performance of the algorithm indicates that GPUs offer a feasible alternative solution over traditional highperformance computing systems. An aggregate simulation on a multiGPU of 1 million trials with 1000 catastrophic events per trial on a typical exposure set and contract structure is performed in less than 5 seconds. The key result is that the multiGPU implementation of the algorithm presented in this paper is approximately 77x times faster than the traditional counterpart and can be used in realtime pricing scenarios. KeywordsGPU computing; highperformance computing; aggregate risk analysis; catastrophe event risk; realtime pricing I.
Multilevel Monte Carlo methods
"... An outline history inspired by undergraduate numerical projects course at Cambridge, and summer projects at RollsRoyce this was one of my first textbooks after 25 years working on CFD, 10 years ago I switched to Monte Carlo methods for computational finance and other application areas ..."
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An outline history inspired by undergraduate numerical projects course at Cambridge, and summer projects at RollsRoyce this was one of my first textbooks after 25 years working on CFD, 10 years ago I switched to Monte Carlo methods for computational finance and other application areas
MULTI INDEX MONTE CARLO: WHEN SPARSITY MEETS SAMPLING
"... Abstract. We propose and analyze a novel Multi Index Monte Carlo (MIMC) method for weak approximation of stochastic models that are described in terms of differential equations either driven by random measures or with random coefficients. The MIMC method is both a stochastic version of the combinati ..."
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Abstract. We propose and analyze a novel Multi Index Monte Carlo (MIMC) method for weak approximation of stochastic models that are described in terms of differential equations either driven by random measures or with random coefficients. The MIMC method is both a stochastic version of the combination technique introduced by Zenger, Griebel and collaborators and an extension of the Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles. Inspired by Giles’s seminal work, instead of using firstorder differences as in MLMC, we use in MIMC highorder mixed differences to reduce the variance of the hierarchical differences dramatically. This in turn yields new and improved complexity results, which are natural generalizations of Giles’s MLMC analysis, and which increase the domain of problem parameters for which we achieve the optimal convergence, O(TOL−2). Moreover, we motivate the systematic construction of optimal sets of indices for MIMC based on properly defined profits that in turn depend
Unlimited Release
, 2014
"... A multilevel stochastic collocation method for partial differential equations with random input data ..."
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A multilevel stochastic collocation method for partial differential equations with random input data
Acknowledgements
, 2014
"... First and foremost I wish to express my deepest gratitude to my supervisor, Prof. Mike Giles for being unfailingly supportive and for being an inspiration. Without his expertise, his dedication and outstanding guidance this thesis would never have been possible. I am extremely grateful to the Man Gr ..."
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First and foremost I wish to express my deepest gratitude to my supervisor, Prof. Mike Giles for being unfailingly supportive and for being an inspiration. Without his expertise, his dedication and outstanding guidance this thesis would never have been possible. I am extremely grateful to the Man Group plc for their financial backing and for providing me with such an amazing work environment at the OxfordMan Institute. Special thanks go to William Chesters for his understanding and stimulation in the final stages of my thesis. Thanks to the University of Oxford, Lady Margaret Hall, the common rooms and clubs for making these years so unique and enriching. Thanks to the many unsung heroes of free software without whom I wouldn’t have had the tools for writing this thesis. On a more personal level, I would also like to mention the very special people I have the privilege to know both in Oxford and across the globe. I am greatly indebted to all of them for their kindness, their joviality, their wisdom and for all the things I have learnt from them. Although not mentioned individually, they will recognise themselves. To all of them: “Thanks for being part of my life”. Finally I want to thank my family for their love and for always supporting me in times of doubt. All I have and will accomplish is only possible thanks to them, the importance of their sacrifices could never be overstated. This work is for them.