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  • 1.
    Bierkens, Joris
    et al.
    TU Delft.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Schlottke, Mikola
    Eindhoven University of Technology.
    Large deviations for the empirical measure of the zig-zag process2021In: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737, Vol. 31, no 6Article in journal (Refereed)
    Abstract [en]

    The zig-zag process is a piecewise deterministic Markov process in position and velocity space. The process can be designed to have an arbitrary Gibbs type marginal probability density for its position coordinate, which makes it suitable for Monte Carlo simulation of continuous probability distributions. An important question in assessing the efficiency of this method is how fast the empirical measure converges to the stationary distribution of the process. In this paper we provide a partial answer to this question by characterizing the large deviations of the empirical measure from the stationary distribution. Based on the Feng-Kurtz approach, we develop an abstract framework aimed at encompassing piecewise deterministic Markov processes in position-velocity space. We derive explicit conditions for the zig-zag process to allow the Donsker-Varadhan variational formulation of the rate function, both for a compact setting (the torus) and one-dimensional Euclidean space. Finally we derive an explicit expression for the Donsker-Varadhan functional for the case of a compact state space and use this form of the rate function to address a key question concerning the optimal choice of the switching rate of the zig-zag process.

  • 2.
    Bierkens, Joris
    et al.
    Vrije Universiteit.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Schlottke, Mikola C.
    Eindhoven university of technology.
    Large deviations for the empirical measure of the zig-zag processManuscript (preprint) (Other academic)
    Abstract [en]

    The zig-zag process is a piecewise deterministic Markov process in position and velocity space. The process can be designed to have an arbitrary Gibbs type marginal probability density for its position coordinate, which makes it suitable for Monte Carlo simulation of continuous probability distributions. An important question in assessing the efficiency of this method is how fast the empirical measure converges to the stationary distribution of the process. In this paper we provide a partial answer to this question by characterizing the large deviations of the empirical measure from the stationary distribution. Based on the Feng-Kurtz approach, we develop an abstract framework aimed at encompassing piecewise deterministic Markov processes in position-velocity space. We derive explicit conditions for the zig-zag process to allow the Donsker-Varadhan variational formulation of the rate function, both for a compact setting (the torus) and one-dimensional Euclidean space. Finally we derive an explicit expression for the Donsker-Varadhan functional for the case of a compact state space and use this form of the rate function to address a key question concerning the optimal choice of the switching rate of the zig-zag process.

  • 3.
    Budhiraja, Amarjit
    et al.
    Univ N Carolina, Dept Stat & Operat Res, Chapel Hill, NC 27599 USA..
    Dupuis, Paul
    Brown Univ, Div Appl Math, Providence, RI 02912 USA..
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Wu, Guo-Jhen
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Quasistationary distributions and ergodic control problems2022In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 145, p. 143-164Article in journal (Refereed)
    Abstract [en]

    We introduce and study the basic properties of two ergodic stochastic control problems associated with the quasistationary distribution (QSD) of a diffusion process X relative to a bounded domain. The two problems are in some sense dual, with one defined in terms of the generator associated with X and the other in terms of its adjoint. Besides proving wellposedness of the associated Hamilton-Jacobi- Bellman equations, we describe how they can be used to characterize important properties of the QSD. Of particular note is that the QSD itself can be identified, up to normalization, in terms of the cost potential of the control problem associated with the adjoint.

  • 4.
    Budhiraja, Amarjit
    et al.
    University of North Carolina at Chapel Hill United States.
    Nyquist, Pierre
    Brown University, United States.
    Large deviations for multidimensional state-dependent shot noise processes2015In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 52, no 4, p. 1097-1114Article in journal (Refereed)
    Abstract [en]

    Shot-noise processes are used in applied probability to model a variety of physical systems in, for example, teletraffic theory, insurance and risk theory, and in the engineering sciences. In this paper we prove a large deviation principle for the sample-paths of a general class of multidimensional state-dependent Poisson shot-noise processes. The result covers previously known large deviation results for one-dimensional state-independent shot-noise processes with light tails. We use the weak convergence approach to large deviations, which reduces the proof to establishing the appropriate convergence of certain controlled versions of the original processes together with relevant results on existence and uniqueness.

  • 5.
    Djehiche, Boualem
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Importance sampling for a Markovian intensity model with applications to credit riskManuscript (preprint) (Other academic)
    Abstract [en]

    This paper considers importance sampling for estimation of rare-event probabilities in a Markovian intensity model for credit risk. The main contribution is the design of efficient importance sampling algorithms using subsolutions of a certain Hamilton-Jacobi equation. For certain instances of the credit risk model the proposed algorithm is proved to be asymptotically optimal. The computational gain compared to standard Monte Carlo is illustrated by numerical experiments.

  • 6.
    Djehiche, Boualem
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Importance Sampling for a Simple Markovian Intensity Model Using Subsolutions2022In: ACM Transactions on Modeling and Computer Simulation, ISSN 1049-3301, E-ISSN 1558-1195, Vol. 32, no 2, p. 1-25, article id 14Article in journal (Refereed)
    Abstract [en]

    This article considers importance sampling for estimation of rare-event probabilities in a specific collection of Markovian jump processes used for, e.g., modeling of credit risk. Previous attempts at designing importance sampling algorithms have resulted in poor performance and the main contribution of the article is the design of efficient importance sampling algorithms using subsolutions. The dynamics of the jump processes cause the corresponding Hamilton-Jacobi equations to have an intricate state-dependence, which makes the design of efficient algorithms difficult. We provide theoretical results that quantify the performance of importance sampling algorithms in general and construct asymptotically optimal algorithms for some examples. The computational gain compared to standard Monte Carlo is illustrated by numerical examples.

  • 7.
    Djehiche, Boualem
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Min-max representations of viscosity solutions of Hamilton-Jacobi equations and applications in rare-event simulationManuscript (preprint) (Other academic)
    Abstract [en]

    In this paper a duality relation between the Mañé potential and Mather's action functional is derived in the context of convex and state-dependent Hamiltonians. The duality relation is used to obtain min-max representations of viscosity solutions of first order Hamilton-Jacobi equations. These min-max representations naturally suggest classes of subsolutions of Hamilton-Jacobi equations that arise in the theory of large deviations. The subsolutions, in turn, are good candidates for designing efficient rare-event simulation algorithms.

  • 8. Doll, Jim
    et al.
    Dupuis, Paul
    Nyquist, Pierre
    A large deviation analysis of certain qualitative properties of parallel tempering and infinite swapping algorithms2018In: Applied mathematics and optimization, ISSN 0095-4616, E-ISSN 1432-0606, Vol. 78, no 1, p. 103-144Article in journal (Refereed)
    Abstract [en]

    Parallel tempering, or replica exchange, is a popular method for simulating complex systems. The idea is to run parallel simulations at different temperatures, and at a given swap rate exchange configurations between the parallel simulations. From the perspective of large deviations it is optimal to let the swap rate tend to infinity and it is possible to construct a corresponding simulation scheme, known as infinite swapping. In this paper we propose a novel use of large deviations for empirical measures for a more detailed analysis of the infinite swapping limit in the setting of continuous time jump Markov processes. Using the large deviations rate function and associated stochastic control problems we consider a diagnostic based on temperature assignments, which can be easily computed during a simulation. We show that the convergence of this diagnostic to its a priori known limit is a necessary condition for the convergence of infinite swapping. The rate function is also used to investigate the impact of asymmetries in the underlying potential landscape, and where in the state space poor sampling is most likely to occur.

  • 9. Doll, Jim
    et al.
    Dupuis, Paul
    Nyquist, Pierre
    Thermodynamic integration methods, infinite swapping and the calculation of generalized averages2017In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 146Article in journal (Refereed)
    Abstract [en]

    In the present paper we examine the risk-sensitive and sampling issues associated with the problem of calculating generalized averages. By combining thermodynamic integration and Stationary Phase Monte Carlo techniques, we develop an approach for such problems and explore its utility for a prototypical class of applications.

  • 10.
    Eriksson, Olivia
    et al.
    KTH, School of Electrical Engineering and Computer Science (EECS), Computer Science, Computational Science and Technology (CST). KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Kramer, Andrei
    KTH, School of Electrical Engineering and Computer Science (EECS), Computer Science, Computational Science and Technology (CST). KTH, Centres, Science for Life Laboratory, SciLifeLab.
    Milinanni, Federica
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics. KTH, Centres, Science for Life Laboratory, SciLifeLab.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Sensitivity Approximation by the Peano-Baker SeriesManuscript (preprint) (Other academic)
    Abstract [en]

    In this paper we develop a new method for numerically approximating sensitivitiesin parameter-dependent ordinary differential equations (ODEs). Our approach,intended for situations where the standard forward and adjoint sensitivity analysisbecome too computationally costly for practical purposes, is based on the PeanoBaker series from control theory. We give a representation, using this series, for thesensitivity matrix S of an ODE system and use the representation to construct anumerical method for approximating S. We prove that, under standard regularityassumptions, the error of our method scales as O(∆t2max), where ∆tmax is the largesttime step used when numerically solving the ODE. We illustrate the performanceof the method in several numerical experiments, taken from both the systemsbiology setting and more classical dynamical systems. The experiments show thesought-after improvement in running time of our method compared to the forwardsensitivity approach. For example, in experiments involving a random linear system,the forward approach requires roughly √n longer computational time, where n isthe dimension of the parameter space, than our proposed method.

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  • 11.
    Gavish, Nir
    et al.
    Technion.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Peletier, Mark
    Eindhoven University of Technology.
    Large deviations and gradient flows for the Brownian one-dimensional hard-rod systemManuscript (preprint) (Other academic)
    Abstract [en]

    We study a system of hard rods of finite size in one space dimension, which move by Brownian noise while avoiding overlap. We consider a scaling in which the number of particles tends to infinity while the volume fraction of the rods remains constant; in this limit the empirical measure of the rod positions converges almost surely to a deterministic limit evolution. We prove a large-deviation principle on path space for the empirical measure, by exploiting a one-to-one mapping between the hard-rod system and a system of non-interacting particles on a shorter domain. The large-deviation principle naturally identifies a gradient-flow structure for the limit evolution, with clear interpretations for both the driving functional (an `entropy') and the dissipation, which in this case is the Wasserstein dissipation. This study is inspired by recent developments in the continuum modelling of multiple-species interacting particle systems with finite-size effects; for such systems many different modelling choices appear in the literature, raising the question how one can understand such choices in terms of more microscopic models. The results of this paper give a clear answer to this question, albeit for the simpler onedimensional hard-rod system. For this specific system this result provides a clear understanding of the value and interpretation of different modelling choices, while giving hints for more general systems.

  • 12.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindhe, Adam
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    On the projected Aubry set of the rate function associated with large deviations for stochastic approximationsManuscript (preprint) (Other academic)
    Abstract [en]

    In this article, we look at the problem of minimizing an action potential that arises from large deviation theory for stochastic approximations. The solutions to the minimising problem satisfy, in the sense of a viscosity solution, a Hamilton-Jacobi equation. From weak KAM theory, we know that these viscosity solutions are characterised by the projected Aubryset. The main result of this paper is that, for a specific rate function corresponding to the astochastic approximation algorithm, we prove that the projected Aubry set is equal to the forward limit set to the limit ODE.

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  • 13.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindhe, Adam
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Wu, Guo-Jhen
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    A weak convergence approach to large deviations for stochastic approximationsManuscript (preprint) (Other academic)
    Abstract [en]

    Large deviations for stochastic approximations is a well-studied field that yields convergence properties for many useful algorithms in statistics, machine learning and statistical physics. In this article, we prove, under certain assumptions, a large deviation principle for a stochastic approximation with state-dependent Markovian noise and with decreasing step size. Common algorithms that satisfy these conditions include stochastic gradient descent, persistent contrastive divergence and the Wang-Landau algorithm. The proof is based don't he weak convergence approach to the theory of large deviations and uses a representation formula to rewrite the problem into a stochastic control problem. The resulting rate function is an action potential over a local rate function that is the Fenchel-Legendre transform of a limiting Hamiltonian.

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  • 14.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Large deviations for weighted empirical measures arising in importance sampling2016In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 126, no 1Article in journal (Refereed)
    Abstract [en]

    Importance sampling is a popular method for efficient computation of various properties of a distribution such as probabilities, expectations, quantiles etc. The output of an importance sampling algorithm can be represented as a weighted empirical measure, where the weights are given by the likelihood ratio between the original distribution and the sampling distribution. In this paper the efficiency of an importance sampling algorithm is studied by means of large deviations for the weighted empirical measure. The main result, which is stated as a Laplace principle for the weighted empirical measure arising in importance sampling, can be viewed as a weighted version of Sanov's theorem. The main theorem is applied to quantify the performance of an importance sampling algorithm over a collection of subsets of a given target set as well as quantile estimates. The proof of the main theorem relies on the weak convergence approach to large deviations developed by Dupuis and Ellis.

  • 15. Kramer, Andrei
    et al.
    Milinanni, Federica
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics. KTH, Centres, Science for Life Laboratory, SciLifeLab.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Jauhiainen, Alexandra
    Eriksson, Olivia
    KTH, School of Electrical Engineering and Computer Science (EECS), Computer Science, Computational Science and Technology (CST). KTH, Centres, SeRC - Swedish e-Science Research Centre.
    UQSA - An R-Package for Uncertainty Quantification and Sensitivity Analysis for Biochemical Reaction Network ModelsManuscript (preprint) (Other academic)
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  • 16.
    Milinanni, Federica
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics. KTH, Centres, Science for Life Laboratory, SciLifeLab.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    A large deviation principle for the empirical measures of Metropolis-Hastings chainsManuscript (preprint) (Other academic)
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  • 17.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Large deviations for weighted empirical measures and processes arising in importance sampling2013Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of two papers related to large deviation results associated with importance sampling algorithms. As the need for efficient computational methods increases, so does the need for theoretical analysis of simulation algorithms. This thesis is mainly concerned with algorithms using importance sampling. Both papers make theoretical contributions to the development of a new approach for analyzing efficiency of importance sampling algorithms by means of large deviation theory.

    In the first paper of the thesis, the efficiency of an importance sampling algorithm is studied using a large deviation result for the sequence of weighted empirical measures that represent the output of the algorithm. The main result is stated in terms of the Laplace principle for the weighted empirical measure arising in importance sampling and it can be viewed as a weighted version of Sanov's theorem. This result is used to quantify the performance of an importance sampling algorithm over a collection of subsets of a given target set as well as quantile estimates. The method of proof is the weak convergence approach to large deviations developed by Dupuis and Ellis.

    The second paper studies moderate deviations of the empirical process analogue of the weighted empirical measure arising in importance sampling. Using moderate deviation results for empirical processes the moderate deviation principle is proved for weighted empirical processes that arise in importance sampling. This result can be thought of as the empirical process analogue of the main result of the first paper and the proof is established using standard techniques for empirical processes and Banach space valued random variables. The moderate deviation principle for the importance sampling estimator of the tail of a distribution follows as a corollary. From this, moderate deviation results are established for importance sampling estimators of two risk measures: The quantile process and Expected Shortfall. The results are proved using a delta method for large deviations established by Gao and Zhao (2011) together with more classical results from the theory of large deviations.

    The thesis begins with an informal discussion of stochastic simulation, in particular importance sampling, followed by short mathematical introductions to large deviations and importance sampling.

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  • 18.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics. Brown Univ, USA.
    MODERATE DEVIATION PRINCIPLES FOR IMPORTANCE SAMPLING ESTIMATORS OF RISK MEASURES2017In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 54, no 2, p. 490-506Article in journal (Refereed)
    Abstract [en]

    Importance sampling has become an important tool for the computation of extreme quantiles and tail-based risk measures. For estimation of such nonlinear functionals of the underlying distribution, the standard efficiency analysis is not necessarily applicable. In this paper we therefore study importance sampling algorithms by considering moderate deviations of the associated weighted empirical processes. Using a delta method for large deviations, combined with classical large deviation techniques, the moderate deviation principle is obtained for importance sampling estimators of two of the most common risk measures: value at risk and expected shortfall.

  • 19.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Moderate deviation principles for importance sampling estimators of risk measures2017In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072Article in journal (Refereed)
    Abstract [en]

    Importance sampling has become an important tool for the computation of tail-based risk measures. Since such quantities are often determined mainly by rare events standard Monte Carlo can be inefficient and importance sampling provides a way to speed up computations. This paper considers moderate deviations for the weighted empirical process, the process analogue of the weighted empirical measure, arising in importance sampling. The moderate deviation principle is established as an extension of existing results. Using a delta method for large deviations established by Gao and Zhao (Ann. Statist., 2011) together with classical large deviation techniques, the moderate deviation principle for the weighted empirical process is extended to functionals of the weighted empirical process which correspond to risk measures. The main results are moderate deviation principles for importance sampling estimators of the quantile function of a distribution and Expected Shortfall.

  • 20.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    On large deviations and design of efficient importance sampling algorithms2014Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of four papers, presented in Chapters 2-5, on the topics large deviations and stochastic simulation, particularly importance sampling. The four papers make theoretical contributions to the development of a new approach for analyzing efficiency of importance sampling algorithms by means of large deviation theory, and to the design of efficient algorithms using the subsolution approach developed by Dupuis and Wang (2007).

    In the first two papers of the thesis, the random output of an importance sampling algorithm is viewed as a sequence of weighted empirical measures and weighted empirical processes, respectively. The main theoretical results are a Laplace principle for the weighted empirical measures (Paper 1) and a moderate deviation result for the weighted empirical processes (Paper 2). The Laplace principle for weighted empirical measures is used to propose an alternative measure of efficiency based on the associated rate function.The moderate deviation result for weighted empirical processes is an extension of what can be seen as the empirical process version of Sanov's theorem. Together with a delta method for large deviations, established by Gao and Zhao (2011), we show moderate deviation results for importance sampling estimators of the risk measures Value-at-Risk and Expected Shortfall.

    The final two papers of the thesis are concerned with the design of efficient importance sampling algorithms using subsolutions of partial differential equations of Hamilton-Jacobi type (the subsolution approach).

    In Paper 3 we show a min-max representation of viscosity solutions of Hamilton-Jacobi equations. In particular, the representation suggests a general approach for constructing subsolutions to equations associated with terminal value problems and exit problems. Since the design of efficient importance sampling algorithms is connected to such subsolutions, the min-max representation facilitates the construction of efficient algorithms.

    In Paper 4 we consider the problem of constructing efficient importance sampling algorithms for a certain type of Markovian intensity model for credit risk. The min-max representation of Paper 3 is used to construct subsolutions to the associated Hamilton-Jacobi equation and the corresponding importance sampling algorithms are investigated both theoretically and numerically.

    The thesis begins with an informal discussion of stochastic simulation, followed by brief mathematical introductions to large deviations and importance sampling. 

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  • 21.
    Peletier, M.
    et al.
    Faculteit Wiskunde en Informatica, Technische Universiteit Eindhoven, Eindhoven, Noord-Brabant, Netherlands.
    Gavish, N.
    Department of Mathematics, Technion - IIT, Haifa, 3200003, Israel.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Large Deviations and Gradient Flows for the Brownian One-Dimensional Hard-Rod System2021In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929XArticle in journal (Refereed)
    Abstract [en]

    We study a system of hard rods of finite size in one space dimension, which move by Brownian noise while avoiding overlap. We consider a scaling in which the number of particles tends to infinity while the volume fraction of the rods remains constant; in this limit the empirical measure of the rod positions converges almost surely to a deterministic limit evolution. We prove a large-deviation principle on path space for the empirical measure, by exploiting a one-to-one mapping between the hard-rod system and a system of non-interacting particles on a contracted domain. The large-deviation principle naturally identifies a gradient-flow structure for the limit evolution, with clear interpretations for both the driving functional (an ‘entropy’) and the dissipation, which in this case is the Wasserstein dissipation. This study is inspired by recent developments in the continuum modelling of multiple-species interacting particle systems with finite-size effects; for such systems many different modelling choices appear in the literature, raising the question how one can understand such choices in terms of more microscopic models. The results of this paper give a clear answer to this question, albeit for the simpler one-dimensional hard-rod system. For this specific system this result provides a clear understanding of the value and interpretation of different modelling choices, while giving hints for more general systems.

  • 22.
    Ringqvist, Carl
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Infinite Swapping Algorithm for Training Restricted Boltzmann Machines2020In: Monte Carlo and Quasi-Monte Carlo Methods, Springer Nature , 2020, p. 285-307Conference paper (Refereed)
    Abstract [en]

    Given the important role latent variable models play, for example in statistical learning, there is currently a growing need for efficient Monte Carlo methods for conducting inference on the latent variables given data. Recently, Desjardins et al. (JMLR Workshop and Conference Proceedings: AISTATS 2010, pp. 145–152, 2010 [3]) explored the use of the parallel tempering algorithm for training restricted Boltzmann machines, showing considerable improvement over the previous state-of-the-art. In this paper we continue their efforts by comparing previous methods, including parallel tempering, with the infinite swapping algorithm, an MCMC method first conceived when attempting to optimise performance of parallel tempering (Dupuis et al. in J. Chem. Phys. 137, 2012 [7]), for the training task. We implement a Gibbs-sampling version of infinite swapping and evaluate its performance on a number of test cases, concluding that the algorithm enjoys better mixing properties than both persistent contrastive divergence and parallel tempering for complex energy landscapes associated with restricted Boltzmann machines.

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