Local Gaussian Process Approximation for Large Computer Experiments

We provide a new approach to approximate emulation of large computer experiments. By focusing expressly on desirable properties of the predictive equations, we derive a family of local sequential design schemes that dynamically define the support of a Gaussian process predictor based on a local subset of the data. We further derive expressions for fast sequential updating of all needed quantities as the local designs are built up iteratively. Then we show how independent application of our local design strategy across the elements of a vast predictive grid facilitates a trivially parallel implementation. The end result is a global predictor able to take advantage of modern multicore architectures, providing a nonstationary modeling feature as a bonus. We demonstrate our method on two examples using designs with thousands of data points, and compare to the method of compactly supported covariances. Supplementary materials for this article are available online.     

A Multiresolution Gaussian Process Model for the Analysis of Large Spatial Datasets

We develop a multiresolution model to predict two-dimensional spatial fields based on irregularly spaced observations. The radial basis functions at each level of resolution are constructed using a Wendland compactly supported correlation function with the nodes arranged on a rectangular grid. The grid at each finer level increases by a factor of two and the basis functions are scaled to have a constant overlap. The coefficients associated with the basis functions at each level of resolution are distributed according to a Gaussian Markov random field (GMRF) and take advantage of the fact that the basis is organized as a lattice. Several numerical examples and analytical results establish that this scheme gives a good approximation to standard covariance functions such as the Matérn and also has flexibility to fit more complicated shapes. The other important feature of this model is that it can be applied to statistical inference for large spatial datasets because key matrices in the computations are sparse. The computational efficiency applies to both the evaluation of the likelihood and spatial predictions.     

Local Gaussian Process Model for Large-Scale Dynamic Computer Experiments

The recent accelerated growth in the computing power has generated popularization of experimentation with dynamic computer models in various physical and engineering applications. Despite the extensive statistical research in computer experiments, most of the focus had been on the theoretical and algorithmic innovations for the design and analysis of computer models with scalar responses. In this article, we propose a computationally efficient statistical emulator for a large-scale dynamic computer simulator (i.e., simulator which gives time series outputs). The main idea is to first find a good local neighborhood for every input location, and then emulate the simulator output via a singular value decomposition (SVD) based Gaussian process (GP) model. We develop a new design criterion for sequentially finding this local neighborhood set of training points. Several test functions and a real-life application have been used to demonstrate the performance of the proposed approach over a naive method of choosing local neighborhood set using the Euclidean distance among design points. The supplementary material, which contains proof of the theoretical results, detailed algorithms, additional simulation results, and R codes, are available online.     

Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations

In this paper we establish lower and upper Gaussian bounds for the solutions to the heat and wave equations driven by an additive Gaussian noise, using the techniques of Malliavin calculus and recent density estimates obtained by Nourdin and Viens in [17]. In particular, we deal with the one-dimensional stochastic heat equation in [0, 1] driven by the space-time white noise, and the stochastic heat and wave equations in R^d${\mathbb{R}}^d$ (d≥1$d\ge 1$ and d≤3$d\le 3$, respectively) driven by a Gaussian noise which is white in time and has a general spatially homogeneous correlation.     

Approximate solution for some stochastic differential equations involving both Gaussian and Poissonian white noises

By combining the Kramers-Moyal expansion with fractional Brownian motion of order n, in a formal symbolic calculus, one can obtain an approximation for the solution of some stochastic differential equations involving both Gaussian and Poissonian white noises, in terms of rotating Gaussian white noises on the grid defined by the complex roots of the unity. Illustrative examples are outlined.     

Stochastic stability of FitzHugh-Nagumo systems in infinite lattice perturbed by Gaussian white noise

The current paper is devoted to the study of stochastic stability of FitzHugh-Nagumo systems in infinite lattice perturbed by Gaussian white noise. We first study the dynamics of stochastic FitzHugh-Nagumo systems, then prove the existence and uniqueness of their equilibriums, which mix exponentially. Finally, we investigate asymptotic behavior of equilibriums when the size of noise gets to zero.     

A Remark on the Equivalence between Poisson and Gaussian Stochastic Partial Differential Equations

We discuss the connection between Gaussian and Poisson noise Wick-type stochastic partial differential equations.     

一种产生广义高斯分布随机数的新算法

本文提出了一种生成广义高斯分布 (GGD)随机数的通用算法 .该算法针对GGD密度函数衰减性的特点 ,采用变步长的方法 ,综合运用了逆函数法、近似复合抽样法及变换抽样法 .通过调整分布参数的数值 ,就能产生具有任何形状参数和任何方差的GGD随机数 ,简单易于实现 .最后将仿真实验结果与已有算法的结果做比较 ,并用 χ2检验法和Kolmogorov Smirnov检验法 (K S检验法 )验证该方法的有效性 .     

一类随机大系统的稳定性

本文对随机大系统dx(t)=b(t,x(t))dt σ(t,x(t)dω(t)采用较文[1]更一般的分解形式，利用比较法和分解－集结法得到该随机大系统随机（一致）稳定性和渐近（一致）稳定性的比较准则。     

随机利率下服从指数O-U过程的可转换债券的鞅定价

从定量的角度分析了随机利率下有股利分配的可转换债券的价值构成,并在股价服从广义O-U过程的条件下,利用鞅定价方法推导出可转换债券的定价公式.     

大垂度柔索的动力学建模与仿真

用多刚体-球铰模型对大垂度柔索进行离散化建模,利用多刚体系统动力学理论建立了该模型的动力学方程.采用矩阵广义逆理论对其位移和速度进行修正,以消除该微分-代数方程在数值分析中的违约现象.数值仿真证明了该方法的正确性.     

Efficient Algorithms for Bayesian Nearest Neighbor Gaussian Processes

We consider alternate formulations of recently proposed hierarchical nearest neighbor Gaussian process (NNGP) models for improved convergence, faster computing time, and more robust and reproducible Bayesian inference. Algorithms are defined that improve CPU memory management and exploit existing high-performance numerical linear algebra libraries. Computational and inferential benefits are assessed for alternate NNGP specifications using simulated datasets and remotely sensed light detection and ranging data collected over the U.S. Forest Service Tanana Inventory Unit (TIU) in a remote portion of Interior Alaska. The resulting data product is the first statistically robust map of forest canopy for the TIU. Supplemental materials for this article are available online.     

Large deviations for stochastic generalized porous media equations

The large deviation principle is established for the distributions of a class of generalized stochastic porous media equations for both small noise and short time.     

战略导弹研制风险分析的仿真技术

本文以战略导弹研制风险分析为背景，对随机网络数据仿真技术原理进行了探讨，分析了曾广泛应用的仿真软件ＶＥＲＴＳＬＡＭ等的不足，并提出了改进方法，文中给出一种含信息流多层次综合网络仿真的构模框架，并用该构模框架对战略导弹风险分析的仿真模型作了研究。     

Modeling and solution strategies for unconstrained stochastic optimization problems

We review some modeling alternatives for handling risk in decision-making processes for unconstrained stochastic optimization problems. Solution strategies are discussed and compared.Invited lecture at the International Institute on Stochastics and Optimization, Gargnano, Italy, September 1–10, 1982.Supported in part by a Guggenheim Fellowship and a grant of the National Science Foundation.     

Nonconforming finite element method for stochastic Stokes equations

In this paper, we study nonconforming finite element method for stochastic Stokes equation driven by white noise. We apply “green function framework” and standard duality technique to study the error estimate for velocity in L²$L^2$-norm and for pressure in H^-1$H^{-1}$-norm. Finally, numerical experiment proves our theoretical results.     

随机扰动下的非自治种群的随机持久性(英文)

本文考虑非线性随机扰动下的生态种群的问题.首先给出全局正解的存在性.其次,在合理的条件下讨论随机最终有界和随机持久问题,同时也给出解的渐近估计.     

Central limit theorems for non-linear functionals of Gaussian fields

In the present paper it is shown that the central limit theorem holds for some non-linear functionals of stationary Gaussian fields if the correlation function of the underlying field tends fast enough to zero. The results are formulated in terms of the Hermite rank of the functional and of the rate of the correlation function. Then we show an example when the limit field is self-similar and Gaussian but not necessarily consisting of independent elements.     

Existence for dynamic contact of a stochastic viscoelastic Gao Beam

This work presents and analyzes a model for the vibrations of a viscoelastic Gao Beam, which may come in contact with a deformable random foundation and allows for stochastic inputs. The body force involves a stochastic integral that includes Brownian motion. In addition, the gap between the beam and the foundation is a stochastic process, which is one of the novelties in the paper, and contact is described with the normal compliance condition. The existence and uniqueness of strong solutions to the model is established and it is shown that the solutions are adapted to the filtration determined by a given Wiener process for the stochastic force noise term.