Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Test Models for Statistical Inference: Two-Dimensional Reaction Systems Displaying Limit Cycle Bifurcations and Bistability Read chapter Read first chapter

2017 | OriginalPaper | Chapter

Importance Sampling for Metastable and Multiscale Dynamical Systems

Author : K. Spiliopoulos

Published in: Stochastic Processes, Multiscale Modeling, and Numerical Methods for Computational Cellular Biology

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this article, we address the issues that come up in the design of importance sampling schemes for rare events associated to stochastic dynamical systems. We focus on the issue of metastability and on the effect of multiple scales. We discuss why seemingly reasonable schemes that follow large deviations optimal paths may perform poorly in practice, even though they are asymptotically optimal. Pre-asymptotic optimality is important when one deals with metastable dynamics and we discuss possible ways as to how to address this issue. Moreover, we discuss how the effect of the multiple scales (either in periodic or random environments) on the efficient design of importance sampling should be addressed. We discuss the mathematical and practical issues that come up, how to overcome some of the issues and discuss future challenges.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Test Models for Statistical Inference: Two-Dimensional Reaction Systems Displaying Limit Cycle Bifurcations and Bistability
next chapter Multiscale Simulation of Stochastic Reaction-Diffusion Networks
Literature
1.
S. Asmussen, P.W. Glynn, Stochastic Simulation: Algorithms and Analysis (Springer, New York, 2007)
2.
P. Banushkina, M. Meuwly, Diffusive dynamics on multidimensional rough free energy surfaces. J. Chem. Phys. 127, 135101 (2007)CrossRef
3.
A. Bensoussan, J.L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, vol. 5, Studies in Mathematics and its Applications (North-Holland Publishing Co., Amsterdam, 1978)
4.
J.H. Blanchet, P. Glynn, Efficient rare-event simulation for the maximum of heavy-tailed random walks. Ann. Appl. Prob. 18, 1351–1378 (2008)MathSciNetCrossRefMATH
5.
J.H. Blanchet, J.C. Liu, State-dependent importance sampling for regularly varying random walks. Adv. Appl. Probab. 40, 1104–1128 (2008)MathSciNetCrossRefMATH
6.
J.H. Blanchet, P. Glynn, J.C. Liu, Fluid heuristics, Lyapunov bounds and efficient importance sampling for a heavy-tailed G/G/1 queue. Queueing Syst. 57, 99–113 (2007)MathSciNetCrossRefMATH
7.
J.H. Blanchet, P. Glynn, K. Leder, On Lyapunov inequalities and subsolutions for efficient importance sampling. ACM TOMACS 22(3), Artical No. 13 (2012)
8.
A. Bovier, M. Eckhoff, V. Gayrard, M. Klein, Metastability in reversible diffusion processes I. Sharp estimates for capacities and exit times. J. Eur. Math. Soc. 6, 399–424 (2004)CrossRefMATH
9.
A. Bovier, V. Gayrard, M. Klein, Metastability in reversible diffusion processes II. Precise estimates for small eigenvalues. J. Eur. Math. Soc. 7, 69–99 (2005)CrossRefMATH
10.
11.
12.
M. Day, T. Darden, Some regularity results on the Ventcel-Freidlin quasi-potential function. Appl. Math Opt. 13 259–282 (1985)CrossRefMATH
13.
A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, vol. 38, 2nd ed., Applications of Mathematics (Springer, New York, 1998)
14.
P. Dupuis, K. Spiliopoulos, Large deviations for multiscale problems via weak convergence methods. Stochastic Process. Appl. 122, 1947–1987 (2012)MathSciNetCrossRefMATH
15.
P. Dupuis, K. Spiliopoulos, Rare event simulation in the neighborhood of a rest point, in 2014 Winter Simulation Conference (IEEE, 2014), pp. 564–573
16.
P. Dupuis, H. Wang, Importance sampling, large deviations and differential games. Stochastics Stochastics Rep. 76, 481–508 (2004)MathSciNetCrossRefMATH
17.
P. Dupuis, H. Wang, Subsolutions of an Isaacs equation and efficient schemes of importance sampling. Math. Oper. Res. 32, 723–757 (2007)MathSciNetCrossRefMATH
18.
P. Dupuis, K. Leder, H. Wang, Large deviations and importance sampling for a tandem network with slow-down. Queueing Syst. 57, 71–83 (2007)MathSciNetCrossRefMATH
19.
20.
21.
P. Dupuis, K. Spiliopoulos, H. Wang, Rare event simulation in rough energy landscapes, in 2011 Winter Simulation Conference (2011), pp. 504–515
22.
23.
P. Dupuis, K. Spiliopoulos, X. Zhou, Escape from an equilibrium: importance sampling and rest points I. Ann. Appl. Probab. 25(5), 2909–2958 (2015)MathSciNetCrossRefMATH
24.
25.
W.H. Fleming, H.M. Soner, Controlled Markov Processes and Viscosity Solutions, 2nd edn. (Springer, Berlin, 2006)MATH
26.
M. Freidlin, R. Sowers, A comparison of homogenization and large deviations with applications to wavefront propagation. Stochastic Process Appl. 82, 23–52 (1999)MathSciNetCrossRefMATH
27.
M.I. Freidlin, A.D. Wentzell, Random Perturbations of Dynamical Systems, 2nd edn. (Springer, New York, 1988)MATH
28.
P. Glasserman, Monte Carlo Methods in Financial Engineering (Springer, New York, 2004)MATH
29.
P. Glasserman, S. Kou, Analysis of an important sampling estimator for tandem queues. ACM Trans. Model. Comput. Simul. 4, 22–42 (1995)CrossRefMATH
30.
P. Glasserman, Y. Wang, Counter examples in importance sampling for large deviations probabilities. Ann. Appl. Probab. 7, 731–746 (1997)MathSciNetCrossRefMATH
31.
P.W. Glynn, D.L. Iglehart, Simulation methods for queues: an overview. Queueing Syst.: Theory Appl. 3, 221–256 (1988)
32.
R.C. Griffiths, S. Tavaré, Simulating probability distributions in the coalescent. Theor. Popul. Biol. 46, 131–159 (1994)CrossRefMATH
33.
W. Janke, Rugged Free-Energy Landscapes, Lecture Notes in Physics, vol. 736/2008 (Springer, Berlin, 2008)
34.
T. Komorowski, C. Landim, S. Olla, Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation (Springer, Berlin, 2012)CrossRefMATH
35.
E. Kosygina, F. Rezakhanlou, S.R.S. Varadhan, Stochastic homogenization of Hamilton-Jacobi-Bellman equations, Commun. Pure Appl. Math. LIX, 0001–0033 (2006)
36.
R.D. Levine, Monte Carlo, maximum entropy and importance sampling. Chem. Phys. 228, 255–264
37.
S. Lifson, J.L. Jackson, On the self-diffusion of ions in a polyelectrolyte solution. J. Chem. Phys. 36, 2410–2414 (1962)CrossRef
38.
R.S. Maier, D.L. Stein, Escape problem for irreversible systems. Phys. Rev. E 48(2), 931–938 (1993)CrossRef
39.
R.S. Maier, D.L. Stein, Limiting exit location distributions in the stochastic exit problem. SIAM J. Appl. Math. 57(3), 752–790 (1997)MathSciNetCrossRefMATH
40.
O. Mazonka, C. Jarzynski, J. Blocki, Computing probabilities of very rare events for Langevin processes: a new method based on importance sampling. Nucl. Phys. A 641, 335–354 (1998)CrossRef
41.
42.
H. Osada, Homogenization of diffusion processes with random stationary coefficients, in Probability Theory and Mathematical Statistics. Lecture Notes in Mathematics, vol. 1021 (Springer, Berlin, 1983), pp. 507–517
43.
G.A. Pavliotis, A.M. Stuart, Multiscale Methods: Averaging and Homogenization (Springer, Berlin, 2007)MATH
44.
G. Papanicolaou, S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, in Colloquia Mathematica Societatis Janos Bolyai 27, Random Fields, Esztergom (Hungary) 1979, North Holland (1982), pp. 835–873
45.
M. Salins, K. Spiliopoulos, Rare event simulation via importance sampling for linear SPDE’s. stochostics and Partial Differential Equation: Analysis and computations (accepted, 2017)
46.
47.
48.
K. Spiliopoulos, Non-asymptotic performance analysis of importance sampling schemes for small noise diffusions. J. Appl. Probab. 52, 1–14 (2015)MathSciNetCrossRefMATH
49.
K. Spiliopoulos, Quenched large deviations for multiscale diffusion processes in random environments. Electron. J. Probab. 20(15), 1–29 (2015)MathSciNetMATH
50.
K. Spiliopoulos, Rare event simulation for multiscale diffusions in random environments. SIAM Multiscale Model. Simul. 13(4), 1290–1311 (2015)MathSciNetCrossRefMATH
51.
E. Vanden-Eijnden, J. Weare, Rare event simulation with vanishing error for small noise diffusions. Commun. Pure Appl. Math. 65(12), 1770–1803 (2012)CrossRefMATH
52.
S.R.S. Varadhan, Large Deviations and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 46 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1984)
53.
A. Viel, M.V. Patel, P. Niyaz, K. Whaley, Importance sampling in rigid body diffusion Monte Carlo. Comput. Phys. Commun. 145, 24–47 (2002)CrossRefMATH
54.
D. Zuckerman, T. Woolf, Efficient dynamic importance sampling of rare events in one dimension. Phys. Rev. E 63(016702), 1–10 (2000)
55.
Metadata
Title
Importance Sampling for Metastable and Multiscale Dynamical Systems
Author
K. Spiliopoulos
Copyright Year
2017
Book
Stochastic Processes, Multiscale Modeling, and Numerical Methods for Computational Cellular Biology

Print ISBN: 978-3-319-62626-0

Electronic ISBN: 978-3-319-62627-7

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-62627-7_2

Premium Partner

    Image Credits