Importance Sampling for Slow-Fast Diffusions Based on Moderate Deviations

Spiliopoulos, Konstantinos; Morse, Matthew R.

doi:10.1137/18m1192962

Cited by 8 publications

(16 citation statements)

References 17 publications

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“…is a cylindrical Wiener process under the measure P . From (16) we see that the second moment of the estimator can be written as an exponential functional of the driving noise and, as such, it admits the variational representation (14) (see (2.5) in [53] as well as (14) in [57] for the finite-dimensional case).…”

Section: Moderate Deviations Importance Sampling and Asymptotic Theorymentioning

confidence: 99%

“…The simulation outcomes are then weighted by likelihood ratios so that the importance sampling estimators remain unbiased under the new probability measures. Importance sampling schemes for events in the large and moderate deviation regimes have been developed for finite-dimensional systems in [27,29,56,57,59]. In [27,57], the authors observed that moderate-deviation based schemes provide a viable and simpler alternative to their largedeviation based counterparts, in cases where both are applicable.…”

Section: Introductionmentioning

confidence: 99%

“…Importance sampling schemes for events in the large and moderate deviation regimes have been developed for finite-dimensional systems in [27,29,56,57,59]. In [27,57], the authors observed that moderate-deviation based schemes provide a viable and simpler alternative to their largedeviation based counterparts, in cases where both are applicable. This is due to the fact that the MDP action functional, which characterizes exponential decay rates of probabilities, takes a much simpler form.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Importance sampling for stochastic reaction-diffusion equations in the moderate deviation regime

Gasteratos¹,

Salins²,

Spiliopoulos³

2022

Preprint

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We develop a provably efficient importance sampling scheme that estimates exit probabilities of solutions to small-noise stochastic reaction-diffusion equations from scaled neighborhoods of a stable equilibrium. The moderate deviation scaling allows for a local approximation of the nonlinear dynamics by their linearized version. In addition, we identify a finite-dimensional subspace where exits take place with high probability. Using stochastic control and variational methods we show that our scheme performs well both in the zero noise limit and pre-asymptotically. Simulation studies for stochastically perturbed bistable dynamics illustrate the theoretical results.

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Section: Moderate Deviations Importance Sampling and Asymptotic Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Importance sampling for stochastic reaction-diffusion equations in the moderate deviation regime

Gasteratos¹,

Salins²,

Spiliopoulos³

2022

Preprint

Self Cite

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show abstract

“…On a computational level, the solution to the stochastic control problem gives vital information for the design of efficient Monte Carlo methods for the approximation of rare event probabilities on the moderate deviation range. In particular, the fact that the limiting equation is affine in η ǫ,u is expected to make moderate deviation-based importance sampling for stochastic PDE easier to implement than its large deviation-based counterpart, see [32] for the related situation in finite dimensions. We plan to explore this in a future work.…”

Section: S(φ) + λ(φ)mentioning

confidence: 99%

“…where P x t denotes the transition semigroup corresponding to the fast process Y x,y (see (31), (32)). Now, for each fixed x ∈ H, the map…”

Section: Proof Of Proposition 51mentioning

confidence: 99%

Moderate deviations for systems of slow-fast stochastic reaction-diffusion equations

Gasteratos¹,

Salins²,

Spiliopoulos³

2021

Preprint

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The goal of this paper is to study the Moderate Deviation Principle (MDP) for a system of stochastic reaction-diffusion equations with a time-scale separation in slow and fast components and small noise in the slow component. Based on weak convergence methods in infinite dimensions and related stochastic control arguments, we obtain an exact form for the moderate deviations rate function in different regimes as the small noise and time-scale separation parameters vanish. Many issues that come up due to the infinite dimensionality of the problem are completely absent in their finite-dimensional counterpart. In comparison to corresponding Large Deviation Principles, the moderate deviation scaling necessitates a more delicate approach to establishing tightness and properly identifying the limiting behavior of the underlying controlled problem. The latter involves regularity properties of a solution of an associated elliptic Kolmogorov equation on Hilbert space along with a finite-dimensional approximation argument.

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Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

Bezemek,

Heldman

2023

Appl Math Optim

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We construct an importance sampling method for computing statistics related to rare events for weakly interacting diffusions. Standard Monte Carlo methods behave exponentially poorly with the number of particles in the system for such problems. Our scheme is based on subsolutions of a Hamilton-Jacobi-Bellman (HJB) equation on Wasserstein space which arises in the theory of mean-field (McKean-Vlasov) control. We identify conditions under which such a scheme is asymptotically optimal. In the process, we make connections between the large deviations principle for the empirical measure of weakly interacting diffusions, mean-field control, and the HJB equation on Wasserstein space. We also provide evidence, both analytical and numerical, that with sufficient regularity of the HJB equation, our scheme can have vanishingly small relative error in the many particle limit.

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Importance Sampling for Slow-Fast Diffusions Based on Moderate Deviations

Cited by 8 publications

References 17 publications

Importance sampling for stochastic reaction-diffusion equations in the moderate deviation regime

Importance sampling for stochastic reaction-diffusion equations in the moderate deviation regime

Moderate deviations for systems of slow-fast stochastic reaction-diffusion equations

Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

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