roe solver with entropy corrector for uncertain hyperbolic systems this paper deals with intrusive galerkin projection methods with a roe type solver for treating uncertain hyperbolic systems using a finite volume discretization in physical space and a piecewise continuous representation at the stochastic level the aim of this paper is to design a cost effective adaptation of the deterministic dubois and mehlman corrector to avoid entropy violating shocks in the presence of sonic points the adaptation relies on an estimate of the eigenvalues and eigenvectors of the galerkin jacobian matrix of the deterministic system of the stochastic modes of the solution and on a correspondence between these approximate eigenvalues and eigenvectors for the intermediate states considered at the interface we derive some indicators that can be used to decide where a correction is needed thereby reducing the computational costs considerably the effectiveness of the proposed corrector is assessed on the burgers and euler equations including sonic points

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

multiobjective programming in optimization of interval objective functions a generalized approach abstract we generalize known concepts those introduced by ishihuchi and tanaka 1 and rommerlfanger et al 2 of the solution of the linear programming problem with interval coefficients in the objective function based on preference relations between intervals we unify all the discussed concepts as well as the corresponding solution methods into one general framework

numerical homogenization of the acoustic wave equations with a continuum of scales in this paper we consider numerical homogenization of acoustic wave equations with heterogeneous coefficients namely when the bulk modulus and the density of the medium are only bounded we show that under a cordes type condition the second order derivatives of the solution with respect to harmonic coordinates are l 2 instead h 1 with respect to euclidean coordinates and the solution itself is in l 0 t h 2 instead of l 0 t h 1 with respect to euclidean coordinates then we propose an implicit time stepping method to solve the resulted linear system on coarse spatial scales and present error estimates of the method it follows that by pre computing the associated harmonic coordinates it is possible to numerically homogenize the wave equation without assumptions of scale separation or ergodicity

long term behavior of polynomial chaos in stochastic flow simulations in this paper we focus on the long term behavior of generalized polynomial chaos gpc and multi element generalized polynomial chaos me gpc for partial differential equations with stochastic coefficients first we consider the one dimensional advection equation with a uniform random transport velocity and derive error estimates for gpc and me gpc discretizations subsequently we extend these results to other random distributions and high dimensional random inputs with numerical verification using the algebraic convergence rate of me gpc finally we apply our results to noisy flow past a stationary circular cylinder simulation results demonstrate that me gpc is effective in improving the accuracy of gpc for a long term integration whereas high order gpc cannot capture the correct asymptotic behavior

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

analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients we consider the problem of numerically approximating statistical moments of the solution of a time dependent linear parabolic partial differential equation pde whose coefficients and or forcing terms are spatially correlated random fields the stochastic coefficients of the pde are approximated by truncated karhunen loeve expansions driven by a finite number of uncorrelated random variables after approximating the stochastic coefficients the original stochastic pde turns into a new deterministic parametric pde of the same type the dimension of the parameter set being equal to the number of random variables introduced after proving that the solution of the parametric pde problem is analytic with respect to the parameters we consider global polynomial approximations based on tensor product total degree or sparse polynomial spaces and constructed by either a stochastic galerkin or a stochastic collocation approach we derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional monte carlo method for this class of problems copyright 2009 john wiley sons ltd

modeling uncertainty in steady state diffusion problems via generalized polynomial chaos abstract we present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations subject to uncertain inputs in particular we focus on the solution of the poisson equation with random diffusivity forcing and boundary conditions the stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the askey scheme as a generalization of the original polynomial chaos idea of wiener amer j math 60 1938 897 a galerkin projection in random space is applied to derive the equations in the weak form the resulting set of deterministic equations for each random mode is solved iteratively by a block gauss seidel iteration technique both discrete and continuous random distributions are considered and convergence is verified in model problems and against monte carlo simulations

solving elliptic boundary value problems with uncertain coefficients by the finite element method the stochastic formulation this work studies a linear elliptic problem with uncertainty the introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations the major emphasis

difference approximations for the second order wave equation difference approximations are derived for the second order wave equation in one and two space dimensions without first writing it as a first order system both the dirichlet and the neumann problems are treated for the one dimensional case relations between the boundary error and the interior phase error are derived for a fully second order accurate discretization as well as a scheme that is fourth order accurate in the interior and second order accurate at the boundary general two dimensional domains are considered for the dirichlet problem where the domain is embedded in a cartesian grid and the boundary conditions are approximated by interpolation a stable conservative scheme is derived where the time step is determined only by the interior discretization formula discretization cells cut by the boundary are treated implicitly but the resulting scheme becomes explicit because the implicit dependence only is pointwise numerical examples are provided to verify the stability and accuracy of the proposed method

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

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

on comparing interval numbers abstract this paper first presents a brief survey of the existing works on comparing and ranking any two interval numbers on the real line and then on the basis of this gives two approaches to compare any two interval numbers the first one describes a value judgement index along with a discussion on its strength and weakness over the other approaches this index has also been called an optimistic decision maker s preference index the other approach defines strict and fuzzy preference ordering between any two interval numbers from a pessimistic decision maker s point of view choice of different decision makers with different degrees of pessimism has also been considered

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

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

fuzzy programming in multiobjective linear programming a parametric approach abstract zimmermann s fuzzy approach to the compromise solution concept in the multi objective linear programming problem is considered it is shown that given a class of membership functions of fuzzy goals assigned to objective functions in the problem wider than primarily proposed by zimmermann the use of classical linear programming methods to solve and analyse the problem is also possible as a result of application of the method based on the parametric programming technique one can obtain a fuzzy solution of the problem this solution is a certain fuzzy subset of the set of weakly efficient solutions of the problem the solution which belongs to this set to the highest degree is a maximizing one in bellman and zadeh s terminology it may also be obtained by use of the usual non parametric simplex method

intrusive galerkin methods with upwinding for uncertain nonlinear hyperbolic systems this paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws we consider problems with parametric uncertainty in initial conditions and model coefficients whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables the stochastic spectral method relies on multi resolution schemes where the stochastic domain is discretized using tensor product stochastic elements supporting local polynomial bases a galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution hyperbolicity of the resulting galerkin system is analyzed a finite volume scheme with a roe type solver is used for discretization of the spatial and time variables an original technique is introduced for the fast evaluation of approximate upwind matrices which is particularly well adapted to local polynomial bases efficiency and robustness of the overall method are assessed on the burgers and euler equations with shocks

linear programming with fuzzy objectives abstract this paper presents a new method for solving linear programming problems with fuzzy parameters in the objective function to determine a compromise solution the authors do not reduce the set of infinitely many objective functions to a single compromise objective function as done in classical deterministic or stochastic procedures here the problem is only reduced to a few extreme objective functions the information contained in the membership functions can be used to any extent by a method called level related pair formation moreover if different level sets are considered the stability of the optimal solution can be tested

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

boundary element discretization of poincare steklov operators the construction of a discretization of poincar e steklov operators with the boundary element method is given providing the same mapping properties as the finite element discretization of these operators

convex fuzzy mapping with differentiability and its application in fuzzy optimization abstract the concepts of differentiability convexity generalized convexity and minimization of a fuzzy mapping are known in the literature the purpose of this present paper is to extend and generalize these concepts to fuzzy mappings of several variables using buckley feuring approach for fuzzy differentiation and derive karush kuhn tucker condition for the constrained fuzzy minimization problem

computation of the optimal tolls on the traffic network abstract the present paper is devoted to the computation of optimal tolls on a traffic network that is described as fuzzy bilevel optimization problem as a fuzzy bilevel optimization problem we consider bilinear optimization problem with crisp upper level and fuzzy lower level an effective algorithm for computation optimal tolls for the upper level decision maker is developed under assumption that the lower level decision maker chooses the optimal solution as well the algorithm is based on the membership function approach this algorithm provides us with a global optimal solution of the fuzzy bilevel optimization problem

a stochastic collocation method for the second order wave equation with a discontinuous random speed in this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions the speed is piecewise smooth in the physical space and depends on a finite number of random variables the numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space this approach leads to the solution of uncoupled deterministic problems as in the monte carlo method we consider both full and sparse tensor product spaces of orthogonal polynomials we provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data in particular we show that unlike in elliptic and parabolic problems the solution to hyperbolic problems is not in general analytic with respect to the random variables therefore the rate of convergence may only be algebraic an exponential fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data we present numerical examples which confirm the analysis and show that the collocation method is a valid alternative to the more traditional monte carlo method for this class of problems

uncertainty quantification for systems of conservation laws uncertainty quantification through stochastic spectral methods has been recently applied to several kinds of non linear stochastic pdes in this paper we introduce a formalism based on kinetic theory to tackle uncertain hyperbolic systems of conservation laws with polynomial chaos pc methods the idea is to introduce a new variable the entropic variable in bijection with our vector of unknowns which we develop on the polynomial basis by performing a galerkin projection we obtain a deterministic system of conservation laws we state several properties of this deterministic system in the case of a general uncertain system of conservation laws we then apply the method to the case of the inviscid burgers equation with random initial conditions and we present some preliminary results for the euler system we systematically compare results from our new approach to results from the stochastic galerkin method in the vicinity of discontinuities the new method bounds the oscillations due to gibbs phenomenon to a certain range through the entropy of the system without the use of any adaptative random space discretizations it is found to be more precise than the stochastic galerkin method for smooth cases but above all for discontinuous cases

a general model for fuzzy linear programming abstract in the current literature there are several models of fuzzy linear programming problems the aim of this paper is first to describe the more important problems in fuzzy linear programming and second to give a general model of fuzzy linear programming problems involving all of the above ones finally a resolution method for that general model is proposed

optimization of linear problems subjected to the intersection of two fuzzy relational inequalities defined by dubois prade family of t norms abstract in this paper optimization of a linear objective function with fuzzy relational inequality constraints is investigated in doing so the feasible region is formed by the intersection of two inequality fuzzy systems and dubois prade family of t norms which are considered as fuzzy composition the most well known continuous t norms are archimedean such as frank yager hamacher sugeno weber and schweizer sklar family an interesting family of t norms that is not archimedean has been introduced by dubois and prade in this paper the resolution of the feasible region of the problem is initialy investigated when it is defined with max duboise prade composition a necessary and sufficient condition along with three other necessary conditions are derived for determining the feasibility of the problem moreover two procedures have also been presented with the aim of simplifying the current linear problems a method is proposed to generate random feasible max dubois prade fuzzy relational inequalities and an algorithm is accordingly presented to solve the problem finally an example is described to illustrate this algorithm

fast parallel solvers for symmetric boundary element domain decomposition equations the boundary element method bem is of advantage in many applications including far field computations in magnetostatics and solid mechanics as well as accurate computations of singularities since the numerical approximation is essentially reduced to the boundary of the domain under consideration the mesh generation and handling is simpler than for example in a finite element discretization of the domain in this paper we discuss fast solution techniques for the linear systems of equations obtained by the bem be equations utilizing the non overlapping domain decomposition dd we study parallel algorithms for solving large scale galerkin be equations approximating linear potential problems in plane bounded domains with piecewise homogeneous material properties we give an elementary spectral equivalence analysis of the bem schur complement that provides the tool for constructing and analysing appropriate preconditioners finally we present numerical results obtained on a massively parallel machine using up to 128 processors and we sketch further applications to elasticity problems and to the coupling of the finite element method fem with the boundary element method as shown theoretically and confirmed by the numerical experiments the methods are of o h 2 algebraic complexity and of high parallel efficiency where h denotes the usual discretization parameter

extended procedure for computing the values of the membership function of a fuzzy solution to a class of fuzzy linear optimization problems in 1994 chanas and kuchta suggested one approach to determine the membership function of a fuzzy solution to the fuzzy linear optimization problem with fuzzy coefficients in the objective function based on computing the sum of lengths of certain intervals in 2012 dempe and ruziyeva introduced a methodology for realizing chanas and kuchta s idea and derived explicit formulas for computing the endpoints of the suggested intervals in the particular case of triangular fuzzy numbers the purpose of this paper is to extend dempe and ruziyeva s approach by handling the possible degeneracy of basic feasible solutions and derive new formulas for computing the values of the membership function the special example considered by dempe and ruziyeva is again used to illustrate the relevance of the extended solving procedure and new formulas

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

a reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks abstract in this paper we propose a reduced basis hybrid method rbhm for the approximation of partial differential equations in domains represented by complex networks where topological features are recurrent the rbhm is applied to stokes equations in domains which are decomposable into smaller similar blocks that are properly coupled the rbhm is built upon the reduced basis element method rbem and it takes advantage from both the reduced basis methods rb and the domain decomposition method we move from the consideration that the blocks composing the computational domain are topologically similar to a few reference shapes on the latter representative solutions corresponding to the same governing partial differential equations are computed for different values of some parameters of interest representing for example the deformation of the blocks a generalized transfinite mapping is used in order to produce a global map from the reference shapes of each block to any deformed configuration the desired solution on the given original computational domain is recovered as projection of the previously precomputed solutions and then glued across subdomain interfaces by suitable coupling conditions the geometrical parametrization of the domain by transfinite mapping induces non affine parameter dependence an empirical interpolation technique is used to recover an approximate affine parameter dependence and a subsequent offline online decomposition of the reduced basis procedure this computational decomposition yields a considerable reduction of the problem complexity results computed on some combinations of 2d and 3d geometries representing cardiovascular networks show the advantage of the method in terms of reduced computational costs and the quality of the coupling to guarantee continuity of both stresses pressure and velocity at subdomain interfaces

an anisotropic sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms input data of the model the method consists of a galerkin approximation in the space variables and a collocation in probability space on sparse tensor product grids utilizing either clenshaw curtis or gaussian knots even in the presence of nonlinearities the collocation approach leads to the solution of uncoupled deterministic problems just as in the monte carlo method this work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem these procedures seem to be very effective for the problems under study the proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation the first approach is effective for problems depending on random variables which weigh approximately equally in the solution while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables as in the case where input random variables are karhunen loeve truncations of smooth random fields this work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates sub exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime with respect to the total number of collocation points it also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems numerical examples illustrate the theoretical results and are used to compare this approach with several others including the standard monte carlo in particular for moderately large dimensional problems the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods

a sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

the use of parametric programming in fuzzy linear programming in this paper the possibility of the identification of a complete fuzzy decision not only the maximizing alternative in fuzzy linear programming by use of the parametric programming technique is presented also it is shown that this fact can be useful in the zimmermann approach to multiple objective linear programming the presented remarks are illustrated by some numerical examples

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

a stochastic collocation method for elliptic partial differential equations with random input data this work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms these input data are assumed to depend on a finite number of random variables the method consists of 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 treats easily a wide 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 of the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method finally we include a section with developments posterior to the original publication of this work there we review sparse grid stochastic collocation methods which are effective collocation strategies for problems that depend on a moderately large number of random variables

galerkin methods for linear and nonlinear elliptic stochastic partial differential equations abstract stationary systems modelled by elliptic partial differential equations linear as well as nonlinear with stochastic coefficients random fields are considered the mathematical setting as a variational problem existence theorems and possible discretisations in particular with respect to the stochastic part are given and investigated with regard to stability different and increasingly sophisticated computational approaches involving both wiener s polynomial chaos as well as the karhunen loeve expansion are addressed in conjunction with stochastic galerkin procedures and stability within the galerkin framework is established new and effective algorithms to compute the mean and covariance of the solution are proposed the similarities and differences with better known monte carlo methods are exhibited as well as alternatives to integration in high dimensional spaces hints are given regarding the numerical implementation and parallelisation numerical examples serve as illustration

a recursive sparse grid collocation method for differential equations with white noise we consider a sparse grid collocation method in conjunction with a time discretiza tion of the differential equations for computing expectations of functionals of solutions to differential equations perturbed by time dependent white noise we first analyze the error of smolyak s sparse grid collocation used to evaluate expectations of functionals of solutions to stochastic differential equations discretized by the euler scheme we show theoretically and numerically that this algo rithm can have satisfactory accuracy for small noise magnitude or small integration time however it does not converge either with decrease of the euler scheme s time step size or with increase of smolyak s sparse grid level subsequently we use this method as a building block for proposing a new algorithm by combining sparse grid collocation with a recursive procedure this approach allows us to numerically integrate linear stochastic partial differential equations over longer times which is illustrated in numerical tests on a stochastic advection diffusion equation

on the calculation of a membership function for the solution of a fuzzy linear optimization problem in the present paper the fuzzy linear optimization problem with fuzzy coefficients in the objective function is considered recent concepts of fuzzy solution to the fuzzy optimization problem based on the level cut and the set of pareto optimal solutions of a multiobjective optimization problem are applied chanas and kuchta suggested one approach to determine the membership function values of fuzzy optimal solutions of the fuzzy optimization problem which is based on calculating the sum of lengths of certain intervals the purpose of this paper is to determine a method for realizing this idea we derive explicit formulas for the bounds of these intervals in the case of triangular fuzzy numbers and show that only one interval needs to be considered

numerical methods for differential equations in random domains physical phenomena in domains with rough boundaries play an important role in a variety of applications often the topology of such boundaries cannot be accurately described in all of its relevant detail due to either insufficient data or measurement errors or both this topological uncertainty can be efficiently handled by treating rough boundaries as random fields so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains to deal with this class of problems we propose a novel computational framework which is based on using stochastic mappings to transform the original deterministic stochastic problem in a random domain into a stochastic problem in a deterministic domain the latter problem has been studied more extensively and existing analytical numerical techniques can be readily applied in this paper we employ both a stochastic galerkin method and monte carlo simulations to solve the transformed stochastic problem we demonstrate our approach by applying it to an elliptic problem in single and double connected random domains and comment on the accuracy and convergence of the numerical methods

analyticity of solutions for a generalized euler equation abstract we consider the so called lake and great lake equations which are shallow water equations that describe the long time motion of an inviscid incompressible fluid contained in a shallow basin with a slowly spatially varying bottom a free upper surface and vertical side walls under the influence of gravity and in the limit of small characteristic velocities and very small surface amplitude if these equations are posed on a space periodic domain and the initial data are real analytic the solution remains real analytic for all times the proof is based on a characterization of gevrey classes in terms of decay of fourier coefficients in particular our result recovers known results for the euler equations in two and three spatial dimensions we believe the proof is new

a domain decomposition method based on bem and fem for linear exterior boundary value problems abstract we develop the finite dimensional analysis of a new domain decomposition method for linear exterior boundary value problems arising in potential theory and heat conductivity our approach uses a dirichlet to neumann mapping to transform the exterior problem into an equivalent boundary value problem on a bounded domain then the domain is decomposed into a finite number of annular subregions and the local steklov poincare operators are expressed conveniently either by bem or fem in order to obtain a symmetric interface problem the global steklov poincare problem is solved by using both a richardson type scheme and the preconditioned conjugate gradient method which yield iteration by subdomain algorithms well suited for parallel processing finally contractivity results and finite dimensional approximations are provided

uncertainty quantification for hyperbolic conservation laws with flux coefficients given by spatiotemporal random fields in this paper hyperbolic partial differential equations pdes with random coefficients are discussed we consider the challenging problem of flux functions with coefficients modeled by spatiotemporal random fields those fields are given by correlated gaussian random fields in space and ornstein uhlenbeck processes in time the resulting system of equations consists of a stochastic differential equation for each random parameter coupled to the hyperbolic conservation law we define an appropriate solution concept in this setting and analyze errors and convergence of discretization methods a novel discretization framework based on monte carlo finite volume methods is presented for the robust computation of moments of solutions to those random hyperbolic pdes we showcase the approach on two examples which appear in applications the magnetic induction equation and linear acoustics both with a spatiotemporal random background velocity field

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

multiobjective optimization through a series of single objective formulations this work deals with bound constrained multiobjective optimization mop of nonsmooth functions for problems where the structure of the objective functions either cannot be exploited or are absent typical situations arise when the functions are computed as the result of a computer simulation we first present definitions and optimality conditions as well as two families of single objective formulations of mop next we propose a new algorithm called for the biobjective optimization bop problem i e mop with two objective functions the property that pareto points may be ordered in bop and not in mop is exploited by our algorithm generates an approximation of the pareto front by solving a series of single objective formulations of bop these single objective problems are solved using the recent mesh adaptive direct search algorithm for nonsmooth optimization the pareto front approximation is shown to satisfy some first order necessary optimality conditions based on the clarke calculus finally is tested on problems from the literature designed to illustrate specific difficulties encountered in biobjective optimization such as a nonconvex or disjoint pareto front local pareto fronts or a nonuniform pareto front

an offline online algorithm for a class of stochastic multiple obstacle scattering configurations in the half plane efficient computational models are required to understand quantities of interest qoi such as the intensity of sound radiated by non deterministic configurations comprising multiple obstacles in the half plane the stochastic nature of the configurations requires tens of thousands of realizations to compute the expected value and standard deviation of the random qoi the acoustic wave scattering configuration usually comprises multiple deterministic structures whose locations in the half plane are non deterministic thus understanding the nature of the deterministic structures is intrinsically an offline process whilst understanding the nature of the resulting qoi is an online process the online process includes interactions between the structures and the half plane boundary in accordance with the physical offline online process we develop a computational t matrix based offline online reduced order model for large realizations of stochastic acoustic wave scattering configurations in the half plane we present results with more than 60 000 monte carlo mc quasi mc qmc and generalized polynomial chaos gpc simulations which are evaluated using our efficient online scheme for the acoustic scattering half plane model our offline online deterministic and stochastic framework can be used in conjunction with any numerical method for the offline simulations of scattering from a single obstacle in free space

a modified pso algorithm for linear optimization problem subject to the generalized fuzzy relational inequalities with fuzzy constraints fri fc in this paper optimization of a linear objective function subject to a generalized fuzzy relational inequalities is investigated in which an arbitrary continuous t norm is considered as fuzzy composition and fuzzy inequality replaces ordinary inequality in the constraints unlike most optimization algorithms in fuzzy relational inequalities with fuzzy constraints fri fc we find a near feasible solution having a better objective value than those resulting from the resolution of the similar problems with crisp ordinary inequality constraints such solutions are called super optima in this paper subsequently an algorithm is proposed to find a super optimum with pre specified desirable infeasibility for this purpose we firstly study some structural properties of the fri fc problem and present a new formulization that is independent of t norms used in the constraints of the problem this new formulation converts the primary problem into an equivalent problem with simple constraints without considering any penalty parameters however it is proved that the transformed equivalent problem enables our algorithm to distinguish unfavorable points and generate a sequence of solutions converging to a super optimum under a certain sufficient condition finally a modified pso is presented in which the ability of pso to solve unconstrained problems with continuous domain the structure of the transformed problem and some fuzzy structural modifications are combined and form an efficient algorithm to solve the generalized fri fc problems the modified pso algorithm has been applied to the generalized fri fc problem defined with ten well known continuous t norms additionally an idea of the fri fc problems as outer approximators for fri problems with ordinary inequalities is also investigated

joint solution to fuzzy programming problems we propose a new solution concept for fuzzy programming problems it is based on our new method of solving fuzzy equations 101 for simplicity we discuss in detail only fuzzy linear programming in this paper we define and obtain the basic properties of the joint solution a fuzzy vector in rn and the optimal value of the objective function a fuzzy number three examples are presented illustrating these concepts

eqsm an efficient high quality surface grid generation method based on remeshing the eqsm method for the generation of unstructured triangular surface grids is presented the method is based on remeshing techniques and avoids the need for traditional mesh triangulation approaches such as the advancing front and the delaunay methods normalized edge lengths based on a metric derived from curvature or from a user specified spacing are employed as the remeshing criterion it is assumed that the geometry is input in the form of composite parametric surfaces with analytic definition or ferguson or nurbs type multiple patch representation examples based upon typical aircraft geometries are included to demonstrate how high quality grids can be efficiently generated on surfaces that exhibit a high degree of geometric complexity