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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Notes
- 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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Acknowledgement
This research has been partially funded by Deutsche Forschungsgemeinschaft (DFG) through the grant CRC 1114 “Scaling Cascades in Complex Systems”, Project A05 “Probing scales in equilibrated systems by optimal nonequilibrium forcing”. Omar Kebiri received funding from the EU-METALIC II Programme.
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Kebiri, O., Neureither, L., Hartmann, C. (2019). Adaptive Importance Sampling with Forward-Backward Stochastic Differential Equations. In: Giacomin, G., Olla, S., Saada, E., Spohn, H., Stoltz, G. (eds) Stochastic Dynamics Out of Equilibrium. IHPStochDyn 2017. Springer Proceedings in Mathematics & Statistics, vol 282. Springer, Cham. https://doi.org/10.1007/978-3-030-15096-9_7
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