Scalable methods for computing sharp extreme event probabilities in infinite-dimensional stochastic systems

Schorlepp, Timo; Tong, Shanyin; Grafke, Tobias; Stadler, Georg

doi:10.1007/s11222-023-10307-2

“…The "diffusive" model fixes α = 0.50, while the "subdiffusive" model adapts α to minimize the cost. E et al 2004;Plotkin et al 2019;Woillez & Bouchet 2020;Schorlepp et al 2023) rare event schemes, aided by machine-learning predictor functions (Ma & Dinner 2005;Chattopadhyay et al 2020;Finkel et al 2021;Miloshevich et al 2023;Finkel et al 2023).…”

Section: Discussionmentioning

confidence: 99%

Mercury’s Chaotic Secular Evolution as a Subdiffusive Process

Abbot,

Webber,

Hernandez

et al. 2024

ApJ

0

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Mercury’s orbit can destabilize, generally resulting in a collision with either Venus or the Sun. Chaotic evolution can cause g 1 to decrease to the approximately constant value of g 5 and create a resonance. Previous work has approximated the variation in g 1 as stochastic diffusion, which leads to a phenomological model that can reproduce the Mercury instability statistics of secular and N-body models on timescales longer than 10 Gyr. Here we show that the diffusive model significantly underpredicts the Mercury instability probability on timescales less than 5 Gyr, the remaining lifespan of the solar system. This is because g 1 exhibits larger variations on short timescales than the diffusive model would suggest. To better model the variations on short timescales, we build a new subdiffusive phenomological model for g 1. Subdiffusion is similar to diffusion but exhibits larger displacements on short timescales and smaller displacements on long timescales. We choose model parameters based on the behavior of the g 1 trajectories in the N-body simulations, leading to a tuned model that can reproduce Mercury instability statistics from 1–40 Gyr. This work motivates fundamental questions in solar system dynamics: why does subdiffusion better approximate the variation in g 1 than standard diffusion? Why is there an upper bound on g 1, but not a lower bound that would prevent it from reaching g 5?

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“…For example, some directions of perturbation (singular vectors) grow much faster than others, a fact which has informed ensemble design in operational weather forecasting (Palmer & Zanna, 2013), and could be used to further improve the algorithm. Methods such as conditional nonlinear optimal perturbation (Wang et al., 2020, and references therein), generalized stability theory (Farrell & Ioannou, 1996), and large deviation theory (Dematteis et al., 2018, 2019; Schorlepp et al., 2023) may prove useful for this task. Related to the previous point, it is desirable to have greater efficiency with samples in order to deploy rare event algorithms at scale. For example, we should not simply discard rejected samples, but rather learn from their “mistakes” to design better perturbations.…”

Section: Discussionmentioning

confidence: 99%

“…For example, some directions of perturbation (singular vectors) grow much faster than others, a fact which has informed ensemble design in operational weather forecasting (Palmer & Zanna, 2013), and could be used to further improve the algorithm. Methods such as conditional nonlinear optimal perturbation (Wang et al., 2020, and references therein), generalized stability theory (Farrell & Ioannou, 1996), and large deviation theory (Dematteis et al., 2018, 2019; Schorlepp et al., 2023) may prove useful for this task.…”

Section: Discussionmentioning

confidence: 99%

Bringing Statistics to Storylines: Rare Event Sampling for Sudden, Transient Extreme Events

Finkel,

O’Gorman

2024

J Adv Model Earth Syst

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A leading goal for climate science and weather risk management is to accurately model both the physics and statistics of extreme events. These two goals are fundamentally at odds: the higher a computational model's resolution, the more expensive are the ensembles needed to capture accurate statistics in the tail of the distribution. Here, we focus on events that are localized in space and time, such as heavy precipitation events, which can start suddenly and decay rapidly. We advance a method for sampling such events more efficiently than straightforward climate model simulation. Our method combines elements of two existing approaches: adaptive multilevel splitting (AMS), a rare event algorithm that generates rigorous statistics but fails to enhance the sampling of sudden, transient extremes; and “ensemble boosting,” which generates physically plausible storylines of these events but not their statistics. We modify AMS by splitting trajectories well in advance of the event's onset, following the approach of ensemble boosting. Early splitting requires a rejection step that reduces efficiency, but it is critical for amplifying and diversifying simulated events in tests with the Lorenz‐96 model, for which we demonstrate improved sampling of extreme local energy fluctuations by approximately a factor of 10 relative to direct sampling. Our method is related to previous algorithms, including subset simulation and anticipated AMS, but is distinctly tailored to handle bursting events caused by chaotic traveling waves. Our work makes progress toward the goal of efficiently sampling such transient local extremes in atmospheric models.

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“…The gradient of the action functional is evaluated exactly on a discrete level ('discretize, then optimize'). A Python source code which illustrates the optimization methods in a simple toy problem can be found in a public GitHub repository [50] and it is explained in [51].…”

Section: J Stat Mech (2023) 123202mentioning

confidence: 99%

Short-time large deviations of the spatially averaged height of a Kardar–Parisi–Zhang interface on a ring

Schorlepp,

Sasorov,

Meerson

2023

J. Stat. Mech.

Self Cite

3

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Using the optimal fluctuation method, we evaluate the short-time probability distribution P ( H ˉ , L , t = T ) of the spatially averaged height H ˉ = ( 1 / L ) ∫ 0 L h ( x , t = T ) d x of a one-dimensional interface h ( x , t ) governed by the Kardar–Parisi–Zhang equation ∂ t h = ν ∂ x 2 h + λ 2 ∂ x h 2 + D ξ x , t on a ring of length L. The process starts from a flat interface, h ( x , t = 0 ) = 0 . Both at λ H ˉ < 0 and at sufficiently small positive λ H ˉ the optimal (that is, the least-action) path h ( x , t ) of the interface, conditioned on H ˉ , is uniform in space, and the distribution P ( H ˉ , L , T ) is Gaussian. However, at sufficiently large λ H ˉ > 0 the spatially uniform solution becomes sub-optimal and gives way to non-uniform optimal paths. We study these, and the resulting non-Gaussian distribution P ( H ˉ , L , T ) , analytically and numerically. The loss of optimality of the uniform solution occurs via a dynamical phase transition of either first or second order, depending on the rescaled system size ℓ = L / ν T , at a critical value H ˉ = H ˉ c ( ℓ ) . At large but finite ℓ the transition is of first order. Remarkably, it becomes an ‘accidental’ second-order transition in the limit of ℓ → ∞ , where a large-deviation behavior − ln P ( H ¯ , L , T ) ≃ ( L / T ) f ( H ¯ ) (in the units λ = ν = D = 1 ) is observed. At small ℓ the transition is of second order, while at ℓ = O ( 1 ) transitions of both types occur.

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Scalable methods for computing sharp extreme event probabilities in infinite-dimensional stochastic systems

Cited by 6 publications

References 81 publications

Mercury’s Chaotic Secular Evolution as a Subdiffusive Process

Mercury’s Chaotic Secular Evolution as a Subdiffusive Process

Bringing Statistics to Storylines: Rare Event Sampling for Sudden, Transient Extreme Events

Short-time large deviations of the spatially averaged height of a Kardar–Parisi–Zhang interface on a ring

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