@misc{SchuetteKlusHartmann, author = {Sch{\"u}tte, Christof and Klus, Stefan and Hartmann, Carsten}, title = {Overcoming the Timescale Barrier in Molecular Dynamics: Transfer Operators, Variational Principles, and Machine Learning}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-88637}, abstract = {One of the main challenges in molecular dynamics is overcoming the "timescale barrier", a phrase used to describe that in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, not even on the largest or specifically dedicated supercomputers. This article discusses how to circumvent the timescale barrier by a collection of transfer operator-based techniques that have emerged from dynamical systems theory, numerical mathematics, and machine learning over the last two decades. We will focus on how transfer operators can be used to approximate the dynamical behavior on long timescales, review the introduction of this approach into molecular dynamics, and outline the respective theory as well as the algorithmic development from the early numerics-based methods, via variational reformulations, to modern data-based techniques utilizing and improving concepts from machine learning. Furthermore, its relation to rare event simulation techniques will be explained, revealing a broad equivalence of variational principles for long-time quantities in MD. The article will mainly take a mathematical perspective and will leave the application to real-world molecular systems to the more than 1000 research articles already written on this subject.}, language = {en} } @incollection{HartmannRichter, author = {Hartmann, Carsten and Richter, Lorenz}, title = {Transgressing the Boundaries: Towards a Rigorous Understanding of Deep Learning and Its (Non )Robustness}, series = {AI - Limits and Prospects of Artificial Intelligence}, volume = {4}, booktitle = {AI - Limits and Prospects of Artificial Intelligence}, editor = {Klimczak, Peter and Petersen, Christer}, publisher = {transcript Verlag}, doi = {10.1515/9783839457320}, pages = {43 -- 82}, abstract = {The recent advances in machine learning in various fields of applications can be largely attributed to the rise of deep learning (DL) methods and architectures. Despite being a key technology behind autonomous cars, image processing, speech recognition, etc., a notorious problem remains the lack of theoretical understanding of DL and related interpretability and (adversarial) robustness issues. Understanding the specifics of DL, as compared to, say, other forms of nonlinear regression methods or statistical learning, is interesting from a mathematical perspective, but at the same time it is of crucial importance in practice: treating neural networks as mere black boxes might be sufficient in certain cases, but many applications require waterproof performance guarantees and a deeper understanding of what could go wrong and why it could go wrong. It is probably fair to say that, despite being mathematically well founded as a method to approximate complicated functions, DL is mostly still more like modern alchemy that is firmly in the hands of engineers and computer scientists. Nevertheless, it is evident that certain specifics of DL that could explain its success in applications demands systematic mathematical approaches. In this work, we review robustness issues of DL and particularly bridge concerns and attempts from approximation theory to statistical learning theory. Further, we review Bayesian Deep Learning as a means for uncertainty quantification and rigorous explainability.}, language = {en} } @misc{HartmannBanischSarichetal., author = {Hartmann, Carsten and Banisch, Ralf and Sarich, Marco and Badowski, Thomas and Sch{\"u}tte, Christof}, title = {Characterization of Rare Events in Molecular Dynamics}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42410}, abstract = {A good deal of molecular dynamics simulations aims at predicting and quantifying rare events, such as the folding of a protein or a phase transition. Simulating rare events is often prohibitive, especially if the equations of motion are high-dimensional, as is the case in molecular dynamics. Various algorithms have been proposed for efficiently computing mean first passage times, transition rates or reaction pathways. This article surveys and discusses recent developments in the field of rare event simulation and outlines a new approach that combines ideas from optimal control and statistical mechanics. The optimal control approach described in detail resembles the use of Jarzynski's equality for free energy calculations, but with an optimized protocol that speeds up the sampling, while (theoretically) giving variance-free estimators of the rare events statistics. We illustrate the new approach with two numerical examples and discuss its relation to existing methods.}, language = {en} } @misc{ZhangHartmannSchuette, author = {Zhang, Wei and Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {Effective Dynamics Along Given Reaction Coordinates, and Reaction Rate Theory}, issn = {1438-0064}, doi = {10.1039/C6FD00147E}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-59706}, abstract = {In molecular dynamics and related fields one considers dynamical descriptions of complex systems in full (atomic) detail. In order to reduce the overwhelming complexity of realistic systems (high dimension, large timescale spread, limited computational resources) the projection of the full dynamics onto some reaction coordinates is examined in order to extract statistical information like free energies or reaction rates. In this context, the effective dynamics that is induced by the full dynamics on the reaction coordinate space has attracted considerable attention in the literature. In this article, we contribute to this discussion: We first show that if we start with an ergodic diffusion processes whose invariant measure is unique then these properties are inherited by the effective dynamics. Then, we give equations for the effective dynamics, discuss whether the dominant timescales and reaction rates inferred from the effective dynamics are accurate approximations of such quantities for the full dynamics, and compare our findings to results from approaches like Zwanzig-Mori, averaging, or homogenization. Finally, by discussing the algorithmic realization of the effective dynamics, we demonstrate that recent algorithmic techniques like the "equation-free" approach and the "heterogeneous multiscale method" can be seen as special cases of our approach.}, language = {en} } @misc{HartmannRichterSchuetteetal., author = {Hartmann, Carsten and Richter, Lorenz and Sch{\"u}tte, Christof and Zhang, Wei}, title = {Variational characterization of free energy: Theory and algorithms}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-65045}, abstract = {The article surveys and extends variational formulations of the thermodynamic free energy and discusses their information-theoretic content from the perspective of mathematical statistics. We revisit the well-known Jarzynski equality for nonequilibrium free energy sampling within the framework of importance sampling and Girsanov change-of-measure transformations. The implications of the different variational formulations for designing efficient stochastic optimization and nonequilibrium simulation algorithms for computing free energies are discussed and illustrated.}, language = {en} } @misc{ZhangWangHartmannetal., author = {Zhang, Wei and Wang, Han and Hartmann, Carsten and Weber, Marcus and Sch{\"u}tte, Christof}, title = {Applications of the cross-entropy method to importance sampling and optimal control of diffusions}, issn = {1438-0064}, doi = {10.1137/14096493X}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-49720}, abstract = {We study the cross-entropy method for diffusions. One of the results is a versatile cross-entropy algorithm that can be used to design efficient importance sampling strategies for rare events or to solve optimal control problems. The approach is based on the minimization of a suitable cross-entropy functional, with a parametric family of exponentially tilted probability distributions. We illustrate the new algorithm with several numerical examples and discuss algorithmic issues and possible extensions of the method.}, language = {en} } @article{SchuetteKlusHartmann, author = {Sch{\"u}tte, Christof and Klus, Stefan and Hartmann, Carsten}, title = {Overcoming the Timescale Barrier in Molecular Dynamics: Transfer Operators, Variational Principles, and Machine Learning}, series = {Acta Numerica}, volume = {32}, journal = {Acta Numerica}, doi = {10.1017/S0962492923000016}, pages = {517 -- 673}, abstract = {One of the main challenges in molecular dynamics is overcoming the 'timescale barrier': in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, even on the largest or specifically dedicated supercomputers. This article discusses how to circumvent the timescale barrier by a collection of transfer operator-based techniques that have emerged from dynamical systems theory, numerical mathematics and machine learning over the last two decades. We will focus on how transfer operators can be used to approximate the dynamical behaviour on long timescales, review the introduction of this approach into molecular dynamics, and outline the respective theory, as well as the algorithmic development, from the early numerics-based methods, via variational reformulations, to modern data-based techniques utilizing and improving concepts from machine learning. Furthermore, its relation to rare event simulation techniques will be explained, revealing a broad equivalence of variational principles for long-time quantities in molecular dynamics. The article will mainly take a mathematical perspective and will leave the application to real-world molecular systems to the more than 1000 research articles already written on this subject.}, language = {en} } @misc{SarichBanischHartmannetal., author = {Sarich, Marco and Banisch, Ralf and Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {Markov State Models for Rare Events in Molecular Dynamics}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42420}, abstract = {Rare but important transition events between long lived states are a key feature of many molecular systems. In many cases the computation of rare event statistics by direct molecular dynamics (MD) simulations is infeasible even on the most powerful computers because of the immensely long simulation timescales needed. Recently a technique for spatial discretization of the molecular state space designed to help overcome such problems, so-called Markov State Models (MSMs), has attracted a lot of attention. We review the theoretical background and algorithmic realization of MSMs and illustrate their use by some numerical examples. Furthermore we introduce a novel approach to using MSMs for the efficient solution of optimal control problems that appear in applications where one desires to optimize molecular properties by means of external controls.}, language = {en} } @article{LatorreMetznerHartmannetal.2011, author = {Latorre, J. and Metzner, Ph. and Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {A Structure-preserving numerical discretization of reversible diffusions}, series = {Commun. Math. Sci.}, volume = {9}, journal = {Commun. Math. Sci.}, number = {4}, pages = {1051 -- 1072}, year = {2011}, language = {en} } @article{ZhangHartmannSchuette, author = {Zhang, Wei and Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {Effective dynamics along given reaction coordinates, and reaction rate theory}, series = {Faraday Discussions}, journal = {Faraday Discussions}, number = {195}, doi = {10.1039/C6FD00147E}, pages = {365 -- 394}, language = {en} } @misc{WangHartmannSchuette, author = {Wang, Han and Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {Linear response theory and optimal control for a molecular system under nonequilibrium conditions}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-18944}, abstract = {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.}, language = {en} } @article{ZhangHartmannvonKleist, author = {Zhang, Wei and Hartmann, Carsten and von Kleist, Max}, title = {Optimal control of Markov jump processes: Asymptotic analysis, algorithms and applications to the modeling of chemical reaction systems}, series = {Communications in Mathematical Sciences}, journal = {Communications in Mathematical Sciences}, doi = {10.4310/CMS.2018.v16.n2.a1}, pages = {293 -- 331}, abstract = {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.}, language = {en} } @article{DonatiHartmannKeller, author = {Donati, Luca and Hartmann, Carsten and Keller, Bettina G.}, title = {Girsanov reweighting for path ensembles and Markov state models}, series = {The Journal of Chemical Physics}, volume = {146}, journal = {The Journal of Chemical Physics}, number = {24}, doi = {10.1063/1.4989474}, pages = {244112}, abstract = {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.}, language = {en} } @article{HartmannSchuetteZhang2018, author = {Hartmann, Carsten and Sch{\"u}tte, Christof and Zhang, Wei}, title = {Jarzynski's equality, fluctuation theorems, and variance reduction: Mathematical analysis and numerical algorithms}, series = {Journal of Statistical Physics}, volume = {175}, journal = {Journal of Statistical Physics}, number = {6}, doi = {10.1007/s10955-019-02286-4}, pages = {1214 -- 1261}, year = {2018}, abstract = {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.}, language = {en} } @article{WangHartmannSchuetteetal.2013, author = {Wang, Han and Hartmann, Carsten and Sch{\"u}tte, Christof and Site, Luigi Delle}, title = {Grand-canonical-like molecular-dynamics simulations by using an adaptive-resolution technique}, series = {Phys. Rev. X}, volume = {3}, journal = {Phys. Rev. X}, doi = {10.1103/PhysRevX.3.011018}, pages = {011018}, year = {2013}, language = {en} } @article{HorenkoHartmannSchuetteetal.2007, author = {Horenko, Illia and Hartmann, Carsten and Sch{\"u}tte, Christof and No{\´e}, Frank}, title = {Data-based Parameter Estimation of Generalized Multidimensional Langevin Processes}, series = {Phys. Rev. E}, volume = {76}, journal = {Phys. Rev. E}, number = {01}, doi = {10.1103/PhysRevE.76.016706}, pages = {016706}, year = {2007}, language = {en} } @article{HartmannSchuetteKalibaevaetal.2009, author = {Hartmann, Carsten and Sch{\"u}tte, Christof and Kalibaeva, G. and Pierro, M. and Ciccotti, Giovanni}, title = {Fast Simulation of Polymer Chains}, series = {J. Chem. Phys.}, volume = {130}, journal = {J. Chem. Phys.}, doi = {10.1063/1.3110603}, pages = {144101}, year = {2009}, language = {en} } @article{HartmannSchuette2012, author = {Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {Efficient rare event simulation by optimal nonequilibrium forcing}, series = {J. Stat. Mech. Theor. Exp.}, volume = {2012}, journal = {J. Stat. Mech. Theor. Exp.}, doi = {10.1088/1742-5468/2012/11/P11004}, pages = {P11004}, year = {2012}, language = {en} } @article{HartmannSchuetteCiccotti2010, author = {Hartmann, Carsten and Sch{\"u}tte, Christof and Ciccotti, Giovanni}, title = {On the linear response of mechanical systems with constraints}, series = {J. Chem. Phys.}, volume = {132}, journal = {J. Chem. Phys.}, number = {11}, doi = {10.1063/1.3354126}, pages = {111103}, year = {2010}, language = {en} } @article{HartmannVulcanovSchuette2010, author = {Hartmann, Carsten and Vulcanov, V.-M. and Sch{\"u}tte, Christof}, title = {Balanced Truncation of Second Order Systems}, series = {Multiscale Model. Simul.}, volume = {8}, journal = {Multiscale Model. Simul.}, number = {4}, doi = {10.1137/080732717}, pages = {1348 -- 1367}, year = {2010}, language = {en} } @article{HartmannSchuette2005, author = {Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {A Geometric Approach to Constrained Molecular Dynamics and Free Energy}, series = {Comm. Math. Sci.}, volume = {3}, journal = {Comm. Math. Sci.}, number = {1}, pages = {1 -- 20}, year = {2005}, language = {en} } @article{SchuetteNoeMeerbachetal.2009, author = {Sch{\"u}tte, Christof and No{\´e}, Frank and Meerbach, E. and Metzner, Ph. and Hartmann, Carsten}, title = {Conformation Dynamics}, series = {Proceedings of the 6th International Congress on Industrial and Applied Mathematics, I. Jeltsch and G. Wanner (eds.),}, journal = {Proceedings of the 6th International Congress on Industrial and Applied Mathematics, I. Jeltsch and G. Wanner (eds.),}, publisher = {EMS publishing house}, doi = {10.4171/056-1/15}, pages = {297 -- 335}, year = {2009}, language = {en} } @article{SchuetteWalterHartmannetal.2004, author = {Sch{\"u}tte, Christof and Walter, J. and Hartmann, Carsten and Huisinga, Wilhelm}, title = {An Averaging Principle for Fast Degrees of Freedom Exhibiting Long-Term Correlations}, series = {Multiscale Model. Simul.}, volume = {2}, journal = {Multiscale Model. Simul.}, number = {3}, doi = {10.1137/030600308}, pages = {501 -- 526}, year = {2004}, language = {en} } @article{SchuetteWinkelmannHartmann2012, author = {Sch{\"u}tte, Christof and Winkelmann, Stefanie and Hartmann, Carsten}, title = {Optimal control of molecular dynamics using Markov state models}, series = {Math. Program. (Series B)}, volume = {134}, journal = {Math. Program. (Series B)}, number = {1}, doi = {10.1007/s10107-012-0547-6}, pages = {259 -- 282}, year = {2012}, language = {en} } @article{HartmannSchuette2008, author = {Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {Balancing of partially-observed stochastic differential equations}, series = {47th IEEE Conference on Decision and Control}, journal = {47th IEEE Conference on Decision and Control}, doi = {10.1109/CDC.2008.4739161}, pages = {4867 -- 4872}, year = {2008}, language = {en} } @article{LatorreHartmannSchuette2010, author = {Latorre, J. and Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {Free energy computation by controlled Langevin processes}, series = {Procedia Computer Science}, volume = {1}, journal = {Procedia Computer Science}, number = {1}, doi = {10.1016/j.procs.2010.04.179}, pages = {1591 -- 1600}, year = {2010}, language = {en} } @article{SchaeferBungHartmannSchmidtetal.2011, author = {Sch{\"a}fer-Bung, B. and Hartmann, Carsten and Schmidt, Burkhard and Sch{\"u}tte, Christof}, title = {Dimension reduction by balanced truncation}, series = {J. Chem. Phys.}, volume = {135}, journal = {J. Chem. Phys.}, number = {1}, pages = {014112}, year = {2011}, language = {en} } @article{HartmannSchuette2007, author = {Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {Comment on Two Distinct Notions of Free Energy}, series = {Physica D}, volume = {228}, journal = {Physica D}, number = {1}, doi = {10.1016/j.physd.2007.02.006}, pages = {59 -- 63}, year = {2007}, language = {en} } @article{HartmannSchuette2005, author = {Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {A Constrained Hybrid Monte-Carlo Algorithm and the Problem of Calculating the Free Energy in Several Variables}, series = {Z. Angew. Math. Mech.}, volume = {85}, journal = {Z. Angew. Math. Mech.}, number = {10}, doi = {10.1002/zamm.200410218}, pages = {700 -- 710}, year = {2005}, language = {en} } @article{HartmannSchuetteWeberetal., author = {Hartmann, Carsten and Sch{\"u}tte, Christof and Weber, Marcus and Zhang, Wei}, title = {Importance sampling in path space for diffusion processes with slow-fast variables}, series = {Probability Theory and Related Fields}, journal = {Probability Theory and Related Fields}, doi = {10.1007/s00440-017-0755-3}, pages = {1 -- 52}, abstract = {Importance sampling is a widely used technique to reduce the variance of a Monte Carlo estimator by an appropriate change of measure. In this work, we study importance sampling in the framework of diffusion process and consider the change of measure which is realized by adding a control force to the original dynamics. For certain exponential type expectation, the corresponding control force of the optimal change of measure leads to a zero-variance estimator and is related to the solution of a Hamilton-Jacobi-Bellmann equation. We focus on certain diffusions with both slow and fast variables, and the main result is that we obtain an upper bound of the relative error for the importance sampling estimators with control obtained from the limiting dynamics. We demonstrate our approximation strategy with an illustrative numerical example.}, language = {en} } @article{HartmannSchuetteZhang, author = {Hartmann, Carsten and Sch{\"u}tte, Christof and Zhang, Wei}, title = {Model reduction algorithms for optimal control and importance sampling of diffusions}, series = {Nonlinearity}, volume = {29}, journal = {Nonlinearity}, number = {8}, doi = {10.1088/0951-7715/29/8/2298}, pages = {2298 -- 2326}, abstract = {We propose numerical algorithms for solving optimal control and importance sampling problems based on simplified models. The algorithms combine model reduction techniques for multiscale diffusions and stochastic optimization tools, with the aim of reducing the original, possibly high-dimensional problem to a lower dimensional representation of the dynamics, in which only a few relevant degrees of freedom are controlled or biased. Specifically, we study situations in which either a reaction coordinate onto which the dynamics can be projected is known, or situations in which the dynamics shows strongly localized behavior in the small noise regime. No explicit assumptions about small parameters or scale separation have to be made. We illustrate the approach with simple, but paradigmatic numerical examples.}, language = {en} } @article{ZhangWangHartmannetal., author = {Zhang, Wei and Wang, Han and Hartmann, Carsten and Weber, Marcus and Sch{\"u}tte, Christof}, title = {Applications of the cross-entropy method to importance sampling and optimal control of diffusions}, series = {Siam Journal on Scientific Computing}, volume = {36}, journal = {Siam Journal on Scientific Computing}, number = {6}, doi = {10.1137/14096493X}, pages = {A2654 -- A2672}, language = {en} } @article{HartmannBanischSarichetal., author = {Hartmann, Carsten and Banisch, Ralf and Sarich, Marco and Badowski, Thomas and Sch{\"u}tte, Christof}, title = {Characterization of Rare Events in Molecular Dynamics}, series = {Entropy (Special Issue)}, volume = {16}, journal = {Entropy (Special Issue)}, number = {1}, doi = {10.3390/e16010350}, pages = {350 -- 376}, language = {en} } @article{WangHartmannSchuette, author = {Wang, Han and Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {Linear response theory and optimal control for a molecular system under nonequilibrium conditions}, series = {Molecular Physics}, volume = {111}, journal = {Molecular Physics}, doi = {10.1080/00268976.2013.844370}, pages = {3555 -- 3564}, language = {en} } @article{SarichBanischHartmannetal., author = {Sarich, Marco and Banisch, Ralf and Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {Markov State Models for Rare Events in Molecular Dynamics}, series = {Entropy (Special Issue)}, volume = {16}, journal = {Entropy (Special Issue)}, number = {1}, doi = {10.3390/e16010258}, pages = {258 -- 286}, language = {en} } @article{HartmannRichterSchuetteetal., author = {Hartmann, Carsten and Richter, Lorenz and Sch{\"u}tte, Christof and Zhang, Wei}, title = {Variational characterization of free energy: theory and algorithms}, series = {Entropy}, volume = {19}, journal = {Entropy}, number = {11}, doi = {10.3390/e19110626}, pages = {626}, language = {en} } @article{HartmannRichter, author = {Hartmann, Carsten and Richter, Lorenz}, title = {Nonasymptotic bounds for suboptimal importance sampling}, series = {SIAM/ASA Journal on Uncertainty Quantification}, volume = {12}, journal = {SIAM/ASA Journal on Uncertainty Quantification}, number = {2}, doi = {10.1137/21M1427760}, pages = {309 -- 346}, abstract = {Importance sampling is a popular variance reduction method for Monte Carlo estimation, where an evident question is how to design good proposal distributions. While in most cases optimal (zero-variance) estimators are theoretically possible, in practice only suboptimal proposal distributions are available and it can often be observed numerically that those can reduce statistical performance significantly, leading to large relative errors and therefore counteracting the original intention. Previous analysis on importance sampling has often focused on asymptotic arguments that work well in a large deviations regime. In this article, we provide lower and upper bounds on the relative error in a nonasymptotic setting. They depend on the deviation of the actual proposal from optimality, and we thus identify potential robustness issues that importance sampling may have, especially in high dimensions. We particularly focus on path sampling problems for diffusion processes with nonvanishing noise, for which generating good proposals comes with additional technical challenges. We provide numerous numerical examples that support our findings and demonstrate the applicability of the derived bounds.}, language = {en} }