Coalescing binaries—Probe of the universe

At present, coalescing binary systems containing neutron stars or black holes are thought to be the most likely sources of gravitational waves to be detected by long baseline interferometers being currently designed. In this essay we calculate the characteristics of the signal from a coalescing binary to the first post-Newtonian order. We show that at coalescence the eccentricity of the orbit, tidal effects, and magnetic interactions can be neglected. We also consider the effects of the expansion of the universe on the signal. We show that observations of gravitational waves from coalescing binaries by a network of detectors will provide a wealth of astrophysical information, e.g., determination of the Hubble constant, new rungs on the cosmic distance ladder, estimates of the masses of components of the binary systems, information about the mass distribution in the universe, highly accurate tests of general relativity, and constraints on neutron-star equations of state. Further development of laser interferometers may enable determination of the deceleration parameter, provide new information about evolution of the universe, and even enable observation of such effects as gravitational lensing.This essay received the second award from the Gravity Research Association for the year 1987-Ed.On leave of absence from Mathematical Institute, Polish Academy of Science, Warsaw, Poland.     

Exploring the need for localization in ensemble data assimilation using a hierarchical ensemble filter

Good performance with small ensemble filters applied to models with many state variables may require ‘localizing’ the impact of an observation to state variables that are ‘close’ to the observation. As a step in developing nearly generic ensemble filter assimilation systems, a method to estimate ‘localization’ functions is presented. Localization is viewed as a means to ameliorate sampling error when small ensembles are used to sample the statistical relation between an observation and a state variable. The impact of spurious sample correlations between an observation and model state variables is estimated using a ‘hierarchical ensemble filter’, where an ensemble of ensemble filters is used to detect sampling error. Hierarchical filters can adapt to a wide array of ensemble sizes and observational error characteristics with only limited heuristic tuning. Hierarchical filters can allow observations to efficiently impact state variables, even when the notion of ‘distance’ between the observation and the state variables cannot be easily defined. For instance, defining the distance between an observation of radar reflectivity from a particular radar and beam angle taken at 1133 GMT and a model temperature variable at 700 hPa 60 km north of the radar beam at 1200 GMT is challenging. The hierarchical filter estimates sampling error from a ‘group’ of ensembles and computes a factor between 0 and 1 to minimize sampling error. An a priori notion of distance is not required. Results are shown in both a low-order model and a simple atmospheric GCM. For low-order models, the hierarchical filter produces ‘localization’ functions that are very similar to those already described in the literature. When observations are more complex or taken at different times from the state specification (in ensemble smoothers for instance), the localization functions become increasingly distinct from those used previously. In the GCM, this complexity reaches a level that suggests that it would be difficult to define efficient localization functions a priori. There is a cost trade-off between running hierarchical filters or running a traditional filter with larger ensemble size. Hierarchical filters can be run for short training periods to develop localization statistics that can be used in a traditional ensemble filter to produce high quality assimilations at reasonable cost, even when the relation between observations and state variables is not well-known a priori. Additional research is needed to determine if it is ever cost-efficient to run hierarchical filters for large data assimilation problems instead of traditional filters with the corresponding total number of ensemble members.     

Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter

Data assimilation is an iterative approach to the problem of estimating the state of a dynamical system using both current and past observations of the system together with a model for the system’s time evolution. Rather than solving the problem from scratch each time new observations become available, one uses the model to “forecast” the current state, using a prior state estimate (which incorporates information from past data) as the initial condition, then uses current data to correct the prior forecast to a current state estimate. This Bayesian approach is most effective when the uncertainty in both the observations and in the state estimate, as it evolves over time, are accurately quantified. In this article, we describe a practical method for data assimilation in large, spatiotemporally chaotic systems. The method is a type of “ensemble Kalman filter”, in which the state estimate and its approximate uncertainty are represented at any given time by an ensemble of system states. We discuss both the mathematical basis of this approach and its implementation; our primary emphasis is on ease of use and computational speed rather than improving accuracy over previously published approaches to ensemble Kalman filtering. We include some numerical results demonstrating the efficiency and accuracy of our implementation for assimilating real atmospheric data with the global forecast model used by the US National Weather Service.     

Exploiting dynamical coherence: A geometric approach to parameter estimation in nonlinear models

Parameter estimation in nonlinear models is a common task, and one for which there is no general solution at present. In the case of linear models, the distribution of forecast errors provides a reliable guide to parameter estimation, but in nonlinear models the facts that predictability may vary with location in state space, and that the distribution of forecast errors is expected not to be Normal, means that parameter estimation based on least squares methods will result in systematic errors. A new approach to parameter estimation is presented which focuses on the geometry of trajectories of the model rather than the distribution of distances between model forecast and the observation at a given lead time. Specifically, we test a number of candidate trajectories to determine the duration for which they can shadow the observations, rather than evaluating a forecast error statistic at any specific lead time(s). This yields insights into both the parameters of the dynamical model and those of the observational noise model. The advances reported here are made possible by extracting more information from the dynamical equations, and thus improving the balance between information gleaned from the structural form of the equations and that from the observations. The technique is illustrated for both flows and maps, applied in 2-, 3-, and 8-dimensional dynamical systems, and shown to be effective in a case of incomplete observation where some components of the state are not observed at all. While the demonstration of effectiveness is strong, there remain fundamental challenges in the problem of estimating model parameters when the system that generated the observations is not a member of the model class. Parameter estimation appears ill defined in this case.     

A Boolean Delay Equation Model of Colliding Cascades. Part II: Prediction of Critical Transitions

We consider here prediction of abrupt overall changes (critical transitions) in the behavior of hierarchical complex systems, using the model developed in the first part of this study. The model merges the physical concept of colliding cascades with the mathematical framework of Boolean delay equations. It describes critical transitions that are due to the interaction between direct cascades of loading and inverse cascades of failures in a hierarchical system. This interaction is controlled by distinct delays between switching of elements from one state to another: loaded vs. unloaded and intact vs. failed. We focus on the earthquake prediction problem; accordingly, the model's heuristic constraints are taken from the dynamics of seismicity. The model exhibits four major types of premonitory seismicity patterns (PSPs), which have been previously identified in seismic observations: (i) rise of earthquake clustering; (ii) rise of the earthquakes' intensity; (iii) rise of the earthquake correlation range; and (iv) certain changes in the size distribution of earthquakes (Gutenberg–Richter relation). The model exhibits new features of individual PSPs and their collective behavior, to be tested in turn on observations. There are indications that the premonitory phenomena considered are not seismicity-specific, but may be common to hierarchical systems of a more general nature.     

Sequential reconstruction of driving-forces from nonlinear nonstationary dynamics

This paper describes a functional analysis-based method for the estimation of driving-forces from nonlinear dynamic systems. The driving-forces account for the perturbation inputs induced by the external environment or the secular variations in the internal variables of the system. The proposed algorithm is applicable to the problems for which there is too little or no prior knowledge to build a rigorous mathematical model of the unknown dynamics. We derive the estimator conditioned on the differentiability of the unknown system’s mapping, and smoothness of the driving-force. The proposed algorithm is an adaptive sequential realization of the blind prediction error method, where the basic idea is to predict the observables, and retrieve the driving-force from the prediction error. Our realization of this idea is embodied by predicting the observables one-step into the future using a bank of echo state networks (ESN) in an online fashion, and then extracting the raw estimates from the prediction error and smoothing these estimates in two adaptive filtering stages. The adaptive nature of the algorithm enables to retrieve both slowly and rapidly varying driving-forces accurately, which are illustrated by simulations. Logistic and Moran-Ricker maps are studied in controlled experiments, exemplifying chaotic state and stochastic measurement models. The algorithm is also applied to the estimation of a driving-force from another nonlinear dynamic system that is stochastic in both state and measurement equations. The results are judged by the posterior Cramer-Rao lower bounds. The method is finally put into test on a real-world application; extracting sun’s magnetic flux from the sunspot time series.     

Adaptive Information Sharing with Ontological Relevance Computation for Decentralized Self-Organization Systems

Decentralization is a peculiar characteristic of self-organizing systems such as swarm intelligence systems, which function as complex collective responsive systems without central control and operates based on contextual local coordination among relatively simple individual systems. The decentralized particularity of self-organizing systems lies in their capacity to spontaneously respond to accommodate environmental changes in a cooperative manner without external control. However, if members cannot obtain observations of the state of the whole team and environment, they have to share their knowledge and policies with each other through communication in order to adapt to the environment appropriately. In this paper, we propose an information sharing mechanism as an independent decision phase to improve individual members’ joint adaption to the world to fulfill an optimal self-organization in general. We design the information sharing decision analogous to human information sharing mechanisms. In this case, information can be shared among individual members by evaluating the semantic relationship of information based on ontology graph and their local knowledge. That is, if individual member collects more relevant information, the information will be used to update its local knowledge and improve sharing relevant information by measuring the ontological relevance. This will enable more related information to be acquired so that their models will be reinforced for more precise information sharing. Our simulations and experimental results show that this design can share information efficiently to achieve optimal adaptive self-organizing systems.     

A numerical method for the simultaneous determination of term values and molecular constants

A method is presented for the simulatneous determination of term values and molecular constants. The method is an extension of an earlier method, the so-called term value approach, for determining such constants. Whereas the latter is a two step procedure, the new method is a one step procedure. In this way a simplifying assumption inherent in the original term value approach (that, in the second step, the term values are regarded as independent measurements) is circumvented.Further the new method avoids some difficulties that may arise when the former approach is applied to systems where the term values form disconnecting sets. All the features of the original term value approach are preserved, including the one that also very large systems can be dealt with on computers of moderate size.Some comparative tests performed with the new method indicate that, in certain cases, the above approximation has a significant effect on the error estimates of the unknown molecular constants. This confirms observations reported previously by Albritton et al. J. Mol. Spectrosc. 46, 67 (1973). On the other hand, observations made by the same authors that also the values of the molecular constants may be significantly affected are not confirmed. It is found that the reported differences are explained by other causes than the removal of the above assumption.It is emphasized that the error estimates obtained with the new method may still be inadequate. Another simplifying assumption, that the line measurements have equal weights, remains to be removed. This objection applies to other methods as well. Reference is made to work in progress to cope with this problem by means of new measuring techniques.     

Deep information from limited observation of robust yet fragile systems

We show how one can completely reconstruct even moderately optimized configurations of the Forest Fire model with relatively few observations. We discuss the relationship between the deep information from limited observations (DILO) to the robust-yet-fragile (RYF) property of the Forest Fire model and propose that DILO may be a general property of RYF complex systems.     

观测窗中多元非线性时间序列的分析

从观察的角度去了解原有系统的特性，不同程度上会受到观测手段的限制．针对一个观测窗口获得的多元时间序列，利用一种能够更好地体现多元时间序列复杂的时空结构的多元奇异系统分析方法，研究多元序列的相关维数和最大李雅普诺夫指数的变化．利用多层感知器神经网络模型分析其可预测性．研究发现，随着观测窗中多元时间序列的变量数目的增加，反映出的复杂性有所提高，稳定性会产生波动，且相应的可预测性也会产生一些差异． 关键词：     

Tracking noisy limit cycle oscillation with nonlinear filters

For engineering systems, the dynamic state estimates provide valuable information for the detection and prediction of failure due to noise and vibration. From this perspective, nonlinear filtering techniques are applied to the problem of state estimation of dynamical systems undergoing noisy limit cycle oscillation. Specifically, the extended Kalman filter, ensemble Kalman filter and particle filter are used to track the limit cycle oscillations of a Duffing oscillator using noisy observational data. The noisy limit cycle oscillations feature highly non-Gaussian trends. The efficiency and limitations of the extended Kalman filter, ensemble Kalman filter and particle filter in tracking limit cycle oscillations are examined with respect to the model and measurement noise and sparsity of measurement data. For the limit cycle oscillations considered here, it is demonstrated that the ensemble Kalman filter and particle filter outperform the extended Kalman filter in the presence of sparse observational data or strong measurement noise. For moderate measurement noise and frequent measurement data, the ensemble Kalman filter and particle filter perform equally well in comparison to the extended Kalman filter.     

Sequential data assimilation with multiple models

Data assimilation is an essential tool for predicting the behavior of real physical systems given approximate simulation models and limited observations. For many complex systems, there may exist several models, each with different properties and predictive capabilities. It is desirable to incorporate multiple models into the assimilation procedure in order to obtain a more accurate prediction of the physics than any model alone can provide. In this paper, we propose a framework for conducting sequential data assimilation with multiple models and sources of data. The assimilated solution is a linear combination of all model predictions and data. One notable feature is that the combination takes the most general form with matrix weights. By doing so the method can readily utilize different weights in different sections of the solution state vectors, allow the models and data to have different dimensions, and deal with the case of a singular state covariance. We prove that the proposed assimilation method, termed direct assimilation, minimizes a variational functional, a generalized version of the one used in the classical Kalman filter. We also propose an efficient iterative assimilation method that assimilates two models at a time until all models and data are assimilated. The mathematical equivalence of the iterative method and the direct method is established. Numerical examples are presented to demonstrate the effectiveness of the new method.     

Response to Disturbance and Abundance of Final State： a Measure for Complexity？

We propose a new definition of complexity. The definition shows that when a system evolves to a final state via a transient state, its complexity depends on the abundance of both the final state and transient state. The abundance of the transient state may be described by the diversity of the response to disturbance. We hope that this definition can describe a clear boundary between simple systems and complex systems by showing that all the simple systems have zero complexity, and all the complex systems have positive complexity. Some examples of the complexity calculations are presented, which supports our hope.     

论动态信息理论

本文综述了作者的研究成果.近十年,作者将现有静态统计信息理论拓展至动态过程,建立了以表述动态信息演化规律的动态信息演化方程为核心的动态统计信息理论.基于服从随机性规律的动力学系统(如随机动力学系统和非平衡态统计物理系统)与遵守确定性规律的动力学系统(如电动力学系统)的态变量概率密度演化方程都可看成是其信息符号演化方程,推导出了动态信息(熵)演化方程.它们表明:对于服从随机性规律的动力学系统,动态信息密度随时间的变化率是由其在系统内部的态变量空间和传递过程的坐标空间的漂移、扩散和耗损三者引起的,而动态信息熵密度随时间的变化率则是由其在系统内部的态变量空间和传递过程的坐标空间的漂移、扩散和产生三者引起的.对于遵守确定性规律的动力学系统,动态信息(熵)演化方程与前者的相比,除动态信息(熵)密度在系统内部的态变量空间仅有漂移外,其余皆相同.信息和熵已与系统的状态和变化规律结合在一起,信息扩散和信息耗损同时存在.当空间噪声可略去时,将会出现信息波.若仅研究系统内部的信息变化,动态信息演化方程就约化为与表述上述动力学系统变化规律的动力学方程相对应的信息方程,它既可看成是表述动力学系统动态信息的演化规律,亦可看成是动力学系统的变化规律都可由信息方程表述.进而给出了漂移和扩散信息流公式、信息耗散率公式和信息熵产生率公式及动力学系统退化和进化的统一信息表述公式.得到了反映信息在传递过程中耗散特性的动态互信息公式和动态信道容量公式,它们在信道长度和信号传递速度之比趋于零的极限情况下变为现有的静态互信息公式和静态信道容量公式.所有这些新的理论公式和结果都是从动态信息演化方程统一推导出的.     

On the intrinsic time scales involved in synchronization: a data-driven approach

We address the problem of detecting, from scalar observations, the time scales involved in synchronization of complex oscillators with several spectral components. Using a recent data-driven procedure for analyzing nonlinear and nonstationary signals [Huang, Proc. R. Soc. London A 454, 903 (1998)], we decompose a time series in distinct oscillation modes which may display a time varying spectrum. When applied to coupled oscillators with multiple time scales, we found that motions are captured in a finite number of phase-locked oscillations. Further, in the synchronized state distinct phenomena as phase slips, anti-phase or perfect phase locking can be simultaneously observed at specific time scales. This fully data-driven approach (without a priori choice of filters or basis functions) is tested on numerical examples and illustrated on electric intracranial signals recorded from an epileptic patient. Implications for the study of the build-up of synchronized states in nonstationary and noisy systems are pointed out.     

Conditional Granger causality of diffusion processes

The statistical concept of Granger causality is defined by prediction improvement, i.e. the causing time series contains unique information about the future of the caused one. Recently we proposed extending this concept to bivariate diffusion processes by defining Granger causality for each point of the state space as the Granger causality of a process obtained by local linearisation. This provides a Granger causality map, well-defined at least in the vicinity of stable fixed points of the deterministic part of the dynamics. This extension has convenient properties, but carries several important limitations. In the current paper we show how the Granger causality of diffusion processes can be further generalized, incorporating in particular the concept of conditional causality. Moreover, we demonstrate the application potential to systems with a more complex attractor structure such as limit cycles or bistability of fixed points.     

Schwinger Functions and Light-Quark Bound States

We examine the applicability and viability of methods to obtain knowledge about bound states from information provided solely in Euclidean space. Rudimentary methods can be adequate if one only requires information about the ground and first excited state and assumptions made about analytic properties are valid. However, to obtain information from Schwinger functions about higher mass states, something more sophisticated is necessary. A method based on the correlator matrix can be dependable when operators are carefully tuned and errors are small. This method is nevertheless not competitive when an unambiguous analytic continuation of even a single Schwinger function to complex momenta is available.     

Synchronization of spatiotemporal chaos in a class of complex dynamical networks

This paper studies the synchronization of complex dynamical networks constructed by spatiotemporal chaotic systems with unknown parameters. The state variables in the systems with uncertain parameters are used to construct the parameter recognizers, and the unknown parameters are identified. Uncertain spatiotemporal chaotic systems are taken as the nodes of complex dynamical networks, connection among the nodes of all the spatiotemporal chaotic systems is of nonlinear coupling. The structure of the coupling functions between the connected nodes and the control gain are obtained based on Lyapunov stability theory. It is seen that stable chaos synchronization exists in the whole network when the control gain is in a certain range. The Gray--Scott models which have spatiotemporal chaotic behaviour are taken as examples for simulation and the results show that the method is very effective.