TY - GEN A1 - Wang, Han A1 - Hartmann, Carsten A1 - Schütte, Christof T1 - Linear response theory and optimal control for a molecular system under nonequilibrium conditions N2 - In this paper, we propose a straightforward generalization of linear response theory to systems in nonequilibrium that are subject to nonequilibrium driving. We briefly revisit the standard linear response result for equilibrium systems, where we consider Langevin dynamics as a special case, and then give an alternative derivation using a change-of-measure argument that does not rely on any stationarity or reversibility assumption. This procedure moreover easily enables us to calculate the second order correction to the linear response formula (which may or may not be useful in practice). Furthermore, we outline how the novel nonequilibirum linear response formula can be used to compute optimal controls of molecular systems for cases in which one wants to steer the system to maximize a certain target expectation value. We illustrate our approach with simple numerical examples. T3 - ZIB-Report - 13-33 KW - nonequilibrium molecular dynamics KW - linear response KW - Girsanov transformation KW - stochastic control Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-18944 ER - TY - JOUR A1 - Zhang, Wei A1 - Hartmann, Carsten A1 - von Kleist, Max T1 - Optimal control of Markov jump processes: Asymptotic analysis, algorithms and applications to the modeling of chemical reaction systems JF - Communications in Mathematical Sciences N2 - Markov jump processes are widely used to model natural and engineered processes. In the context of biological or chemical applications one typically refers to the chemical master equation (CME), which models the evolution of the probability mass of any copy-number combination of the interacting particles. When many interacting particles (“species”) are considered, the complexity of the CME quickly increases, making direct numerical simulations impossible. This is even more problematic when one aims at controlling the Markov jump processes defined by the CME. In this work, we study both open loop and feedback optimal control problems of the Markov jump processes in the case that the controls can only be switched at fixed control stages. Based on Kurtz’s limit theorems, we prove the convergence of the respective control value functions of the underlying Markov decision problem as the copy numbers of the species go to infinity. In the case of the optimal control problem on a finite time-horizon, we propose a hybrid control policy algorithm to overcome the difficulties due to the curse of dimensionality when the copy number of the involved species is large. Two numerical examples demonstrate the suitability of both the analysis and the proposed algorithms. Y1 - 2018 U6 - https://doi.org/10.4310/CMS.2018.v16.n2.a1 SP - 293 EP - 331 ER - TY - JOUR A1 - Donati, Luca A1 - Hartmann, Carsten A1 - Keller, Bettina G. T1 - Girsanov reweighting for path ensembles and Markov state models JF - The Journal of Chemical Physics N2 - The sensitivity of molecular dynamics on changes in the potential energy function plays an important role in understanding the dynamics and function of complex molecules. We present a method to obtain path ensemble averages of a perturbed dynamics from a set of paths generated by a reference dynamics. It is based on the concept of path probability measure and the Girsanov theorem, a result from stochastic analysis to estimate a change of measure of a path ensemble. Since Markov state models (MSMs) of the molecular dynamics can be formulated as a combined phase-space and path ensemble average, the method can be extended to reweight MSMs by combining it with a reweighting of the Boltzmann distribution. We demonstrate how to efficiently implement the Girsanov reweighting in a molecular dynamics simulation program by calculating parts of the reweighting factor “on the fly” during the simulation, and we benchmark the method on test systems ranging from a two-dimensional diffusion process and an artificial many-body system to alanine dipeptide and valine dipeptide in implicit and explicit water. The method can be used to study the sensitivity of molecular dynamics on external perturbations as well as to reweight trajectories generated by enhanced sampling schemes to the original dynamics. Y1 - 2017 U6 - https://doi.org/10.1063/1.4989474 VL - 146 IS - 24 SP - 244112 ER - TY - JOUR A1 - Hartmann, Carsten A1 - Schütte, Christof A1 - Zhang, Wei T1 - Jarzynski's equality, fluctuation theorems, and variance reduction: Mathematical analysis and numerical algorithms JF - Journal of Statistical Physics N2 - In this paper, we study Jarzynski's equality and fluctuation theorems for diffusion processes. While some of the results considered in the current work are known in the (mainly physics) literature, we review and generalize these nonequilibrium theorems using mathematical arguments, therefore enabling further investigations in the mathematical community. On the numerical side, variance reduction approaches such as importance sampling method are studied in order to compute free energy differences based on Jarzynski's equality. Y1 - 2018 U6 - https://doi.org/10.1007/s10955-019-02286-4 VL - 175 IS - 6 SP - 1214 EP - 1261 ER - TY - JOUR A1 - Wang, Han A1 - Hartmann, Carsten A1 - Schütte, Christof A1 - Site, Luigi Delle T1 - Grand-canonical-like molecular-dynamics simulations by using an adaptive-resolution technique JF - Phys. Rev. X Y1 - 2013 U6 - https://doi.org/10.1103/PhysRevX.3.011018 VL - 3 SP - 011018 ER - TY - JOUR A1 - Horenko, Illia A1 - Hartmann, Carsten A1 - Schütte, Christof A1 - Noé, Frank T1 - Data-based Parameter Estimation of Generalized Multidimensional Langevin Processes JF - Phys. Rev. E Y1 - 2007 UR - http://publications.imp.fu-berlin.de/31/ U6 - https://doi.org/10.1103/PhysRevE.76.016706 VL - 76 IS - 01 SP - 016706 ER - TY - JOUR A1 - Hartmann, Carsten A1 - Schütte, Christof A1 - Kalibaeva, G. A1 - Pierro, M. A1 - Ciccotti, Giovanni T1 - Fast Simulation of Polymer Chains JF - J. Chem. Phys. Y1 - 2009 UR - http://publications.imp.fu-berlin.de/18/ U6 - https://doi.org/10.1063/1.3110603 VL - 130 SP - 144101 ER - TY - JOUR A1 - Hartmann, Carsten A1 - Schütte, Christof T1 - Efficient rare event simulation by optimal nonequilibrium forcing JF - J. Stat. Mech. Theor. Exp. Y1 - 2012 UR - http://publications.imp.fu-berlin.de/1153/ U6 - https://doi.org/10.1088/1742-5468/2012/11/P11004 VL - 2012 SP - P11004 ER - TY - JOUR A1 - Hartmann, Carsten A1 - Schütte, Christof A1 - Ciccotti, Giovanni T1 - On the linear response of mechanical systems with constraints JF - J. Chem. Phys. Y1 - 2010 UR - http://publications.imp.fu-berlin.de/794/ U6 - https://doi.org/10.1063/1.3354126 VL - 132 IS - 11 SP - 111103 ER - TY - JOUR A1 - Hartmann, Carsten A1 - Vulcanov, V.-M. A1 - Schütte, Christof T1 - Balanced Truncation of Second Order Systems JF - Multiscale Model. Simul. Y1 - 2010 UR - http://publications.imp.fu-berlin.de/28/ U6 - https://doi.org/10.1137/080732717 VL - 8 IS - 4 SP - 1348 EP - 1367 ER -