Auto-correlated behavior of WTI crude oil volatilities: A multiscale perspective

In this paper, we investigate the long-range auto-correlated behavior of WTI crude oil volatility series employing multifractal detrended fluctuation analysis. Our findings show that the for small time scales, the auto-correlations of volatilities were multifractal while for large time scales, the auto-correlations were nearly monofractal. Based on multiscale analysis, we also investigate the dynamics of auto-correlations for different intervals of time scales and find that several shocks could make significant effects on the auto-correlated behaviors for small time scales. Analyzing the dynamics of multifractality degrees of auto-correlations for small time scales, we find that the stronger auto-correlations were always related to the lower degrees of multifractality. At last, we have discussions on the determination factors of price behavior, the predictive implications of scaling behavior in volatilities for oil markets and the reasons why long-range auto-correlations of volatility were always strong for both small time scales and large time scales. Our results are very important theoretically and practically.     

Multiscale combustion and turbulence

Multiscale physics is the interaction of different physical processes occurring at largely separated scales. In combustion, many elementary reactions combine to only a few, but still have separated time scales. In flames, owing to the presence of diffusion, time scales manifest themselves as length scales, i.e. thicknesses of reaction layers embedded within each other. For premixed flames there results a single velocity scale, the laminar burning velocity, which in turn defines a flame thickness and a flame time as global length and time scales, respectively. The laminar burning velocity represents the simplest microscale model to be used at a premixed combustion interface.While combustion is a multiscale process, this is not so evident for turbulence. Based on the picture of a cascade process traditional turbulent closure approximations treat turbulence as a single-scale problem. Attempts to model turbulent combustion in the same way by using methods developed for non-reacting turbulent flows therefore must fail, because they ignore the multiscale nature of combustion.There is, however, a long tradition and much progress in multiscale modeling of combustion, both on the macroscale as well as on the microscale level. Unfortunately much of that work is conceived only in its particular context, not as part of a multiscale approach. For instance, papers in the TURBULENT FLAMES Colloquium and the FIRE RESEARCH Colloquium at this and at previous Combustion Symposia often take the viewpoint of macroscale modeling only, while REACTION KINETICS and LAMINAR FLAMES concentrate on microscale aspects. What seems to be needed is a more explicit reference to the needs of models developed in the other parts of the community. Furthermore, research is needed to develop suitable definitions of the interface between macroscale and microscale models.     

A multiscale coupling method for the modeling of dynamics of solids with application to brittle cracks

We present a multiscale model for numerical simulations of dynamics of crystalline solids. The method combines the continuum nonlinear elasto-dynamics model, which models the stress waves and physical loading conditions, and molecular dynamics model, which provides the nonlinear constitutive relation and resolves the atomic structures near local defects. The coupling of the two models is achieved based on a general framework for multiscale modeling – the heterogeneous multiscale method (HMM). We derive an explicit coupling condition at the atomistic/continuum interface. Application to the dynamics of brittle cracks under various loading conditions is presented as test examples.     

Normal form transforms separate slow and fast modes in stochastic dynamical systems

Modelling stochastic systems has many important applications. Normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from insignificant detailed microscale dynamics. We explore such coordinate transforms of stochastic differential systems when the dynamics have both slow modes and quickly decaying modes. The thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems. Thus we not only must reduce the dimensionality of the dynamics, but also endeavour to separate all slow processes from all fast time processes, both deterministic and stochastic. Quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes. Some examples demonstrate that the coordinate transform may be only locally valid or may be globally valid depending upon the dynamical system. The results will help us accurately model, interpret and simulate multiscale stochastic systems.     

A hierarchical fracture model for the iterative multiscale finite volume method

An iterative multiscale finite volume (i-MSFV) method is devised for the simulation of multiphase flow in fractured porous media in the context of a hierarchical fracture modeling framework. Motivated by the small pressure change inside highly conductive fractures, the fully coupled system is split into smaller systems, which are then sequentially solved. This splitting technique results in only one additional degree of freedom for each connected fracture network appearing in the matrix system. It can be interpreted as an agglomeration of highly connected cells; similar as in algebraic multigrid methods. For the solution of the resulting algebraic system, an i-MSFV method is introduced. In addition to the local basis and correction functions, which were previously developed in this framework, local fracture functions are introduced to accurately capture the fractures at the coarse scale. In this multiscale approach there exists one fracture function per network and local domain, and in the coarse scale problem there appears only one additional degree of freedom per connected fracture network. Numerical results are presented for validation and verification of this new iterative multiscale approach for fractured porous media, and to investigate its computational efficiency. Finally, it is demonstrated that the new method is an effective multiscale approach for simulations of realistic multiphase flows in fractured heterogeneous porous media.     

Analysis of complex time series using refined composite multiscale entropy

Multiscale entropy (MSE) is an effective algorithm for measuring the complexity of a time series that has been applied in many fields successfully. However, MSE may yield an inaccurate estimation of entropy or induce undefined entropy because the coarse-graining procedure reduces the length of a time series considerably at large scales. Composite multiscale entropy (CMSE) was recently proposed to improve the accuracy of MSE, but it does not resolve undefined entropy. Here we propose a refined composite multiscale entropy (RCMSE) to improve CMSE. For short time series analyses, we demonstrate that RCMSE increases the accuracy of entropy estimation and reduces the probability of inducing undefined entropy.     

Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes

We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes, which can provide a good balance between the numerical accuracy and computational cost. The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions. The multiscale basis functions have abilities to capture originally perturbed information in the local problem, as a result, our method is capable of reducing the boundary layer errors remarkably on graded meshes, where the layer-adapted meshes are generated by a given parameter. Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L² norm and first order convergence in the energy norm on graded meshes, which is independent of ε. In contrast with the conventional methods, our method is much more accurate and effective.     

A finite difference approach with velocity interfacial conditions for multiscale computations of crystalline solids

We propose a class of velocity interfacial conditions and formulate a finite difference approach for multiscale computations of crystalline solids with relatively strong nonlinearity and large deformation. Full atomistic computations are performed in a selected small subdomain only. With a coarse grid cast over the whole domain and the coarse scale dynamics computed by finite difference schemes, we perform a fast average of the fine scale solution in the atomistic subdomain to force agreement between scales. During each coarse scale time step, we adopt a linear wave approximation around the interface, with the wave speed updated using the coarse grid information. We then develop a class of velocity interfacial conditions with different order of accuracy. The interfacial conditions are straightforward to formulate, easy to implement, and effective for reflection reduction in crystalline solids with strong nonlinearity. The nice features are demonstrated through numerical tests.     

Anomalous diffusion in folding dynamics of minimalist protein landscape

A novel method is proposed to quantify collectivity at different space and time scales in multiscale dynamics of proteins. This is based on the combination of the principal component (PC) and the concept recently developed for multiscale dynamical systems called the finite size Lyapunov exponent. The method can differentiate the well-known apparent correlation along the low-indexed PCs in multidimensional Brownian systems from the correlated motion inherent to the system. As an illustration, we apply the method to a model protein of 46 amino beads with three different types of residues. We show how the motion of the model protein changes depending on the space scales and the choices of degrees of freedom. In particular, anomalous superdiffusion is revealed along the low-indexed PC in the unfolded state. The implication of superdiffusion in the process of folding is also discussed.     

Modified DFA and DCCA approach for quantifying the multiscale correlation structure of financial markets

We use multiscale detrended fluctuation analysis (MSDFA) and multiscale detrended cross-correlation analysis (MSDCCA) to investigate auto-correlation (AC) and cross-correlation (CC) in the US and Chinese stock markets during 1997–2012. The results show that US and Chinese stock indices differ in terms of their multiscale AC structures. Stock indices in the same region also differ with regard to their multiscale AC structures. We analyze AC and CC behaviors among indices for the same region to determine similarity among six stock indices and divide them into four groups accordingly. We choose S&P500, NQCI, HSI, and the Shanghai Composite Index as representative samples for simplicity. MSDFA and MSDCCA results and average MSDFA spectra for local scaling exponents (LSEs) for individual series are presented. We find that the MSDCCA spectrum for LSE CC between two time series generally tends to be greater than the average MSDFA LSE spectrum for individual series. We obtain detailed multiscale structures and relations for CC between the four representatives. MSDFA and MSDCCA with secant rolling windows of different sizes are then applied to reanalyze the AC and CC. Vertical and horizontal comparisons of different window sizes are made. The MSDFA and MSDCCA results for the original window size are confirmed and some new interesting characteristics and conclusions regarding multiscale correlation structures are obtained.     

Slow integral manifolds for Lagrangian fluid dynamics in unsteady geophysical flows

The authors consider Lagrangian motion of fluid particles in unsteady gravity currents in geophysical flows. The vertical motion of fluid particles, especially the induced vertical mixing in these currents, is partially responsible for the ocean thermohaline circulation, and thus plays a role in the global climate dynamics.First, a reduced dynamic system for slow variables is derived for a nonautonomous multiscale system. The reduced system, still nonautonomous, is the original system restricted to a centre-like nonautonomous invariant manifold (so-called slow manifold) which holds slow motions of the system. An algorithm is also presented to obtain an approximation of the nonautonomous slow manifold. A novelty here is that the reduction principle applies to nonautonomous multiscale systems which satisfy conditions that are true only locally in space (as in many physical cases). This makes the reduction principle applicable to real physical systems.Then, this invariant manifold reduction principle is applied to an approximate conceptual Lagrangian model of gravity currents and a reduced nonautonomous system for slow vertical motion is obtained. This reduced system may be useful as a conceptual tractable tool for understanding some features of vertical mixing in unsteady gravity currents.     

纳米体系的计算表征

纳米系统在许多应用中发挥着重要作用. 由于纳米体系的复杂性,准确表征其结构和性质很具挑战性. 一种重要的表征手段是基于第一性原理电子结构计算的理论模拟. 近年来低标度和高精度的电子结构算法得到了极大的发展,特别是,适用于周期性体系的杂化密度泛函计算效率得到了显著的提高. 利用电子结构信息,可以发展模拟算法直接获得可与实验对比的数据. 例如,扫描隧道显微谱现在可以使用先进的算法高效地模拟. 当感兴趣的系统与环境存在强耦合时,例如在近藤效应中,求解级联运动方程被证明是一个非常有效的计算表征方法;此外,激发态动力学的第一性原理模拟近年来进展迅速,其中非绝热分子动力学方法发挥了重要作用. 对于涉及化学反应的纳米系统,例如石墨烯生长体系,往往需要发展多尺度模拟方法来表征其原子细节. 本文综述了纳米系统计算表征算法的一些最新进展,先进的算法和软件对于我们更好地了解纳米世界至关重要.     

Bridging the scales in nano engineering and science

This review article describes various multiscale approaches, development of which was spurred by the emergence of nanotechnology. The multiscale approaches are grouped into two main categories: information-passing and concurrent. In the concurrent multiscale methods both, the discrete and continuum scales are simultaneously resolved, whereas in the information-passing schemes, the discrete scale is modelled and its gross response is infused into the continuum scale. Most of the information-passing approaches provide sublinear computational complexity, (i.e., scales sublinearly with the cost of solving a fine scale problem), but the quantities of interest are limited to or defined only on the coarse scale. The issues of appropriate scale selection and uncertainty quantification are also reviewed.     

A general strategy for designing seamless multiscale methods

We present a new general framework for designing multiscale methods. Compared with previous work such as Brandt’s systematic up-scaling, the heterogeneous multiscale method (HMM) and the “equation-free” approach, this new framework has the distinct feature that it does not require reinitializing the microscale model at each macro time step or each macro iteration step. In the new strategy, the macro- and micro-models evolve simultaneously using different time steps (and therefore different clocks), and they exchange data at every step. The micro-model uses its own appropriate time step. The macro-model runs at a slower pace than required by accuracy and stability considerations for the macroscale dynamics, in order for the micro-model to relax. Examples are discussed and application to modeling complex fluids is presented.     

Amplification of intrinsic fluctuations by the Lorenz equations

Macroscopic systems (e.g., hydrodynamics, chemical reactions, electrical circuits, etc.) manifest intrinsic fluctuations of molecular and thermal origin. When the macroscopic dynamics is deterministically chaotic, the intrinsic fluctuations may become amplified by several orders of magnitude. Numerical studies of this phenomenon are presented in detail for the Lorenz model. Amplification to macroscopic scales is exhibited, and quantitative methods (binning and a difference-norm) are presented for measuring macroscopically subliminal amplification effects. In order to test the quality of the numerical results, noise induced chaos is studied around a deterministically nonchaotic state, where the scaling law relating the Lyapunov exponent to noise strength obtained for maps is confirmed for the Lorenz model, a system of ordinary differential equations.     

Bearing Fault Diagnosis Using Refined Composite Generalized Multiscale Dispersion Entropy-Based Skewness and Variance and Multiclass FCM-ANFIS

Bearing vibration signals typically have nonlinear components due to their interaction and coupling effects, friction, damping, and nonlinear stiffness. Bearing faults affect the signal complexity at various scales. Hence, measuring signal complexity at different scales is helpful to diagnosis of bearing faults. Numerous studies have investigated multiscale algorithms; nevertheless, multiscale algorithms using the first moment lose important complexity data. Accordingly, generalized multiscale algorithms have been recently introduced. The present research examined the use of refined composite generalized multiscale dispersion entropy (RCGMDispEn) based on the second moment (variance) and third moment (skewness) along with refined composite multiscale dispersion entropy (RCMDispEn) in bearing fault diagnosis. Moreover, multiclass FCM-ANFIS, which is a combination of adaptive network-based fuzzy inference systems (ANFIS), was developed to improve the efficiency of rotating machinery fault classification. According to the results, it is recommended that generalized multiscale algorithms based on variance and skewness be examined for diagnosis, along with multiscale algorithms, and be used to achieve an improvement in the results. The simultaneous usage of the multiscale algorithm and generalized multiscale algorithms improved the results in all three real datasets used in this study.     

基于多尺度加权主成分分析的SF6红外光谱分析

利用红外光谱法分析SF6气体及其衍生物是判断气体绝缘组合电器(GIS)运行状态和故障的一种重要手段。传统的诊断方法过程繁琐、效率低下,而且受主观因素的影响较大。本文指出可以采用机器学习的方法实现GIS设备的故障诊断,并提出了多尺度加权主成分分析的特征提取方法。多尺度加权主成分分析结合了主成分分析和多尺度分解的特点,保证了尺度特征信息的最大化,并且修正了特征向量在数据分类时的权重。通过对广西电力研究院提供的SF6及其衍生物的红外光谱进行分析,证明了多尺度加权主成分分析算法对训练样本的分类效果要比标准的主成分分析算法好3～4倍。     

Numerical methods for stochastic partial differential equations with multiple scales

A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Commun. Pure Appl. Math., 58(11) (2005) 1544–1585]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blömker, M. Hairer, G.A. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20(7) (2007) 1721–1744]. For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method.     

Multistability in coupled different-dimensional dynamical systems

Multistability or coexistence of different chaotic attractors for a given set of parameters depending on the initial condition only is one of the most exciting phenomenon in dynamical systems. The schemes to design multistability systems via coupling two identical or non-identical but the same-dimensional systems have been proposed earlier. Coupled different-dimensional systems are very useful to describe the real-world physical and biological systems. In this paper, a scheme for designing a multistable system by coupling two different-dimensional dynamical systems has been proposed. Coupled Lorenz and Lorenz–Stenflo systems have been considered to illustrate the scheme. The efficiency of the scheme is shown numerically, by presenting phase diagrams, bifurcation diagrams and variation of maximum Lyapunov exponents.