| 1. |
- Are, Sasanka, et al.
(author)
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Coarse-Grained Langevin Approximations and Spatiotemporal Acceleration for Kinetic Monte Carlo Simulations of Diffusion of Interacting Particles
- 2009
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In: Chinese Annals of Mathematics. Series B. - : Springer Science and Business Media LLC. - 0252-9599 .- 1860-6261. ; 30:6, s. 653-682
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Journal article (peer-reviewed)abstract
- Kinetic Monte Carlo methods provide a powerful computational tool for the simulation of microscopic processes such as the diffusion of interacting paxticles oil a surface, at a detailed atomistic level. However such algorithms are typically computationally expensive and are restricted to fairly small spatiotemporal scales. One approach towards overcoming this problem was the development of coarse-gained Monte Carlo algorithms. In recent literature, these methods were shown to be capable of efficiently describing much larger length scales while still incorporating information on microscopic interactions and fluctuations. In this paper, a coarse-grained Langevin system of stochastic differential equations as approximations of diffusion of interacting particles is derived, based on these earlier coarse-grained models. The authors demonstrate the asymptotic equivalence of transient and long time behavior of the Langevin approximation and the underlying microscopic process, using asymptotics methods such as large deviations for interacting particles systems, and furthermore, present corresponding numerical simulations, comparing statistical quantities like mean paths, auto correlations and power spectra of the microscopic and the approximating Langevin processes. Finally, it is shown that the Langevin approximations presented here are much more computationally efficient than conventional Kinetic Monte Carlo methods, since in addition to the reduction in the number of spatial degrees of freedom in coarse-grained Monte Carlo methods, the Langevin system of stochastic differential equations allows for multiple particle moves in a single timestep.
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| 2. |
- Björk, T., et al.
(author)
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Monte Carlo Euler approximations of HJM term structure financial models
- 2013
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In: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 53:2, s. 341-383
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Journal article (peer-reviewed)abstract
- We present Monte Carlo-Euler methods for a weak approximation problem related to the Heath-Jarrow-Morton (HJM) term structure model, based on Itô stochastic differential equations in infinite dimensional spaces, and prove strong and weak error convergence estimates. The weak error estimates are based on stochastic flows and discrete dual backward problems, and they can be used to identify different error contributions arising from time and maturity discretization as well as the classical statistical error due to finite sampling. Explicit formulas for efficient computation of sharp error approximation are included. Due to the structure of the HJM models considered here, the computational effort devoted to the error estimates is low compared to the work to compute Monte Carlo solutions to the HJM model. Numerical examples with known exact solution are included in order to show the behavior of the estimates.
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| 3. |
- Katsoulakis, Markos A., et al.
(author)
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Stochastic hydrodynamical limits of particle systems
- 2006
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In: Communications in Mathematical Sciences. - 1539-6746 .- 1945-0796. ; 4:3, s. 513-549
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Journal article (peer-reviewed)abstract
- Even small noise can have substantial influence on the dynamics of differential equations, e.g. for nucleation/coarsening and interface dynamics in phase transformations. The aim of this work is to establish accurate models for the noise in macroscopic differential equations, related to phase transformations/reactions, derived from more fundamental microscopic master equations. For this purpose the mathematical paradigm of the dynamic Ising model is considered in the relatively tractable case of stochastic spin flip dynamics and long range spin/spin interactions. More specifically, this paper shows that localized spatial averages, with width epsilon, of solutions to such Ising systems with long range interaction of range O(1), are approximated with error O(epsilon + (gamma/epsilon)(2d)) in distribution by a solution of an Ito stochastic differential equation, with drift as in the corresponding mean field model and a small diffusion coefficient of order (gamma/epsilon)(d/2), generating noise with spatial correlation length epsilon, where gamma is the distance between neighboring spin sites on a uniform periodic lattice in R-d. To determine the correct noise is subtle in the sense that there are expected values, i.e. observables, that require different noise: the expected values that can be accurately approximated by the Einstein-diffusion and the expected values that need an alternative diffusion related to large deviation theory are identified; for instance dendrite dynamics up to a bounded time needs Einstein diffusion while transition rates need a different diffusion model related to invariant measures. The elementary proofs use O((gamma/epsilon)(2d)) consistency of the Kolmogorov-backward equations for the averaged spin and the stochastic differential equation and show that the long range interaction yields smoothing, which contributes with the O(epsilon) error. A new aspect of the derivation is that the error, based on residuals and weights, is computable and suitable for adaptive refinements and modeling.
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