Averaging in parabolic systems subject to weakly dependent random actions. The L2-approach

The first initial-boundary problem for a parabolic equation with a small parameter under external action described by some random process satisfying an arbitrary condition of weak dependence is considered. Averaging of the coefficients over a time variable is carried out. The existence of a generalized solution for the initial stochastic problem as well as for the problem with an averaged equation which turns out to be deterministic is assumed. Exponential bounds of the type of the well-known Bernstein inequalities for a sum of independent random variables are established for the probability of the deviation of the solution of the initial equation from the solution of the averaged problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 315–322, March, 1991.     

Averaging in parabolic systems,subjected to weakly dependent random perturbations. The L1-approach

One considers an averaging method in equations of parabolic type, situated under the action of centered, weakly dependent random perturbations so that their integrals, normalized in an appropriate manner, satisfy S. N. Bernshtein's exponential estimate. For normalized fluctuations of the solution of the initial equation relative to the solution of the averaged equation, which turns out to be deterministic, one has established S. N. Bernshtein's exponential estimates. On the basis of the obtained inequalities, for an arbitrary prescribed confidence level, one can indicate a confidence band, whose bounds are determined by the solving of the averaged equation, which contains the solution of the initial problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 167–172, February, 1991.     

Averaging in hyperbolic systems subject to weakly dependent random perturbations

The first initial boundary value problem is considered for a hyperbolic equation with a small parameter for an external action described by some stochastic process satisfying some of the conditions of weak dependence. Averaging of the coefficients over the temporal variable is conducted. The existence is assumed of a unique generalized solution both for the initial stochastic problem and for the problem with an averaged equation, which turns out to be deterministic. For the probability of deviation of a solution of the initial equation from the solution of the averaged problem, exponential bounds are established of the type of S. N. Bernshtein inequalities for the sums of independent random variables.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 8, pp. 1011–1020, August, 1992.     

Estimation of rated influence in parabolic systems.L 2-approach

By using observations of solutions of the first initial boundary-value problem for a parabolic quasilinear equation with fast random oscillations, we estimate the nonlinear term of the equation. In the metric of the space L₂, we study large deviations of a nonparametric estimate of nonlinear influence.     

Averaging stochastic systems under weakly dependent perturbations

An averaging procedure for stochastic systems with dependence on the whole past subject to an action described by a random process satisfying the strong mixing condition is investigated. For the probability of deviation beyond the level of normalized fluctuations of the solution of the initial stochastic equation relative to a solution of the average equation which turns out to be deterministic, exponential bounds of the type of the well-known S. N. Bernstein inequalities for the sum of independent random variables are constructed. These bounds can be used for construction of tolerance bands for a solution of the initial equation whose boundaries are determined by the deterministic solution of the averaged equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 593–600, May, 1990.     

LIL behavior for weakly dependent random variables in Banach spaces

Given a sequence of identically distributed ψ-mixing random variables {X _n ; n ≧ 1} with values in a type 2 Banach space B, under certain conditions, the law of the iterated logarithm for this sequence is obtained without second moment.     

Parameter estimation for partially observable systems subject to random failure

In this paper, we present a parameter estimation procedure for a condition‐based maintenance model under partial observations. Systems can be in a healthy or unhealthy operational state, or in a failure state. System deterioration is driven by a continuous time homogeneous Markov chain and the system state is unobservable, except the failure state. Vector information that is stochastically related to the system state is obtained through condition monitoring at equidistant sampling times. Two types of data histories are available — data histories that end with observable failure, and censored data histories that end when the system has been suspended from operation but has not failed. The state and observation processes are modeled in the hidden Markov framework and the model parameters are estimated using the expectation–maximization algorithm. We show that both the pseudolikelihood function and the parameter updates in each iteration of the expectation–maximization algorithm have explicit formulas. A numerical example is developed using real multivariate spectrometric oil data coming from the failing transmission units of 240‐ton heavy hauler trucks used in the Athabasca oil sands of Alberta, Canada. Copyright © 2012 John Wiley & Sons, Ltd.     

Principal eigenvalues and equilibrium states corresponding to weakly coupled parabolic systems of PDE

This paper extends the theory from [K2] to some weakly coupled parabolic systems of PDE which generate Markov processes with switching at random times between a finite number of diffusion processes. Dedicated to Professor Shmuel Agmon Partially supported by the US-Israel Binational Science Foundation.     

Correction to “Principal eigenvalues and equilibrium states corresponding to weakly coupled parabolic systems of PDE”

The online version of the original article can be found at     

Availability and maintenance modeling for systems subject to dependent hard and soft failures

This paper develops availability and maintenance models for single‐unit systems subject to dependent hard and soft failures. A hard failure stops the system immediately, whereas a soft failure only reduces the performance capacity of the system. Dependence between these 2 types of failures is reflected in the fact that each soft failure directly increases the hazard rate of the hard failure. On the basis of such interaction, we derive recursive equations for the system reliability and availability functions. To detect both types of failures, inspections are executed periodically. Furthermore, we investigate the optimal inspection policy via the minimization of the expected cost per unit time. The applicability of the developed availability and maintenance models is validated by a case study on an electrical distribution system.     

Stability of autoresonance models subject to random perturbations for systems of nonlinear oscillation equations

Systems of differential equations arising in the theory of nonlinear oscillations in resonance-related problems are considered. Of special interest are solutions whose amplitude increases without bound with time. Specifically, such solutions correspond to autoresonance. The stability of autoresonance solutions with respect to random perturbations is analyzed. The classes of admissible perturbations are described. The results rely on information on Lyapunov functions for the unperturbed equations.     

The numerical solution of a periodic parabolic problem subject to a nonlinear boundary condition. II

Approximating the differential equation by the standard fully implicit finite difference approximation reduces the problem to that of solving a set of non-linear algebraic equations. In the first part of this paper it was shown that Newton's method provided a satisfactory technique for the solution of this set of equations. In this paper the implementation of Newton's method is discussed, and it is shown that a device due toBickley andMacNamee can be used very successfully. A more accurate difference formula is also considered and a summary of numerical results presented.