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## Improving MLMC for SDEs with application to the Langevin (2014)

### Citations

1398 |
Numerical solution of stochastic differential equations
- Kloeden, Platen
- 1992
(Show Context)
Citation Context ...nctions C, sup 0≤tn≤T E [ φ(X(tn)) ]− E[φ(Xn)] = O(h). If f and G are sufficiently smooth, C contains all infinitely differentiable functions whose derivatives are polynomially bounded; for example, (=-=Kloeden & Platen, 1992-=-, Theorem 14.5.1). In some cases (Shardlow, 2006; Zygalakis, 2011), it is possible to find a second SDE, called the modified SDE with solution Xh(t), such that Xn is a second-order weak approximation ... |

262 | Geometric Numerical Integration - Hairer, Lubich, et al. - 2002 |

227 |
Simulating Hamiltonian Dynamics
- Leimkuhler, Reich
- 2004
(Show Context)
Citation Context ...r field to be broken down into meaningful parts and integrated separately over a single time step, before combining into an integrator for the full vector field. See for example (Hairer et al., 2010; =-=Leimkuhler & Reich, 2004-=-). The Langevin equation breaks down into the sum of a Hamiltonian system and a linear SDE for an Ornstein–Uhlenbeck (OU) process. Then, for a splitting method, we define integrators for the Hamiltoni... |

187 | Multilevel Monte Carlo path simulation - Giles - 2008 |

167 |
Expansion of the global error for numerical schemes solving stochastic differential equations.
- Talay, Tubaro
- 1991
(Show Context)
Citation Context ...n the MLMC method and then evaluate the moments on the coarsest level directly at a dramatically reduced cost in comparison to Monte Carlo. The final enhancement that we investigate is extrapolation (=-=Talay & Tubaro, 1990-=-). It is a natural addition to MLMC methods, already mentioned in the original work (Giles, 2008) and studied in more detail in (Lemaire & Pagès, 2013). It reduces the bias in the numerical approximat... |

142 | Multilevel Monte Carlo Algorithms for Lévy-driven SDEs with Gaussian Correction
- Dereich
- 2009
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Citation Context ...d has been successfully applied to a wide class of problems in stochastic simulation and in uncertainty quantification; for example, (Anderson & Higham, 2011; Barth et al., 2011; Cliffe et al., 2011; =-=Dereich & Heidenreich, 2011-=-; Giles & Reisinger, 2012; Giles & Szpruch, 2013; Hoel et al., 2012; Mishra et al., 2012). The variance reduction in MLMC is achieved by computing approximations of the solution on different “levels” ... |

49 |
N.: Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients
- Barth, Schwab, et al.
- 2011
(Show Context)
Citation Context ...method that is well established by now and has been successfully applied to a wide class of problems in stochastic simulation and in uncertainty quantification; for example, (Anderson & Higham, 2011; =-=Barth et al., 2011-=-; Cliffe et al., 2011; Dereich & Heidenreich, 2011; Giles & Reisinger, 2012; Giles & Szpruch, 2013; Hoel et al., 2012; Mishra et al., 2012). The variance reduction in MLMC is achieved by computing app... |

48 |
Stochastic boundary conditions for molecular dynamics simulations of ST2
- Brunger, Brooks, et al.
- 1984
(Show Context)
Citation Context ...P TP + V (Q), and kBT = σ2/2λ. As usual, kB denotes the Boltzmann constant and T temperature. Numerical integrators for the Langevin equation are well developed for example in (Beard & Schlick, 2000; =-=Brunge et al., 1984-=-; Wang & Skeel, 2003). Recently, there has been a strong push to understand the invariant measure associated to the integrators (Bou-Rabee & Owhadi, 2010; Debussche & Faou, 2012; Kopec, 2013; Zygalaki... |

44 | L.: Multilevel Monte Carlo methods for applications in finance
- Giles, Szpruch
(Show Context)
Citation Context ...established by now and has been successfully applied to a wide class of problems in stochastic simulation and in uncertainty quantification; for example, (Anderson & Higham, 2011; Barth et al., 2011; =-=Cliffe et al., 2011-=-; Dereich & Heidenreich, 2011; Giles & Reisinger, 2012; Giles & Szpruch, 2013; Hoel et al., 2012; Mishra et al., 2012). The variance reduction in MLMC is achieved by computing approximations of the so... |

38 | Stochastic Lagrangian models of turbulent diffusion - Rodean - 1996 |

23 | Multi-level Monte Carlo for continuous time Markov chains with applications in biochemical kinetics
- Anderson, Higham
(Show Context)
Citation Context ...rtant variance-reduction method that is well established by now and has been successfully applied to a wide class of problems in stochastic simulation and in uncertainty quantification; for example, (=-=Anderson & Higham, 2011-=-; Barth et al., 2011; Cliffe et al., 2011; Dereich & Heidenreich, 2011; Giles & Reisinger, 2012; Giles & Szpruch, 2013; Hoel et al., 2012; Mishra et al., 2012). The variance reduction in MLMC is achie... |

22 | Long-run accuracy of variational integrators in the stochastic context. - Bou-Rabee, Owhadi - 2010 |

19 | Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions - Mishra, Schwab, et al. |

18 | C.: Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance
- Giles, Reisinger
- 2012
(Show Context)
Citation Context ...ed to a wide class of problems in stochastic simulation and in uncertainty quantification; for example, (Anderson & Higham, 2011; Barth et al., 2011; Cliffe et al., 2011; Dereich & Heidenreich, 2011; =-=Giles & Reisinger, 2012-=-; Giles & Szpruch, 2013; Hoel et al., 2012; Mishra et al., 2012). The variance reduction in MLMC is achieved by computing approximations of the solution on different “levels” consisting, in the SDE ca... |

14 | Modified equation for stochastic differential equation - Shardlow - 2004 |

9 | Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations
- Hutzenthaler, Jentzen, et al.
- 1966
(Show Context)
Citation Context ... on the time-step size. This is the same for SDEs and such stability constraints may severely restrict the number of coarse levels that can be employed in the MLMC method (Abdulle & Blumenthal, 2013; =-=Hutzenthaler et al., 2013-=-) and thus its efficiency. However, there are explicit integrators that are unconditionally stable. We discuss a class of such integrators for an important model in molecular dynamics and atmospheric ... |

9 | Analysis of a few numerical integration methods for the Langevin equation
- Wang, Skeel
- 2003
(Show Context)
Citation Context ... = σ2/2λ. As usual, kB denotes the Boltzmann constant and T temperature. Numerical integrators for the Langevin equation are well developed for example in (Beard & Schlick, 2000; Brunge et al., 1984; =-=Wang & Skeel, 2003-=-). Recently, there has been a strong push to understand the invariant measure associated to the integrators (Bou-Rabee & Owhadi, 2010; Debussche & Faou, 2012; Kopec, 2013; Zygalakis, 2011). Second-ord... |

8 | Weak backward error analysis for SDEs.
- Debussche, Faou
- 2012
(Show Context)
Citation Context ...n (Beard & Schlick, 2000; Brunge et al., 1984; Wang & Skeel, 2003). Recently, there has been a strong push to understand the invariant measure associated to the integrators (Bou-Rabee & Owhadi, 2010; =-=Debussche & Faou, 2012-=-; Kopec, 2013; Zygalakis, 2011). Second-order modified equations are available for the most important integrators for the Langevin equation. In particular, we study splitting methods based on exact sa... |

5 | Stabilized multilevel Monte Carlo method for stiff stochastic differential equations - Abdulle, Blumenthal |

5 |
Weak backward error analysis for Langevin process.
- Kopec
- 2013
(Show Context)
Citation Context ...; Brunge et al., 1984; Wang & Skeel, 2003). Recently, there has been a strong push to understand the invariant measure associated to the integrators (Bou-Rabee & Owhadi, 2010; Debussche & Faou, 2012; =-=Kopec, 2013-=-; Zygalakis, 2011). Second-order modified equations are available for the most important integrators for the Langevin equation. In particular, we study splitting methods based on exact sampling of an ... |

4 | Adaptive multilevel Monte Carlo simulation, in Numerical Analysis of Multiscale Computations - Hoel, Schwerin, et al. - 2012 |

3 |
Inertial stochastic dynamics: I. Long timestep methods for Langevin dynamics
- Beard, Schlick
- 2000
(Show Context)
Citation Context ...constant, H(Q,P ) := 12P TP + V (Q), and kBT = σ2/2λ. As usual, kB denotes the Boltzmann constant and T temperature. Numerical integrators for the Langevin equation are well developed for example in (=-=Beard & Schlick, 2000-=-; Brunge et al., 1984; Wang & Skeel, 2003). Recently, there has been a strong push to understand the invariant measure associated to the integrators (Bou-Rabee & Owhadi, 2010; Debussche & Faou, 2012; ... |

3 | Multilevel richardson-romberg extrapolation. arXiv preprint arXiv:1401.1177
- Lemaire, Pagès
- 2014
(Show Context)
Citation Context ... enhancement that we investigate is extrapolation (Talay & Tubaro, 1990). It is a natural addition to MLMC methods, already mentioned in the original work (Giles, 2008) and studied in more detail in (=-=Lemaire & Pagès, 2013-=-). It reduces the bias in the numerical approximation of the solution due to time stepping and relies on having a sharp estimate for the bias error. If such an estimate is available, it is possible to... |

2 | On the Acceleration of the Multi–Level Monte Carlo Method, ArXiv preprint 1301.7650 - Debrabant, Rößler - 2013 |

1 | Multilevel Monte Carlo Methods. Pages 58–67 of: Large-scale scientific computing - Heinrich |