A STOCHASTIC GALERKIN METHOD FOR MAXWELL EQUATIONS WITH UNCERTAINTY

In this article,we investigate a stochastic Galerkin method for the Maxwell equations with random inputs.The generalized Polynomial Chaos(gPC)expansion technique is used to obtain a deterministic system of the gPC expansion coefficients.The regularity of the solution with respect to the random is analyzed.On the basis of the regularity results,the optimal convergence rate of the stochastic Galerkin approach for Maxwell equations with random inputs is proved.Numerical examples are presented to support the theoretical analysis.

A FAST STOCHASTIC GALERKIN METHOD FOR A CONSTRAINED OPTIMAL CONTROL PROBLEM GOVERNED BY A RANDOM FRACTIONAL DIFFUSION EQUATION

We develop a fast stochastic Galerkin method for an optimal control problem governed by a random space-fractional diffusion equation with deterministic constrained control. Optimal control problems governed by a fractional diffusion equation tends to provide a better description for transport or conduction processes in heterogeneous media. Howev- er, the fractional control problem introduces significant computation complexity due to the nonlocal nature of fractional differential operators, and this is further worsen by the large number of random space dimensions to discretize the probability space. We ap- proximate the optimality system by a gradient algorithm combined with the stochastic Galerkin method through the discretization with respect to both the spatial space and the probability space. The resulting linear system can be decoupled for the random and spatial variable, and thus solved separately. A fast preconditioned Bi-Conjugate Gradient Stabilized method is developed to efficiently solve the decoupled systems derived from the fractional diffusion operators in the spatial space. Numerical experiments show the utility of the method.

A Stochastic Galerkin Method for Stochastic Control Problems

In an interdisciplinary field on mathematics and physics,we examine a physical problem,fluid flow in porous media,which is represented by a stochastic partial differential equation(SPDE).We first give a priori error estimates for the solutions to an optimization problem constrained by the physical model under lower regularity assumptions than the literature.We then use the concept of Galerkin finite element methods to establish a new numerical algorithm to give approximations for our stochastic optimal physical problem.Finally,we develop original computer programs based on the algorithm and use several numerical examples of various situations to see how well our solver works by comparing its outputs to the priori error estimates.

Transient stochastic response of quasi integerable Hamiltonian systems

The approximate transient response of quasi in- tegrable Hamiltonian systems under Gaussian white noise excitations is investigated. First, the averaged It6 equa- tions for independent motion integrals and the associated Fokker-Planck-Kolmogorov （FPK） equation governing the transient probability density of independent motion integrals of the system are derived by applying the stochastic averag- ing method for quasi integrable Hamiltonian systems. Then, approximate solution of the transient probability density of independent motion integrals is obtained by applying the Galerkin method to solve the FPK equation. The approxi- mate transient solution is expressed as a series in terms of properly selected base functions with time-dependent coeffi- cients. The transient probability densities of displacements and velocities can be derived from that of independent mo- tion integrals. Three examples are given to illustrate the ap- plication of the proposed procedure. It is shown that the re- suits for the three examples obtained by using the proposed procedure agree well with those from Monte Carlo simula- tion of the original systems.

ON EFFECTIVE STOCHASTIC GALERKIN FINITE ELEMENT METHOD FOR STOCHASTIC OPTIMAL CONTROL GOVERNED BY INTEGRAL-DIFFERENTIAL EQUATIONS WITH RANDOM COEFFICIENTS

In this paper, we apply stochastic Galerkin finite element methods to the optimal control problem governed by an elliptic integral-differential PDEs with random field. The control problem has the control constraints of obstacle type. A new gradient algorithm based on the pre-conditioner conjugate gradient algorithm （PCG） is developed for this optimal control problem. This algorithm can transform a part of the state equation matrix and co-state equation matrix into block diagonal matrix and then solve the optimal control systems iteratively. The proof of convergence for this algorithm is also discussed. Finally numerical examples of a medial size are presented to illustrate our theoretical results.

Nonlinear Stochastic Galerkin and Collocation Methods:Application to a Ferromagnetic Cylinder Rotating at High Speed

The stochastic Galerkin and stochastic collocation method are two state-ofthe-art methods for solving partial differential equations(PDE)containing random coefficients.While the latter method,which is based on sampling,can straightforwardly be applied to nonlinear stochastic PDEs,this is nontrivial for the stochastic Galerkin method and approximations are required.In this paper,both methods are used for constructing high-order solutions of a nonlinear stochastic PDE representing the magnetic vector potential in a ferromagnetic rotating cylinder.This model can be used for designing solid-rotor induction machines in various machining tools.A precise design requires to take ferromagnetic saturation effects into account and uncertainty on the nonlinear magnetic material properties.Implementation issues of the stochastic Galerkin method are addressed and a numerical comparison of the computational cost and accuracy of both methods is performed.The stochastic Galerkin method requires in general less stochastic unknowns than the stochastic collocation approach to reach a certain level of accuracy,however at a higher computational cost.

Numerical comparison of three stochastic methods for nonlinear PN junction problems

We apply the Monte Carlo, stochastic Galerkin, and stochastic collocation methods to solving the drift-diffusion equations coupled with the Poisson equation arising in semiconductor devices with random rough surfaces. Instead of dividing the rough surface into slices, we use stochastic mapping to transform the original deterministic equations in a random domain into stochastic equations in the corresponding deterministic domain. A finite element discretization with the help of AFEPack is applied to the physical space, and the equations obtained are solved by the approximate Newton iterative method. Comparison of the three stochastic methods through numerical experiment on different PN junctions are given. The numerical results show that, for such a complicated nonlinear problem, the stochastic Galerkin method has no obvious advantages on efficiency except accuracy over the other two methods, and the stochastic collocation method combines the accuracy of the stochastic Galerkin method and the easy implementation of the Monte Carlo method.

Nonstationary probability densities of system response of strongly nonlinear single-degree-of-freedom system subject to modulated white noise excitation

The nonstationary probability densities of system response of a single-degree- of-freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to modulated white noise excitation are studied. Using the stochastic averaging method based on the generalized harmonic functions, the averaged Fokl^er-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude is derived. The solution of the equation is approximated by the series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. According to the Galerkin method, the time-dependent coefficients can be solved from a set of first-order linear differential equations. Then, the semi-analytical formulae of the nonstationary probability density of the amplitude response as well as the nonstationary probability density of the state response and the statistic moments of the amplitude response can be obtained. A van der Pol-Duffing oscillator subject to modulated white noise is given as an example to illustrate the proposed procedures. The effects of the system parameters, such as the linear damping coefficient and the nonlinear stiffness coefficient, on the system response are discussed.

Newton-Multigrid for Biological Reaction-Diffusion Problems with Random Coefficients

An algebraic Newton-multigrid method is proposed in order to efficiently solve systems of nonlinear reaction-diffusion problems with stochastic coefficients.These problems model the conversion of starch into sugars in growing apples.The stochastic system is first converted into a large coupled system of deterministic equations by applying a stochastic Galerkin finite element discretization.This method leads to high-order accurate stochastic solutions.A stable and high-order time discretization is obtained by applying a fully implicit Runge-Kutta method.After Newton linearization,a point-based algebraic multigrid solution method is applied.In order to decrease the computational cost,alternative multigrid preconditioners are presented.Numerical results demonstrate the convergence properties,robustness and efficiency of the proposed multigrid methods.