Exact asymptotics in eigenproblems for fractional Brownian covariance operators

Many results in the theory of Gaussian processes rely on the eigenstructure of the covariance operator. However, eigenproblems are notoriously hard to solve explicitly and closed form solutions are known only in a limited number of cases. In this paper we set up a framework for the spectral analysis of the fractional type covariance operators, corresponding to an important family of processes, which includes the fractional Brownian motion and its noise. We obtain accurate asymptotic approximations for the eigenvalues and the eigenfunctions. Our results provide a key to several problems, whose solution is long known in the standard Brownian case, but was missing in the more general fractional setting. This includes computation of the exact limits of $L^2$-small ball probabilities and asymptotic analysis of singularly perturbed integral equations, arising in mathematical physics and applied probability.     

Réduction a priori de modèles thermomécaniquesAn a priori model reduction method for thermomechanical problems

A model reduction method is proposed for finite element models. A previous computation of the state of the structure is not necessary. Residuals defined over the entire time interval and the Karhunen–Loève method provide basis functions. A non-incremental algorithm, from the LATIN method, is used to compute this basis functions. Because of the non-incremental feature, the reduced order model is representative for a large evolution of the state of the structure. To cite this article: D. Ryckelynck, C. R. Mecanique 330 (2002) 499–505.     

湍流大气中哈特曼传感器的模式波前复原误差

 分别采用Zernike和Karhunen-loeve两种模式波前复原法,分析了子孔径斜率测量不受噪声影响的理想情况下,哈特曼传感器对大气湍流畸变波前的模式复原误差与大气湍流相干长度、传感器的结构尺寸、模式复原阶数等的关系。结果表明Karhunen-loeve模式法比Zernike模式法的波前复原误差更小些。     

A new regularized stochastic approximation framework for stochastic inverse problems

We study the nonlinear inverse problem of estimating stochastic parameters in the fourth-order partial differential equation with random data. The primary focus is on developing a novel stochastic approximation framework for inverse problems consisting of three key components. As a first step, we reformulate the inverse problem into a stochastic convex optimization problem. The second step includes developing a new regularized stochastic extragradient framework for a nonlinear variational inequality, which subsumes the optimality conditions for the optimization formulation of the inverse problem. The third step involves modeling random variables by a Karhunen–Loève type finite-dimensional noise representation, allowing the direct and the inverse problems to be conveniently discretized. We show that the regularized extragradient methods are strongly convergent in a Hilbert space setting, and we also provide several auxiliary results for the inverse problem, including Lipschitz continuity and a derivative characterization of the solution map. We provide the outcome of computational experiments to estimate stochastic and deterministic parameters. The numerical results demonstrate the feasibility and effectiveness of the developed framework and validate stochastic approximation as an effective method for stochastic inverse problems.     

Karhunen Loeve变换用于光学相关识别旋转目标研究

提出利用Karhunen Loeve变换（K-L变换）识别旋转畸变目标的方法.利用K-L变换法对不同旋转目标进行处理得到不同特征值,取前两个最大的特征值所对应的特征向量分别进行滤波器编码,把得到的匹配滤波器分别加载到VLC相关器上进行光路识别.针对两个滤波器产生的相关峰值曲线波动性较大的缺点,采用线性叠加法进行改进.实验结果表明,该方法使峰值的波动性得到降低.     

Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function

This paper deals with the randomized heat equation defined on a general bounded interval [L₁, L₂] and with nonhomogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic process of the random heat equation defined on [0,1] with homogeneous boundary conditions. Results in the extant literature establish conditions under which the probability density function of the solution process to the random heat equation on [0,1] with homogeneous boundary conditions can be approximated. Via the changes of variable and the Random Variable Transformation technique, we set mild conditions under which the probability density function of the solution process to the random heat equation on a general bounded interval [L₁, L₂] and with nonhomogeneous boundary conditions can be approximated uniformly or pointwise. Furthermore, we provide sufficient conditions in order that the expectation and the variance of the solution stochastic process can be computed from the proposed approximations of the probability density function. Numerical examples are performed in the case that the initial condition process has a certain Karhunen‐Loève expansion, being Gaussian and non‐Gaussian.     

Optimal sampling points in reproducing kernel Hilbert spaces

The recent development of compressed sensing seeks to extract information from as few samples as possible. In such applications, since the number of samples is restricted, one should deploy the sampling points wisely. We are motivated to study the optimal distribution of finite sampling points in reproducing kernel Hilbert spaces, the natural background function spaces for sampling. Formulation under the framework of optimal reconstruction yields a minimization problem. In the discrete measure case, we estimate the distance between the optimal subspace resulting from a general Karhunen–Loève transform and the kernel space to obtain another algorithm that is computationally favorable. Numerical experiments are then presented to illustrate the effectiveness of the algorithms for the searching of optimal sampling points.     

Karhunen–Loève expansion for additive Slepian processes

For Gaussian random fields defined as additive processes based on Slepian processes, we study their Karhunen–Loève expansions and obtain the Pythagorean type distribution identities. As applications, the corresponding small deviation estimates are given.     

Microstructural characterization of materials by neural network technique

Ultrasonic signals received by pulse echo technique from plane parallel Zircaloy 2 samples of fixed thickness and of three different microstructures, were subjected to signal analysis, as conventional parameters like velocity and attenuation could not reliably discriminate them. The signals, obtained from these samples, were first sampled and digitized. Modified Karhunen Loeve Transform was used to reduce their dimensionality. A multilayered feed forward Artificial Neural Network was trained using a few signals in their reduced domain from the three different microstructures. The rest of the signals from the three samples with different microstructures were classified satisfactorily using this network.     

一种色彩向量的降维表示方法

提出了一种依据人眼的颜色视觉机制对真彩色显微生物图像进行分割的色彩向量的降维表示方法。该方法将由红绿蓝三色彩刺激经Karhunen Lo埁ｖｅ变换得到的Ｉ1I2 I3 色彩空间中的I1强度信息摒弃 ,将色彩特征由三维I1I2 I3 色彩空间向量降维为二维I2 I3 色彩向量。并以此I2 I3 色彩向量作为图像的色彩表示特征 ,运用反向传播网络对真彩色显微生物图像进行了分割实验。实验表明该方法应用于分割真彩色显微生物图像时 ,与采用I1I2色彩向量的降维方法相比 ,可优化色彩表示特征 ,加快网络的收敛速度 ,提高图像分割的正确率