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Erschienen in:
Stochastic Mean-Field Dynamics and Applications to Life Sciences Kapitel lesen Erstes Kapitel lesen

2019 | OriginalPaper | Buchkapitel

Adaptive Importance Sampling with Forward-Backward Stochastic Differential Equations

verfasst von : Omar Kebiri, Lara Neureither, Carsten Hartmann

Erschienen in: Stochastic Dynamics Out of Equilibrium

Verlag: Springer International Publishing

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Abstract

We describe an adaptive importance sampling algorithm for rare events that is based on a dual stochastic control formulation of a path sampling problem. Specifically, we focus on path functionals that have the form of cumulate generating functions, which appear relevant in the context of, e.g. molecular dynamics, and we discuss the construction of an optimal (i.e. minimum variance) change of measure by solving a stochastic control problem. We show that the associated semi-linear dynamic programming equations admit an equivalent formulation as a system of uncoupled forward-backward stochastic differential equations that can be solved efficiently by a least squares Monte Carlo algorithm. We illustrate the approach with a suitable numerical example and discuss the extension of the algorithm to high-dimensional systems.

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Vorheriges Kapitel On Optimal Decay Estimates for ODEs and PDEs with Modal Decomposition
Nächstes Kapitel Ergodic Properties of Quasi-Markovian Generalized Langevin Equations with Configuration Dependent Noise and Non-conservative Force
Fußnoten
1
More precisely, \(\varphi |_{\mathcal {F}_{\tau }}\) is understood as the restriction of the measure \(Q^*\) defined by \(dQ^*=\varphi dP\) to the \(\sigma \)-algebra \({\mathcal {F}_{\tau }}\) that contains all measurable sets \(E\in {\mathcal E}\), with the property that for every \(t\geqslant 0\) the set \(E\cap \{\tau \leqslant t\}\) is an element of the \(\sigma \)-algebra \(\mathcal {F}_{t}=\sigma (X_s:0\leqslant s\leqslant t)\) that is generated by all trajectories \((X_s)_{0\leqslant s\leqslant t}\) of length t.
 
2
For the numerical computation, we add reflecting boundary conditions at \(x=-L\) for some \(L>0\), the precise value of which does not affect the results (assuming that it is sufficiently large, say, \(L>3\)) since the potential has a 4-th order growth.
 
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Metadaten
Titel
Adaptive Importance Sampling with Forward-Backward Stochastic Differential Equations
verfasst von
Omar Kebiri
Lara Neureither
Carsten Hartmann
Copyright-Jahr
2019
Buch
Stochastic Dynamics Out of Equilibrium

Print ISBN: 978-3-030-15095-2

Electronic ISBN: 978-3-030-15096-9

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-030-15096-9_7