Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees

A domain decomposition method is developed for the numerical solution of nonlinear parabolic partial differential equations in any space dimension, based on the probabilistic representation of solutions as an average of suitable multiplicative functionals. Such a direct probabilistic representation requires generating a number of random trees, whose role is that of the realizations of stochastic processes used in the linear problems. First, only few values of the sought solution inside the space-time domain are computed (by a Monte Carlo method on the trees). An interpolation is then carried out, in order to approximate interfacial values of the solution inside the domain. Thus, a fully decoupled set of sub-problems is obtained. The algorithm is suited to massively parallel implementation, enjoying arbitrary scalability and fault tolerance properties. Pruning the trees is shown to increase appreciably the efficiency of the algorithm. Numerical examples conducted in 2D, including some for the KPP equation, are given.     

Scattering of a plane wave from a periodic random surface: a probabilistic approach

This paper deals with a probabilistic formulation of the wave scattering from a periodic random surface. When a plane wave is incident on a random surface described by a periodic stationary stochastic process, it is shown by a group-theoretic consideration that the scattered wave may have a stochastic Floquet form, i.e. a product of a periodic stationary random function and an exponential phase factor. Such a periodic stationary random function is then written by a harmonic series representation similar to a Fourier series, where Fourier coefficients are mutually correlated stationary processes instead of constants. The mutually correlated stationary processes are represented by Wiener - Hermite functional series with unknown coefficient functions called Wiener kernels. In case of a slightly rough surface and TE wave incidence, low-order Wiener kernels are determined from the boundary condition. Several statistical properties of the scattering are calculated and illustrated in figures.     

Stochastic matching problem

The matching problem plays a basic role in combinatorial optimization and in statistical mechanics. In its stochastic variants, optimization decisions have to be taken given only some probabilistic information about the instance. While the deterministic case can be solved in polynomial time, stochastic variants are worst-case intractable. We propose an efficient method to solve stochastic matching problems which combines some features of the survey propagation equations and of the cavity method. We test it on random bipartite graphs, for which we analyze the phase diagram and compare the results with exact bounds. Our approach is shown numerically to be effective on the full range of parameters, and to outperform state-of-the-art methods. Finally we discuss how the method can be generalized to other problems of optimization under uncertainty.     

Pairwise correlations in a three-partite quantum system from a prequantum random field

Prequantum classical statistical field theory (PCSFT) is a model that provides the possibility to represent the averages of quantum observables (including correlations of observables on subsystems of a composite system) as averages with respect to fluctuations of classical random fields. In view of the PCSFT terminology, quantum states are classical random fields. The aim of our approach is to represent all quantum probabilistic quantities by means of classical random fields. We obtain the classical-random-field representation for pairwise correlations in three-partite quantum systems. The three-partite case (surprisingly) differs substantially from the bipartite case. As an important first step, we generalized the theory developed for pure quantum states of bipartite systems to the states given by density operators.     

On the probabilistic treatment of fields

Some basic problems of the probabilistic treatment of fields are considered, proceeding from the fundamentals of the complete probability theory. Two essentially equivalent definitions of random fields related to continuous objects are suggested. It is explained why the conventional classical probabilistic treatment generally is inapplicable to fields in principle. Two types of finite-dimensional random variables created by random fields are compared. Some general regularities related to Lagrangian and Hamiltonian partial equations, obtainable proceeding from the corresponding sets of ordinary differential equations, are revealed by using the functional derivative defined anew. It is shown that Hamiltonian random fields give rise to two types of Hamiltonian random variables, variables of the second type being those considered in the author's previous paper and immediately suited to the quantum approach. The results obtained are illustrated by some general examples. Critical remarks concerning second quantization are made, demonstrating the artificiality of this method. It is emphasized that the given probabilistic consideration of fields cannot be directly applied to, for instance, the electromagnetic field, which needs a special treatment.     

Numerical analysis of uncertain temperature field by stochastic finite difference method

Based on the combination of stochastic mathematics and conventional finite difference method,a new numerical computing technique named stochastic finite difference for solving heat conduction problems with random physical parameters,initial and boundary conditions is discussed.Begin with the analysis of steady-state heat conduction problems,difference discrete equations with random parameters are established,and then the computing formulas for the mean value and variance of temperature field are derived by the second-order stochastic parameter perturbation method.Subsequently,the proposed random model and method are extended to the field of transient heat conduction and the new analysis theory of stability applicable to stochastic difference schemes is developed.The layer-by-layer recursive equations for the first two probabilistic moments of the transient temperature field at different time points are quickly obtained and easily solved by programming.Finally,by comparing the results with traditional Monte Carlo simulation,two numerical examples are given to demonstrate the feasibility and effectiveness of the presented method for solving both steady-state and transient heat conduction problems.     

A uniform asymptotic integral representation for the fourth moment of the field of a wave propagating in a statistically homogeneous medium

In this paper, a uniform integral representation has been obtained for the fourth moment of the field of a wave propagating in a medium with random large-scale irregularities. The solution to the equation was obtained using a method of integral transformations and Maslov's complex WKB method. The representation obtained differs in its form from those reported thus far and in particular from those given by the method of two-scale expansions and the interference integral method. First, the paper considers the case of a plane wave incident on a layer with irregularities, followed by a treatment of the general case of an arbitrary source.     

Synthesis of multivariate random vibration systems: A two-stage least squares AR-MA model approach

A parametric time series model procedure for the synthesis of multivariate stationary time series random vibrations is shown. The vibrations are assumed to be the outputs of a regularly sampled, random noise excited, differential equation model of a vivration system. The procedure is a two-stage least squares method for realizing a multivariate disrcrete time mixed autoregressive-moving average (AR-MA) model from a given stationary process matrix covariance function. The synthesis procedure and the problem of the minimal representation of multivariate output systems and the overparameterization of AR-MA models are discussed and illustrated.     

Random walks in generalized delayed recursive trees

Recently a great deal of effort has been made to explicitly determine the mean first-passage time （MFPT） between two nodes averaged over all pairs of nodes on a fractal network. In this paper, we first propose a family of generalized delayed recursive trees characterized by two parameters, where the existing nodes have a time delay to produce new nodes. We then study the MFPT of random walks on this kind of recursive tree and investigate the effect of the time delay on the MFPT. By relating random walks to electrical networks, we obtain an exact formula for the MFPT and verify it by numerical calculations. Based on the obtained results, we further show that the MFPT of delayed recursive trees is much shorter, implying that the efficiency of random walks is much higher compared with the non-delayed counterpart. Our study provides a deeper understanding of random walks on delayed fractal networks.     

Stochastic interval analysis of natural frequency and mode shape of structures with uncertainties

In this paper, natural frequencies and mode shapes of structures with mixed random and interval parameters are investigated by using a hybrid stochastic and interval approach. Expressions for the mean value and variance of natural frequencies and mode shapes are derived by using perturbation method and random interval moment method. The bounds of these probabilistic characteristics are then determined by interval arithmetic. Two examples are given first to illustrate the feasibility of the presented method and the results are verified by Monte Carlo Simulations. The presented approach is also applicable to solve pure random and pure interval problems. This capability is demonstrated in the third and fourth examples through the comparisons with the peer research outcomes.     

Probability Distribution on Full Rooted Trees

The recursive and hierarchical structure of full rooted trees is applicable to statistical models in various fields, such as data compression, image processing, and machine learning. In most of these cases, the full rooted tree is not a random variable; as such, model selection to avoid overfitting is problematic. One method to solve this problem is to assume a prior distribution on the full rooted trees. This enables the optimal model selection based on Bayes decision theory. For example, by assigning a low prior probability to a complex model, the maximum a posteriori estimator prevents the selection of the complex one. Furthermore, we can average all the models weighted by their posteriors. In this paper, we propose a probability distribution on a set of full rooted trees. Its parametric representation is suitable for calculating the properties of our distribution using recursive functions, such as the mode, expectation, and posterior distribution. Although such distributions have been proposed in previous studies, they are only applicable to specific applications. Therefore, we extract their mathematically essential components and derive new generalized methods to calculate the expectation, posterior distribution, etc.     

关于推广的组态混合法(Ⅰ)

本文提出了一个借助无规位相近似法的解,应用推广的组态混合法以求得更精确解的方法,导出了有关矩阵元的一组明显表达式。此外并证明了推广的组态混合法是自洽的,给出了有关的格林函数所满足的运动方程及展开式,根据后者对推广的组态混合法的解的性质进行了讨论。     

Quantum Mechanics Based on Probability Wave Functions Induced by the Minimum Mean Deviation from Statistical Equilibrium. I

The maximum entropy probability distributionsare generally used as probabilistic models fordescribing statistical equilibrium subject to given meanvalues of some random variables. The paper deals with the construction of probability wave functionsby minimizing the mean deviation from statisticalequilibrium subject to generalized correlations, inducedby random fluctuations, whose values are determined looking for stationary points of the meanenergy of the system. The results are applied to thestudy of several quantum systems (the free particle ina box, two independent particles in a box, the harmonic oscillator, and the hydrogen atom, withouthaving to solve the corresponding Schrodingerequation.     

A Necklace of Wulff Shapes

In a probabilistic model of a film over a disordered substrate, Monte-Carlo simulations show that the film hangs from peaks of the substrate. The film profile is well approximated by a necklace of Wulff shapes. Such a necklace can be obtained as the infimum of a collection of Wulff shapes resting on the substrate. When the random substrate is given by iid heights with exponential distribution, we prove estimates on the probability density of the resulting peaks, at small density.AMS subject classification: 60K35, 60K37, 82B24, 82B41     

A functional phase-integral method and applications to the laser beam propagation in random media

The problem of propagation of a high-intensity light beam in a half-space with random inhomogeneities is treated. An exact solution is constructed through a functional integral representation. For a Gaussian random field, the exact moments of solution are given explicitly. A functional phase-integral method is developed to provide an asymptotic evaluation of the moment integrals. The method is applied to two problems in a stochastic laser beam propagation in random media with a homogeneous background or with a focusing effect.     

Witness Trees in the Moser–Tardos Algorithmic Lovász Local Lemma and Penrose Trees in the Hard-Core Lattice Gas

We point out a close connection between the Moser–Tardos algorithmic version of the Lovász local lemma, a central tool in probabilistic combinatorics, and the cluster expansion of the hard-core lattice gas in statistical mechanics. We show that the notion of witness trees given by Moser and Tardos is essentially coincident with that of Penrose trees in the Cluster expansion scheme of the hard-core gas. Such an identification implies that the Moser–Tardos algorithm is successful in a polynomial time if the cluster expansion converges.     

PFC: A Novel Perceptual Features-Based Framework for Time Series Classification

Time series classification (TSC) is a significant problem in data mining with several applications in different domains. Mining different distinguishing features is the primary method. One promising method is algorithms based on the morphological structure of time series, which are interpretable and accurate. However, existing structural feature-based algorithms, such as time series forest (TSF) and shapelet traverse, all features through many random combinations, which means that a lot of training time and computing resources are required to filter meaningless features, important distinguishing information will be ignored. To overcome this problem, in this paper, we propose a perceptual features-based framework for TSC. We are inspired by how humans observe time series and realize that there are usually only a few essential points that need to be remembered for a time series. Although the complex time series has a lot of details, a small number of data points is enough to describe the shape of the entire sample. First, we use the improved perceptually important points (PIPs) to extract key points and use them as the basis for time series segmentation to obtain a combination of interval-level and point-level features. Secondly, we propose a framework to explore the effects of perceptual structural features combined with decision trees (DT), random forests (RF), and gradient boosting decision trees (GBDT) on TSC. The experimental results on the UCR datasets show that our work has achieved leading accuracy, which is instructive for follow-up research.     

Verification of the optimal probabilistic basis of aural processing in pitch of complex tones

Periodicity pitch for complex tones has been quantitatively accounted for by a two-stage process of Fourier-frequency analysis subject to random errors and significant nonlinearities, followed by an harmonic pattern recognizer that makes an optimum probabilistic estimate of the fundamental period of musical and speech sounds. The theory predicts that periodicity pitch is a multimodal probabilistic function of a given stimulus. A clear and empirically supported distinction is made between limitations on the pitch mechanism caused by the stochastic nature of aural frequency representation and by the deterministic resolution bandwidths of aural frequency analysis. This model was developed earlier [J. L. Goldstein, J. Acoust. Soc. Am 54, 1496-1516 (1973)] to account for probabilistic data on pitch errors [A. J. M. Houtsma and J. L. Goldstein, J. Acoust. Soc. Am. 51, 520 (1972)] measured with periodic stimuli comprising two successive harmonics. This paper presents new predictions by the theory that were calculated, with computer simulation where needed, for known probabilistic pitch data from stimuli comprising three to six successive harmonics. Predicted pitch errors increase with increasing errors in estimating the frequencies of stimulus harmonics and decrease as more harmonics are added to the stimulus. Optimum processor theory fully accounts for the multicomponent pitch data on the basis of similar errors in estimating component stimulus frequencies as reported earlier, thus providing further evidence for the optimum probabilistic basis of aural signal processing in pitch of complex tones.     

Asymptotic representations of elastic wave fields in permeable strata

An analytical method has been developed for studying filtration wave fields in oil-gas collectors, which makes it possible to reduce wave conjugation problems to simple problems for the aymptotic expansion coefficients. Simple analytic dependences have been found to calculate the fields in inhomogeneous anisotropic layers for the zero and first coefficients. Such dependences can serve as the basis for a fundamentally new and more complete study of wave fields as applied to acoustic logging and seismic surveying. The reliability of this method is substantiated by comparing the obtained asymptotic solutions with the expansion coefficients of an exact solution to the parameterized problem expanded into a Maclaurin series for the formal parameter.