### Abstract

Milestoning is a computational procedure that reduces the dynamics of complex systems to memoryless jumps between intermediates, or milestones, and only retains some information about the probability of these jumps and the time lags between them. Here we analyze a variant of this procedure, termed optimal milestoning, which relies on a specific choice of milestones to capture exactly some kinetic features of the original dynamical system. In particular, we prove that optimal milestoning permits the exact calculation of the mean first passage times (MFPT) between any two milestones. In so doing, we also analyze another variant of the method, called exact milestoning, which also permits the exact calculation of certain MFPTs, but at the price of retaining more information about the original system's dynamics. Finally, we discuss importance sampling strategies based on optimal and exact milestoning that can be used to bypass the simulation of the original system when estimating the statistical quantities used in these methods.

Original language | English (US) |
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Journal | Communications on Pure and Applied Mathematics |

DOIs | |

State | Accepted/In press - 2017 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**A Mathematical Theory of Optimal Milestoning (with a Detour via Exact Milestoning).** / Lin, Ling; Lu, Jianfeng; Vanden Eijnden, Eric.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - A Mathematical Theory of Optimal Milestoning (with a Detour via Exact Milestoning)

AU - Lin, Ling

AU - Lu, Jianfeng

AU - Vanden Eijnden, Eric

PY - 2017

Y1 - 2017

N2 - Milestoning is a computational procedure that reduces the dynamics of complex systems to memoryless jumps between intermediates, or milestones, and only retains some information about the probability of these jumps and the time lags between them. Here we analyze a variant of this procedure, termed optimal milestoning, which relies on a specific choice of milestones to capture exactly some kinetic features of the original dynamical system. In particular, we prove that optimal milestoning permits the exact calculation of the mean first passage times (MFPT) between any two milestones. In so doing, we also analyze another variant of the method, called exact milestoning, which also permits the exact calculation of certain MFPTs, but at the price of retaining more information about the original system's dynamics. Finally, we discuss importance sampling strategies based on optimal and exact milestoning that can be used to bypass the simulation of the original system when estimating the statistical quantities used in these methods.

AB - Milestoning is a computational procedure that reduces the dynamics of complex systems to memoryless jumps between intermediates, or milestones, and only retains some information about the probability of these jumps and the time lags between them. Here we analyze a variant of this procedure, termed optimal milestoning, which relies on a specific choice of milestones to capture exactly some kinetic features of the original dynamical system. In particular, we prove that optimal milestoning permits the exact calculation of the mean first passage times (MFPT) between any two milestones. In so doing, we also analyze another variant of the method, called exact milestoning, which also permits the exact calculation of certain MFPTs, but at the price of retaining more information about the original system's dynamics. Finally, we discuss importance sampling strategies based on optimal and exact milestoning that can be used to bypass the simulation of the original system when estimating the statistical quantities used in these methods.

UR - http://www.scopus.com/inward/record.url?scp=85019234701&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85019234701&partnerID=8YFLogxK

U2 - 10.1002/cpa.21725

DO - 10.1002/cpa.21725

M3 - Article

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

ER -