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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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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
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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Cited by 2 (1 self)
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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
Multilevel path simulation for jumpdiffusion
"... Abstract We investigate the extension of the multilevel Monte Carlo path simulation method to jumpdiffusion SDEs. We consider models with finite rate activity using a jumpadapted discretisation in which the jump times are computed and added to the standard uniform discretisation times. The key com ..."
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Abstract We investigate the extension of the multilevel Monte Carlo path simulation method to jumpdiffusion SDEs. We consider models with finite rate activity using a jumpadapted discretisation in which the jump times are computed and added to the standard uniform discretisation times. The key component in multilevel analysis is the calculation of an expected payoff difference between a coarse path simulation and a fine path simulation with twice as many timesteps. If the Poisson jump rate is constant, the jump times are the same on both paths and the multilevel extension is relatively straightforward, but the implementation is more complex in the case of statedependent jump rates for which the jump times naturally differ 1
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.