Stochastic Boundary Research Articles

A variational approach to stochastic nonlinear diffusion problems with dynamical boundary conditions

We study a nonlinear partial differential equation of the calculus of variation in a bounded domain, perturbed by noise; we allow stochastic boundary conditions that depend on the time derivative of the solution on the boundary. This work provides the existence and uniqueness of the solution and it shows the existence of an ergodic invariant measure for the corresponding transition semigroup; furthermore, under suitable additional assumptions, uniqueness and strong asymptotic stability of the invariant measure are proved.

A Lagrangian particle random walk model for simulating a deep-sea hydrothermal plume with both buoyant and non-buoyant features

This paper presents a computational model of simulating a deep-sea hydrothermal plume based on a Lagrangian particle random walk algorithm. This model achieves the efficient process to calculate a numerical plume developed in a fluid-advected environment with the characteristics such as significant filament intermittency and significant plume meander due to flow variation with both time and location. Especially, this model addresses both non-buoyant and buoyant features of a deep-sea hydrothermal plume in three dimensions, which significantly challenge a strategy for tracing the deep-sea hydrothermal plume and localizing its source. This paper also systematically discusses stochastic initial and boundary conditions that are critical to generate a proper numerical plume. The developed model is a powerful tool to evaluate and optimize strategies for the tracking of a deep-sea hydrothermal plume via an autonomous underwater vehicle (AUV).

Edge and divertor plasma measurements with ion sensitive and Mach probes in LHD

Abstract Spatial profiles of plasma flow and Mach number in the stochastic magnetic boundary layer as well as ion temperature ( T i ) and electron temperature ( T e ) in the divertor region in Large Helical Device (LHD) have been studied by a movable multiple functions probe, which consists of Mach probes and an ion sensitive probe. The results of ion saturation current measurements indicated plasma flow direction is alternated in the stochastic magnetic boundary. Mach number profiles for different plasma densities have been evaluated experimentally which compared with 3-D transport code. T i and T e in the divertor region measured by the ion sensitive probe decreased with increasing line-averaged density. Although T i was higher than T e in the low density plasma, both temperatures became almost the same at higher density.

A Proposed Stochastic Finite Difference Approach Based on Homogenous Chaos Expansion

This paper proposes a stochastic finite difference approach, based on homogenous chaos expansion (SFDHC). The said approach can handle time dependent nonlinear as well as linear systems with deterministic or stochastic initial and boundary conditions. In this approach, included stochastic parameters are modeled as second-order stochastic processes and are expanded using Karhunen-Loève expansion, while the response function is approximated using homogenous chaos expansion. Galerkin projection is used in converting the original stochastic partial differential equation (PDE) into a set of coupled deterministic partial differential equations and then solved using finite difference method. Two well-known equations were used for efficiency validation of the method proposed. First one being the linear diffusion equation with stochastic parameter and the second is the nonlinear Burger's equation with stochastic parameter and stochastic initial and boundary conditions. In both of these examples, the probability distribution function of the response manifested close conformity to the results obtained from Monte Carlo simulation with optimized computational cost.

Stochastic Optimal Robin Boundary Control Problems of Advection-Dominated Elliptic Equations

In this work we deal with a stochastic optimal Robin boundary control problem constrained by an advection-diffusion-reaction elliptic equation with advection-dominated term. We assume that the uncertainty comes from the advection field and consider a stochastic Robin boundary condition as control function. A stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem, based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables. Stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system. A global error estimate in the product of physical space and stochastic space for the numerical approximation is derived. Illustrative numerical experiments are provided.

No-slip boundary conditions and forced flow in multiparticle collision dynamics

Multiparticle collision dynamics (MPCD) is a particle-based fluid simulation technique that is becoming increasingly popular for mesoscale fluid modeling. However, some confusion and conflicting results persist in literature regarding several important methodological details, in particular the enforcement of the no-slip condition and thermostatting in forced flow. These issues persist in simple flows past stationary boundaries, which we exclusively focus on here. We discuss the parametrization of MPCD fluids and its consequences for fluid-solid boundaries in great detail, and show that the method of virtual particles proposed by Lamura et al. and adopted by many others is required only for parameter choices that lead to viscosities dominated by collisional contributions. We test several implementations of the virtual particle method and discuss how to completely eliminate slip at stationary boundaries. We also show that stochastic boundary reflection rules are inherently problematic for forced flow and suggest a possible remedy. Finally, we discuss the most robust way to achieve forced flow and evaluate several thermostatting methods in the process. All discussion is limited to solid objects that do not move as a result of collisions with MPCD particles (i.e., walls). However, the results can be extended to solutes that experience forces and torques due to interactions with MPCD particles (e.g., colloids). The detailed analysis presented for this simple case provides the level of rigor and accuracy to the MPCD method required for the study of more complex systems.

Numerical solution of the Stratonovich- and Ito–Euler equations: Application to the stochastic piston problem

We consider a piston with a velocity perturbed by Brownian motion moving into a straight tube filled with a perfect gas at rest. The shock generated ahead of the piston can be located by solving the one-dimensional Euler equations driven by white noise using the Stratonovich or Ito formulations. We approximate the Brownian motion with its spectral truncation and subsequently apply stochastic collocation using either sparse grid or the quasi-Monte Carlo (QMC) method. In particular, we first transform the Euler equations with an unsteady stochastic boundary into stochastic Euler equations over a fixed domain with a time-dependent stochastic source term. We then solve the transformed equations by splitting them up into two parts, i.e., a ‘deterministic part’ and a ‘stochastic part’. Numerical results verify the Stratonovich–Euler and Ito–Euler models against stochastic perturbation results, and demonstrate the efficiency of sparse grid and QMC for small and large random piston motions, respectively. The variance of shock location of the piston grows cubically in the case of white noise in contrast to colored noise reported in [1], where the variance of shock location grows quadratically with time for short times and linearly for longer times.

Reformulation of the Stochastic BEM to improve the computational efficiency in the prediction of the vibro-acoustic behaviour of structures with uncertainties

Non-determinism plays a crucial role in the prediction of the vibro-acoustic behaviour of structures. The Stochastic Boundary Element Method (SBEM) allows computing the average response of systems with geometrical uncertainties. The method consists in including such perturbations in the standard BEM formulation and adding a set of equations representative of the energetic behaviour. Due to the presence of unknown cross-products, the addition of a set of auxiliary equation is necessary to solve the problem. As a consequence, the system dimension becomes very large and the application of the methodology is limited to small models.This paper deals with a reformulation of the SBEM approach such that the system dimension is drastically decreased and the methodology can be applied to more complex cases. Moreover, the approach is extended to predict the variance of the field variable at no additional computational cost.After a theoretical introduction, the SBEM is applied to two-dimensional examples with pressure and velocity boundary conditions. The comparison with Monte Carlo simulations shows a good agreement of the results and a reduction in the computational load is achieved.

Optimality conditions for stochastic boundary control problems governed by semilinear parabolic equations

We study the boundary control problems for stochastic parabolic equations with Neumann boundary conditions. Imposing super-parabolic conditions, we establish the existence and uniqueness of the solution of state and adjoint equations with non-homogeneous boundary conditions by the Galerkin approximations method. We also find that, in this case, the adjoint equation (BSPDE) has two boundary conditions (one is non-homogeneous, the other is homogeneous). By these results we derive necessary optimality conditions for the control systems under convex state constraints by the convex perturbation method.

Stochastic optimal control analysis of a piezoelectric shell subjected to stochastic boundary perturbations

The stochastic optimal control for a piezoelectric spherically symmetric shell subjected to stochastic boundary perturbations is constructed, analyzed and evaluated. The stochastic optimal control problem on the boundary stress output reduction of the piezoelectric shell subjected to stochastic boundary displacement perturbations is presented. The electric potential integral as a function of displacement is obtained to convert the differential equations for the piezoelectric shell with electrical and mechanical coupling into the equation only for displacement. The displacement transformation is constructed to convert the stochastic boundary conditions into homogeneous ones, and the transformed displacement is expanded in space to convert further the partial differential equation for displacement into ordinary differential equations by using the Galerkin method. Then the stochastic optimal control problem of the piezoelectric shell in partial differential equations is transformed into that of the multi-degree-of-freedom system. The optimal control law for electric potential is determined according to the stochastic dynamical programming principle. The frequency-response function matrix, power spectral density matrix and correlation function matrix of the controlled system response are derived based on the theory of random vibration. The expressions of mean-square stress, displacement and electric potential of the controlled piezoelectric shell are finally obtained to evaluate the control effectiveness. Numerical results are given to illustrate the high relative reduction in the root-mean-square boundary stress of the piezoelectric shell subjected to stochastic boundary displacement perturbations by the optimal electric potential control.

Stochastic Boundary Methods of Fundamental Solutions for solving PDEs

Abstract A mesh-free Stochastic Boundary Method (SBM) based on randomized versions of the Method of Fundamental Solutions (MFS) is suggested. The randomization is used in the following steps of MFS: (1) the singular source positions are randomly distributed outside the domain, (2) the large system of linear equations for the weights in the expansion over the fundamental solutions is resolved by a randomized SVD method we introduced in [56] , or the randomized projection method we developed in [54] . We construct also a new method of stochastic boundary method based on the inversion of the Poisson formula representing the solution in a disc (a sphere, in R 3 ). We present a series of applications of the suggested SBM: we combine SBM with the Random Walk on Spheres and Random Walk on Boundary algorithms which results in methods giving the solution in any set of arbitrary points, without introducing any mesh in the domain. The Laplace, biharmonic, and the system of elasticity equations are involved in our analysis. We present some numerical results and give a brief discussion of the performance of the suggested methods. The numerical experiments carried out for the Laplace and Lame equations confirm our conclusion that the best results are obtained with the overdetermined systems generated by MFS where the number of source points is considerably smaller than the number of collocation points.

Markov modulation of a two-sided reflected Brownian motion with application to fluid queues

In this paper, we study a reflected Markov-modulated Brownian motion with a two sided reflection in which the drift, diffusion coefficient and the two boundaries are (jointly) modulated by a finite state space irreducible continuous time Markov chain. The goal is to compute the stationary distribution of this Markov process, which in addition to the complication of having a stochastic boundary can also include jumps at state change epochs of the underlying Markov chain because of the boundary changes. We give the general theory and then specialize to the case where the underlying Markov chain has two states.

Stochastic spline fictitious boundary element method in elastostatic problems with random fields

Mathematical formulation and computational implementation of the stochastic spline fictitious boundary element method (SFBEM) are presented for the analysis of plane elasticity problems with material parameters modeled with random fields. Two sets of governing differential equations with respect to the means and deviations of structural responses are derived by including the first order terms of deviations. These equations, being in similar forms to those of deterministic elastostatic problems, can be solved using deterministic fundamental solutions. The calculation is conducted with SFBEM, a modified indirect boundary element method (IBEM), resulting in the means and covariances of responses. The proposed method is validated by comparing the solutions obtained with Monte Carlo simulation for a number of example problems and a good agreement of results is observed.

Analytical solution of the stochastic steady-state creep boundary value problem for a thick-walled tube

A non-linear steady-state creep stochastic boundary value problem is solved for a thick-walled tube acted upon by an internal pressure for the case of a plane strain state. It is assumed that the properties of the tube material are described by a random function of its radius. The constitutive creep relations are taken in accordance with non-linear viscous flow theory in a stochastic form. A recurrent form of the system of stochastic differential equations is obtained by expanding the radial stress in a series in powers of a small parameter, from which the components of the radial stress can be found to any degree of accuracy. The random stress field and strain rate field are analised statistically as a function of the non-linearity exponent and the degree of inhomogeneity of the material. A comparative analysis of the solutions of the stochastic steady-state creep boundary value problem for a thick-walled tube, obtained is the fourth approximation of the small parameter method and the Monte carlo method, is performed.

Stochastic boundary collocation and spectral methods for solving PDEs

We develop a stochastic boundary method (SBM) which can be considered as a randomized version of the method of fundamental solutions (MFS). We suggest solving the large system of linear equations for the weights in the expansion over the fundamental solutions by a randomized SVD method introduced by Sabelfeld and Mozartova (2011). In addition, we also deal with solving inhomogeneous problems where we use the integral representation through the Green integral formula. The relevant volume integrals are calculated by a Monte Carlo integration technique which uses the symmetry of the Green function. We construct also a stochastic boundary method based on the spectral inversion of the Poisson formula representing the solution in a disc. This is done for the Laplace equation, and the system of elasticity equations. We stress that the stochastic boundary method proposed is of high generality; it can be applied to any bounded and unbounded domain with any boundary condition provided the existence and uniqueness of the solution are proven. We present a series of numerical results which illustrate the performance of the suggested methods.

A Simulation Environment for Deep-Sea Hydrothermal Plume Tracing with Autonomous Underwater Vehicles

For using an autonomous underwater vehicle(AUV) to localize seafloor hydrothermal vents fast and accurately, a biomimetic approach to hydrothermal plume tracing is proposed.And to support the investigation of AUV's biologicallyinspired control strategies,a graphical simulation environment is designed and developed.In this paper,hydrothermal plume tracing with AUVs and the modular architecture of the simulation environment are firstly introduced and described.Then,the modules of hydrothermal plume and flow field in simulation environment are modeled,and the efficient numerical solution algorithms are given.To facilitate Monte Carlo simulation,a set of stochastic initial and boundary conditions are set.And the modular behavior-based AUV control system architecture employed in the control system module is mainly described. The simulation environment addresses the key factors that complicate the investigation of AUV's biologically-inspired control strategies for the hydrothermal plume tracing,including the current field being non-uniform and non-steady,the plume distribution being irregular,intermittent and of large scale,the plume centreline being meandrous,and the plume containing buoyant part and nonconservative tracer.In addition,this simulation environment achieves good visual effect and high computational efficiency,which allows it to be suitable for real-time simulation and Monte Carlo simulation studies.The presented simulation environment provides sufficient support for the investigation of AUVs biologically-inspired control strategies for hydrothermal plume tracing.

Löwner evolution driven by a stochastic boundary point

We consider evolution in the unit disk in which the sample paths are represented by the trajectories of points evolving randomly under the generalized Loewner equation. The driving mechanism differs from the SLE evolution, but nevertheless solutions possess similar invariance properties.

A variational‐inequality approach to stochastic boundary value problems with inequality constraints and its application to contact and elastoplasticity

SUMMARYThis paper is concerned with stochastic boundary value problems (SBVPs) whose formulation involves inequality constraints. A class of stochastic variational inequalities (SVIs) is defined, which is well adapted to characterize the solution of specified inequality‐constrained SBVPs. A methodology for solving such SVIs is proposed, which involves their discretization by projection onto polynomial chaos and collocation of the inequality constraints, followed by the solution of a finite‐dimensional constrained optimization problem. Simulation studies in contact and elastoplasticity are provided to demonstrate the proposed framework. Copyright © 2011 John Wiley & Sons, Ltd.

Stochastic boundary conditions for grand-canonical-like molecular dynamics simulations of open systems

In this paper we develop a stochastic boundary conditions (SBC) for event-driven molecular dynamics simulations of a finite volume embedded within an infinite environment. In this method, we first collect the statistics of injection/ejection events in periodic boundary conditions (PBC). Once sufficient statistics are collected, we remove the PBC and turn on the SBC. In the SBC simulations, we allow particles leaving the system to be truly ejected from the simulation, and randomly inject particles at the boundaries by resampling from the injection/ejection statistics collected from the current or previous simulations. With the SBC, we can measure thermodynamic quantities within the grand canonical ensemble, based on the particle number and energy fluctuations. To demonstrate how useful the SBC algorithm is, we simulated a hard disk gas and measured the pair distribution function, the compressibility and the specific heat, comparing them against literature values.

Reliability analysis of plane elasticity problems by stochastic spline fictitious boundary element method

In this paper a stochastic spline fictitious boundary element method (SFBEM) is proposed for reliability analysis of plane elasticity problems in conjunction with the advanced first-order-second-moment (AFOSM) method. The AFOSM method has been demonstrated to be a reliable and practical approach to the structural reliability analysis, yielding results of reasonable accuracy for the engineering applications. And as a modified method for the conventional indirect boundary element method, SFBEM can provide accurate numerical solutions at high efficiency in deterministic analyses. For the purpose of structural reliability analysis, SFBEM is introduced during the iteration process of the AFOSM method, to obtain the required values of structural responses and their derivations with respect to the random variables considered. The use of SFBEM in the formulation of the AFOSM method makes it unnecessary to construct an explicit expression to the implicit limit state function of the problem, leading to a higher efficiency and better accuracy. The present approach is validated by comparing calculated solutions with those of Monte Carlo simulation for a number of example problems and a good agreement of the results is achieved.