stochastic analysis of transport in tubes with rough walls flow and transport in tubes with rough surfaces play an important role in a variety of applications often the topology of such surfaces cannot be accurately described in all of its relevant details due to either insufficient data or measurement errors or both in such cases this topological uncertainty can be efficiently handled by treating rough boundaries as random fields so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains to deal with this class of problems we use a computational framework which is based on stochastic mappings to transform the original deterministic stochastic problem in a random domain into a stochastic problem in a deterministic domain the latter problem has been studied more extensively and existing analytical numerical techniques can be readily applied in this paper we employ both a generalized polynomial chaos and monte carlo simulations to solve the transformed stochastic problem we use our approach to describe transport of a passive scalar in stokes flow and to quantify the corresponding predictive uncertainty

multi level monte carlo weak galerkin method with nested meshes for stochastic brinkman problem abstract this paper is devoted to the numerical analysis of a multi level monte carlo weak galerkin mlmcwg approximation with nested meshes for solving stochastic brinkman equations with two dimensional spatial domain with weak gradient operator and a stabilizer at hand the weak galerkin wg technique is a high order accurate and stable method which can easily handle deterministic partial differential equations with complex geometries flows with jump fluid viscosity coefficients or high contrast permeability fields given by each sample the multi level monte carlo mlmc technique with nested meshes balances the sampling error and the spatial approximation error where the computational cost can be sharply reduced to log linear complexity with respect to the degree of freedom in spatial direction the nested meshes requirement is introduced here in order to simplify the analysis which can be generalized to mlmc with non nested meshes error estimates are derived in terms of the spatial meshsize and the number of samples the numerical tests are provided to illustrate the behavior of the mlmcwg method and verify our theoretical results regarding optimal convergence of the approximate solutions

sparse second moment analysis for elliptic problems in stochastic domains we consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations assuming normal perturbations with small amplitude and known mean field and two point correlation function we derive using a second order shape calculus deterministic equations for the mean field and the two point correlation function of the random solution for a model dirichlet problem which are 3rd order accurate in the boundary perturbation size using a variational boundary integral equation formulation on the unperturbed nominal boundary and a wavelet discretization we present and analyze an algorithm to approximate the random solution s mean and its two point correlation function at essentially optimal order in essentially mathcal o n work and memory where n denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain

on the stabilizing control of a class of nonlinear stochastic systems a class of discrete time nonlinear stochastic control systems is considered by using previously established proper mean square stabilizability and observability conditions the author shows the relationship between the stabilization conditions for the moving and infinite horizon quadratic optimal controllers in this way the existence of the mean square stabilizing infinite horizon controller implies the existence of a multitude of moving finite horizon controllers with different horizon lengths with the same stability property the results allow the use of finite stage solutions of a riccati like matrix equation in a stabilizing control design which can drastically reduce the computation time

some unconditionally stable time stepping methods for the 3d maxwell s equations almost all the difficulties that arise in the numerical solution of maxwell s equations are due to material interfaces in case that their geometrical features are much smaller than a typical wavelength one would like to use small space steps with large time steps the first time stepping method which combines a very low cost per time step with unconditional stability was the adi fdtd method introduced in 1999 the present discussion starts with this method and with an even more recent crank nicolson based split step method with similar properties we then explore how these methods can be made even more efficient by combining them with techniques that increase their temporal accuracies

karhunen loeve approximation of random fields by generalized fast multipole methods kl approximation of a possibly instationary random field a x l2 dp l d subject to prescribed meanfield ea x a x dp and covariance va x x a x ea x a x ea x dp in a polyhedral domain d rd is analyzed we show how for stationary covariances va x x ga x x with ga z analytic outside of z 0 an m term approximate kl expansion am x of a x can be computed in log linear complexity the approach applies in arbitrary domains d and for nonseparable covariances ca it involves galerkin approximation of the kl eigenvalue problem by discontinuous finite elements of degree p 0 on a quasiuniform possibly unstructured mesh of width h in d plus a generalized fast multipole accelerated krylov eigensolver the approximate kl expansion am x of a x has accuracy o exp bm1 d if ga is analytic at z 0 and accuracy o m k d if ga is ck at zero it is obtained in o mn logn b operations where n o h d

regularity analysis of metamaterial maxwell s equations with random coefficients and initial conditions abstract in this paper we develop and analyze the stochastic collocation method for solving the time dependent metamaterial maxwell s equations subject to random coefficients and random initial conditions we provide a rigorous regularity analysis of the solution with respect to the random variables to our best knowledge this is the first theoretical results derived for the stochastic metamaterial maxwell equations the rate of convergence is proved depending on the regularity of the solution numerical results are presented to confirm the theoretical analysis we also demonstrate that the backward wave propagation phenomenon still exists in random metamaterial

high order fdtd methods via derivative matching for maxwell s equations with material interfaces this paper introduces a series of novel hierarchical implicit derivative matching methods to restore the accuracy of high order finite difference time domain fdtd schemes of computational electromagnetics cem with material interfaces in one 1d and two spatial dimensions 2d by making use of fictitious points systematic approaches are proposed to locally enforce the physical jump conditions at material interfaces in a preprocessing stage to arbitrarily high orders of accuracy in principle while often limited by numerical instability orders up to 16 and 12 are achieved respectively in 1d and 2d detailed stability analyses are presented for the present approach to examine the upper limit in constructing embedded fdtd methods as natural generalizations of the high order fdtd schemes the proposed derivative matching methods automatically reduce to the standard fdtd schemes when the material interfaces are absent an interesting feature of the present approach is that it encompasses a variety of schemes of different orders in a single code another feature of the present approach is that it can be robustly implemented with other high accuracy time domain approaches such as the multiresolution time domain method and the local spectral time domain method to cope with material interfaces numerical experiments on both 1d and 2d problems are carried out to test the convergence examine the stability access the efficiency and explore the limitation of the proposed methods it is found that operating at their best capacity the proposed high order schemes could be over 2000 times more efficient than their fourth order versions in 2d in conclusion the present work indicates that the proposed hierarchical derivative matching methods might lead to practical high order schemes for numerical solution of time domain maxwell s equations with material interfaces

convergence analysis of finite element methods for h curl ω elliptic interface problems in this article we investigate the analysis of a finite element method for solving h curl elliptic interface problems in general three dimensional polyhedral domains with smooth interfaces the continuous problems are discretized by means of the first family of lowest order nedelec h curl conforming finite elements on a family of tetrahedral meshes which resolve the smooth interface in the sense of sufficient approximation in terms of a parameter that quantifies the mismatch between the smooth interface and the triangulation optimal error estimates in the h curl norm are obtained for the first time the analysis is based on a strip argument a new extension theorem for h 1 curl functions across smooth interfaces a novel non standard interface aware interpolation operator and a perturbation argument for degrees of freedom for h curl conforming finite elements numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution

further results on stabilizing controllers for discrete non linear stochastic systems more relationships that exist among stabilizability conditions of controllers designed for a class of discrete non linear stochastic systems are described the conditions are in terms of both mean square and almost sure asymptotic stability of the controlled systems and provide converse results to those presented earlier

an energy conserving method for stochastic maxwell equations with multiplicative noise abstract in this paper it is shown that three dimensional stochastic maxwell equations with multiplicative noise are stochastic hamiltonian partial differential equations possessing a geometric structure i e stochastic multi symplectic conservation law and the energy of system is a conservative quantity almost surely we propose a stochastic multi symplectic energy conserving method for the equations by using the wavelet collocation method in space and stochastic symplectic method in time numerical experiments are performed to verify the excellent abilities of the proposed method in providing accurate solution and preserving energy the mean square convergence result of the method in temporal direction is tested numerically and numerical comparisons with finite difference method are also investigated

some existence principles and some general results for singular nonlinear two point boundary value problems abstract existence results for singular second order nonlinear two point boundary value problems of the form 1 p t p t y q t f t y 0 0 where limy 0 f t y are presented here we may allow singularities at y 0 t 0 and or t 1 also existence principles are presented for the general two point boundary value problem 1 p py qf t y py 0 0 where f 0 1 r 2 r is continuous the proof uses the nonlinear alternative of leray and schauder

existence theorems for certain classes of singular boundary value problems abstract existence results for nonlinear singular second order two point boundary value problems of the form 1 p py f t y 0 satisfies appropriate boundary conditions are presented the class of problems considered includes both the thomas fermi and singular emden fowler equations

a fictitious domain approach to the numerical solution of pdes in stochastic domains we present an efficient method for the numerical realization of elliptic pdes in domains depending on random variables domains are bounded and have finite fluctuations the key feature is the combination of a fictitious domain approach and a polynomial chaos expansion the pde is solved in a larger fixed domain the fictitious domain with the original boundary condition enforced via a lagrange multiplier acting on a random manifold inside the new domain a generalized wiener expansion is invoked to convert such a stochastic problem into a deterministic one depending on an extra set of real variables the stochastic variables discretization is accomplished by standard mixed finite elements in the physical variables and a galerkin projection method with numerical integration which coincides with a collocation scheme in the stochastic variables a stability and convergence analysis of the method as well as numerical results are provided the convergence is spectral in the polynomial chaos order in any subdomain which does not contain the random boundaries

symplectic and multisymplectic numerical methods for maxwell s equations in this paper we compare the behaviour of one symplectic and three multisymplectic methods for maxwell s equations in a simple medium this is a system of pdes with symplectic and multisymplectic structures we give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments we also show that these numerical methods preserve the divergence furthermore we extend the discussion on dispersion for multi symplectic methods applied to pdes with one spatial dimension to include anisotropy when applying multi symplectic methods to maxwell s equations in two spatial dimensions lastly we demonstrate how varying the courant friedrichs lewy cfl number can cause the multi symplectic methods in our comparison to behave differently which can be explained by the study of backward error analysis for pdes

an upwinding boundary condition capturing method for maxwell s equations in media with material interfaces by using ghost points on either side of the interfaces a global second order accurate upwinding boundary condition capturing method for time domain maxwell s equations in media with material interfaces is proposed the equations are discretized on a uniform cartesian grid and the interfaces are allowed to intersect the grid in an arbitrary fashion the method is then obtained by combining central finite difference schemes with applicable nodes being replaced by the ghost points and upwinding technique with jump conditions across the interfaces being captured in a manner that the upwind property is always satisfied the resulting discretization has the desirable property that the allowed time step size is independent of the locations and the shapes of the interfaces numerical examples are then given to demonstrate the second order accuracy as well as the stability of the method where it is used to study wave equations with various types of material interfaces including electromagnetic scattering of a plane incident wave by a dielectric circular cylinder

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

a control scheme for a class of discrete nonlinear stochastic systems a control scheme is presented for effective stabilization of discrete time nonlinear stochastic systems where the nonlinearity involves a zero mean independent random sequence the control is of a constant feedback type and makes use of finite time solutions of a riccati like matrix difference equation stability results are both in terms of mean square and almost sure stochastic stability moreover a matrix inequality is given to check the existence of weighting matrices which would result in a stable regulator problem

computation of moments for maxwell s equations with random interfaces via pivoted low rank approximation abstract the aim of this paper is to compute the mean field and variance of solutions to three dimensional maxwell s equations with random interfaces via shape calculus and pivoted low rank approximation based on the perturbation theory and shape calculus we characterize the statistical moments of solutions to maxwell s equations with random interfaces in terms of the perturbation magnitude via the first order shape taylor expansion in order to capture oscillations with high resolution close to the interface an adaptive finite element method using nedelec s third order edge elements of the first kind is employed to solve the deterministic maxwell s equations with the mean interface to approximate the expectation of solutions for the second moment computation an efficient low rank approximation of the pivoted cholesky decomposition is proposed to compute the two point correlation function to approximate the variance of solutions numerical experiments are presented to demonstrate our theoretical results

a note on existence uniqueness results for a class of doubly singular boundary value problems abstract in this paper we establish the existence and uniqueness of solution for nonlinear singular boundary value problem p x y x q x f x y for 0 x b and lim x 0 p x y x 0 1 y b 1 p b y b 1 here p 0 0 and also q is not assumed to be continuous at 0 so the problem may be doubly singular in this note we discuss the cases when y 0 0 at the singular end is not possible but lim x 0 p x y x 0 regions for multiple solutions have also been determined

multi level monte carlo weak galerkin method for elliptic equations with stochastic jump coefficients in this paper we present a multi level monte carlo weak galerkin method for solving elliptic equations with stochastic jump coefficients the multi level monte carlo technique balances the spatial approximation error and the sampling error the weak galerkin technique is a stable and high order accurate method which can easily handle deterministic partial differential equations with complex geometries or jump coefficients given by each sample and this method is also able to capture highly complex solutions exhibiting discontinuities or oscillations with high resolution comparing with the standard monte carlo method by using the multi level monte carlo weak galerkin method the computational cost can be sharply reduced to log linear complexity with respect to the degree of freedom in spatial direction the numerical experiments verify the efficiency of our algorithms

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

stochastic conformal schemes for damped stochastic klein gordon equation with additive noise abstract in this article stochastic conformal schemes of the damped stochastic klein gordon equation with additive noise are studied it is shown that this equation possesses the stochastic conformal multi symplectic conservation law under appropriate boundary conditions the global momentum evolution law and the global energy evolution law are proposed we chiefly develop the stochastic conformal preissman scheme the stochastic conformal discrete gradient scheme and the stochastic conformal euler box scheme to preserve the geometric structures of the original system specifically we make theoretical discussions on the three proposed schemes to obtain corresponding discrete conservation laws or discrete evolution laws then the damped stochastic linear klein gordon equation and the damped stochastic nonlinear klein gordon equation with cubic nonlinearity are taken as examples to demonstrate the validity of the proposed schemes through numerical experiments and comparisons the superiorities of the proposed schemes are fully shown which are consistent with our theoretical analysis moreover the mean square convergence orders of the three stochastic conformal schemes in time direction and space direction are tested numerically

preservation of physical properties of stochastic maxwell equations with additive noise via stochastic multi symplectic methods stochastic maxwell equations with additive noise are a system of stochastic hamiltonian partial differential equations intrinsically possessing the stochastic multi symplectic conservation law it is shown that the averaged energy increases linearly with respect to the evolution of time and the flow of stochastic maxwell equations with additive noise preserves the divergence in the sense of expectation moreover we propose three novel stochastic multi symplectic methods to discretize stochastic maxwell equations in order to investigate the preservation of these properties numerically we make theoretical discussions and comparisons on all of the three methods to observe that all of them preserve the corresponding discrete version of the averaged divergence meanwhile we obtain the corresponding dissipative property of the discrete averaged energy satisfied by each method especially the evolution rates of the averaged energies for all of the three methods are derived which are in accordance with the continuous case numerical experiments are performed to verify our theoretical results

a stochastic multi symplectic scheme for stochastic maxwell equations with additive noise abstract in this paper we investigate a stochastic multi symplectic method for stochastic maxwell equations with additive noise based on the stochastic version of variational principle we find a way to obtain the stochastic multi symplectic structure of three dimensional 3 d stochastic maxwell equations with additive noise we propose a stochastic multi symplectic scheme and show that it preserves the stochastic multi symplectic conservation law and the local and global stochastic energy dissipative properties which the equations themselves possess numerical experiments are performed to verify the numerical behaviors of the stochastic multi symplectic scheme

numerical simulation of focusing stochastic nonlinear schrodinger equations abstract in this paper we numerically investigate nonlinear schrodinger equations with a stochastic contribution which is of white noise type and acts either as a potential multiplicative noise or as a forcing term additive noise in the subcritical case we recover similar results as in the case of the korteweg de vries equation in the critical or supercritical case we observe that depending on its smoothness the noise may have different effects spatially smooth noises amplify blow up phenomena whereas delta correlated multiplicative noises prevent blow up formation note that in this latter case very few results are known both from a theoretical and a numerical point of view

stochastic symplectic and multi symplectic methods for nonlinear schrdinger equation with white noise dispersion we indicate that the nonlinear schrdinger equation with white noise dispersion possesses stochastic symplectic and multi symplectic structures based on these structures we propose the stochastic symplectic and multi symplectic methods which preserve the continuous and discrete charge conservation laws respectively moreover we show that the proposed methods are convergent with temporal order one in probability numerical experiments are presented to verify our theoretical results

splitting multisymplectic integrators for maxwell s equations in the paper we describe a novel kind of multisymplectic method for three dimensional 3 d maxwell s equations splitting the 3 d maxwell s equations into three local one dimensional lod equations then applying a pair of symplectic runge kutta methods to discretize each resulting lod equation it leads to splitting multisymplectic integrators we say this kind of schemes to be lod multisymplectic scheme lod ms the discrete conservation laws convergence dispersive relation dissipation and stability are investigated for the schemes theoretical analysis shows that the schemes are unconditionally stable non dissipative and of first order accuracy in time and second order accuracy in space as a reduction we also consider the application of lod ms to 2 d maxwell s equations numerical experiments match the theoretical results well they illustrate that lod ms is not only efficient and simple in coding but also has almost all the nature of multisymplectic integrators

an octree multigrid method for quasi static maxwell s equations with highly discontinuous coefficients in this paper we develop an octree discretization for maxwell s equations in the quasi static regime we then use this discretization in order to develop a multigrid method for maxwell s equations with highly discontinuous coefficients we test our algorithms and compare it to other multilevel algorithms

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

stochastic wave propagation in maxwell s equations we use the spectral representation to describe the random electromagnetic fields which are coupled by maxwell s equations with a random source term the covariance matrix functions of the random electromagnetic fields are expressed in terms of the spectral matrix function of the random source term and the green function of maxwell s equations when the random source term is gaussian the electromagnetic fields exhibit diffusive scaling limits in the sense of distributions

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

relationships between several novel control schemes proposed for a class of non linear stochastic systems several different novel controllers are proposed to stabilize a class of discrete time non linear stochaslic systems where the non linearity involves the state and input of the system as well as a white noise sequence vector it is shown that the conditions of stabilizability by these controllers are closely related

exponential integrators for stochastic maxwell s equations driven by ito noise abstract this article presents explicit exponential integrators for stochastic maxwell s equations driven by both multiplicative and additive noises by utilizing the regularity estimate of the mild solution we first prove that the strong order of the numerical approximation is 1 2 for general multiplicative noise combining a proper decomposition with the stochastic fubini s theorem the strong order of the proposed scheme is shown to be 1 for additive noise moreover for linear stochastic maxwell s equation with additive noise the proposed time integrator is shown to preserve exactly the symplectic structure the evolution of the energy as well as the evolution of the divergence in the sense of expectation several numerical experiments are presented in order to verify our theoretical findings

infinite horizon quadratic optimal control of a class of nonlinear stochastic systems the author considers a class of discrete time nonlinear stochastic control systems whose nonlinearities are described by statistical means by introducing the proper mean square stabilizability and detectability properties of such systems and giving a characterization of these properties in terms of system parameters he examines the steady state properties of quadratically optimal controllers a robustness property of these controllers is pointed out

finite element methods with matching and nonmatching meshes for maxwell equations with discontinuous coefficients we investigate the finite element methods for solving time dependent maxwell equations with discontinuous coefficients in general three dimensional lipschitz polyhedral domains both matching and nonmatching finite element meshes on the interfaces are considered and optimal error estimates for both cases are obtained the analysis of the latter case is based on an abstract framework for nested saddle point problems along with a characterization of the trace space for h curl d a new extension theorem for h curl d functions in any lipschitz domain d and a novel compactness argument for deriving discrete inf sup conditions

numerical approximation of some linear stochastic partial differential equations driven by special additive noises this paper is concerned with the numerical approximation of some linear stochastic partial differential equations with additive noises a special representation of the noise is considered and it is compared with general representations of noises in the infinite dimensional setting convergence analysis and error estimates are presented for the numerical solution based on the standard finite difference and finite element methods the effects of the noises on the accuracy of the approximations are illustrated results of the numerical experiments are provided

the existence of solutions to second order singular boundary value problems abstract this article analyzes qualitative properties of solutions to two point boundary value problems for singular ordinary differential equations in particular we form new approaches that ensure that all possible solutions satisfy certain a priori bounds the methods involve differential inequalities the ideas are then applied to generate new existence results for solutions to the boundary value problems under consideration many of the results are novel for both the singular and the nonsingular cases the main concepts are further illustrated via applications to various equations arising in modeling such as the poisson boltzmann equation and the thomas fermi equation with an ionized atom

edge element methods for maxwell s equations with strong convergence for gauss laws in this paper we propose and investigate some edge element approximations for three maxwell systems in three dimensions the stationary maxwell equations the time harmonic maxwell equations and the time dependent maxwell equations these approximations have three novel features first the resulting discrete edge element systems can be solved by some existing preconditioned solvers with optimal convergence rate independent of nite element meshes including the stationary maxwell equations second they ensure the optimal strong convergence of the gauss laws in some appropriate norm in addition to the standard optimal convergence in energy norm under the general weak regularity assumptions that hold for both convex and non convex polyhedral domains and for the discontinuous coecients that may have large jumps across the interfaces between dierent media finally no saddle point discrete systems are needed to solve for the stationary maxwell equations unlike most existing edge element schemes