Stochastic Boundary Value Problem Research Articles

SOLVING STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS BASED ON THE EXPERIMENTAL DATA

We consider a stochastic linear elliptic boundary value problem whose stochastic coefficient a(x, ω) is expressed by a finite number NKL of mutually independent random variables, and transform this problem into a deterministic one. We show how to choose a suitable NKL which should be as low as possible for practical reasons, and we give the a priori estimates for modeling error when a(x, ω) is completely known. When a random function a(x, ω) is selected to fit the experimental data, we address the estimation of the error in this selection due to insufficient experimental data. We present a simple model problem, simulate the experiments, and give the numerical results and error estimates.

Solving parabolic Wick-stochastic boundary value problems using a finite element method

A class of linear parabolic stochastic boundary value problems of Wick-type is studied. The equations are understood in a weak sense on a suitable stochastic distribution space, and existence and uniqueness results are provided. The paper continues to discuss a numerical method for this type of problem, based on a Galerkin type of approximation. Estimates showing linear convergence in time and space are derived, and rate of convergence results for the stochastic dimension are reported.

ASYMPTOTIC BEHAVIOR OF A STOCHASTIC EVOLUTION PROBLEM IN A VARYING DOMAIN

We study the asymptotic behaviour of the initial boundary value problem for a stochastic partial differential equation in a sequence of perforated domains. We prove that the sequence of solutions of the problem converges in appropriate topologies to the solution of a limit stochastic initial boundary value problem of the same type as the original problem, but containing an additional term expressed in terms of some characteristics of the perforated domain.

A Stochastic Two-Point Boundary Value Problem

We investigate the two-point stochastic boundary-value problem on $[0,1]$: \begin{equation}\label{1} \begin{split} U'' &= f(U)\dot W + g(U,U')\\ U(0) &= \xi\\ U(1)&= \eta. \end{split} \tag{1} \end{equation} where $\dot W$ is a white noise on $[0,1]$, $\xi$ and $\eta$ are random variables, and $f$ and $g$ are continuous real-valued functions. This is the stochastic analogue of the deterministic two point boundary-value problem, which is a classical example of bifurcation. We find that if $f$ and $g$ are affine, there is no bifurcation: for any r.v. $\xi$ and $\eta$, (1) has a unique solution a.s. However, as soon as $f$ is non-linear, bifurcation appears. We investigate the question of when there is either no solution whatsoever, a unique solution, or multiple solutions. We give examples to show that all these possibilities can arise. While our results involve conditions on $f$ and $g$, we conjecture that the only case in which there is no bifurcation is when $f$ is affine.

Solution of the stochastic boundary-value problem of elasticity theory for composites with disordered structures in the correlative approximation of the method of quasi-periodic components

The method of quasi-periodic components, a new method of statistical mechanics of composites, is presented. In correlative approximation, this method makes it possible to reduce the problem of solving the stochastic boundary-value problem of elasticity theory for composites with disordered structures to a simpler problem for an individual cell with one inclusion in a homogeneous elastic medium. The generalized volumetric forces on the cell boundary take into account the random distribution of inclusions in the composite fragment studied. The problem for one inclusion cell can be solved, for example, by the boundary element method. The numerical solution in the correlative approximation of the method of quasi-periodic components for inhomogeneous interphase stress fields in the matrix of an epoxy composite containing randomly distributed unidirectional fibers is given. A comparison with the known analytical solutions obtained by other authors confirms the high accuracy of the correlative approximation.

Synthesis of averaging methods in stochastic boundary-value problems of the mechanics of microheterogeneous solids

The method of periodic components is further developed, which allows us, on unified positions, to predict the effective characteristics and structural strain fields in partially or fully disordered composites. The stochastic boundary-value problem of elasticity theory for microheterogeneous solids with a statistically homogeneous structure is treated. The heterogeneous solid is considered to be macroscopically homogeneous and macroanisotropic (or quasi-isotropic) with geometric form and properties of the structural components determined and given. The structural elements are assumed to have perfect interfaces, i.e., the displacements and tractions are continuous across the interface. The boundary-value problem was solved by the method of local approximation. Numerical results were obtained for composites with a stochastic structure.

Simulation of nonhomogeneous random fields for structural applications

In this paper, an original approach to the simulation of discretized, nonhomogeneous, spatial random fields is presented. To solve the simulation problem, an effective version of the rejection method and conditional probability distributions is implemented. A superior role in the method plays a propagation scheme consisting of a growing number of points, covering sequentially the nodes of a regular or irregular mesh. The results of the simulation of different types of second-order field (Brown, Wiener, Shinozuka) are presented. Estimators of local and global characteristics of these fields show agreement with theoretical Wishart (gamma) distributions. The proposed method can be used for the solution of two-dimensional stochastic nonlinear boundary value problems by computer simulation.

A boundary element method for stochastic flow problems in a semiconfined aquifer with random boundary conditions

A boundary element technique will be developed to deal with steady state flows through a shallow aquifer with random boundary conditions. The aquifer is bounded by a leaky layer above, and an impervious stratum below. The flow problem is formulated in terms of a stochastic boundary value problem involving a Helmholtz equation. A formulation of the boundary element method for the stochastic mixed boundary value problem is presented. The moments of unknown random hydraulic head can be obtained by application of an expectation operation to the boundary element formulations. A numerical method to evaluate hydraulic head moments on boundaries and inside the domain is then developed. The influence of the spatial structure of the boundary conditions on the hydraulic head statistics is discussed and the results are compared with some analytical solutions for one-dimensional flow problems for the purpose of verification of the numerical algorithm. The significance of the presence of the leaky layer on the hydraulic head statistics is also investigated. A numerical example for more general steady state flows through a semiconfined aquifer is presented.

Probabilistic analysis of flow in random porous media by stochastic boundary elements

The mathematical and numerical modeling of groundwater flows in random porous media is studied assuming that the formation's hydraulic log-transmissivity is a statistically homogeneous, Gaussian, random field with given mean and covariance function. In the model, log-transmissivity may be conditioned to take exact field values measured at a few locations. Our method first assumes that the log-transmissivity may be expanded in a Fourier-type series with random coefficients, known as the Karhunen-Loéve (KL) expansion. This expansion has optimal properties and is valid for both homogeneous and nonhomogeneous fields. By combining the KL expansion with a small parameter perturbation expansion, we transform the original stochastic boundary value problem into a hierarchy of deterministic problems. To the first order of perturbation, the hydraulic head is expanded on the same set of random variables as in the KL representation of log-transmissivity. To solve for the corresponding coefficients of this expansion, we adopt a boundary integral formulation whose numerical solution is carried out by using boundary elements and dual reciprocity (DRBEM). To illustrate and validate our scheme, we solve three test problems and compare the numerical solutions against Monte Carlo simulations based on a finite difference formulation of the original flow problem. In all three cases we obtain good quantitative agreement and the present approach is shown to provide both a more efficient and accurate way of solving the problem.

Dynamic stiffness matrix for axially vibrating stochastic rods using Stratonovich’s averaging principle

The problem of determining the dynamic stiffness matrix of a rod with broad band randomly varying mass and stiffness properties is considered. The governing stochastic boundary value problem is solved. First, a general solution to the field equation is obtained by using Stratonovich’s stochastic averaging theorem. Subsequently, the elements of dynamic stiffness coefficients are evaluated by choosing appropriately the arbitrary constants of the general solution. The analytically determined statistics of the amplitude and phase of the stiffness coefficients are shown to compare favorably with digital simulation solutions.

Stochastic systems governed by B-evolutions on Hilbert spaces

SynopsisIn this paper, we consider the question of existence of solutions and their regularity properties for a large class of stochastic evolution equations governed by B-evolutions involving two different Hilbert spaces. This allows dynamic boundary conditions together with noisy boundary data. They cover also stochastic boundary value problems. Our results are illustrated by two practical examples.

Markov field property for stochastic differential equations with boundary conditions

In this paper we show that the solution of a stochastic boundary value problem with additive noise and with a completely nonlinear drift is a Markov field if only if the boundary condition is an initial or a final type condition

SH-Waves in a Medium Containing a Disordered Periodic Array of Cracks

Reflection and transmission of an SH-wave by a disordered periodic array of coplanar cracks is investigated, and subsequently its application to the dispersion and attenuation of an SH-wave in a disorderedly cracked medium is also treated. This is a stochastic boundary value problem. The formulation largely follows Mikata and Achenbach (1988b). The problem is formulated for an averaged scattered field, and the governing singular integral equation is derived for a conditionally averaged crack-opening displacement using a quasi-crystalline-like approximation. Unlike our previous study (Mikata and Achenbach, 1988b) where a point scatterer approximation was used for the regular part of the integral kernel, however, no further approximation is introduced. The singular integral equation is solved by an eigenfunction expansion involving Chebyschev polynomials. Numerical results are presented for the averaged reflection and transmission coefficients of zeroth order as a function of the wave number for normal incidence, a completely disordered crack spacing, and various values of the ratio of crack length and average crack spacing. Numerical results are also presented for the dispersion and attenuation of an SH-wave in a disorderedly cracked medium.

Stochastic boundary value problems: a white noise functional approach

We give a program for solving stochastic boundary value problems involving functionals of (multiparameter) white noise. As an example we solve the stochastic Schrodinger equation {ie391-1} whereV is a positive, noisy potential. We represent the potentialV by a white noise functional and interpret the product of the two distribution valued processesV andu as a Wick productV ◊u. Such an interpretation is in accordance with the usual interpretation of a white noise product in ordinary stochastic differential equations. The solutionu will not be a generalized white noise functional but can be represented as anL 1 functional process.

Micromechanics as a basis of random elastic continuum approximations

The problem of the determination of stochastic constitutive laws for input to continuum-type boundary value problems is analyzed from the standpoint of the micromechanics of polycrystals and matrix-inclusion composites. Passage to a sought-for random continuum is based on a scale dependent window playing the role of a Representative Volume Element (RVE). It turns out that an elastic microstructure with piecewise continuous realizations of random tensor fields of stiffness cannot be uniquely approximated by a random field of stiffness with continuous realizations. Rather, two random continuum fields may be introduced to bound the material response from above and from below. As the size of the RVE relative to the crystal size increases to infinity, both fields converge to a deterministic continuum with a progressively decreasing strength of fluctuations. Since the RVE corresponds to a single finite element cell not infinitely larger than the crystal size, two random fields are to be used to bound the solution of a given boundary value problem at a given scale of resolution. The method applies to a number of other elastic microstructures, and provides the basis for stochastic finite differences and elements. The latter point is illustrated by an example of a stochastic boundary value problem of a heterogeneous membrane.

Boundary element solution for stochastic groundwater flow: Random boundary condition and recharge

This paper presents an integral equation technique for solving stochastic boundary value problems in groundwater flow. The aquifer considered has deterministic hydraulic conductivity but is subject to random boundary condition and domain recharge. Using the distribution of fictitious sources and dipoles, stochastic integral equations for the mean and covariance of head and flux are derived. An iterative boundary element technique is applied for numerical solution. Two one‐dimensional examples are examined and compared with exact solution. A two‐dimensional problem is then presented.

A change of variables formula for stratonovich integrals and existence of solutions for two-point stochastic boundary value problems

Existence results for a class of two point non-adaptive stochastic boundary value problems are derived by combining a (new) change of variables formula with optimal control techniques. The class of equations treated is significantly wider than that for which previously known existence results apply.

Probability Distribution of the Eigenvalues of the Random String Equation

The probability distribution of the eigenvalues of a second-order stochastic boundary value problem is considered. The solution is characterized in terms of the zeros of an associated initial value problem. It is further shown that the probability distribution is related to the solution of a first-order nonlinear stochastic differential equation. Solutions of this equation based on the theory of Markov processes and also on the closure approximation are presented. A string with stochastic mass distribution is considered as an example for numerical work. The theoretical probability distribution functions are compared with digital simulation results. The comparison is found to be reasonably good.

Abstract stochastic evolution equations on banach spaces

In this paper we consider two basic problems. We discuss the questions of abstract formulation of stochastic initial boundary value problems based on semi-group theory and study the regularity properties of solutions. We present this in section 2. In section 3,we consider, under various generalities, the questions of existence and uniqueness of solutions and their regularity properties for semilinear stochastic evolution equations on Banach spaces of the form For illustration, we present an example of a semilinear parabolic system in which distributed White noise appears under spatial derivatives

Stochastic boundary value problems

Stochastic boundary value problems