the three dimensional cavity resonator abstract the response of a three dimensional cavity to an external excitation is examined by using the technique of matched asymptotic expansions the full response curve is predicted and the low frequency helmholtz mode is studied in some detail the results are compared with those of rayleigh and others to reveal some interesting differences

sparse finite elements for elliptic problems with stochastic loading we formulate elliptic boundary value problems with stochastic loading in a bounded domain d d we show well posedness of the problem in stochastic sobolev spaces and we derive a deterministic elliptic pde in d d for the spatial correlation of the random solution we show well posedness and regularity results for this pde in a scale of weighted sobolev spaces with mixed highest order derivatives discretization with sparse tensor products of any hierarchic finite element fe spaces in d yields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data multilevel preconditioning in d d allows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equation

algorithm and architecture for a galois field multiplicative arithmetic processor we present a new algorithm for generic multiplicative computations of the form ab c in gf p sup m including multiplication inversion squaring and division the algorithm is based on solving a sequence of congruences that are derived from the theory of grobner bases in modules over the polynomial ring gf p x its corresponding hardware and software architectures can be successfully used in applications such as error control coding and cryptography we describe a versatile circuit associated with the algorithm for the most important case p 2 the same hardware can be used for a range of field sizes thus permitting applications in which different levels of error control or of security are required by different classes of user the operations listed are all performed by the hardware in the same number of clock cycles which prevents certain side channel attacks the loss in performance by having 2m iterations for multiplication is compensated by the full parameterization of the galois field and the ability to perform division and multiplication in parallel

fast rational interpolation reed solomon decoding and the linear complexity profiles of sequences an asymptotically fast algorithm for solving the generalized rational interpolation problem is presented this problem has been studied as part of system theory and is related to the solution of the classical and welch berlekamp 1983 key equations which arise in reed solomon decoding the algorithm can also be used to compute the linear complexity profile of a binary sequence of length m using only o m log m sup 2 log log m bit operations

a fast monte carlo method with a reduced basis of control variates applied to uncertainty propagation and bayesian estimation abstract the reduced basis control variate monte carlo method was introduced recently in s boyaval t lelievre a variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm commun math sci 8 2010 735 762 special issue mathematical issues on complex fluids as an improved monte carlo method for the fast estimation of many parametrized expected values at many parameter values we provide here a more complete analysis of the method including precise error estimates and convergence results we also numerically demonstrate that it can be useful to some parametric frameworks in uncertainty quantification in particular i the case where the parametrized expectation is a scalar output of the solution to a partial differential equation pde with stochastic coefficients an uncertainty propagation problem and ii the case where the parametrized expectation is the bayesian estimator of a scalar output in a similar pde context moreover in each case a pde has to be solved many times for many values of its coefficients this is costly and we also use a reduced basis of pde solutions like in s boyaval c le bris y maday n nguyen a patera a reduced basis approach for variational problems with stochastic parameters application to heat conduction with variable robin coefficient comput methods appl mech eng 198 2009 3187 3206 to our knowledge this is the first combination of various reduced basis ideas with a view to reducing as much as possible the computational cost of a simple versatile monte carlo approach to uncertainty quantification

strong and weak error estimates for elliptic partial differential equations with random coefficients we consider the problem of numerically approximating the solution of an elliptic partial differential equation with random coefficients and homogeneous dirichlet boundary conditions we focus on the case of a lognormal coefficient and deal with the lack of uniform coercivity and uniform boundedness with respect to the randomness this model is frequently used in hydrogeology we approximate this coefficient by a finite dimensional noise using a truncated karhunen loeve expansion we give estimates of the corresponding error on the solution both a strong error estimate and a weak error estimate that is an estimate of the error commited on the law of the solution we obtain a weak rate of convergence which is twice the strong one in addition we give a complete error estimate for the stochastic collocation method in this case where neither coercivity nor boundedness is stochastically uniform to conclude we apply these results of strong and weak convergence to two classical cases of covariance kernel choices the case of an exponential covariance kernel on a box and the case of an analytic covariance kernel yielding explicit weak and strong convergence rates

energy conserving galerkin approximation of two dimensional wave equations with random coefficients abstract wave propagation problems for heterogeneous media are known to have many applications in physics and engineering recently there has been an increasing interest in stochastic effects due to the uncertainty which may arise from impurities of the media this work considers a two dimensional wave equation with random coefficients which may be discontinuous in space generalized polynomial chaos method is used in conjunction with stochastic galerkin approximation and local discontinuous galerkin method is used for spatial discretization our method is shown to be energy preserving in semi discrete form as well as in fully discrete form when leap frog time discretization is used its convergence rate is proved to be optimal and the error grows linearly in time the theoretical properties of the proposed scheme are validated by numerical tests

a reduced basis approach for variational problems with stochastic parameters application to heat conduction with variable robin coefficient in this work a reduced basis rb approach is used to solve a large number of boundary value problems bvps parametrized by a stochastic input expressed as a karhunen loeve expansion in order to compute outputs that are smooth functionals of the random solution fields the rb method proposed here for variational problems parametrized by stochastic coefficients bears many similarities to the rb approach developed previously for deterministic systems however the stochastic framework requires the development of new a posteriori estimates for statistical outputs such as the first two moments of integrals of the random solution fields these error bounds in turn permit efficient sampling of the input stochastic parameters and fast reliable computation of the outputs in particular in the many query context

grobner basis solutions of constrained interpolation problems this paper extends the previous work of the authors on recursive grobner basis techniques in coding theory pade approximation partial realization interpolation and modelling discrete time behaviours we present a general algorithm applicable to a wide range of constrained interpolation problems in coding theory and systems theory including list decoding and m pade approximation

numerical methods for the discretization of random fields by means of the karhunen loeve expansion the computational efficiency of random field representations with the karhunen loeve kl expansion relies on the solution of a fredholm integral eigenvalue problem this contribution compares different methods that solve this problem focus is put on methods that apply to arbitrary shaped domains and arbitrary autocovariance functions these include the nystrom method as well as collocation and galerkin projection methods among the galerkin methods we investigate the finite element method fem and propose the application of the finite cell method fcm this method is based on an extension to the fem but avoids mesh generation on domains of complex geometric shape the fcm was originally presented in parvizian et al 2007 17 for the solution of elliptic boundary value problems as an alternative to the l2l2 projection of the covariance function used in the galerkin method h1 2h1 2 projection and discrete projection are investigated it is shown that the expansion optimal linear estimation eole method proposed in li and der kiureghian 1993 18 constitutes a special case of the nystrom method it is found that the eole method is most efficient for the numerical solution of the kl expansion the fem and the fcm are more efficient than the eole method in evaluating a realization of the random field and therefore are suitable for problems in which the time spent in the evaluation of random field realizations has a major contribution to the overall runtime e g in finite element reliability analysis

general error locator polynomials for binary cyclic codes with t le 2 and n in this paper we show that a recently proposed algorithm for decoding cyclic codes may be applied efficiently to all binary cyclic codes with tles2 and n 63 this is accomplished by providing structure theorems for the codes in this range and classifying the relevant cases

some remarks on an algorithm of fitzpatrick fitzpatrick s algorithm for solving the classical reed solomon key equation is shown to be related to an algorithm for solving the welch berlekamp key equation it can be made more efficient when decoding binary bch codes its efficiency is about the same as for the berlekamp massey algorithm

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

long time existence for the heat equation with a noise term we consider the equationu t u xx u w forx on a finite interval with dirichlet boundary conditions w is spacetime white noise the initial condition is continuous and nonnegative we show existence and uniqueness for all time provided 1 3 2

on the equivalence between berlekamp s and euclid s algorithms corresp it is shown that berlekamp s iterative algorithm can be derived from a normalized version of euclid s extended algorithm simple proofs of the results given recently by cheng are also presented

general principles for the algebraic decoding of cyclic codes this paper provides two theorems for decoding all types of cyclic codes it is shown that from a polynomial ideal point of view the decoding problems of cyclic codes are closely related to the monic generators of certain polynomial ideals this conclusion is also generalized to the decoding problems of algebraic geometry codes

generalized spectral decomposition for stochastic nonlinear problems we present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems the method consists in the construction of a reduced basis approximation of the galerkin solution and is independent of the stochastic discretization selected polynomial chaos stochastic multi element or multi wavelets two algorithms are proposed for the sequential construction of the successive generalized spectral modes they involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems compared to the classical galerkin method the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes the methodology is detailed and tested on two model problems the one dimensional steady viscous burgers equation and a two dimensional nonlinear diffusion problem these examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

a commutative algebra approach to linear codes abstract recently some methods have been proposed to find the distance and weight distribution of cyclic codes using grobner bases we identify a class of codes for which these methods can be generalized we show that this class contains all interesting linear codes and we provide variants and improvements this approach sometimes reveals an unexpected algebraic structure in the code we also investigate the decoding for a subclass proving the existence of general error locator polynomials

on solving elliptic stochastic partial differential equations abstract a model elliptic boundary value problem of second order with stochastic coefficients described by the karhunen loeve expansion is addressed this problem is transformed into an equivalent deterministic one the perturbation method and the method of successive approximations is analyzed rigorous error estimates in the framework of sobolev spaces are given

hamming metric decoding of alternant codes over galois rings the standard decoding procedure for alternant codes over fields centers on solving a key equation which relates an error locator polynomial and an error evaluator polynomial by a syndrome sequence we extend this technique to decode alternant codes over galois rings we consider the module m a b as spl equiv b mod x sup r of all solutions to the key equation where s is the syndrome polynomial and r is the number of rows in a parity check matrix for the code in decoding we seek a particular solution spl sigma spl omega spl isin m which we prove can be found in a grobner basis for m we present an iterative algorithm which generates a grobner basis modulo x sup k 1 from a given basis modulo x sup k at the rth step a grobner basis for m is found and the required solution recovered

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

strong and weak convergence order of finite element methods for stochastic pdes with spatial white noise we investigate the strong and weak convergence order of piecewise linear finite element methods for a class of semilinear elliptic equations with additive spatial white noise using a spectral truncation of white noise taking the number of modes for spectral approximation of the white noise at the order of 1 h d 1 hd we show that the strong convergence order of the finite element approximation is h 2 d 2 h2 d 2 where h is the element size and d le 3 d 3 is the dimension moreover we show that the weak convergence order is h min 4 d 2 hmin 4 d 2 we then consider a fourth order equation and an advection reaction equation and show that a spectral approximation of the white noise can lead to higher convergence order in both strong and weak sense when the solutions are smooth numerical results confirm our prediction for one and two dimensional elliptic problems

generalized brownian functionals and the solution to a stochastic partial differential equation abstract a class of generalized wiener functionals related to those of hida and watanabe is introduced and the malliavin calculs associated with these functionals is developed these notions are applied to the derivation of a solution to the stochastic partial differential equation y t ly y where l is a second order partial differential operator in the n space variables denotes a white noise in the n 1 space time variables y denotes a skorohod type stochastic integral and is a non random function of the n 1 independent variables

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

sound penetration through a thin circular ring abstract the penetration of sound through a circular ring aperture in a rigid screen is investigated in the low frequency limit the approximate solution for the relevant integral equations is found in the limit of a very thin ring the solution which is valid for arbitrary right hand members is applied to the integral equations which occur in the determination of the first three terms of the low frequency expansion special attention has been given to the transmission cross section which is of interest for the study of acoustical powerleakage through peripheral cracks the results show good agreement with those obtained by numerical methods not taken into account are the influence of the frictional resistance at the edge of an imperfectly rigid screen and the non linear effects of viscosity which will render the normal velocity finite at the edges

low frequency scattering through an aperture in a rigid screen some numerical results abstract the penetration of a plane wave through an aperture in a rigid screen is investigated the first three terms of the low frequency expansion are evaluated numerically for both simply connected and annular apertures in the numerical method use is made of a division of the aperture into arbitrary triangles the singular behaviour at the edges is incorporated in the shape functions which yields good accuracy and fast convergence special attention has been given to the thin ring which can serve as a model for e g imperfectly shielded windows

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

on the key equation we consider the set m a b a bh mod x2t of all solutions of the key equation for alternant codes where h is the syndrome polynomial in decoding these codes a particular solution m is sought subject to and being relatively prime and satisfying certain degree conditions we prove that these requirements specify uniquely as the minimal element of m analogous to the monic polynomial of minimal degree generating an ideal of f x with respect to a certain term order and that as such may be determined from an appropriate grobner basis of m motivated by this and other variations of the key equation such as that appropriate to errors and erasures decoding we derive a general algorithm for solving the congruence a bg mod xn for a range of term orders defined by the conditions on the particular solution required our techniques provide a unified approach to the solution of these key equations

finite dimensional models for random functions abstract truncated karhunen loeve kl representations are used to construct finite dimensional fd models for non gaussian functions with finite variances the second moment specification of the random coefficients of these representations are enhanced to full probabilistic characterization by using translation polynomial chaos and translated polynomial chaos models referred to as t pc and pct models following theoretical considerations on kl representations and t pc and pct models three numerical examples are presented to illustrate the implementation and performance of these models the pct models inherit the desirable features of both t and pc models it approximates accurately all quantities of interest considered in these examples

stochastic heat equation driven by fractional noise and local time the aim of this paper is to study the d dimensional stochastic heat equation with a multiplicative gaussian noise which is white in space and has the covariance of a fractional brownian motion with hurst parameter h 0 1 in time two types of equations are considered first we consider the equation in the ito skorohod sense and later in the stratonovich sense an explicit chaos expansion for the solution is obtained on the other hand the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the brownian motion

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

partial inverses mod m x and reverse berlekamp massey decoding this semi tutorial paper introduces the partial inverse problem for polynomials and develops its application to decoding reed solomon codes and some related codes the most natural algorithm to solve the partial inverse problem is very similar to but more general than the berlekamp massey algorithm two additional algorithms are obtained as easy variations of the basic algorithm the first variation is entirely new while the second variation may be viewed as a version of the euclidean algorithm decoding reed solomon codes and some related codes can be reduced to the partial inverse problem both via the standard key equation and more naturally via an alternative key equation with a new converse shortened and singly extended reed solomon codes are automatically included using the properties of the partial inverse problem two further key equations with attractive properties are obtained the paper also points out a variety of options for interpolation

an efficient spde approach for el nino abstract we consider the numerical approximation of stochastic partial differential equations spdes based models for a quasi periodic climate pattern in the tropical pacific ocean known as el nino phenomenon we show that for these models the mean and the covariance are given by a deterministic partial differential equation and by an operator differential equation respectively in this context we provide a numerical framework to approximate these parameters directly we compare this method to stochastic differential equations and spdes based models from the literature solved by taylor methods and stochastic galerkin methods respectively numerical results for different scenarios taking as a reference measured data of the years 2014 and 2015 last nino event validate the efficiency of our approach

efficient iterative solvers for stochastic galerkin discretizations of log transformed random diffusion problems we consider the numerical solution of a steady state diffusion problem where the diffusion coefficient is the exponent of a random field the standard stochastic galerkin formulation of this problem is computationally demanding because of the nonlinear structure of the uncertain component of it we consider a reformulated version of this problem as a stochastic convection diffusion problem with random convective velocity that depends linearly on a fixed number of independent truncated gaussian random variables the associated galerkin matrix is nonsymmetric but sparse and allows for fast matrix vector multiplications with optimal complexity we construct and analyze two block diagonal preconditioners for this galerkin matrix for use with krylov subspace methods such as the generalized minimal residual method we test the efficiency of the proposed preconditioning approaches and compare the iterative solver performance for a model problem posed in both diffusion and convection diffusion formulations

efficient gaussian process regression for large datasets gaussian processes are widely used in nonparametric regression classification and spatiotemporal modelling facilitated in part by a rich literature on their theoretical properties however one of their practical limitations is expensive computation typically on the order of n3 where n is the number of data points in performing the necessary matrix inversions for large datasets storage and processing also lead to computational bottlenecks and numerical stability of the estimates and predicted values degrades with increasing n various methods have been proposed to address these problems including predictive processes in spatial data analysis and the subset of regressors technique in machine learning the idea underlying these approaches is to use a subset of the data but this raises questions concerning sensitivity to the choice of subset and limitations in estimating fine scale structure in regions that are not well covered by the subset motivated by the literature on compressive sensing we propose an alternative approach that involves linear projection of all the data points onto a lower dimensional subspace we demonstrate the superiority of this approach from a theoretical perspective and through simulated and real data examples

a space time multiscale method for computing statistical moments in strongly heterogeneous poroelastic media of evolving scales summary a new multiscale procedure is proposed to compute flow in compressible heterogeneous porous media with geology characterized by power law covariance structure at the fine scale the deformable medium is modeled by the partially coupled formulation of poroelasticity with young s modulus and permeability treated as stationary random fields represented by their karhunen loeve decompositions the framework underlying the multiscale procedure is based on mapping these random parameters to an auxiliary domain and constructing a family of equivalent stochastic processes at different length scales characterized by the same ensemble mean and covariance function the poromechanical variables inherit a space time version of the scaling relations of the random input parameters which allows for constructing a set of multiscale solutions of the same governing equations posed at different space and time scales a notable feature of the multiscale method proposed herein is the feasibility of solving both the poroelastic model and the fredholm integral equation for the eigenpairs of the karhunen loeve expansion in an auxiliary domain with much lower computational effort and then derive the long term behavior at a coarser scale from a straightforward rescaling of the auxiliary solution within the framework of the finite element approximation in conjunction with the monte carlo algorithm numerical simulations of fluid withdrawal and injection problems in a heterogeneous poroelastic reservoir are performed to illustrate the potential of the method in drastically reducing the computational burden in the computation of the statistical moments of the poromechanical unknowns in large scale simulations copyright 2012 john wiley sons ltd

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

efficient stochastic galerkin methods for random diffusion equations we discuss in this paper efficient solvers for stochastic diffusion equations in random media we employ generalized polynomial chaos gpc expansion to express the solution in a convergent series and obtain a set of deterministic equations for the expansion coefficients by galerkin projection although the resulting system of diffusion equations are coupled we show that one can construct fast numerical methods to solve them in a decoupled fashion the methods are based on separation of the diagonal terms and off diagonal terms in the matrix of the galerkin system we examine properties of this matrix and show that the proposed method is unconditionally stable for unsteady problems and convergent for steady problems with a convergent rate independent of discretization parameters numerical examples are provided for both steady and unsteady random diffusions to support the analysis

decoding a class of lee metric codes over a galois ring we investigate a class of lee 1958 metric alternant codes with symbols in z sub pn establishing a lower bound on the minimum lee distance where certain restrictions are placed on the code parameters corresponding to this bound we have devised a decoding algorithm which is implemented over a finite field the algorithm proceeds by finding a grobner basis of the module m of solutions to a key equation we obtain a necessary characterization of the solution module by solving iteratively a linear sequence over a galois ring and show that the particular solution sought by the decoder is minimal in m the required solution can then be found in an appropriate grobner basis of m

the burgers equation with a random force and a general model for directed polymers in random environments the study of the burgers equation with a random force leads via a hopf cole type transformation to a stochastic heat equation having a white noise with spatial parameters type potential the latter can be studied by means of a general model of directed polymers in random environments with two point random potentials these models exhibit a gaussian behavior at large times and have certain stationary distributions which yield the corresponding results for the above stochastic heat and burgers equations

wiener chaos solutions of linear stochastic evolution equations a new method is described for constructing a generalized solution of a stochastic evolution equation existence uniqueness regularity and a probabilistic representation of this wiener chaos solution are established for a large class of equations as an application of the general theory new results are obtained for several types of the passive scalar equation

accurate and efficient evaluation of failure probability for partial different equations with random input data several computational challenges arise when evaluating the failure probability of a given system in the context of risk prediction or reliability analysis when the dimension of the uncertainties becomes high well established direct numerical methods can not be employed because of the curse of dimensionality many surrogate models have been proposed with the aim of reducing computational effort however most of them fail in computing an accurate failure probability when the limit state surface defined by the failure event in the probability space lacks smoothness in addition for a stochastic system modeled by partial differential equations pdes with random input only a limited number of the underlying pdes order of a few tens are affordable to solve in practice due to the considerable computational effort therefore preventing the application of many numerical methods especially for high dimensional random inputs in this work we develop hybrid and goal oriented adaptive reduced basis methods to tackle these challenges by accurately and efficiently computing the failure probability of a stochastic pde the curse of dimensionality is significantly alleviated by reduced basis approximation whose bases are constructed by goal oriented adaptation moreover an accurate evaluation of the failure probability for pde system with solution of low regularity in probability space is guaranteed by the certified a posteriori error bound for the output approximation error at the end of this paper we suitably extend our proposed method to deal with more general pde models finally we perform several numerical experiments to illustrate its computational accuracy and efficiency c 2013 elsevier b v all rights reserved

use of grobner bases to decode binary cyclic codes up to the true minimum distance a general algebraic method for decoding all types of binary cyclic codes is presented it is shown that such a method can correct t d 1 2 errors where d is the true minimum distance of the given cyclic code the key idea behind this decoding technique is a systematic application of the algorithmic procedures of grobner bases to obtain the error locator polynomial l z the discussion begins from a set of syndrome polynomials f and the ideal t f generated by f it is proved here that the process of transforming f to the normalized reduced grobner basis of i f with respect to the purely lexicographical ordering automatically converges to l z furthermore it is shown that l z can be derived from any normalized grobner basis of i f with respect to any admissible total ordering to illustrate this new approach the procedures for decoding certain bch codes and quadratic residue codes are demonstrated

rough burgers like equations with multiplicative noise we construct solutions to vector valued burgers type equations perturbed by a multiplicative space time white noise in one space dimension due to the roughness of the driving noise solutions are not regular enough to be amenable to classical methods we use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution subject to the choice of the correct reference rough path we prove unique solvability for the equation and we show that our solutions are stable under smooth approximations of the driving noise

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

on rational interpolation based list decoding and list decoding binary goppa codes we derive the wu list decoding algorithm for generalized reed solomon grs codes by using grobner bases over modules and the euclidean algorithm as the initial algorithm instead of the berlekamp massey algorithm we present a novel method for constructing the interpolation polynomial fast we give a new application of the wu list decoder by decoding irreducible binary goppa codes up to the binary johnson radius finally we point out a connection between the governing equations of the wu algorithm and the guruswami sudan algorithm immediately leading to equality in the decoding range and a duality in the choice of parameters needed for decoding both in the case of grs codes and in the case of goppa codes

stability and convergence of a finite element method for a semilinear elliptical problem with small viscosity abstract in this paper we propose and analyze a new finite element method for a semilinear elliptical problem with small viscosity firstly we prove the existence and uniqueness of solution for its variational problem then we obtain the stability and convergence of the corresponding finite element galerkin approximation furthermore we derive the optimal error estimates in the l 2 and h 1 norms for the finite element approximations respectively finally several numerical experiments are provided to confirm the above theoretical results

parallel domain decomposition methods for stochastic elliptic equations we present parallel schwarz type domain decomposition preconditioned recycling krylov subspace methods for the numerical solution of stochastic elliptic problems whose coefficients are assumed to be a random field with finite variance karhunen loeve kl expansion and double orthogonal polynomials are used to reformulate the stochastic elliptic problem into a large number of related but uncoupled deterministic equations the key to an efficient algorithm lies in recycling computed subspaces based on a careful analysis of the kl expansion we propose and test a grouping algorithm that tells us when to recycle and when to recompute some components of the expensive computation we show theoretically and experimentally that the schwarz preconditioned recycling gmres method is optimal for the entire family of linear systems a fully parallel implementation is provided and scalability results are reported in the paper

on unbiased stochastic navier stokes equations a random perturbation of a deterministic navier stokes equation is considered in the form of an spde with wick type nonlinearity the nonlinear term of the perturbation can be characterized as the highest stochastic order approximation of the original nonlinear term u nabla u this perturbation is unbiased in that the expectation of a solution of the perturbed equation solves the deterministic navier stokes equation the perturbed equation is solved in the space of generalized stochastic processes using the cameron martin version of the wiener chaos expansion it is shown that the generalized solution is a markov process and scales effectively by catalan numbers

minimal grobner bases and the predictable leading monomial property abstract we focus on grobner bases for modules of univariate polynomial vectors over a ring we identify a useful property the predictable leading monomial plm property that is shared by minimal grobner bases of modules in f x q no matter what positional term order is used the plm property is useful in a range of applications and can be seen as a strengthening of the wellknown predictable degree property row reducedness a terminology introduced by forney in the 70 s because of the presence of zero divisors minimal grobner bases over a finite ring of the type z p r where p is a prime integer and r is an integer 1 do not necessarily have the plm property in this paper we show how to derive from an ordered minimal grobner basis a so called minimal grobner p basis that does have a plm property we demonstrate that minimal grobner p bases lend themselves particularly well to derive minimal realization parametrizations over z p r applications are in coding and sequences over z p r

convergence to spdes in stratonovich form we consider the perturbation of parabolic operators of the form t p x d by large amplitude highly oscillatory spatially dependent potentials modeled as gaussian random fields the amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order we show that when the dimension is smaller than the order of the elliptic pseudo differential operator p x d the perturbed parabolic equation admits a solution given by a duhamel expansion moreover as the correlation length of the potential vanishes we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with multiplicative noise that should be interpreted in the stratonovich sense the theory of mild solutions for such stochastic partial differential equations is developed

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

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

finite element error analysis of elliptic pdes with random coefficients and its application to multilevel monte carlo methods we consider a finite element approximation of elliptic partial differential equations with random coefficients such equations arise for example in uncertainty quantification in subsurface flow modeling models for random coefficients frequently used in these applications such as log normal random fields with exponential covariance have only very limited spatial regularity and lead to variational problems that lack uniform coercivity and boundedness with respect to the random parameter in our analysis we overcome these challenges by a careful treatment of the model problem almost surely in the random parameter which then enables us to prove uniform bounds on the finite element error in standard bochner spaces these new bounds can then be used to perform a rigorous analysis of the multilevel monte carlo method for these elliptic problems that lack full regularity and uniform coercivity and boundedness to conclude we give some numerical results that confirm the new bounds

a fitzpatrick algorithm for multivariate rational interpolation abstract in this paper we first apply the fitzpatrick algorithm to osculatory rational interpolation then based on a fitzpatrick algorithm we present a neville like algorithm for cauchy interpolation with this algorithm we can determine the value of the interpolating function at a single point without computing the rational interpolating function

adaptive wavelet methods for elliptic partial differential equations with random operators we apply adaptive wavelet methods to boundary value problems with random coefficients discretized by wavelets in the spatial domain and tensorized polynomials in the parameter domain greedy algorithms control the approximate application of the fully discretized random operator and the construction of sparse approximations to this operator we suggest a power iteration for estimating errors induced by sparse approximations of linear operators

simulation of stochastic systems via polynomial chaos expansions and convex optimization polynomial chaos expansions represent a powerful tool to simulate stochastic models of dynamical systems yet deriving the expansion s coefficients for complex systems might require a significant and nontrivial manipulation of the model or the computation of large numbers of simulation runs rendering the approach too time consuming and impracticable for applications with more than a handful of random variables we introduce a computationally tractable technique for computing the coefficients of polynomial chaos expansions the approach exploits a regularization technique with a particular choice of weighting matrices which allows to take into account the specific features of polynomial chaos expansions the method completely based on convex optimization can be applied to problems with a large number of random variables and uses a modest number of monte carlo simulations while avoiding model manipulations additional information on the stochastic process when available can be also incorporated in the approach by means of convex constraints we show the effectiveness of the proposed technique in three applications in diverse fields including the analysis of a nonlinear electric circuit a chaotic model of organizational behavior and finally a chemical oscillator

grobner bases and generalized pade approximation it is shown how to find general multivariate pade approximation using the grobner basis technique this method is more flexible than previous approaches and several examples are given to illustrate this advantage when the number of variables is small compared to the degree of approximation the grobner basis technique is more efficient than the linear algebra methods in the literature