### Abstract

We investigate the theoretical foundations of the simulated tempering (ST) method and use our findings to design an efficient accelerated sampling algorithm. Employing a large deviation argument first used for replica exchange molecular dynamics (Plattner et al 2011 J. Chem. Phys. 135 134111), we demonstrate that the most efficient approach to simulated tempering is to vary the temperature infinitely rapidly. In this limit, we can replace the equations of motion for the temperature and physical variables by averaged equations for the latter alone, with the forces rescaled according to a position-dependent function defined in terms of temperature weights. The averaged equations are similar to those used in Gao's integrated-over-temperature method, except that we show that it is better to use a continuous rather than a discrete set of temperatures. We give a theoretical argument for the choice of the temperature weights as the reciprocal partition function, thereby relating simulated tempering to Wang-Landau sampling. Finally, we describe a self-consistent algorithm for simultaneously sampling the canonical ensemble and learning the weights during simulation. This infinite switch simulated tempering (ISST) algorithm is tested on three examples of increasing complexity: a system of harmonic oscillators; a continuous variant of the Curie-Weiss model, where ISST is shown to perform better than standard ST and to accurately capture the second-order phase transition observed in this model; and alanine-12 in vacuum, where ISST also compares favorably with standard ST in its ability to calculate the free energy profiles of the root mean square deviation (RMSD) and radius of gyration of the molecule in the 300-500 K temperature range.

Original language | English (US) |
---|---|

Article number | 013207 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Volume | 2019 |

Issue number | 1 |

DOIs | |

State | Published - Jan 16 2019 |

### Fingerprint

### Keywords

- large deviation
- mixing
- molecular dynamics
- sampling algorithms
- stochastic processes

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Journal of Statistical Mechanics: Theory and Experiment*,

*2019*(1), [013207]. https://doi.org/10.1088/1742-5468/aaf323

**The simulated tempering method in the infinite switch limit with adaptive weight learning.** / Martinsson, Anton; Lu, Jianfeng; Leimkuhler, Benedict; Vanden Eijnden, Eric.

Research output: Contribution to journal › Article

*Journal of Statistical Mechanics: Theory and Experiment*, vol. 2019, no. 1, 013207. https://doi.org/10.1088/1742-5468/aaf323

}

TY - JOUR

T1 - The simulated tempering method in the infinite switch limit with adaptive weight learning

AU - Martinsson, Anton

AU - Lu, Jianfeng

AU - Leimkuhler, Benedict

AU - Vanden Eijnden, Eric

PY - 2019/1/16

Y1 - 2019/1/16

N2 - We investigate the theoretical foundations of the simulated tempering (ST) method and use our findings to design an efficient accelerated sampling algorithm. Employing a large deviation argument first used for replica exchange molecular dynamics (Plattner et al 2011 J. Chem. Phys. 135 134111), we demonstrate that the most efficient approach to simulated tempering is to vary the temperature infinitely rapidly. In this limit, we can replace the equations of motion for the temperature and physical variables by averaged equations for the latter alone, with the forces rescaled according to a position-dependent function defined in terms of temperature weights. The averaged equations are similar to those used in Gao's integrated-over-temperature method, except that we show that it is better to use a continuous rather than a discrete set of temperatures. We give a theoretical argument for the choice of the temperature weights as the reciprocal partition function, thereby relating simulated tempering to Wang-Landau sampling. Finally, we describe a self-consistent algorithm for simultaneously sampling the canonical ensemble and learning the weights during simulation. This infinite switch simulated tempering (ISST) algorithm is tested on three examples of increasing complexity: a system of harmonic oscillators; a continuous variant of the Curie-Weiss model, where ISST is shown to perform better than standard ST and to accurately capture the second-order phase transition observed in this model; and alanine-12 in vacuum, where ISST also compares favorably with standard ST in its ability to calculate the free energy profiles of the root mean square deviation (RMSD) and radius of gyration of the molecule in the 300-500 K temperature range.

AB - We investigate the theoretical foundations of the simulated tempering (ST) method and use our findings to design an efficient accelerated sampling algorithm. Employing a large deviation argument first used for replica exchange molecular dynamics (Plattner et al 2011 J. Chem. Phys. 135 134111), we demonstrate that the most efficient approach to simulated tempering is to vary the temperature infinitely rapidly. In this limit, we can replace the equations of motion for the temperature and physical variables by averaged equations for the latter alone, with the forces rescaled according to a position-dependent function defined in terms of temperature weights. The averaged equations are similar to those used in Gao's integrated-over-temperature method, except that we show that it is better to use a continuous rather than a discrete set of temperatures. We give a theoretical argument for the choice of the temperature weights as the reciprocal partition function, thereby relating simulated tempering to Wang-Landau sampling. Finally, we describe a self-consistent algorithm for simultaneously sampling the canonical ensemble and learning the weights during simulation. This infinite switch simulated tempering (ISST) algorithm is tested on three examples of increasing complexity: a system of harmonic oscillators; a continuous variant of the Curie-Weiss model, where ISST is shown to perform better than standard ST and to accurately capture the second-order phase transition observed in this model; and alanine-12 in vacuum, where ISST also compares favorably with standard ST in its ability to calculate the free energy profiles of the root mean square deviation (RMSD) and radius of gyration of the molecule in the 300-500 K temperature range.

KW - large deviation

KW - mixing

KW - molecular dynamics

KW - sampling algorithms

KW - stochastic processes

UR - http://www.scopus.com/inward/record.url?scp=85062537833&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062537833&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/aaf323

DO - 10.1088/1742-5468/aaf323

M3 - Article

AN - SCOPUS:85062537833

VL - 2019

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

SN - 1742-5468

IS - 1

M1 - 013207

ER -