@article{ZhangSchuette, author = {Zhang, Wei and Sch{\"u}tte, Christof}, title = {Reliable approximation of long relaxation timescales in molecular dynamics}, series = {Entropy}, volume = {19}, journal = {Entropy}, number = {7}, doi = {10.3390/e19070367}, language = {en} } @misc{ZhangSchuette, author = {Zhang, Wei and Sch{\"u}tte, Christof}, title = {Reliable approximation of long relaxation timescales in molecular dynamics}, issn = {1438-0064}, doi = {10.3390/e19070367}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-63718}, abstract = {Many interesting rare events in molecular systems like ligand association, protein folding or con- formational changes happen on timescales that often are not accessible by direct numerical simulation. Therefore rare event approximation approaches like interface sampling, Markov state model building or advanced reaction coordinate based free energy estimation have attracted huge attention recently. In this article we analyze the reliability of such approaches: How precise is an estimate of long relaxation timescales of molecular systems resulting from various forms of rare event approximation methods? Our results give a theoretical answer to this question by relating it with the transfer operator approach to molecular dynamics. By doing so they also allow for understanding deep connections between the different approaches.}, language = {en} } @article{vonKleistSchuetteZhang, author = {von Kleist, Max and Sch{\"u}tte, Christof and Zhang, Wei}, title = {Statistical analysis of the first passage path ensemble of jump processes}, series = {Journal of Statistical Physics}, volume = {170}, journal = {Journal of Statistical Physics}, doi = {10.1007/s10955-017-1949-x}, pages = {809 -- 843}, abstract = {The transition mechanism of jump processes between two different subsets in state space reveals important dynamical information of the processes and therefore has attracted considerable attention in the past years. In this paper, we study the first passage path ensemble of both discrete-time and continuous-time jump processes on a finite state space. The main approach is to divide each first passage path into nonreactive and reactive segments and to study them separately. The analysis can be applied to jump processes which are non-ergodic, as well as continuous-time jump processes where the waiting time distributions are non-exponential. In the particular case that the jump processes are both Markovian and ergodic, our analysis elucidates the relations between the study of the first passage paths and the study of the transition paths in transition path theory. We provide algorithms to numerically compute statistics of the first passage path ensemble. The computational complexity of these algorithms scales with the complexity of solving a linear system, for which efficient methods are available. Several examples demonstrate the wide applicability of the derived results across research areas.}, language = {en} } @article{Zhang, author = {Zhang, Wei}, title = {Some new results on relative entropy production, time reversal, and optimal control of time-inhomogeneous diffusion processes}, series = {Journal of Mathematical Physics}, volume = {62}, journal = {Journal of Mathematical Physics}, number = {4}, doi = {10.1063/5.0038740}, pages = {26}, abstract = {This paper studies time-inhomogeneous nonequilibrium diffusion processes, including both Brownian dynamics and Langevin dynamics. We derive upper bounds of the relative entropy production of the time-inhomogeneous process with respect to the transient invariant probability measures. We also study the time reversal of the reverse process in Crooks' fluctuation theorem. We show that the time reversal of the reverse process coincides with the optimally controlled forward process that leads to zero variance importance sampling estimator based on Jarzynski's equality.}, language = {en} } @article{SharmaZhang2020, author = {Sharma, Upanshu and Zhang, Wei}, title = {Non-reversible sampling schemes on submanifolds}, series = {SIAM Journal on Numerical Analysis}, volume = {59}, journal = {SIAM Journal on Numerical Analysis}, number = {6}, doi = {10.1137/20M1378752}, pages = {2989 -- 3031}, year = {2020}, abstract = {Calculating averages with respect to probability measures on submanifolds is often necessary in various application areas such as molecular dynamics, computational statistical mechanics and Bayesian statistics. In recent years, various numerical schemes have been proposed in the literature to study this problem based on appropriate reversible constrained stochastic dynamics. In this paper we present and analyse a non-reversible generalisation of the projection-based scheme developed by one of the authors [ESAIM: M2AN, 54 (2020), pp. 391-430]. This scheme consists of two steps - starting from a state on the submanifold, we first update the state using a non-reversible stochastic differential equation which takes the state away from the submanifold, and in the second step we project the state back onto the manifold using the long-time limit of a ordinary differential equation. We prove the consistency of this numerical scheme and provide quantitative error estimates for estimators based on finite-time running averages. Furthermore, we present theoretical analysis which shows that this scheme outperforms its reversible counterpart in terms of asymptotic variance. We demonstrate our findings on an illustrative test example.}, language = {en} } @article{LelievreZhang2018, author = {Leli{\`e}vre, Tony and Zhang, Wei}, title = {Pathwise estimates for effective dynamics: the case of nonlinear vectorial reaction coordinates}, series = {Multiscale Modeling and Simulation}, journal = {Multiscale Modeling and Simulation}, number = {17}, doi = {10.1137/18M1186034}, pages = {1019 -- 1051}, year = {2018}, abstract = {Effective dynamics using conditional expectation was proposed in [F. Legoll and T. Leli{\`e}vre, Nonlinearity, 2010] to approximate the essential dynamics of high-dimensional diffusion processes along a given reaction coordinate. The approximation error of the effective dynamics when it is used to approximate the behavior of the original dynamics has been considered in recent years. As a continuation of the previous work [F. Legoll, T. Leli{\`e}vre, and S. Olla, Stoch. Process. Appl, 2017], in this paper we obtain pathwise estimates for effective dynamics when the reaction coordinate function is either nonlinear or vector-valued.}, language = {en} } @article{Zhang2018, author = {Zhang, Wei}, title = {Ergodic SDEs on submanifolds and related numerical sampling schemes}, series = {ESAIM: Mathematical Modelling and Numerical Analysis}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis}, year = {2018}, abstract = {In many applications, it is often necessary to sample the mean value of certain quantity with respect to a probability measure \$\mu\$ on the level set of a smooth function ξ:R^d→R^k, 1≤k