An efficient ergodic simulation of multivariate stochastic processes with spectral representation

A simulation formula to generate stationary multivariate stochastic processes is derived from the Fourier-Stieltjes integral of spectral representation. It is proved that the proposed algorithm generates ergodic sample functions in the mean value and in the correlation when the sample length is equal to one period (the generated sample functions are periodic). The algorithm is very efficient computationally since it takes advantage of the fast Fourier transform technique. The simulation of longitudinal wind velocity fluctuations and the simulation of longitudinal and vertical wind fluctuating components on a bridge deck are performed. It has been noted that there are good agreements between the temporal and target auto-/cross-correlation functions of simulated wind velocities.     

基于Hermite插值的简化风场模拟

针对传统Deodatis谐波合成法的模拟效率受Cholesky分解次数制约的问题,通过对互谱密度矩阵分解引入Hermite插值,推导了基于Hermite插值的简化风场模拟方法,将传统谐波合成法中的Cholesky分解次数由n×N次缩减为n×2k次（2k < N）,从而大幅度提升了传统谐波合成法的计算效率。以某大跨度三塔悬索桥主梁风场模拟为例,分别基于传统Deodatis法、三次Lagrange插值法、Hermite插值法模拟了时长为4096 s的脉动风速时程,三者在模拟耗时与模拟精度方面的对比表明：Hermite插值法与Lagrange插值法均能显著提高传统谐波合成法的模拟效率;Hermite插值法的模拟效率略低于三次Lagrange插值法,但其对H矩阵的模拟精度明显高出一个层次,因而Hermite插值法在风场模拟中表现更优。采用基于Hermite插值的简化方法,模拟脉动风速的功率谱与相关函数均能与目标值吻合较好,表明所模拟的脉动风速仍具有较高的保真度。在此基础上,通过插值间距的优化分析给出了插值间距的建议取值区间。     

与反应谱相容的多点完全非平稳地震动随机过程的快速模拟

为了精确评估结构地震响应的概率特性,地震动随机过程的模拟需要考虑时间变异性(频率和强度非平稳)、空间变异性以及与反应谱的相容性。在经典的多点完全非平稳随机过程的模拟方法中,由于频率与时间变量不可分离,演化功率谱矩阵分解效率较低。为了加快谱矩阵的分解,提出了新Cholesky分解方法。该方法的核心是将演化谱矩阵分离为相位和模矩阵,而模矩阵进一步被转化为与时间不相关的延迟相干矩阵。通过与时间相关的演化谱矩阵相比,延迟相干矩阵仅与频率相关,这样就显著提高了矩阵分解的效率;此外,延迟相干矩阵更适合采用插值技术。最后,将新Cholesky分解方法和插值技术应用到生成与反应谱相容的随机方法中。结果表明：新Cholesky分解与插值能够高效地模拟多点完全非平稳并且与反应谱相容的地震动样本;线性插值与三次样条插值均可达到良好的分辨率,少量的插值点即可满足精度的要求。     

Bayesian Local Kriging

We consider the problem of constructing metamodels for computationally expensive simulation codes; that is, we construct interpolators/predictors of functions values (responses) from a finite collection of evaluations (observations). We use Gaussian process (GP) modeling and kriging, and combine a Bayesian approach, based on a finite set GP models, with the use of localized covariances indexed by the point where the prediction is made. Our approach is not based on postulating a generative model for the unknown function, but by letting the covariance functions depend on the prediction site, it provides enough flexibility to accommodate arbitrary nonstationary observations. Contrary to kriging prediction with plug-in parameter estimates, the resulting Bayesian predictor is constructed explicitly, without requiring any numerical optimization, and locally adjusts the weights given to the different models according to the data variability in each neighborhood. The predictor inherits the smoothness properties of the covariance functions that are used and its superiority over plug-in kriging, sometimes also called empirical-best-linear-unbiased predictor, is illustrated on various examples, including the reconstruction of an oceanographic field over a large region from a small number of observations. Supplementary materials for this article are available online.     

A spectral representation based model for Monte Carlo simulation

A new model is proposed for generating samples of real-valued stationary Gaussian processes. The model is based on the spectral representation theorem stating that a weakly stationary process can be viewed as a superposition of harmonics with random properties. The classical use of this theorem for Monte Carlo simulation is based on models consisting of a superposition of harmonics with fixed frequencies but random amplitude and phase. The resulting samples have the same period depending on the discretization of the frequency band. In contrast, the proposed model consists of a superposition of harmonics with random amplitude, phase, and frequency so that different samples have different periods depending on the particular sample values of the harmonic frequencies.

A band limited Gaussian white noise process is used to illustrate the proposed Monte Carlo simulation algorithm and demonstrate that the estimates of the covariance function based on the samples of the proposed model are not periodic.     

On the determination of the power spectrum of randomly excited oscillators via stochastic averaging: An alternative perspective

An approximate formula which utilizes the concept of conditional power spectral density (PSD) has been employed by several investigators to determine the response PSD of stochastically excited nonlinear systems in numerous applications. However, its derivation has been treated to date in a rather heuristic, even “unnatural” manner, and its mathematical legitimacy has been based on loosely supported arguments. In this paper, a perspective on the veracity of the formula is provided by utilizing spectral representations both for the excitation and for the response processes of the nonlinear system; this is done in conjunction with a stochastic averaging treatment of the problem. Then, the orthogonality properties of the monochromatic functions which are involved in the representations are utilized. Further, not only stationarity but ergodicity of the system response are invoked. In this context, the nonlinear response PSD is construed as a sum of the PSDs which correspond to equivalent response amplitude dependent linear systems. Next, relying on classical excitation-response PSD relationships for these linear systems leads, readily, to the derivation of the formula for the determination of the PSD of the nonlinear system. Related numerical results are also included.     

Spectral density estimation of stochastic vector processes

A spectral density matrix estimator for stationary stochastic vector processes is studied. As the duration of the analyzed data tends to infinity, the probability distribution for this estimator at each frequency approaches a complex Wishart distribution with mean equal to an aliased version of the power spectral density at that frequency. It is shown that the spectral density matrix estimators corresponding to different frequencies are asymptotically statistically independent. These properties hold for general stationary vector processes, not only Gaussian processes, and they allow efficient calculation of updated probabilities when formulating a Bayesian model updating problem in the frequency domain using response data. A three-degree-of-freedom Duffing oscillator is used to verify the results.     

基于本征正交分解的谱表示法模拟风场的误差

推导了本征正交分解(Proper Orthogonal Decomposition,POD)型谱表示法模拟所得平稳正态脉动风场的偏度误差和随机误差.从POD型谱表示法的模拟公式出发,推导了Ⅳ变量风场模拟结果序列的样本均值、相关函数、功率谱函数和根方差等前二阶矩统计特征的时域估计表达式;并证明了时域估计相关函数是正态过程,功率谱函数为非正态随机过程.进一步,计算上述样本时域估计二阶矩特征的均值和根方差,即得到了POD型谱表示法模拟所得风场的各统计量时域估计的偏度误差和随机误差,并以此给出了误差计算的通式.算例中统计误差和理论误差值的对比验证了所推导的解析解.     

具有桥塔风效应的桥梁风场数值模拟

将特征正交分解型谱表示法运用于模拟具有桥塔风效应的桥梁风场中。首先介绍桥塔风效应和桥梁风场的概率描述,然后结合模态截断技术,介绍特征正交分解（Proper Orthogonal Decomposition,POD）型谱表示法,该方法是对常用的原型谱表示法的继承和提高,且物理概念更加清晰。通过引入对风速谱矩阵的显式预分解,推导模拟具有桥塔风效应的桥梁风场的简化计算公式,将对付目标功率谱矩阵的特征值分解运算简化为对实矩阵的运算。该方法可用FFT加速,相对于原有的模拟方法具有较高的计算效率。最后,以模拟龙潭河特大桥施工最大双悬臂阶段的脉动风速场为算例,解释了脉动风速过程特征正交分解模态的物理意义,说明该方法的可靠性。在算例中,观察到复杂相关结构下,特征正交分解发生振型交换的现象,并分析其原因。     

Models for space–time random functions

Models are developed for random functions of space and time from samples of these functions and any other information when available. Most of the models in the paper can be viewed as extensions of Karhunen–Loève (KL) representations for random fields. Their samples are linear forms of basis functions with random coefficients which are extracted from samples of by singular value decomposition. The coefficients of these forms are stochastic processes rather than random variables. The proposed models can be used to generate large sets of samples whose statistics are similar to those of target random functions. Theoretical arguments and numerical examples are presented to establish properties of the proposed models, assess their accuracy, and illustrate their implementation.     

一种新的求解零空间线性鉴别分析的快速算法

利用随机矩阵相乘是最近提出的一种求解零空间线性鉴别分析的算法,但是此算法需要对一个n×n的矩阵进行特征值分解(n指的是训练样本数),使得其算法复杂度依然较高。为了进一步提高零空间线性鉴别分析算法的求解速度,本文提出了一种新的利用随机矩阵相乘的求解零空间线性鉴别分析的快速算法。本文的算法不需要对n×n的矩阵进行特征值分解,使得其算法复杂度比现有的零空间线性鉴别分析求解算法要低得多。理论分析和在人脸数据库上的实验表明,本文算法的计算速度远比现有的零空间线性鉴别分析求解算法要快,但是其识别率与现有的零空间线性鉴别分析求解算法相同。     

A comparison of interpolation grids over the triangle or the tetrahedron

A simple strategy for constructing a sequence of increasingly refined interpolation grids over the triangle or the tetrahedron is discussed with the goal of achieving uniform convergence and ensuring high interpolation accuracy. The interpolation nodes are generated based on a one-dimensional master grid comprised of the zeros of the Lobatto, Legendre, Chebyshev, and second-kind Chebyshev polynomials. Numerical computations show that the Lebesgue constant and interpolation accuracy of some proposed grids compare favorably with those of alternative grids constructed by optimization, including the Fekete set. While some sets are clearly preferable to others, no single set can claim uniformly better convergence properties as the number of nodes is raised.     

Spectral density of oscillator with bilinear stiffness and white noise excitation

The power spectral density of an oscillator with bilinear stiffness excited by Gaussian white noise is considered. A method originally proposed by Krenk and Roberts [J Appl Mech 66 (1999) 225] relying on slowly changing energy for lightly damped systems is applied. In this method an approximate solution for the power spectral density at a given energy level is obtained by considering local similarity with the free undamped response. The total spectrum is obtained by integrating over all energy levels weighting each with the stationary probability density of the energy. The accuracy of the approximate analytical solution is demonstrated by comparing with results obtained by stochastic simulation. It is shown how the method successfully captures the broadening of the resonance peak and the presence of higher harmonics in the power spectral density of strongly non-linear systems.