shannon entropy fisher information and uncertainty relations for log periodic oscillators abstract we calculate the time dependent shannon s x and s p entropy and fisher f x and f p information of three log periodic oscillators we obtain a general expression for s x p and f x p in the state n 0 in terms of a c number quantity satisfying a nonlinear differential equation for two out of three oscillators s x p and f x p depend on time but s x s p and f x f p do not the other oscillator behaves as the time independent harmonic oscillator where s x p and f x p are all constants relations among the fisher information and the stam and cramer rao inequalities are also discussed

a new algorithm for high dimensional uncertainty quantification based on dimension adaptive sparse grid approximation and reduced basis methods in this work we develop an adaptive and reduced computational algorithm based on dimension adaptive sparse grid approximation and reduced basis methods for solving high dimensional uncertainty quantification uq problems in order to tackle the computational challenge of curse of dimensionality commonly faced by these problems we employ a dimension adaptive tensor product algorithm 16 and propose a verified version to enable effective removal of the stagnation phenomenon besides automatically detecting the importance and interaction of different dimensions to reduce the heavy computational cost of uq problems modelled by partial differential equations pde we adopt a weighted reduced basis method 7 and develop an adaptive greedy algorithm in combination with the previous verified algorithm for efficient construction of an accurate reduced basis approximation the efficiency and accuracy of the proposed algorithm are demonstrated by several numerical experiments

a multiscale preconditioner for stochastic mortar mixed finite elements the aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media the governing equations are based on darcy s law with a stochastic permeability field starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions a multiscale mortar flux basis is computed for a single permeability called a training permeability that captures the main characteristics of the porous media and is used as a preconditioner for each stochastic realization we prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media

dissipative terms and local time stepping improvements in a spatial high order discontinuous galerkin scheme for the time domain maxwell s equations in this paper we present some improvements in terms of accuracy and speed up for a particular well adapted discontinuous galerkin method devoted to the time domain maxwell equations first to reduce spurious modes on very distorted meshes the addition of dissipative terms as penalization in the numerical scheme is studied and compared on examples second in order to increase the efficiency of the method a multi class local time stepping strategy is presented and its validation and advantages are highlighted on different examples

a data driven stochastic collocation approach for uncertainty quantification in mems this work presents a data driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi physics simulation of micro electromechanical systems mems the proposed framework comprises of two key steps first probabilistic characterization of the input uncertain parameters based on available experimental information and second propagation of these uncertainties through the predictive model to relevant quantities of interest the uncertain input parameters are modeled as independent random variables for which the distributions are estimated based on available experimental observations using a nonparametric diffusion mixing based estimator botev nonparametric density estimation via diffusion mixing technical report 2007 the diffusion based estimator derives from the analogy between the kernel density estimation kde procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent the diffusion model allows for the incorporation of the prior density and leads to an improved density estimate in comparison with the standard kde approach as demonstrated through several numerical examples following the characterization step the uncertainties are propagated to the output variables using the stochastic collocation approach based on sparse grid interpolation smolyak soviet math dokl 1963 4 240 243 the developed framework is used to study the effect of variations in young s modulus induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a mems switch copyright 2010 john wiley sons ltd

minimal multi element stochastic collocation for uncertainty quantification of discontinuous functions we propose a multi element stochastic collocation method that can be applied in high dimensional parameter space for functions with discontinuities lying along manifolds of general geometries the key feature of the method is that the parameter space is decomposed into multiple elements defined by the discontinuities and thus only the minimal number of elements are utilized on each of the resulting elements the function is smooth and can be approximated using high order methods with fast convergence properties the decomposition strategy is in direct contrast to the traditional multi element approaches which define the sub domains by repeated splitting of the axes in the parameter space such methods are more prone to the curse of dimensionality because of the fast growth of the number of elements caused by the axis based splitting the present method is a two step approach firstly a discontinuity detector is used to partition parameter space into disjoint elements in each of which the function is smooth the detector uses an efficient combination of the high order polynomial annihilation technique along with adaptive sparse grids and this allows resolution of general discontinuities with a smaller number of points when the discontinuity manifold is low dimensional after partitioning an adaptive technique based on the least orthogonal interpolant is used to construct a generalized polynomial chaos surrogate on each element the adaptive technique reuses all information from the partitioning and is variance suppressing we present numerous numerical examples that illustrate the accuracy efficiency and generality of the method when compared against standard locally adaptive sparse grid methods the present method uses many fewer number of collocation samples and is more accurate

multi fidelity non intrusive polynomial chaos based on regression abstract in this paper we present a multi fidelity mf extension of non intrusive polynomial chaos based on regression point collocation for uncertainty quantification purposes the proposed method uses the principle of a global correction function from a previous similar method that uses spectral projection to estimate the coefficients due to its usage of regression to estimate the coefficients the present method offers high flexibility in the sampling and generation of the polynomial basis the method takes advantage of a nested sampling plan to create the samples for the low fidelity lf and correction expansions where the high fidelity hf samples are a subset of the lf ones to build the polynomial basis a total order or hyperbolic truncation strategy is used with a highly flexible combination of the lf and correction polynomial expansions the method is demonstrated on some artificial test problems and aerodynamic problems of the euler flow around an airfoil and common three dimensional research models in order to derive the strategies for successful mf approximation the effect of the correlation and the errors between the lf and hf functions is also studied the results show that high correlation and moderately low errors are important to improve the mf approximation s accuracy on a common research model problem the mf approach with partially converged simulations as the lf samples can successfully reduce the computational cost to about 40 for similar accuracy compared to an approach using a single hf expansion

fisher information and shannon entropy of position dependent mass oscillators abstract we calculate the fisher information and the shannon entropy for three position dependent mass oscillators these systems can be seen as deformed harmonic oscillators in the sense that when the deformation parameter goes to zero they are identical to the constant mass harmonic oscillator for two out of the three oscillators we observe that as increases the position fisher information f x increases while the momentum fisher information f p decreases on the other hand the shannon entropy always increases for the three systems with increasing discussion about squeezing effect in either position or momentum due to the variation and a relation between the product of fisher information and the shannon entropy are also presented

a posteriori error estimation for discontinuous galerkin solutions of hyperbolic problems abstract we analyze the spatial discretization errors associated with solutions of one dimensional hyperbolic conservation laws by discontinuous galerkin methods dgms in space we show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a radau polynomial of degree p 1 on each element we also prove that the local and global discretization errors are o x2 p 1 and o x2p 1 at the downwind point of each element this strong superconvergence enables us to show that local and global discretization errors converge as o xp 2 at the remaining roots of radau polynomial of degree p 1 on each element convergence of local and global discretization errors to the radau polynomial of degree p 1 also holds for smooth solutions as p these results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

weak galerkin methods for time dependent maxwell s equations abstract this paper adapts the weak galerkin wg finite element scheme to maxwell s equations in the time domain developed by wang and ye in 2011 the wg scheme is a discontinuous galerkin like method that relies on a new notion of the discrete weak derivative the wg method as applied to maxwell s equations uses a discrete weak curl and an extra stabilization term to enforce a weak continuity of the solution across elements semi discrete and fully discrete schemes are developed and are shown to be unconditionally stable an optimal order of convergence is proven in an appropriate energy norm and confirmed by numerical results for clarity the theoretical analysis is carried out for 3 d case some details showing how to implement this wg scheme in 2 d are provided and numerical results are given in 2 d

a class of discontinuous petrov galerkin methods part iv the optimal test norm and time harmonic wave propagation in 1d the phase error or the pollution effect in the finite element solution of wave propagation problems is a well known phenomenon that must be confronted when solving problems in the high frequency range this paper presents a new method with no phase errors for one dimensional 1d time harmonic wave propagation problems using new ideas that hold promise for the multidimensional case the method is constructed within the framework of the discontinuous petrov galerkin dpg method with optimal test functions we have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm in this paper we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm this is answered in the specific case of the helmholtz equation with l 2 norm as the energy norm we obtain uniform stability with respect to the wave number we illustrate the method with a number of 1d numerical experiments using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally a 1d theoretical stability analysis is also developed

locally implicit discontinuous galerkin method for time domain electromagnetics in the recent years there has been an increasing interest in discontinuous galerkin time domain dgtd methods for the solution of the unsteady maxwell equations modeling electromagnetic wave propagation one of the main features of dgtd methods is their ability to deal with unstructured meshes which are particularly well suited to the discretization of the geometrical details and heterogeneous media that characterize realistic propagation problems such dgtd methods most often rely on explicit time integration schemes and lead to block diagonal mass matrices however explicit dgtd methods are also constrained by a stability condition that can be very restrictive on highly refined meshes and when the local approximation relies on high order polynomial interpolation an implicit time integration scheme is a natural way to obtain a time domain method which is unconditionally stable but at the expense of the inversion of a global linear system at each time step a more viable approach consists of applying an implicit time integration scheme locally in the refined regions of the mesh while preserving an explicit time scheme in the complementary part resulting in an hybrid explicit implicit or locally implicit time integration strategy in this paper we report on our recent efforts towards the development of such a hybrid explicit implicit dgtd method for solving the time domain maxwell equations on unstructured simplicial meshes numerical experiments for 3d propagation problems in homogeneous and heterogeneous media illustrate the possibilities of the method for simulations involving locally refined meshes

parameter and state model reduction for large scale statistical inverse problems a greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters large scale models are too costly to evaluate repeatedly as is required in the statistical setting furthermore these models often have high dimensional parametric input spaces which compounds the difficulty of effectively exploring the uncertainty space we simultaneously address both challenges by constructing a projection based reduced model that accepts low dimensional parameter inputs and whose model evaluations are inexpensive the associated parameter and state bases are obtained through a greedy procedure that targets the governing equations model outputs and prior information the methodology and results are presented for groundwater inverse problems in one and two dimensions

entropy and information of a harmonic oscillator in a time varying electric field in 2d and 3d noncommutative spaces we analyze the noncommutativity effects on the fisher information fr p and shannon entropies sr p of a harmonic oscillator immersed in a time varying electric field in two and three dimensions we find the exact solutions of the respective time dependent schrodinger equation and use them to calculate the fisher information and the shannon entropy for the simplest case corresponding to the lowest lying state of each system while there is no problem in defining the shannon entropy for noncommutating spaces the definition of the fisher information have to be modified to satisfy the cramer rao inequalities for both systems we observe how the fisher information and shannon entropy in position and momentum change due to the noncommutativity of the space we verify that the bialynicki birula mycielski bbm entropic uncertainty relation still holds in the systems considered

probabilistic collocation for period 1 limit cycle oscillations in this paper probabilistic collocation for limit cycle oscillations pclco is proposed probabilistic collocation pc is a non intrusive approach to compute the polynomial chaos description of uncertainty numerically polynomial chaos can require impractical high orders to approximate long term time integration problems due to the fast increase of required polynomial chaos order with time pclco is a pc formulation for modeling the long term stochastic behavior of dynamical systems exhibiting a periodic response i e a limit cycle oscillation lco in the pc method deterministic time series are computed at collocation points in probability space in pclco pc is applied to a time independent parametrization of the periodic response of the deterministic solves instead of to the time dependent functions themselves due to the time independent parametrization the accuracy of pclco is independent of time the approach is applied to period 1 oscillations with one main frequency subject to a random parameter numerical results are presented for the harmonic oscillator a two dof airfoil flutter model and the fluid structure interaction of an elastically mounted cylinder

optimal control with stochastic pde constraints and uncertain controls abstract the optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed the lagrangian that defines the problem is postulated in terms of stochastic functions with the control function possibly decomposed into an unknown deterministic component and a known zero mean stochastic component the extra freedom provided by the stochastic dimension in defining cost functionals is explored demonstrating the scope for controlling statistical aspects of the system response one shot stochastic finite element methods are used to find approximate solutions to control problems it is shown that applying the stochastic collocation finite element method to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional thereby forgoing the primary advantage of the collocation method over the stochastic galerkin method for the considered problem the application of the presented methods is demonstrated through a number of numerical examples the presented framework is sufficiently general to also consider a class of inverse problems and numerical examples of this type are also presented

a multiscale discontinuous galerkin method with the computational structure of a continuous galerkin method proliferation of degrees of freedom has plagued discontinuous galerkin methodology from its inception over 30 years ago this paper develops a new computational formulation that combines the advantages of discontinuous galerkin methods with the data structure of their continuous galerkin counterparts the new method uses local element wise problems to project a continuous finite element space into a given discontinuous space and then applies a discontinuous galerkin formulation the projection leads to parameterization of the discontinuous degrees of freedom by their continuous counterparts and has a variational multiscale interpretation this significantly reduces the computational burden and at the same time little or no degradation of the solution occurs in fact the new method produces improved solutions compared with the traditional discontinuous galerkin method in some situations

on stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity on stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity

a dynamically bi orthogonal method for time dependent stochastic partial differential equations ii adaptivity and generalizations this is part ii of our paper in which we propose and develop a dynamically bi orthogonal method dybo to study a class of time dependent stochastic partial differential equations spdes whose solutions enjoy a low dimensional structure in part i of our paper 9 we derived the dybo formulation and proposed numerical algorithms based on this formulation some important theoretical results regarding consistency and bi orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the dybo method in this paper we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically our complexity analysis shows that the ratio of computational complexities between the dybo method and a generalized polynomial chaos method gpc is roughly of order o m n p 3 for a quadratic nonlinear spde where m is the number of mode pairs used in the dybo method and n p is the number of elements in the polynomial basis in gpc the effective dimensions of the stochastic solutions have been found to be small in many applications so we can expect m is much smaller than n p and computational savings of our dybo method against gpc are dramatic the adaptive strategy plays an essential role for the dybo method to be effective in solving some challenging problems another important contribution of this paper is the generalization of the dybo formulation for a system of time dependent spdes several numerical examples are provided to demonstrate the effectiveness of our method including the navier stokes equations and the boussinesq approximation with brownian forcing

a fast collocation method for solving stochastic integral equations based on sparse grid multiscale piecewise polynomial bases we develop a fast collocation method for solving fredholm integral equations of the second kind with stochastic loading terms it is proved that the proposed method preserves the optimal rate of convergence and has linear up to a logarithmic factor computational complexity the reduction in computational costs come from three aspects the adoption of the collocation principle the use of the sparse grid for the solution space and the employment of a truncation strategy for the coefficient matrix of the resulting linear system numerical experiments confirm the theoretical results on approximation accuracy and computational efficiency of the proposed method

risk averse structural topology optimization under random fields using stochastic expansion methods abstract this work proposes a level set based approach for solving risk averse structural topology optimization problems considering random field loading and material uncertainty the use of random fields increases the dimensionality of the stochastic domain which poses several computational challenges related to the minimization of the excess probability as a measure of risk awareness this problem is addressed both from the theoretical and numerical viewpoints first an existence result under a typical geometrical constraint on the set of admissible shapes is proved second a level set continuous approach to find the numerical solution of the problem is proposed since the considered cost functional has a discontinuous integrand the numerical approximation of the functional and its sensitivity combine an adaptive anisotropic polynomial chaos pc approach with a monte carlo mc sampling method for uncertainty propagation furthermore to address the increment of dimensionality induced by the random field an anisotropic sparse grid stochastic collocation method is used for the efficient computation of the pc coefficients a key point is that the non intrusive nature of such an approach facilitates the use of high performance computing hpc to alleviate the computational burden of the problem several numerical experiments including random field loading and material uncertainty are presented to show the feasibility of the proposal

entropy and complexity analysis of dirac delta like quantum potentials abstract the dirac delta like quantum mechanical potentials are frequently used to describe and interpret numerous phenomena in many scientific fields including atomic and molecular physics condensed matter and quantum computation the entropy and complexity properties of potentials with one and two dirac delta functions are here analytically calculated and numerically discussed in both position and momentum spaces we have studied the information theoretic lengths of fisher renyi and shannon types as well as the cramer rao fisher shannon and lmc shape complexities of the lowest lying stationary states of one delta and twin delta they allow us to grasp and quantify different facets of the spreading of the charge and momentum of the system far beyond the celebrated standard deviation

discontinuous and coupled continuous discontinuous galerkin methods for the shallow water equations we consider the approximation of a simplified model of the depth averaged two dimensional shallow water equations by two approaches in both approaches a discontinuous galerkin dg method is used to approximate the continuity equation in the first approach a continuous galerkin method is used for the momentum equations in the second approach a particular dg method the nonsymmetric interior penalty galerkin method is used to approximate momentum a priori error estimates are derived and numerical results are presented for both approaches

numerical study of signal propagation in corrugated coaxial cables this paper is concerned with high fidelity modeling of signal propagation in corrugated coaxial cables taking advantage of the axisymmetry we reduce the 3 d problem to a 2 d problem by solving a time dependent maxwell s equations in cylindrical coordinates we develop a nodal discontinuous galerkin method for solving our model equations stability and error analysis are proved for the semi discrete scheme extensive numerical results are provided to demonstrate that our algorithm not only converges as our theoretical analysis predicts but also is very effective in solving various signal propagation problems in practical corrugated coaxial cables

interior penalty dg methods for maxwell s equations in dispersive media in this paper we develop a fully discrete interior penalty discontinuous galerkin method for solving the time dependent maxwell s equations in dispersive media the model is described by a vector integral differential equation our scheme is proved to be unconditionally stable and achieve optimal error estimates in both l 2 norm and energy norm the scheme is implemented and numerical results supporting our analysis are presented

a p adaptive method for electromagnetic wave propagation summary the discontinuous galerkin dg finite element method is used for the numerical solution of the three dimensional maxwell equations control of errors in the numerical level for the divergence free constraint of the magnetic field can be obtained through the use of divergence free vector bases in this work the so called perfectly hyperbolic ph formulation of the maxwell equations is employed to retain both divergence free magnetic field and in the presence of charges to satisfy the gauss constraint for the electric field at the numerical level for both approaches it is found that higher order approximations have favorable effect on the preservation of the divergence constraints and that the ph formulations retains these errors to a lower level it is shown that high order accuracy in space and time is achieved in unstructured meshes using implicit time marching for nonuniform meshes local resolution refinement is employed using p type adaptivity to ensure accurate electromagnetic wave propagation thus the potential of the method to reach the required higher resolution in anisotropic meshes and obtain accurate electromagnetic wave propagation with reduced computational effort is demonstrated this article is protected by copyright all rights reserved

a robust bi orthogonal dynamically orthogonal method using the covariance pseudo inverse with application to stochastic flow problems abstract we develop a new robust methodology for the stochastic navier stokes equations based on the dynamically orthogonal do and bi orthogonal bo methods 1 2 3 both approaches are variants of a generalized karhunen loeve kl expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics hence capturing the low dimensional structure of the solution the do and bo formulations are mathematically equivalent 3 but they exhibit computationally complimentary properties specifically the bo formulation may fail due to crossing of the eigenvalues of the covariance matrix while both bo and do become unstable when there is a high condition number of the covariance matrix or zero eigenvalues to this end we combine the two methods into a robust hybrid framework and in addition we employ a pseudo inverse technique to invert the covariance matrix the robustness of the proposed method stems from addressing the following issues in the do bo formulation i eigenvalue crossing we resolve the issue of eigenvalue crossing in the bo formulation by switching to the do near eigenvalue crossing using the equivalence theorem and switching back to bo when the distance between eigenvalues is larger than a threshold value ii ill conditioned covariance matrix we utilize a pseudo inverse strategy to invert the covariance matrix iii adaptivity we utilize an adaptive strategy to add remove modes to resolve the covariance matrix up to a threshold value in particular we introduce a soft threshold criterion to allow the system to adapt to the newly added removed mode and therefore avoid repetitive and unnecessary mode addition removal when the total variance approaches zero we show that the do bo formulation becomes equivalent to the evolution equation of the optimally time dependent modes 4 we demonstrate the capability of the proposed methodology with several numerical examples namely i stochastic burgers equation we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues ii stochastic kovasznay flow we examine the method in the presence of a singular covariance matrix and iii we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen do bo modes are active

approximating infinity dimensional stochastic darcy s equations without uniform ellipticity we consider a stochastic darcy s pressure equation whose coefficient is generated by a white noise process on a hilbert space employing the ordinary rather than the wick product a weak form of this equation involves different spaces for the solution and test functions and we establish a continuous inf sup condition and well posedness of the problem we generalize the numerical approximations proposed in benth and theting stochastic anal appl 20 2002 pp 1191 1223 for wick stochastic partial differential equations to the ordinary product stochastic pressure equation we establish discrete inf sup conditions and provide a priori error estimates for a wide class of norms the proposed numerical approximation is based on wiener chaos finite element methods and yields a positive definite symmetric linear system we also improve and generalize the approximation results of benth and gjerde stochastics stochastics rep 63 1998 pp 313 326 and cao stochastics 78 2006 pp 179 187 when a generalized process is truncated by a finite wiener chaos expansion finally we present numerical experiments to validate the results

a stochastic modeling methodology based on weighted wiener chaos and malliavin calculus in many stochastic partial differential equations spdes involving random coefficients modeling the randomness by spatial white noise may lead to ill posed problems here we consider an elliptic problem with spatial gaussian coefficients and present a methodology that resolves this issue it is based on stochastic convolution implemented via generalized malliavin operators in conjunction with weighted wiener spaces that ensure the ellipticity condition we present theoretical and numerical results that demonstrate the fast convergence of the method in the proper norm our approach is general and can be extended to other spdes and other types of multiplicative noise stochastic elliptic problem spdes uncertainty quantification

anchored anova petrov galerkin projection schemes for parabolic stochastic partial differential equations abstract we present a numerical scheme based on the combination of hoeffding s functional analysis of variance anova decomposition with stochastic galerkin projection for solving a class of high dimensional parabolic stochastic partial differential equations spdes the central idea underpinning the proposed approach is to approximate the spde solution using a hoeffding anova decomposition whose component functions are constrained to be orthogonal with respect to an appropriate measure we show that when a dirac product measure is chosen to enforce the orthogonality constraints and the test functions are chosen appropriately the original stochastic weak formulation can be decoupled into low dimensional subproblems that can be efficiently solved in parallel using polynomial chaos based stochastic projection schemes as a result the proposed approach scales very well to spdes with large number of random variables we theoretically analyze the proposed formulation and provide a priori error estimates as a function of the spatial stochastic and temporal discretization parameters and the anova expansion order that hold under appropriate stochastic regularity assumptions numerical studies are presented for a set of time dependent stochastic diffusion problems with up to 50 random variables to demonstrate the effectiveness of the proposed approach and comparisons are made against classical polynomial chaos stochastic galerkin projection scheme gpc and the generalized spectral decomposition scheme these studies show that the proposed approach provides accuracy comparable to classical gpc galerkin projection schemes while incurring significantly lower computational cost

a new c0 discontinuous galerkin method for kirchhoff plates abstract a general framework of constructing c 0 discontinuous galerkin cdg methods is developed for solving the kirchhoff plate bending problem following some ideas in castillo et al 2000 10 and cockburn 2003 12 the numerical traces are determined based on a discrete stability identity which lead to a class of stable cdg methods a stable cdg method called the lcdg method is particularly interesting in our study it can be viewed as an extension to fourth order problems of the ldg method studied in castillo et al 2000 10 and cockburn 2003 12 for this method optimal order error estimates in certain broken energy norm and h 1 norm are established some numerical results are reported confirming the theoretical convergence orders

weighted stochastic sobolev spaces and bilinear spdes driven by space time white noise in this paper we develop basic elements of malliavin calculus on a weightedl2 this class of generalized wiener functionals is a hilbert space it turns out to be substantially smaller than the space of hida distributions while large enough to accommodate solutions of bilinear stochastic pdes as an example we consider a stochastic advection diffusion equation driven by space time white noise in rd it is known that ford 1 this equation has no solutions inl2 in contrast it is shown in the paper that in an appropriately weightedl2 there is a unique solution to the stochastic advection diffusion equation for anyd 1 in addition we present explicit formulas for the hermite fourier coefficients in the wiener chaos expansion of the solution

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

optimal non dissipative discontinuous galerkin methods for maxwells equations in drude metamaterials simulation of electromagnetic wave propagation in metamaterials leads to more complicated time domain maxwells equations than the standard maxwells equations in free space in this paper we develop and analyze a non dissipative discontinuous galerkin dg method for solving the maxwells equations in drude metamaterials previous discontinuous galerkin methods in the literature for electromagnetic wave propagation in metamaterials were either non dissipative but sub optimal or dissipative and optimal our method uses a different and simple choice of numerical fluxes achieving provable non dissipative stability and optimal error estimates simultaneously we prove the stability and optimal error estimates for both semi and fully discrete dg schemes with the leap frog time discretization for the fully discrete case numerical results are given to demonstrate that the dg method can solve metamaterial maxwells equations effectively

a stochastic mixed finite element heterogeneous multiscale method for flow in porous media a computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales in order to capture the small scale heterogeneity a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method hmm in the spatial domain this new method ensures both local and global mass conservation starting from a specified covariance function the stochastic log permeability is discretized in the stochastic space using a truncated karhunen loeve expansion with several random variables due to the small correlation length of the covariance function this often results in a high stochastic dimensionality therefore a newly developed adaptive high dimensional stochastic model representation technique hdmr is used in the stochastic space this results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method asgc numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method

on the constants in hp finite element trace inverse inequalities we derive inverse trace inequalities for hp finite elements utilizing orthogonal polynomials we show how to derive explicit expressions for the constants when considering triangular and tetrahedral elements we also discuss how to generalize this technique to the general d simplex

discontinuous galerkin finite element method applied to the 1 d spherical neutron transport equation discontinuous galerkin finite element methods are used to estimate solutions to the non scattering 1 d spherical neutron transport equation various trial and test spaces are compared in the context of a few sample problems whose exact solution is known certain trial spaces avoid unphysical behaviors that seem to plague other methods comparisons with diamond differencing and simple corner balancing are presented to highlight these improvements

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

fisher like atomic divergences mathematical grounds and physical applications abstract two different local divergence measures the fisher fd and the jensen fisher jfd ones are compared in this work by applying them to atomic one particle densities in position and momentum spaces they are defined in terms of the absolute and the relative fisher information functionals the analysis here afforded includes not only neutral atoms but also singly charged cations the results are interpreted and justified according to i shell filling patterns ii short and long range behaviors of the atomic densities and iii the value of the atomic ionization potential the strengths of the fd measure as compared to the jfd one are emphasized

discontinuous galerkin methods for dispersive and lossy maxwell s equations and pml boundary conditions in this paper we will present a unified formulation of discontinuous galerkin method dgm for maxwell s equations in linear dispersive and lossy materials of debye type and in the artificial perfectly matched layer pml regions an auxiliary differential equation ade method is used to handle the frequency dependent constitutive relations with the help of auxiliary polarization currents in the computational and pml regions the numerical flux for the dispersive lossy maxwell s equations with the auxiliary polarization current variables is derived various numerical results are provided to validate the proposed formulation

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

efficient posterior exploration of a high dimensional groundwater model from two stage markov chain monte carlo simulation and polynomial chaos expansion 1 this study reports on two strategies for accelerating posterior inference of a highly parameterized and cpu demanding groundwater flow model our method builds on previous stochastic collocation approaches e g marzouk and xiu 2009 and marzouk and najm 2009 and uses generalized polynomial chaos gpc theory and dimensionality reduction to emulate the output of a large scale groundwater flow model the resulting surrogate model is cpu efficient and serves to explore the posterior distribution at a much lower computational cost using two stage mcmc simulation the case study reported in this paper demonstrates a two to five times speed up in sampling efficiency

an implicit leap frog discontinuous galerkin method for the time domain maxwell s equations in metamaterials numerical simulation of metamaterials play a very important role in the design of invisibility cloak and sub wavelength imaging in this paper we propose a leap frog discontinuous galerkin method to solve the time dependent maxwell s equations in metamaterials conditional stability and error estimates are proved for the scheme the proposed algorithm is implemented and numerical results supporting the analysis are provided

local discontinuous galerkin methods for nonlinear dispersive equations we develop local discontinuous galerkin dg methods for solving nonlinear dispersive partial differential equations that have compactly supported traveling waves solutions the so called compactons the schemes we present extend the previous works of yan and shu on approximating solutions for linear dispersive equations and for certain kdv type equations we present two classes of dg methods for approximating solutions of such pdes first we generate nonlinearly stable numerical schemes with a stability condition that is induced from a conservation law of the pde an alternative approach is based on constructing linearly stable schemes i e schemes that are linearly stable to small perturbations the numerical simulations we present verify the desired properties of the methods including their expected order of accuracy in particular we demonstrate the potential advantages of using dg methods over pseudospectral methods in situations where discontinuous fronts and rapid oscillations co exist in a solution

a domain decomposition method of stochastic pdes recent advances in high performance computing systems and sensing technologies motivate computational simulations with extremely high resolution models with capabilities to quantify uncertainties for credible numerical predictions a two level domain decomposition method is reported in this investigation to devise a linear solver for the large scale system in the galerkin spectral stochastic finite element method ssfem in particular a two level scalable preconditioner is introduced in order to iteratively solve the large scale linear system in the intrusive ssfem using an iterative substructuring based domain decomposition solver the implementation of the algorithm involves solving a local problem on each subdomain that constructs the local part of the preconditioner and a coarse problem that propagates information globally among the subdomains the numerical and parallel scalabilities of the two level preconditioner are contrasted with the previously developed one level preconditioner for two dimensional flow through porous media and elasticity problems with spatially varying non gaussian material properties a distributed implementation of the parallel algorithm is carried out using mpi and petsc parallel libraries the scalabilities of the algorithm are investigated in a linux cluster

development of discontinuous galerkin methods for maxwell s equations in metamaterials and perfectly matched layers the discontinuous galerkin method has proved to be an accurate and efficient way to numerically solve many differential equations in this paper we extend this method to solve the time dependent maxwell s equations when metamaterials and perfectly matched layers are involved numerical results are presented to demonstrate that our method is not only simple to implement but also quite effective in solving maxwell s equations in complex media

analysis and application of the nodal discontinuous galerkin method for wave propagation in metamaterials in this paper we develop a nodal discontinuous galerkin method for solving the time dependent maxwell s equations when metamaterials are involved both semi and fully discrete schemes are constructed numerical stability and error estimate are proved for both schemes numerical results are presented to demonstrate that the method is not only efficient but also very effective in solving metamaterial maxwell s equations

acceleration of the spectral stochastic fem using pod and element based discrete empirical approximation for a micromechanical model of heterogeneous materials with random geometry abstract the spectral stochastic fem with local basis functions in the stochastic domain sl fem is one of the most flexible and accurate stochastic methods however also the most computationally expensive these expenses are traditionally associated with the extra large tangent stiffness matrix and a huge number of elements which need to be re integrated in every iteration in this work we incorporate the proper orthogonal decomposition pod into the sl fem thus performing a drastic reduction of the stiffness matrix in order to reduce the integration costs by hyperreduction a novel element based modification of the discrete empirical interpolation the so called element based empirical approximation method edeam is developed and combined with the pod particular advantages of the sl fem for order reduction and hyperreduction compared to other stochastic techniques are discussed the new reduced order sl fem is applied to the computational homogenization of materials with random geometry of the microstructure i e to a general class of problems exhibiting strongly nonlinear non smooth and sometimes discontinuous dependency of the solution on some random parameters the reduced order sl fem demonstrates a high accuracy and a high solution speed whereby the solution time for the reduced order sl fem is comparable to the solution time of only one single monte carlo sample

interior penalty discontinuous galerkin method for maxwell s equations energy norm error estimates we develop the symmetric interior penalty discontinuous galerkin dg method for the time dependent maxwell equations in second order form we derive optimal a priori error estimates in the energy norm for smooth solutions we also consider the case of low regularity solutions that have singularities in space

locally divergence free discontinuous galerkin methods for the maxwell equations in this paper we develop the locally divergence free discontinuous galerkin method for numerically solving the maxwell equations the distinctive feature of the method is the use of approximate solutions that are exactly divergence free inside each element as a consequence this method has a smaller computational cost than that of the discontinuous galerkin method with standard piecewise polynomial spaces we show that in spite of this fact it produces approximations of the same accuracy we also show that this method is more efficient than the discontinuous galerkin method using globally divergence free piecewise polynomial bases finally a post processing technique is used to recover 2k 1 th order of accuracy when piecewise polynomials of degree k are used

a note on stochastic elliptic models there exist two types of commonly studied stochastic elliptic models in literature i a x u x f x and ii a x u x f x where indicates randomness the wick product and a x is a positive random process model i is widely used in engineering and physical applications while model ii is usually studied from the mathematical point of view the difference between the above two stochastic elliptic models has not been fully clarified in this work we discuss the difference between models i and ii when a x is a log normal random process we show that the difference between models i and ii is mainly characterized by a scaling factor which is an exponential function of the degree of perturbation of a x we then construct a new stochastic elliptic model iii a 1 1 u x f x which has the same scaling factor as model i after removing the divergence from the scaling factor models i and iii can be highly comparable for many cases we demonstrate this by a numerical study for a one dimensional problem

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

efficient solvers for a linear stochastic galerkin mixed formulation of diffusion problems with random data we introduce a stochastic galerkin mixed formulation of the steady state diffusion equation and focus on the efficient iterative solution of the saddle point systems obtained by combining standard finite element discretizations with two distinct types of stochastic basis functions so called mean based preconditioners based on fast solvers for scalar diffusion problems are introduced for use with the minimum residual method we derive eigenvalue bounds for the preconditioned system matrices and report on the efficiency of the chosen preconditioning schemes with respect to all the discretization parameters

numerical convergence and physical fidelity analysis for maxwell s equations in metamaterials abstract in this paper we develop a leap frog mixed finite element method for solving maxwell s equations resulting from metamaterials our scheme is similar to the popular yee s fdtd scheme used in electrical engineering community and is preferable for three dimensional large scale modeling since no storage of the large coefficient matrix is needed our scheme is proved to obey the gauss s law automatically if the initial fields satisfy that furthermore the conditional stability and optimal error estimate for the proposed scheme are proved to our best knowledge we are unaware of any other publications devoted to the convergence analysis of this leap frog explicit scheme for maxwell s equations even in a simple medium while our results for metamaterials automatically reduce to the standard maxwell s equations in vacuum by dropping some terms resulting from the constitutive equations numerical results confirming our analysis are presented