Model Error, Information Barriers, State Estimation and Prediction in Complex Multiscale Systems

Abstract

Complex multiscale systems are ubiquitous in many areas. This research expository article discusses the development and applications of a recent information-theoretic framework as well as novel reduced-order nonlinear modeling strategies for understanding and predicting complex multiscale systems. The topics include the basic mathematical properties and qualitative features of complex multiscale systems, statistical prediction and uncertainty quantification, state estimation or data assimilation, and coping with the inevitable model errors in approximating such complex systems. Here, the information-theoretic framework is applied to rigorously quantify the model fidelity, model sensitivity and information barriers arising from different approximation strategies. It also succeeds in assessing the skill of filtering and predicting complex dynamical systems and overcomes the shortcomings in traditional path-wise measurements such as the failure in measuring extreme events. In addition, information theory is incorporated into a systematic data-driven nonlinear stochastic modeling framework that allows effective predictions of nonlinear intermittent time series. Finally, new efficient reduced-order nonlinear modeling strategies combined with information theory for model calibration provide skillful predictions of intermittent extreme events in spatially-extended complex dynamical systems. The contents here include the general mathematical theories, effective numerical procedures, instructive qualitative models, and concrete models from climate, atmosphere and ocean science.

Statistical dynamical model to predict extreme events and anomalous features in shallow water waves with abrupt depth change

“…Understanding and predicting extreme events and their anomalous statistics in complex nonlinear systems are a grand challenge in climate, material, and neuroscience as well as for engineering design. This is a very active contemporary topic in applied mathematics with qualitative and quantitative models ( 1 – 7 ) and novel numerical algorithms, which overcome the curse of dimensionality for extreme event prediction in large complex systems ( 2 , 8 – 11 ). The occurrence of Rogue waves as extreme events within different physical settings of deep water ( 12 – 16 ) and shallow water ( 17 – 19 ) is an important practical topic.…”

Feasibility of Laser Communication Beacon Light Compressed Sensing

“… described the random variation of atmospheric structure and its physical parameters on various time and space scales. With the improvement of modern nonlinear dynamical [ 50 , 51 ], the atmosphere turbulence in various scale and various space-time plays an important role in saltation and its predictability of atmospheric processes. Therefore, it plays an important role in atmospheric turbulence and related problems.…”

Conditional Gaussian Systems for Multiscale Nonlinear Stochastic Systems: Prediction, State Estimation and Uncertainty Quantification

“…While these two path-wise measurements are easy to implement and are able to quantify the data assimilation and prediction skill to some extent, they have fundamental limitations. It has been shown in [42,211] that these two measurements fail to quantify the skill of capturing the extreme events and other non-Gaussian features, which lead to misleading results. In fact, concrete examples even in the Gaussian models [42,211] showed that two different predictions can have the same RMSE and PC, but one is way more skillful than the other in capturing the extreme events.…”