ISALT：局部Lipschitz遍历系统的大时间步长自适应推理算法
ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems
摘要：
在许多应用中，特别是对于需要同时对短时动态和大时统计进行有效仿真的遍历系统，有效仿真SDES是必不可少的。然而，局部Lipschitz SDES往往需要特殊的处理，如隐格式与小的时间步长，以准确地模拟遍历措施。我们引入了一个框架来构建基于推理的计划适应大的时间步长（ISALT）从数据，实现了几个数量级的时间减少。关键是对无限维离散时间流图的近似的统计学习。我们探讨使用数值方案（如欧拉-丸山，混合RK 4，和隐格式），以获得知情的基础函数，导致参数推断问题。我们引入了一个可扩展的算法来估计参数的最小二乘，我们证明了收敛的估计数据大小的增加。我们测试ISALT上的三个非全局Lipschitz SDES：一维双阱势，二维多尺度梯度系统，和3D随机洛伦兹方程退化噪声。数值结果表明，ISALT可以容忍的时间步长幅度大于普通数值格式。当时间步长为中-大时，该算法在再现不变测度时达到最优精度。
英文摘要：
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 measure. 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, a 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 multi-scale gradient system, and the 3D stochastic Lorenz equation with 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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