a stochastic projection method for fluid flow ii random process an uncertainty quantification scheme is developed for the simulation of stochastic thermofluid processes the scheme relies on spectral representation of uncertainty using the polynomial chaos pc system the solver combines a galerkin procedure for the determination of pc coefficients with a projection method for efficiently simulating the resulting system of coupled transport equations implementation of the numerical scheme is illustrated through simulations of natural convection in a 2d square cavity with stochastic temperature distribution at the cold wall the properties of the uncertainty representation scheme are analyzed and the predictions are contrasted with results obtained using a monte carlo approach

stochastic inverse heat conduction using a spectral approach an adjoint based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data process conditions and measurement noise the ill posed stochastic inverse problem is restated as a conditionally well posed l 2 optimization problem the gradient of the objective function is obtained in a distributional sense by defining an appropriate stochastic adjoint field the l 2 optimization problem is solved using a conjugate gradient approach accuracy and effectiveness of the proposed approach is appraised with the solution of several stochastic inverse heat conduction problems

a stochastic lagrangian approach for geometrical uncertainties in electrostatics this work proposes a general framework to quantify uncertainty arising from geometrical variations in the electrostatic analysis the uncertainty associated with geometry is modeled as a random field which is first expanded using either polynomial chaos or karhunen loeve expansion in terms of independent random variables the random field is then treated as a random displacement applied to the conductors defined by the mean geometry to derive the stochastic lagrangian boundary integral equation the surface charge density is modeled as a random field and is discretized both in the random dimension and space using polynomial chaos and classical boundary element method respectively various numerical examples are presented to study the effect of uncertain geometry on relevant parameters such as capacitance and net electrostatic force the results obtained using the proposed method are verified using rigorous monte carlo simulations it has been shown that the proposed method accurately predicts the statistics and probability density functions of various relevant parameters

a multigrid solver for two dimensional stochastic diffusion equations steady and unsteady diffusion equations with stochastic diffusivity coefficient and forcing term are modeled in two dimensions by means of stochastic spectral representations problem data and solution variables are expanded using the polynomial chaos system the approach leads to a set of coupled problems for the stochastic modes spatial finite difference discretization of these coupled problems results in a large system of equations whose dimension necessitates the use of iterative approaches in order to obtain the solution within a reasonable computational time to accelerate the convergence of the iterative technique a multigrid method based on spatial coarsening is implemented numerical experiments show good scaling properties of the method both with respect to the number of spatial grid points and the stochastic resolution level

approximate crank nicolson schemes for the 2 d finite difference time domain method for te sub z waves two implicit finite difference time domain fdtd methods are presented in this paper for a two dimensional te sub z wave which are based on the unconditionally stable crank nicolson scheme to treat pec boundaries efficiently the methods deal with the electric field components rather than the magnetic field the approximate decoupling method solves two tridiagonal matrices and computes only one explicit equation for a full update cycle it has the same numerical dispersion relation as the adi fdtd method the cycle sweep method solves two tridiagonal matrices and computes two equations explicitly for a full update cycle it has the same numerical dispersion relation as the previously reported crank nicolson douglas gunn algorithm which solves for the magnetic field the cycle sweep method has much smaller numerical anisotropy than the approximate decoupling method though the dispersion error is the same along the axes as and larger along the 45 spl deg diagonal than adi fdtd with different formulations two algorithms for the approximate decoupling method and four algorithms for the cycle sweep method are presented all the six algorithms are strictly nondissipative unconditionally stable and are tested by numerical computation in this paper the numerical dispersion relations are validated by numerical experiments and very good agreement between the experiments and the theoretical predication is obtained

sufficiency and duality for multiobjective variational control problems with g invexity antczak introduced vector valued g invex functions in 2009 which is a new class of generalized convex functions for differentiable multiobjective programming problems in this paper we extend the vector valued g invex functions to multiobjective variational control problems by using the new concepts a number of sufficient optimality results and mond weir type duality results are obtained for multiobjective variational control programming problems

kt invex control problem abstract in this paper we introduce a new condition on functions of a control problem for which we define a kt invex control problem we prove that a kt invex control problem is characterized in order that a kuhn tucker point is an optimal solution we generalize optimality results of known mathematical programming problems we illustrate these results with examples

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

an adaptive high dimensional stochastic model representation technique for the solution of stochastic partial differential equations a computational methodology is developed to address the solution of high dimensional stochastic problems it utilizes high dimensional model representation hdmr technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower order to higher order component functions hdmr is efficient at capturing the high dimensional input output relationship such that the behavior for many physical systems can be modeled to good accuracy only by the first few lower order terms an adaptive version of hdmr is also developed to automatically detect the important dimensions and construct higher order terms using only the important dimensions the newly developed adaptive sparse grid collocation asgc method is incorporated into hdmr to solve the resulting sub problems by integrating hdmr and asgc it is computationally possible to construct a low dimensional stochastic reduced order model of the high dimensional stochastic problem and easily perform various statistic analysis on the output several numerical examples involving elementary mathematical functions and fluid mechanics problems are considered to illustrate the proposed method the cases examined show that the method provides accurate results for stochastic dimensionality as high as 500 even with large input variability the efficiency of the proposed method is examined by comparing with monte carlo mc simulation

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

a comparison of performance of three orthogonal polynomials in extraction of wide band response using early time and low frequency data the objective of this paper is to generate a wideband and temporal response of three dimensional composite structures by using a hybrid method that involves generation of early time and low frequency information the data in these two separate time and frequency domains are mutually complementary and contain all the necessary information for a sufficient record length utilizing a set of orthogonal polynomials the time domain signal be it the electric or the magnetic currents or the near far scattered electromagnetic field could be expressed in an efficient way as well as the corresponding frequency domain responses the available data is simultaneously extrapolated in both domains computational load for electromagnetic analysis in either domain time or frequency can be thus significantly reduced three orthogonal polynomial representations including hermite polynomial laguerre function and bessel function are used in this approach however the performance of this new method is sensitive to two important parameters the scaling factor l sub 1 and the expansion order n it is therefore important to find the optimal parameters to achieve the best performance a comparison is presented to illustrate that for the classes of problems dealt with the choice of the laguerre polynomials has the best performance as illustrated by a typical scattering example from a dielectric hemisphere

multi element stochastic reduced basis methods this paper presents multi element stochastic reduced basis methods me srbms for solving linear stochastic partial differential equations in me srbms the domain of definition of the random inputs is decomposed into smaller subdomains or random elements stochastic reduced basis methods srbms are employed in each random element to evaluate the response statistics these elemental statistics are assimilated to compute the overall statistics the effectiveness of the method is demonstrated by solving the stochastic steady state heat transfer equation on two geometries involving different types of boundary conditions numerical studies are conducted to investigate the h convergence rates of global and local preconditioning strategies

simultaneous extrapolation in time and frequency domains using hermite expansions the time domain response of a three dimensional 3 d conducting object is modeled as an associate hermite ah series expansion using the isomorphism of the ah function and its fourier transform the frequency domain response can be expressed as a scaled version of the time domain expansion using early time and low frequency data we demonstrate simultaneous expansion in both domains this approach is attractive because expansions with only 10 20 terms give good extrapolation in both time and frequency domains the computation involved is minimal with this method in this paper we have applied this technique to the problem of extrapolating the current on a scatterer being excited by a uniform plane wave five scatterers were considered a plate disk sphere cube and cone

multiobjective control problem with generalized invexity abstract a multiobjective control problem is introduced the definition of invexity for continuous functions is extended to invexity pseudoinvexity and quasi invexity duality results are established for wolfe as well as mond wier type duals

variational multiscale stabilized fem formulations for transport equations stochastic advection diffusion and incompressible stochastic navier stokes equations an extension of the deterministic variational multiscale vms approach with algebraic subgrid scale sgs modeling is considered for developing stabilized finite element formulations for the stochastic advection and the incompressible stochastic navier stokes equations the stabilized formulations are numerically implemented using the spectral stochastic formulation of the finite element method ssfem generalized polynomial chaos and karhunen loeve expansion techniques are used for representation of uncertain quantities the proposed stabilized method is then applied to various standard advection diffusion and fluid flow examples with uncertainty in essential boundary conditions comparisons are drawn between the numerical solutions and monte carlo analytical solutions wherever possible

an eulerian scheme for the second order approximation of subsurface transport moments the moments of a conservative tracer cloud migrating in a mean uniform flow field are estimated using an operator approximation scheme results are presented for the second third and fourth central moments in the mean flow direction it is assumed that the spatially variable flow field and therefore the tracer migration problem itself is amenable to a probabilistic description the effects of local dispersion on cloud migration are neglected in this study variation in the flow field is assumed to be the result of spatial variation in the hydraulic conductivity spatial variation in porosity is assumed negligible the operator approximation scheme as implemented in this study is second order correct which requires a second order correct approximation of the velocity field correlation structure because estimation of the velocity correlation structure is decidedly the most difficult aspect of second order analysis an ad hoc extension of the imperfectly stratified approximation developed earlier is implemented for this purpose the first order approximation resulting from the operator expansion scheme is equivalent to small perturbation eulerian results presented earlier naff 1990 1992 the infinite order approximation resulting from this scheme is equivalent to the exponential operator results obtained by van kampen 1976

probabilistic characterization of transport in heterogeneous media abstract the mechanics of transport and flow in a random porous medium are addressed in this paper the hydraulic properties of the porous medium are modeled as spatial random processes the random aspect of the problem is treated by introducing a new dimension along which spectral approximations are implemented thus the hydraulic processes are discretized using the spectral karhunen loeve expansion this expansion represents the random spatial functions as deterministic modes of fluctuation with random amplitudes these amplitudes form a basis in the manifold associated with the random processes the concentrations over the whole domain are also random processes with unknown probabilistic structure these processes are represented using the polynomial chaos basis this is a basis in the functional space described by all second order random variables the deterministic coefficients in this expansion are calculated via a weighted residual procedure with respect to the random measure and the inner product specified by the expectation operator once the spatio temporal variation of the concentrations has been specified in terms of the polynomial chaos expansion individual realizations can be readily computed

stochastic model reduction for chaos representations abstract this paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic galerkin schemes for the solution of partial differential equations with stochastic coefficients in particular an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation spde in the hilbert space spanned by wiener chaos a description of the stochastic aspect of the problem on two well separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale with such algorithms at hand the solution of spde s becomes both computationally manageable and efficient moreover a solid foundation is thus provided for the adaptive error control in stochastic galerkin procedures different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics

a computational method for simulating subsurface flow and reactive transport in heterogeneous porous media embedded with flexible uncertainty quantification 1 in chen et al 2013 the fundamental concepts of the modular uq methodology have been introduced for general multiphysics applications in which each physics module can be independently embedded with its internal uq method intrusive or nonintrusive without losing the global uncertainty propagation property in the current paper we extend the modular uq methodology to subsurface flow and reactive transport applications which are characterized by high dimensionality in the stochastic space due to spatially random velocity field in randomly heterogeneous porous media specifically we develop a scheme to reduce the dimension of the stochastic space this is achieved via a doubly nested dimension reduction by applying karhunen loeve expansion to the logarithmic hydraulic conductivity field followed by proper orthogonal decomposition to the velocity field this scheme enables the modular uq framework to handle spatially random models efficiently while maintaining solution accuracy when compared against sampling based nonintrusive uq methods the modular uq method demonstrates a similar accuracy at a fraction of computational cost on designed numerical experiments

on minimax fractional optimality conditions with invexity abstract under different forms of invexity conditions sufficient kuhn tucker conditions and three dual models are presented for the minimax fractional programming

a new efficient algorithm for the unconditionally stable 2 d wlp fdtd method we previously introduced an efficient algorithm for implementing the 2 d laguerre based finite difference time domain fdtd method numerical results indicated that the efficient algorithm can save cpu time and memory storage greatly while maintaining comparable computational accuracy however the splitting error associated with the perturbation term becomes pronounced in regions with larger spatial derivatives of the field in this paper we present a new efficient algorithm based on the use of an iterative procedure to reduce the splitting error the new algorithm does not involve any nonphysical intermediate variables and its update equations are much simpler and more concise than the original ones instead of applying the iterative procedure uniformly to the entire computational domain one can apply an additional number of iterations to regions with relatively large field variation using this approach the overall accuracy can be improved with little computational overhead numerical examples are used to illustrate the effectiveness of the proposed algorithm

an inversion algorithm for a banded matrix in this paper an inversion algorithm for a banded matrix is presented the n twisted decompositions of a banded matrix are given first then the inverse of the matrix is obtained one column at time the method is about two times faster than the standard method based on the lu decomposition as is shown with the analysis of computing complexity and the numerical experiments

modeling diffusion in random heterogeneous media data driven models stochastic collocation and the variational multiscale method in recent years there has been intense interest in understanding various physical phenomena in random heterogeneous media any accurate description simulation of a process in such media has to satisfactorily account for the twin issues of randomness as well as the multilength scale variations in the material properties an accurate model of the material property variation in the system is an important prerequisite towards complete characterization of the system response we propose a general methodology to construct a data driven reduced order model to describe property variations in realistic heterogeneous media this reduced order model then serves as the input to the stochastic partial differential equation describing thermal diffusion through random heterogeneous media a decoupled scheme is used to tackle the problems of stochasticity and multilength scale variations in properties a sparse grid collocation strategy is utilized to reduce the solution of the stochastic partial differential equation to a set of deterministic problems a variational multiscale method with explicit subgrid modeling is used to solve these deterministic problems an illustrative example using experimental data is provided to showcase the effectiveness of the proposed methodology

a spectral stochastic element free galerkin method for the problems with random material parameter this paper presents a spectral stochastic element free galerkin method ssefgm for the problems involving a random material property the random material property and resulting system response quantity are represented by a probabilistic spectral expansion techniques karhunen loeve expansion and polynomical chaos series respectively and implemented into the element free galerkin efg analysis numerical solutions in 1d linear elastic problem with random elastic modulus are introduced and compared with those of monte carlo simulation mcs so as to provide the validation of the proposed approach the present ssefgm approach can produce a probabilistic density distribution as well as a first and second order statistical moments mean and variance of response quantity by a single calculation which is distinguished from an iterative mcs moreover the method is based on an element free analysis so that there is no need of nodal connectivities which usually require more time and labourious task than main calculations thus the proposed ssefgm approach can provide an alternative analysis tool for the problems contains a stochastic material property and demands complex mesh structures copyright 2004 john wiley sons ltd

an explicit and unconditionally stable fdtd method for the analysis of general 3 d lossy problems the root cause of the instability of an explicit finite difference time domain fdtd method is quantitatively identified for the analysis of general lossy problems where both dielectrics and conductors can be lossy and inhomogeneous based on the root cause analysis an efficient algorithm is developed to eradicate the root cause of instability and subsequently achieve unconditional stability in an explicit fdtd based simulation of general lossy problems numerical experiments have demonstrated the unconditional stability accuracy and efficiency of the proposed method

the 3 d unconditionally stable associated hermite finite difference time domain method this article discussed the implementation of 3 d associated hermite ah finite difference time domain fdtd method and its application in the analysis of lumped parameters modeling based on the paralleling in order scheme the basic formula of the 3 d ah fdtd method incorporated with complex frequency shifted cfs perfectly matched layer pml absorption boundary was derived then by using the kronecker product scheme the coefficient matrix of the final matrix equation was fast generated and by using the permutation technology its bandwidth was compressed updating the coefficient matrix and source term of the matrix equation was implemented when dealing with the lumped parameters to solve the final matrix to obtain the ah expanding coefficients the direct or indirect methods mainly by the iterative method could be used finally several numerical results were presented to validate the accuracy and efficiency of the proposed method

stochastic finite difference lattice boltzmann method for steady incompressible viscous flows with the advent of state of the art computers and their rapid availability the time is ripe for the development of efficient uncertainty quantification uq methods to reduce the complexity of numerical models used to simulate complicated systems with incomplete knowledge and data the spectral stochastic finite element method ssfem which is one of the widely used uq methods regards uncertainty as generating a new dimension and the solution as dependent on this dimension a convergent expansion along the new dimension is then sought in terms of the polynomial chaos system and the coefficients in this representation are determined through a galerkin approach this approach provides an accurate representation even when only a small number of terms are used in the spectral expansion consequently saving in computational resource can be realized compared to the monte carlo mc scheme recent development of a finite difference lattice boltzmann method fdlbm that provides a convenient algorithm for setting the boundary condition allows the flow of newtonian and non newtonian fluids with and without external body forces to be simulated with ease also the inherent compressibility effect in the conventional lattice boltzmann method which might produce significant errors in some incompressible flow simulations is eliminated as such the fdlbm together with an efficient uq method can be used to treat incompressible flows with built in uncertainty such as blood flow in stenosed arteries the objective of this paper is to develop a stochastic numerical solver for steady incompressible viscous flows by combining the fdlbm with a ssfem validation against mc solutions of channel couette driven cavity and sudden expansion flows are carried out

uncertainty quantification in chemical systems summary we demonstrate the use of multiwavelet spectral polynomial chaos techniques for uncertainty quantification in non isothermal ignition of a methane air system we employ bayesian inference for identifying the probabilistic representation of the uncertain parameters and propagate this uncertainty through the ignition process we analyze the time evolution of moments and probability density functions of the solution we also examine the role and significance of dependence among the uncertain parameters we finish with a discussion of the role of non linearity and the performance of the algorithm copyright q 2009 john wiley sons ltd

an improved paralleling in order solving scheme for ah fdtd method using eigenvalue transformation this paper describes a modification of the unconditionally stable finite difference time domain fdtd algorithm based on the associated hermite ah functions the original method is involved with solving a banded nested matrix equation to calculate the ah expanding coefficients using lower upper lu decomposition which is the main cost for cpu time and memory storage here the calculation of expansion coefficients is modified as a paralleling in order solving scheme by using the eigenvalue transformation which greatly enhances the computational efficiency and reduces the memory storage the numerical results demonstrate the performance of this improvement

a higher order approximation to effective conductivity in media of anisotropic random structure properties of the effective conductivity tensor keff are studied by deriving the second order terms in its expansion in the variance 2 of normally distributed log conductivity it is shown that for media of anisotropic structure the components of the effective conductivity tensor are expressed by a functional of the log conductivity covariance that is it depends on the shape of the correlation function and not only on anisotropy ratios variance 2 and space dimensions however the trace of keff is independent of the log conductivity autocovariance and for a given mean conductivity depends only on 2 the dependence of the effective conductivity on the correlation structure is illustrated for gaussian and exponential autocovariances of log conductivity and for two and three dimensional flows

efficient implementation for the ah fdtd method with iterative procedure and cfs pml an efficient implementation for the associated hermite finite difference time domain ah fdtd method with iterative procedure is proposed which is an extension of previous works the original and the paralleling in order based ah fdtd methods by introducing the alternating direction implicit method the direct calculation of five point matrix equation is replaced with iterative solving of tridiagonal matrices equations for the 2 d case only two tridiagonal matrixes need to be calculated in a full iterative cycle in addition the complex frequency shifted cfs perfectly matched layer pml is extended to the ah fdtd method and implemented together with this efficient solution scheme numerical examples show the superior computational performance of the proposed iterative procedure by comparison with the conventional fdtd method the adi fdtd method and previous ah fdtd methods the effectiveness of cfs pml in ah fdtd is also verified

an adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations in recent years there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary partial differential equations to this end the spectral stochastic finite element method ssfem is the most popular method due to its fast convergence rate recently the stochastic sparse grid collocation method has emerged as an attractive alternative to ssfem it approximates the solution in the stochastic space using lagrange polynomial interpolation the collocation method requires only repetitive calls to an existing deterministic solver similar to the monte carlo method however both the ssfem and current sparse grid collocation methods utilize global polynomials in the stochastic space thus when there are steep gradients or finite discontinuities in the stochastic space these methods converge very slowly or even fail to converge in this paper we develop an adaptive sparse grid collocation strategy using piecewise multi linear hierarchical basis functions hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region numerical examples especially for problems related to long term integration and stochastic discontinuity are presented comparisons with monte carlo and multi element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method

multiobjective control problem with v invexity abstract a multiobjective control problem is considered duality results are obtained for mond weir type duals under v invexity assumptions and their generalizations

uncertainty propagation in finite deformations a spectral stochastic lagrangian approach this work proposes a method for quantifying uncertainty propagation in finite deformation problems using the spectral stochastic finite element method ssfem a spectral expansion of the current configuration of a deforming body is proposed to compute the stochastic deformation gradient which is in turn used to compute the stochastic analogs of the various quantities which appear in large deformation analysis such as strain and stress measures and consistent moduli a total lagrangian approach to the stochastic large deformation problem is presented model problems in large deformation elasto plasticity are considered highlighting the features of the methodology developed rigorous comparisons with monte carlo solutions are presented it is shown that the proposed approach can estimate the probability density function and response statistics of the field variables with satisfactory accuracy

a scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach experimental evidence suggests that the dynamics of many physical phenomena are significantly affected by the underlying uncertainties associated with variations in properties and fluctuations in operating conditions recent developments in stochastic analysis have opened the possibility of realistic modeling of such systems in the presence of multiple sources of uncertainties these advances raise the possibility of solving the corresponding stochastic inverse problem the problem of designing estimating the evolution of a system in the presence of multiple sources of uncertainty given limited information a scalable parallel methodology for stochastic inverse design problems is developed in this article the representation of the underlying uncertainties and the resultant stochastic dependant variables is performed using a sparse grid collocation methodology a novel stochastic sensitivity method is introduced based on multiple solutions to deterministic sensitivity problems the stochastic inverse design problem is transformed to a deterministic optimization problem in a larger dimensional space that is subsequently solved using deterministic optimization algorithms the design framework relies entirely on deterministic direct and sensitivity analysis of the continuum systems thereby significantly enhancing the range of applicability of the framework for the design in the presence of uncertainty of many other systems usually analyzed with legacy codes various illustrative examples with multiple sources of uncertainty including inverse heat conduction problems in random heterogeneous media are provided to showcase the developed framework

on the construction and analysis of stochastic models characterization and propagation of the errors associated with limited data this paper investigates the predictive accuracy of stochastic models in particular a formulation is presented for the impact of data limitations associated with the calibration of parameters for these models on their overall predictive accuracy in the course of this development a new method for the characterization of stochastic processes from corresponding experimental observations is obtained specifically polynomial chaos representations of these processes are estimated that are consistent in some useful sense with the data the estimated polynomial chaos coefficients are themselves characterized as random variables with known probability density function thus permitting the analysis of the dependence of their values on further experimental evidence moreover the error in these coefficients associated with limited data is propagated through a physical system characterized by a stochastic partial differential equation spde this formalism permits the rational allocation of resources in view of studying the possibility of validating a particular predictive model a bayesian inference scheme is relied upon as the logic for parameter estimation with its computational engine provided by a metropolis hastings markov chain monte carlo procedure

karhunen loeve decomposition of random fields based on a hierarchical matrix approach summary the simulation of the behavior of structures with uncertain properties is a challenging issue because it requires suitable probabilistic models and adequate numerical tools nowadays it is possible to perform probabilistic investigations of the structural performance which take into account a space variant uncertainty characterization of the structures given a structural solver and the probabilistic models the reliability analysis of the structural response depends on the continuous random fields approximation which is carried out by means of a finite set of random variables the paper analyzes the main aspects of discretization in the case of 2d problems the combination of the well known karhunen loeve series expansion the finite element method and the hierarchical matrices approach is proposed in the paper copyright 2013 john wiley sons ltd

an analysis of polynomial chaos approximations for modeling single fluid phase flow in porous medium systems we examine a variety of polynomial chaos motivated approximations to a stochastic form of a steady state groundwater flow model we consider approaches for truncating the infinite dimensional problem and producing decoupled systems we discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature it would be necessary to find new multivariate cubature rules finally we use the acceleration of monte carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than monte carlo for the relatively small domains considered in this work a curse of dimensionality in the series expansion of the log normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work other than the monte carlo method

a stabilized stochastic finite element second order projection method for modeling natural convection in random porous media we consider natural convection in flow saturated porous media with random porosity the porosity is treated as a random field and a stochastic finite element method is developed the stochastic projection method is considered for the solution of the high dimensional stochastic navier stokes equations since it leads to the uncoupling of the velocity and pressure degrees of freedom because of the porosity dependence of the pressure gradient term in the governing flow equations one cannot use the first order projection method a stabilized stochastic finite element second order projection method is presented based on a pressure gradient projection a two dimensional stochastic problem with moderate and large variation in the random porosity field is examined and the results are compared with monte carlo and sparse grid smolyak collocation approaches excellent agreement between these results indicates the effectiveness and accuracy of the proposed methodology

the wlp fdtd method for periodic structures with oblique incident wave a new unconditionally stable finite difference time domain fdtd method for periodic structures is presented which is based on the field transformation method and the weighted laguerre polynomials fdtd wlp fdtd the proposed method uses a field transformation to remove the time gradient across the grid and then uses the concept of the wlp fdtd to get the implicit relationship between the transformed field variables it holds the advantages of the wlp fdtd can eliminate the restriction of the courant friedrich levy cfl stability condition compared with other field transform methods the new method needn t to do special treatment for the additional terms it appears to be much more efficient than the other field transformation fdtd method for solving periodic structures with fine structures and large incident angle to verify the accuracy and the efficiency of the proposed method we compare the results of the split field fdtd method with the proposed method

stochastic spectral methods for efficient bayesian solution of inverse problems we present a reformulation of the bayesian approach to inverse problems that seeks to accelerate bayesian inference by using polynomial chaos pc expansions to represent random variables evaluation of integrals over the unknown parameter space is recast more efficiently as monte carlo sampling of the random variables underlying the pc expansion we evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion the accuracy of posterior estimates is examined with respect to the order of the pc representation the choice of pc basis and the decomposition of the support of the prior the computational cost of the new scheme shows significant gains over direct sampling

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

spectral representation and reduced order modeling of the dynamics of stochastic reaction networks via adaptive data partitioning dynamical analysis tools are well established for deterministic models however for many biochemical phenomena in cells the molecule count is low leading to stochastic behavior that causes deterministic macroscale reaction models to fail the main mathematical framework representing these phenomena is based on jump markov processes that model the underlying stochastic reaction network conventional dynamical analysis tools do not readily generalize to the stochastic setting due to nondifferentiability and absence of explicit state evolution equations we developed a reduced order methodology for dynamical analysis that relies on the karhunen loeve decomposition and polynomial chaos expansions the methodology relies on adaptive data partitioning to obtain an accurate representation of the stochastic process especially in the case of multimodal behavior as a result a mixture model is obtained that represents the reduced order dynamics of the system the schlogl model is used as a prototype bistable process that exhibits time scale separation and leads to multimodality in the reduced order model

a model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations we present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations our method consists of three main ingredients 1 the hybridizable discontinuous galerkin hdg discretization of elliptic partial differential equations pdes which allows us to obtain high order accurate solutions of the governing pde 2 the reduced basis method for a new hdg discretization of the underlying pde to enable real time solution of the parameterized pde in the presence of stochastic parameters and 3 a multilevel variance reduction method that exploits the statistical correlation among the different reduced basis approximations and the high fidelity hdg discretization to accelerate the convergence of the monte carlo simulations the multilevel variance reduction method provides efficient computation of the statistical outputs by shifting most of the computational burden from the high fidelity hdg approximation to the reduced basis approximations furthermore we develop a posteriori error estimates for our approximations of the statistical outputs based on these error estimates we propose an algorithm for optimally choosing both the dimensions of the reduced basis approximations and the sizes of monte carlo samples to achieve a given error tolerance we provide numerical examples to demonstrate the performance of the proposed method

a two stage adaptive stochastic collocation method on nested sparse grids for multiphase flow in randomly heterogeneous porous media a new computational method is proposed for efficient uncertainty quantification of multiphase flow in porous media with stochastic permeability for pressure estimation it combines the dimension adaptive stochastic collocation method on smolyak sparse grids and the kronrod patterson hermite nested quadrature formulas for saturation estimation an additional stage is developed in which the pressure and velocity samples are first generated by the sparse grid interpolation and then substituted into the transport equation to solve for the saturation samples to address the low regularity problem of the saturation numerical examples are presented for multiphase flow with stochastic permeability fields to demonstrate accuracy and efficiency of the proposed two stage adaptive stochastic collocation method on nested sparse grids

duality for a class of multiobjective control problems with generalized invexity abstract b mond and i smart j math anal appl 136 1988 325 333 defined a kind of invexity and discussed the duality and sufficiency in scalar control problems with such invexity d bhatia and p kumar j math anal appl 189 1995 676 692 defined another kind of invexity corresponding generalized invexity and discussed the duality for multiobjective control problems with such generalized invexity in this paper the duality results for multiobjective control problems with mond and smart s generalized invexity are discussed

duality for a class of multiobjective control problems abstract in this paper a class of multiobjective control problems is considered where the objective and constraint functions involved are f t x t t y t z t with x t rn y t rn and z t rm where x t and z t are the control variables and y t is the state variable under the assumption of invexity and its generalization duality theorems are proved through a parametric approach to related properly efficient solutions of the primal and dual problems

sparse grid collocation schemes for stochastic natural convection problems in recent years there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems spectral stochastic methods and monte carlo based sampling methods are two approaches that have been used to analyze these problems as the complexity of the problem or the number of random variables involved in describing the input uncertainties increases these approaches become highly impractical from implementation and convergence points of view this is especially true in the context of realistic thermal flow problems where uncertainties in the topology of the boundary domain boundary flux conditions and heterogeneous physical properties usually require high dimensional random descriptors the sparse grid collocation method based on the smolyak algorithm offers a viable alternate method for solving high dimensional stochastic partial differential equations an extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation we show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured comparison of the present method with the generalized polynomial chaos expansion and monte carlo methods are made

a three dimensional fdtd subgridding algorithm with separated temporal and spatial interfaces and related stability analysis a finite different time domain subgrid algorithm locally refines the mesh at regions requiring higher resolution a novel separation of spatial and temporal subgridding interfaces is presented that allows implementing a novel spatial subgridding method and investigating the stability of each subalgorithm individually details are given for a spatial subgridding algorithm having a 1 3 mesh ratio in the spatial subgridding algorithm the fine mesh is constructed with a recessed interface and the interpolation scheme is designed to be symmetric to maintain the stability of the update process the stability of the spatial subgridding algorithm is analyzed with a matrix method numerical examples showing stability and accuracy are provided

enforcing positivity in intrusive pc uq methods for reactive ode systems we explore the relation between the development of a non negligible probability of negative states and the instability of numerical integration of the intrusive galerkin ordinary differential equation system describing uncertain chemical ignition to prevent this instability without resorting to either multi element local polynomial chaos pc methods or increasing the order of the pc representation in time we propose a procedure aimed at modifying the amplitude of the pc modes to bring the probability of negative state values below a user defined threshold this modification can be effectively described as a filtering procedure of the spectral pc coefficients which is applied on the fly during the numerical integration when the current value of the probability of negative states exceeds the prescribed threshold we demonstrate the filtering procedure using a simple model of an ignition process in a batch reactor this is carried out by comparing different observables and error measures as obtained by non intrusive monte carlo and gauss quadrature integration and the filtered intrusive procedure the filtering procedure has been shown to effectively stabilize divergent intrusive solutions and also to improve the accuracy of stable intrusive solutions which are close to the stability limits

weakly efficient solutions and pseudoinvexity in multiobjective control problems abstract in this paper we introduce new pseudoinvexity conditions on functionals involved in a multiobjective control problem called w kt pseudoinvexity and w fj pseudoinvexity we prove that all kuhn tucker or fritz john points are weakly efficient solutions if and only if these conditions are fulfilled we relate weakly efficient solutions to optimal solutions of weighting control problems we generalize recently obtained optimality results of known mathematical programming problems and control problems we illustrate these results with an example

a new unconditionally stable scheme for fdtd method using associated hermite orthogonal functions an unconditionally stable solution using associated hermite ah functions is proposed for the finite difference time domain fdtd method the electromagnetic fields and their time derivatives in time do main maxwell s equations are expanded by these orthonormal basis func tions by applying galerkin temporal testing procedure to these expanded equations the time variable can be eliminated from the calculations a set of implicit equations is derived to calculate the magnetic filed expansion coefficients of all orders of ah functions for the temporal variable and the electrical field coefficients can be obtained respectively with the ap propriate translation and scale parameters we can find a minimum order basis functions subspace to approach a particular electromagnetic field the numerical results have shown that the proposed method can reduce the cpu time to 0 59 of the traditional fdtd method while maintaining good accuracy

dimensionality reduction and polynomial chaos acceleration of bayesian inference in inverse problems we consider a bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field endowed with a hierarchical gaussian process prior computational challenges in this construction arise from the need for repeated evaluations of the forward model e g in the context of markov chain monte carlo and are compounded by high dimensionality of the posterior we address these challenges by introducing truncated karhunen loeve expansions based on the prior distribution to efficiently parameterize the unknown field and to specify a stochastic forward problem whose solution captures that of the deterministic forward model over the support of the prior we seek a solution of this problem using galerkin projection on a polynomial chaos basis and use the solution to construct a reduced dimensionality surrogate posterior density that is inexpensive to evaluate we demonstrate the formulation on a transient diffusion equation with prescribed source terms inferring the spatially varying diffusivity of the medium from limited and noisy data

embedding uncertainty into ocean acoustic propagation models l a probabilistic formalism is proposed for the direct inclusion of environmental uncertainty into an acoustic model that describes propagation in an ocean waveguide incomplete environmental knowledge is characterized by a spectral representation of uncertainty using expansions of random processes in terms of orthogonal random polynomials a brief summary of the method is presented and a set of coupled differential equations describing the propagation of both the acoustic field and its associated uncertainty is derived for the case where the uncertain environment is attributed to a lack of complete information concerning the waveguide s sound speed distribution

efficient solutions in v kt pseudoinvex multiobjective control problems a characterization in this paper we introduce a new condition on functionals involved in a multiobjective control problem for which we define the v kt pseudoinvex control problem we prove that a v kt pseudoinvex control problem is characterized so that a kuhn tucker point is an efficient solution we generalize recently obtained optimality results of known mathematical programming problems and control problems we illustrate these results with an example

comparative study of projection schemes for stochastic finite element analysis we present a comparison of subspace projection schemes for stochastic finite element analysis in terms of accuracy and computational efficiency more specifically we compare the polynomial chaos projection scheme with reduced basis projection schemes based on the preconditioned stochastic krylov subspace numerical studies are presented for two problems 1 static analysis of a plate with random young s modulus and 2 settlement of a foundation supported on a randomly heterogeneous soil monte carlo simulation results based on exact structural analysis are used to generate benchmark results against which the projection schemes are compared we show that stochastic reduced basis methods require significantly less computer memory and execution time compared to the polynomial chaos approach particularly for large scale problems with many random variables for the class of problems considered we find that stochastic reduced basis methods can be up to orders of magnitude faster while providing results of comparable or better accuracy

numerical approximation of poroelasticity with random coefficients using polynomial chaos and hybrid high order methods abstract in this work we consider the biot problem with uncertain poroelastic coefficients the uncertainty is modeled using a finite set of parameters with prescribed probability distribution we present the variational formulation of the stochastic partial differential system and establish its well posedness we then discuss the approximation of the parameter dependent problem by non intrusive techniques based on polynomial chaos decompositions we specifically focus on sparse spectral projection methods which essentially amount to performing an ensemble of deterministic model simulations to estimate the expansion coefficients the deterministic solver is based on a hybrid high order discretization supporting general polyhedral meshes and arbitrary approximation orders we numerically investigate the convergence of the probability error of the polynomial chaos approximation with respect to the level of the sparse grid finally we assess the propagation of the input uncertainty onto the solution considering an injection extraction problem

an efficient high order perturbation approach for flow in random porous media via karhunen loeve and polynomial expansions in this study we attempt to obtain higher order solutions of the means and co variances of hydraulic head for saturated flow in randomly heterogeneous porous media on the basis of the combination of karhunen loeve decomposition polynomial expansion and perturbation methods we first decompose the log hydraulic conductivity y ln ks as an infinite series on the basis of a set of orthogonal gaussian standard random variables i the coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of log hydraulic conductivity we then write head as an infinite series whose terms h n represent head of nth order in terms of y the standard deviation of y and derive a set of recursive equations for h n we then decompose h n with polynomial expansions in terms of the products of n gaussian random variables the coefficients in these series are determined by substituting decompositions of y and h n into those recursive equations we solve the mean head up to fourth order in y and the head variances up to third order in y2 we conduct monte carlo mc simulation and compare mc results against approximations of different orders from the moment equation approach based on karhunen loeve decomposition klme we also explore the validity of the klme approach for different degrees of medium variability and various correlation scales it is evident that the klme approach with higher order corrections is superior to the conventional first order approximations and is computationally more efficient than the monte carlo simulation

propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach abstract this paper presents an efficient procedure for characterizing the solution of evolution equations with stochastic coefficients these typically model the behavior of physical systems whose properties are modeled as spatially or temporally varying stochastic processes described within the framework of probability theory the concepts of projection orthogonality and weak convergence are exploited in a manner which directly mimics deterministic finite element solutions except that in the stochastic case inner products refer to expectation operations specifically the karhunen loeve expansion is used to discretize these processes into a denumerable set of random variables thus providing a denumerable function space in which the problem is cast the polynomial chaos expansion is then used to represent the solution in this space and the coefficients in the expansion are evaluated as generalized fourier coefficients via a galerkin procedure in the hilbert space of random variables

a stochastic projection method for fluid flow i basic formulation we describe the construction and implementation of a stochastic navier stokes solver the solver combines a spectral stochastic uncertainty representation scheme with a finite difference projection method for flow simulation the uncertainty quantification scheme is adapted from the spectral stochastic finite element method ssfem which is based on regarding uncertainty as generating a new dimension and the solution as being dependent on this dimension in the ssfem formalism the stochastic dependence is represented in terms of the polynomial chaos system and the coefficients in the corresponding spectral representation are obtained using a galerkin approach it is shown that incorporation of the spectral uncertainty representation scheme into the projection method results in a coupled system of advection diffusion equations for the various uncertainty fields and in a decoupled system of pressure projection steps this leads to a very efficient stochastic solver whose advantages are illustrated using steady and transient simulations of transport and mixing in a microchannel

a dynamically adaptive wavelet approach to stochastic computations based on polynomial chaos capturing all scales of random modes on independent grids in stochastic computations or uncertainty quantification methods the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion the size of this system increases drastically when the number of independent random variables and or order of polynomial chaos expansions increases this is invariably the case for large scale simulations and or problems involving steep gradients and other multiscale features such features are variously reflected on each solution component or random uncertainty mode requiring the development of adaptive methods for their accurate resolution in this paper we propose a new approach for treating such problems based on a dynamically adaptive wavelet methodology involving space refinement on physical space that allows all scales of each solution component to be refined independently of the rest we exemplify this using the convection diffusion model with random input data and present three numerical examples demonstrating the salient features of the proposed method thus we establish a new elegant and flexible approach for stochastic problems with steep gradients and multiscale features based on polynomial chaos expansions

a stochastic multiscale framework for modeling flow through random heterogeneous porous media flow through porous media is ubiquitous occurring from large geological scales down to the microscopic scales several critical engineering phenomena like contaminant spread nuclear waste disposal and oil recovery rely on accurate analysis and prediction of these multiscale phenomena such analysis is complicated by inherent uncertainties as well as the limited information available to characterize the system any realistic modeling of these transport phenomena has to resolve two key issues i the multi length scale variations in permeability that these systems exhibit and ii the inherently limited information available to quantify these property variations that necessitates posing these phenomena as stochastic processes a stochastic variational multiscale formulation is developed to incorporate uncertain multiscale features a stochastic analogue to a mixed multiscale finite element framework is used to formulate the physical stochastic multiscale process recent developments in linear and non linear model reduction techniques are used to convert the limited information available about the permeability variation into a viable stochastic input model an adaptive sparse grid collocation strategy is used to efficiently solve the resulting stochastic partial differential equations spdes the framework is applied to analyze flow through random heterogeneous media when only limited statistics about the permeability variation are given

a flexible uncertainty quantification method for linearly coupled multi physics systems abstract this paper presents a novel approach to building an integrated uncertainty quantification uq methodology suitable for modern day component based approach for multi physics simulation development our hybrid uq methodology supports independent development of the most suitable uq method intrusive or non intrusive for each physics module by providing an algorithmic framework to couple these stochastic modules for propagating global uncertainties we address algorithmic and computational issues associated with the construction of this hybrid framework we demonstrate the utility of such a framework on a practical application involving a linearly coupled multi species reactive transport model

uncertainty quantification in hybrid dynamical systems uncertainty quantification uq techniques are frequently used to ascertain output variability in systems with parametric uncertainty traditional algorithms for uq are either system agnostic and slow such as monte carlo or fast with stringent assumptions on smoothness such as polynomial chaos and quasi monte carlo in this work we develop a fast uq approach for hybrid dynamical systems by extending the polynomial chaos methodology to these systems to capture discontinuities we use a wavelet based wiener haar expansion we develop a boundary layer approach to propagate uncertainty through separable reset conditions we also introduce a transport theory based approach for propagating uncertainty through hybrid dynamical systems here the expansion yields a set of hyperbolic equations that are solved by integrating along characteristics the solution of the partial differential equation along the characteristics allows one to quantify uncertainty in hybrid or switching dynamical systems the above methods are demonstrated on example problems

sufficiency and duality for multiobjective variational control problems with generalized f α ρ θ v convexity abstract zalmai introduced generalized f v convex functions a new class of functions that unifies several concepts of generalized convex functions in this article the generalized f v convex functions are extended to variational control problem by utilizing the new concepts we obtain sufficient optimality conditions and prove wolfe type and mond weir type duality results for multiobjective variational control programming problem

a concurrent model reduction approach on spatial and random domains for the solution of stochastic pdes a methodology is introduced for rapid reduced order solution of stochastic partial differential equations on the random domain a generalized polynomial chaos expansion gpce is used to generate a reduced subspace gpce involves expansion of the random variable as a linear combination of basis functions defined using orthogonal polynomials from the askey series a proper orthogonal decomposition pod approach coupled with the method of snapshots is used to generate a reduced solution space from the space spanned by the finite element basis functions on the spatial domain pod methods have been extremely popular in fluid mechanics applications and have subsequently been applied to other interesting areas they have been shown to be capable of representing complicated phenomena with a handful of degrees of freedom this concurrent model reduction on the random and spatial domains is applied to stochastic partial differential equations pdes in natural convection processes involving randomness in the porosity of the medium and the rayleigh number the results indicate that owing to the multiplicative nature of the concurrent model reduction extremely large computational gains are realized without significant loss of accuracy