Adaptive minimum action method for the study of rare events

Zhou, Xiang; Ren, Weiqing; Weinan, E

doi:10.1063/1.2830717

Cited by 106 publications

(134 citation statements)

References 31 publications

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“…In order to capture the correct optimal path, the selection of a large enough time interval [0, t * ] is essential 22,23 . If the absorbing boundary is far from the origin so that the energy differences between points on the boundary and origin are much larger than k B T , the optimal path is not sensitive to the value of t * as long as t * is reasonably large.…”

Section: -Dimensional Casementioning

confidence: 99%

Large deviations of Rouse polymer chain: First passage problem

Cao

Zhu

Wang

et al. 2015

The Journal of Chemical Physics

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Section: -Dimensional Casementioning

confidence: 99%

Large deviations of Rouse polymer chain: First passage problem

Cao

Zhu

Wang

et al. 2015

The Journal of Chemical Physics

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“…In practice, we need to use finite values. A nice thing is that the error due to finite truncation drops very fast (nearly exponentially) with increasing T 2 − T 1 [39]. In our computations for the Lorenz system, we tested different transition times and numbers of mesh points to ensure that our numerical results are robust.…”

Section: The Minimum Action Methodsmentioning

confidence: 99%

“…If the dynamical system is a gradient system, the minimum action path has a simpler geometric characterization (the minimum energy path, or "MEP") and a simplified algorithm, the string method developed in [3,4], works better for this case. There have been some recent improvements of MAM: the adaptive minimum action method ("aMAM", [39]) and the geometric minimum action method ("gMAM", [40]). Both are designed to overcome the difficulties associated with representing the optimal path using the physical time.…”

Section: The Minimum Action Methodsmentioning

confidence: 99%

Study of noise-induced transitions in the Lorenz system using the minimum action method

Zhou¹,

Weinan²

2010

Communications in Mathematical Sciences

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Abstract. We investigate noise-induced transitions in non-gradient systems when complex invariant sets emerge. Our example is the Lorenz system in three representative Rayleigh number regimes. It is found that before the homoclinic explosion bifurcation, the only transition state is the saddle point, and the transition is similar to that in gradient systems. However, when the chaotic invariant set emerges, an unstable limit cycle continues from the homoclinic trajectory. This orbit, which is embedded in a local tube-like manifold around the initial stable stationary point as a relative attractor, plays the role of the most probable exit set in the transition process. This example demonstrates how limit cycles, the next simplest invariant set beyond fixed points, can be involved in the transition process in smooth dynamical systems.

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“…The objective is to find the optimal mesh using carefully chosen monitor functions, as the iteration proceeds. This is very simple, but it proves to be quite effective, see [9].…”

Section: The Adaptive Minimum Action Methodsmentioning

confidence: 99%

Solving Boltzmann and Fokker-Planck Equations Using Sparse Representation

Shen¹,

Weinan²

2011

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Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to Department of Defense, Washington Headquarters Services, Directorate for Information Approved for public release A major issue in modeling and computation is how to handle high dimen-sional problems. We can divide these high dimensional problems into two classes: moderately high dimensional problems or very high dimensional prob-lems. In the former class, we have problems such as the Boltzmann equation and Fokker-Planck equation, whose dimensionality is moderately high but are amendable to sparse grid based methods. In the latter class, we have problems such as exploration of the configuration space of a large molecule. These prob-lems often involve hundreds of thousands of dimensions, and methods based on fixed grids are far from being adequate. We developed various techniques in handling these problems using the hyperbolic cross/sparse representation for the former class, and adaptive sampling for the latter. These developments are aimed at providing a solid foundation for efficient and reliable numerical simulations of Boltzmann and Fokker-Planck equations. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) SPONSOR/MONITOR'S REPORT NUMBER(S)Besides the work presented in this report, a number of other related publi-cations by the PIs were also partially supported by this grant. UU UU UUJie Shen 7654941923 solving boltzmann and fokker-planck equations using sparse representation AFOSR FA9550-08-1-0416Jie Shen Department of Mathematics, Purdue University Weinan E Department of Mathematics, Princeton University Abstract A major issue in modeling and computation is how to handle high dimensional problems. We can divide these high dimensional problems into two classes: moderately high dimensional problems or very high dimensional problems. In the former class, we have problems such as the Boltzmann equation and Fokker-Planck equation, whose dimensionality is moderately high but are amendable to sparse grid based methods. In the latter class, we have problems such as exploration of the configuration space of a large molecule. These problems often involve hundreds of thousands of dimensions, and methods based on fixed grids are far from being adequate. We developed various techniques in handling these problems using the hyperbolic cross/sparse representation for the former class, and adaptive sampling for the latter. These developments are aimed at providing a solid foundation for efficient and reliable numerical simulations of Boltzmann and Fokker-Planck equati...

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Adaptive minimum action method for the study of rare events

Cited by 106 publications

References 31 publications

Large deviations of Rouse polymer chain: First passage problem

Large deviations of Rouse polymer chain: First passage problem

Study of noise-induced transitions in the Lorenz system using the minimum action method

Solving Boltzmann and Fokker-Planck Equations Using Sparse Representation

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