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Title: ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems

Efficient simulation of SDEs is essential in many applications, particularly for ergodic systems that demand efficient simulation of both short-time dynamics and large-time statistics. However, locally Lipschitz SDEs often require special treatments such as implicit schemes with small time-steps to accurately simulate the ergodic measures. We introduce a framework to construct inference-based schemes adaptive to large time-steps (ISALT) from data, achieving a reduction in time by several orders of magnitudes. The key is the statistical learning of an approximation to the infinite-dimensional discrete-time flow map. We explore the use of numerical schemes (such as the Euler-Maruyama, the hybrid RK4, and an implicit scheme) to derive informed basis functions, leading to a parameter inference problem. We introduce a scalable algorithm to estimate the parameters by least squares, and we prove the convergence of the estimators as data size increases.

We test the ISALT on three non-globally Lipschitz SDEs: the 1D double-well potential, a 2D multiscale gradient system, and the 3D stochastic Lorenz equation with a degenerate noise. Numerical results show that ISALT can tolerate time-step magnitudes larger than plain numerical schemes. It reaches optimal accuracy in reproducing the invariant measure when the time-step is medium-large.

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Award ID(s):
1847770
NSF-PAR ID:
10329663
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Discrete & Continuous Dynamical Systems - S
Volume:
15
Issue:
4
ISSN:
1937-1632
Page Range / eLocation ID:
747
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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