A solution of the nonlinear stochastic realization problem

Generalized spectral decomposition for stochastic nonlinear problems

We consider a problem modeling a porous medium with a random perturbation. This model occurs in many applications such as biology, medical sciences, oil exploitation, and chemical engineering. Many authors focused their study mostly on the deterministic case. The more classical one was due to Biot in the 50s, where he suggested to ignore everything that happens at the microscopic level, to apply the principles of the continuum mechanics at the macroscopic level. Here we consider a stochastic problem, that is, a problem with a random perturbation. First we prove a result on the existence and uniqueness of the solution, by making use of the weak formulation. Furthermore, we use a numerical scheme based on finite differences to present numerical results.

The present book includes contributions presented at the Advanced Study Institute (ASI) on Nonlinear Stochastic Problems held in Algarve, Portugal, May 16–28, 1982.

Although many varied techniques have been proposed for handling deterministic non-linear programming problems there apperars to have been little success in solving the more realistic problem of stochastic non-linear programming, despite the many results that have been obtained for stochastic linear programming. In this paper the stochastic non-linear problem is treated by means of an adaptation of a method used by Berkovitz [1] in obtaining an exiatence theorem for a type of inequality constrained variational problem involving one independent variable. The stochastic programming problem of course involves many independent variables. Necessary conditions are obtained for the existence of a solution of a fairly general type of non-linear problem, and these conditions are shown to be also sufficient for the convex problem. A duality theorem is given for the latter problem.

Industrial structure systems may have nonlinearity, and are also sometimes exposed to the danger of random excitation. This paper proposes a method to analyze response and reliability design of a complex nonlinear structure system under random excitation. The nonlinear structure system which is subjected to random process is modeled by finite element method. The nonlinear equations are expanded sequentially using the perturbation theory. Then, the perturbed equations are solved in probabilistic methods. Several statistical properties of random process that are of interest in random vibration applications are reviewed in accordance with the nonlinear stochastic problem.

As has been observed, turbulence can be characterized as a nonlinear stochastic problem. Physically, turbulence is a mixture of laminar motion and certain subsidiary motions whose smallest length scales are comparable to the mean free path of the molecules. Strictly speaking, the Navier-Stokes equation is not applicable to phenomena with such small length scales. However, it has been observed that these types of motion die out quite rapidly due to the action of viscosity. The spectral distribution of their energy decays long before the length scales become comparable to those at the molecular level. This permits the use of the Navier-Stokes equation as a valid dynamical formulation of such phenomena.KeywordsSample SpaceSpectral Density FunctionProbability Generate FunctionSmall Length ScaleCorrelation TensorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Short-term load forecasting (STLF) is challenging due to complex time series (TS) which express three seasonal patterns and a nonlinear trend. This article proposes a novel hybrid hierarchical deep-learning (DL) model that deals with multiple seasonality and produces both point forecasts and predictive intervals (PIs). It combines exponential smoothing (ES) and a recurrent neural network (RNN). ES extracts dynamically the main components of each individual TS and enables on-the-fly deseasonalization, which is particularly useful when operating on a relatively small dataset. A multilayer RNN is equipped with a new type of dilated recurrent cell designed to efficiently model both short and long-term dependencies in TS. To improve the internal TS representation and thus the model's performance, RNN learns simultaneously both the ES parameters and the main mapping function transforming inputs into forecasts. We compare our approach against several baseline methods, including classical statistical methods and machine learning (ML) approaches, on STLF problems for 35 European countries. The empirical study clearly shows that the proposed model has high expressive power to solve nonlinear stochastic forecasting problems with TS including multiple seasonality and significant random fluctuations. In fact, it outperforms both statistical and state-of-the-art ML models in terms of accuracy.

We present an introduction to phase-space techniques (PST) based on a quantum-field-theoretical (QFT) approach. In addition to bridging the gap between PST and QFT, our approach results in a number of generalizations of the PST. First, for problems where the usual PST do not result in a genuine Fokker-Planck equation (even after phase-space doubling) and hence fail to produce a stochastic differential equation (SDE), we show how the system in question may be approximated via stochastic difference equations $(\mathrm{S}\ensuremath{\Delta}\mathrm{E}).$ Second, we show that introducing sources into the SDE's (or $\mathrm{S}\ensuremath{\Delta}\mathrm{E}\mathrm{'}\mathrm{s})$ generalizes them to a full quantum nonlinear stochastic response problem (thus generalizing Kubo's linear reaction theory to a quantum nonlinear stochastic response theory). Third, we establish general relations linking quantum response properties of the system in question to averages of operator products ordered in a way different from time normal. This extends PST to a much wider assemblage of operator products than are usually considered in phase-space approaches. In all cases, our approach yields a very simple and straightforward way of deriving stochastic equations in phase space.

Traffic terminal is a spot of high density for people’s traffic activities,the pedestrian transportation in the traffic terminal can be considered as a bunching density un-sure and nonlinear stochastic dynamics problem of interaction between each individual.pedestrian,as the primary part that composed all activities in the traffic terminal,has a significant influence on traffic behaviors and psychic properties.In this paper,we proposal a series of rationalized improvement measures to relief congestion and promote pedestrian security in the traffic terminal,by analysis on travel environment and traffic behaviors properties of pedestrian and deeply studying their traffic psychic properties

Call admission control with two classes of users is investigated via a nonlinear stochastic knapsack model. The feasibility region represents the subset of the call space, where given constraints on the quality of service have to be satisfied. Admissible strategies are searched for within the class of coordinate-convex policies. Structural properties that the optimal policies belonging to such a class have to satisfy are derived. They are exploited to narrow the search for the optimal solution to the nonlinear stochastic knapsack problem that models call admission control. To illustrate the role played by these properties, the numbers of coordinate-convex policies by which they are satisfied are estimated. A graph-based algorithm to generate all such policies is presented.

An adaptive stepsize algorithm is implemented on a stochastic implicit strong order 1 method, namely a stiffly accurate diagonal implicit stochastic Runge-Kutta method where a modified Wiener increment [?(?W n )] is involved instead of a regular ?W n = ??h, ??N(0,1) to avoid unboundedness. The modified Wiener increment is equal to the regular one only if ??? ? A h , otherwise [?(?W)] = ?A h ?h or A h ?h if ? ? ?A h or ? ? A h , respectively. The parameter A h is determined based on a strong order inequality requirement (Milstein et al., 2002) ?? The variable stepsize algorithm is based on Richardson's extrapolation. For every step change additional work is required to compute [?(?W n )] so that the correct Brownian path is maintained from simulation to simulation. Numerical experiments in solving nonlinear and linear stochastic differential equation problems demonstrate that better approximations are obtained with the modified Wiener increment compared to using the regular Wiener increment.

In this work, an attempt is made to apply the Local Lagged Adapted Generalized Method of Moments (LLGMM) to estimate state and parameters in stochastic differential dynamic models. The development of LLGMM is motivated by parameter and state estimation problems in continuous-time nonlinear and non-stationary stochastic dynamic model validation problems in biological, chemical, engineering, energy commodity markets, financial, medical, military, physical sciences and social sciences. The byproducts of this innovative approach (LLGMM) are the balance between model specification and model prescription of continuous-time dynamic process and the development of discrete-time interconnected dynamic model of local sample mean and variance statistic process (DTIDMLSMVSP). Moreover, LLGMM is a dynamic non-parametric method. The DTIDMLSMVSP is an alternative approach to the GARCH(1,1) model, and it provides an iterative scheme for updating statistic coefficients in a system of generalized method of moment/observation equations. Furthermore, applications of LLGMM to energy commodities price, U.S. Treasury Bill interest rate and the U.S.–U.K. foreign exchange rate data strongly exhibit its unique role, scope and performance, in particular, in forecasting and confidence-interval problems in applied statistics.

Transport hub belongs to the people highly gathering places,the pedestrian traffic within the hub is a nonlinear stochastic dynamic problem with uncertain cluster density. The computer dynamic simulation is one of the effective methods to investigate the dynamic characteristics of pedestrian traffic flow at present within the hub.The article studies the error between pedestrian traffic simulation of the commuter simulation software and the actual pedestrian traffic,and explains the simulation error due to the discrete space, fidelity are introduced to evaluate the accuracy of the simulation model.By comparing the Hohhot underground tunnel experimental data and simulation results and the analysis of pedestrian traffic characteristics, it is concluded that the pedestrian traffic simulation performance is closely related to the environment in the hub and the importance of the fidelity of simulation model.

The stochastic collocation method (Babuska et al. in SIAM J Numer Anal 45(3):1005---1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411---2442, 2008a; SIAM J Numer Anal 46(5):2309---2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118---1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289---294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu , 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229---275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41---44):3187---3206, 2009; Arch Comput Methods Eng 17:435---454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions $$O(1)$$ O ( 1 ) to moderate dimensions $$O(10)$$ O ( 10 ) and to high dimensions $$O(100)$$ O ( 100 ) . The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs.

Green’s function-stochastic methods framework for probing nonlinear evolution problems: Burger’s equation, the nonlinear Schrödinger’s equation, and hydrodynamic organization of near-molecular-scale vorticity