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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.

 

This paper offers a new feature-oriented compression algorithm for flexible reduction of data redundancy commonly found in images and videos streams. Using a combination of image segmentation and face detection techniques as a preprocessing step, we derive a compression framework to adaptively treat `feature' and `ground' while balancing the total compression and quality of `feature' regions. We demonstrate the utility of a feature compliant compression algorithm (FC-SVD), a revised peak signal-to-noise ratio PSNR assessment, and a relative quality ratio to control artificial distortion. The goal of this investigation is to provide new contributions to image and video processing research via multi-scale resolution and the block-based adaptive singular value decomposition.