transient flow in bounded randomly heterogeneous domains 1 exact conditional moment equations and recursive approximations in a recent paper we developed exact nonlocal conditional moment equations for transient flow in bounded domains driven by random forcing terms sources initial head and boundary conditions whereas our conditional mean equations took into account the randomness of forcing terms our conditional second moment equations did not we do so in this brief addendum

a time domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems we study a class of time domain decomposition based methods for the numerical solution of large scale linear quadratic optimal control problems our methods are based on a multiple shooting reformulation of the linear quadratic optimal control problem as a discrete time optimal control dtoc problem the optimality conditions for this dtoc problem lead to a linear block tridiagonal system the diagonal blocks are invertible and are related to the original linear quadratic optimal control problem restricted to smaller time subintervals this motivates the application of block gauss seidel gs type methods for the solution of the block tridiagonal systems numerical experiments show that the spectral radii of the block gs iteration matrices are larger than one for typical applications but that the eigenvalues of the iteration matrices decay to zero fast hence while the gs method is not expected to convergence for typical applications it can be effective as a preconditioner for krylov subspace methods this is confirmed by our numerical tests a byproduct of this research is the insight that certain instantaneous control techniques can be viewed as the application of one step of the forward block gs method applied to the dtoc optimality system

a bayesian approach to multiscale inverse problems with on the fly scale determination abstract a bayesian computational approach is presented to provide a multi resolution estimate of an unknown spatially varying parameter from indirect measurement data in particular we are interested in spatially varying parameters with multiscale characteristics in our work we consider the challenge of not knowing the characteristic length scale s of the unknown a priori and present an algorithm for on the fly scale determination our approach is based on representing the spatial field with a wavelet expansion wavelet basis functions are hierarchically structured localized in both spatial and frequency domains and tend to provide sparse representations in that a large number of wavelet coefficients are approximately zero for these reasons wavelet bases are suitable for representing permeability fields with non trivial correlation structures moreover the intra scale correlations between wavelet coefficients form a quadtree and this structure is exploited to identify additional basis functions to refine the model bayesian inference is performed using a sequential monte carlo smc sampler with a markov chain monte carlo mcmc transition kernel the smc sampler is used to move between posterior densities defined on different scales thereby providing a computationally efficient method for adaptive refinement of the wavelet representation we gain insight from the marginal likelihoods by computing bayes factors for model comparison and model selection the marginal likelihoods provide a termination criterion for our scale determination algorithm the bayesian computational approach is rather general and applicable to several inverse problems concerning the estimation of a spatially varying parameter the approach is demonstrated with permeability estimation for groundwater flow using pressure sensor measurements

stochastic multiscale flux basis for stokes darcy flows abstract three algorithms are developed for uncertainty quantification in modeling coupled stokes and darcy flows the porous media may consist of multiple regions with different properties the permeability is modeled as a non stationary stochastic variable with its log represented as a sum of local karhunen loeve kl expansions the problem is approximated by stochastic collocation on either tensor product or sparse grids coupled with a multiscale mortar mixed finite element method for the spatial discretization a non overlapping domain decomposition algorithm reduces the global problem to a coarse scale mortar interface problem which is solved by an iterative solver for each stochastic realization in the traditional domain decomposition implementation each subdomain solves a local dirichlet or neumann problem in every interface iteration to reduce this cost two additional algorithms based on deterministic or stochastic multiscale flux basis are introduced the basis consists of the local flux or velocity trace responses from each mortar degree of freedom it is computed for each subdomain independently before the interface iteration begins the use of the multiscale flux basis avoids the need for subdomain solves on each iteration the deterministic basis is computed at each stochastic collocation and used only at this realization the stochastic basis is formed by further looping over all local realizations of a subdomain s kl region before the stochastic collocation begins it is reused over multiple realizations numerical tests are presented to illustrate the performance of the three algorithms with the stochastic multiscale flux basis showing significant savings in computational cost compared to the other two algorithms

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

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

a nystrm method for integral equations with fixed singularities of mellin type in weighted lp spaces we consider integral equations of the second kind with fixed singularities of mellin type according to the behavior of the mellin kernel we first determine suitable weighted lp spaces where we look for the solution then for its approximation we propose a numerical method of nystrm type based on a gaussjacobi quadratura formula actually a slight modification of the classical procedure is introduced in order to prove convergence results in weighted lp spaces moreover a preconditioning technique allows us to solve well conditioned linear systems we show the efficiency of the proposed method through some numerical tests

problem formulation for multidisciplinary optimization this paper is about multidisciplinary design optimization or mdo the coupling of two or more analysis disciplines with numerical optimization the paper has three goals first it is an expository introduction to mdo aimed at those who do research on optimization algorithms since the optimization community has much to contribute to this important class of computational engineering problems second this paper presents to the mdo research community a new abstraction for multidisciplinary analysis and design problems as well as new decomposition formulations for these problems third the individual discipline feasible idf approaches introduced here make use of existing specialized analysis codes and they introduce significant opportunities for coarse grained computational parallelism particularly well suited to heterogeneous computing environments the key distinguishing characteristic of the three fundamental approaches to mdo formulation discussed here is the kind of disciplinary feasibility that

identification of large scale spatial trends in hydrologic data it is often useful to distinguish different scales of variability in hydrologic properties such as hydraulic conductivity in the simplest two scale case large scale variations can be viewed as a trend while small scale fluctuations about this trend can be viewed as a random residual this paper describes a method for estimating spatial trends from scattered field measurements the basic concept is to treat both the trend and the residual as stationary random functions these functions are distinguished by their spatial spectral or covariance properties which may be estimated from available data or simply hypothesized we present two versions of a general algorithm for estimating spatial trends 1 a discrete version which is useful in practical applications where data are limited and irregularly spaced and 2 a continuous version which can be used to study the effects of using incorrect spectral parameters applications of the discrete algorithm to both synthetically generated data and field measurements yield satisfactory trend estimates an analysis based on the continuous algorithm shows that the estimation error lower bound for these applications depends on two dimensionless ratios that is the scale disparity ratio of the trend and residual correlation scales and the signal to noise ratio ratio of the trend and residual variances these ratios may be used to evaluate the feasibility of trend estimation before field samples are actually collected

a spatial domain decomposition method for parabolic optimal control problems we present a non overlapping spatial domain decomposition method for the solution of linear quadratic parabolic optimal control problems the spatial domain is decomposed into non overlapping subdomains the original parabolic optimal control problem is decomposed into smaller problems posed on space time cylinder subdomains with auxiliary state and adjoint variables imposed as dirichlet boundary conditions on the space time interface boundary the subdomain problems are coupled through robin transmission conditions this leads to a schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space time interface boundary the schur complement operator is the sum of space time subdomain schur complement operators the application of these subdomain schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem the subdomain schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem we introduce a new family of neumann neumann type preconditioners for the schur complement system including several different coarse grid corrections we compare the numerical performance of our preconditioners with an alternative approach recently introduced by benamou

numerical methods for integral equations of mellin type we present a survey of numerical methods based on piecewise polynomial approximation for integral equations of mellin type including examples arising in boundary integral methods for partial differential equations on polygonal domains

the collocation method for mixed boundary value problems on domains with curved polygonal boundaries we consider an indirect boundary integral equation formulation for the mixed dirichlet neumann boundary value problem for the laplace equation on a plane domain with a polygonal boundary the resulting system of integral equations is solved by a collocation method which uses a mesh grading transformation and a cosine approximating space the mesh grading transformation method yields fast convergence of the collocation solution by smoothing the singularities of the exact solution a complete stability and solvability analysis of the transformed integral equations is given by use of a mellin transform technique in a setting in which each arc of the polygon has associated with it a periodic sobolev space

domain decomposition methods for advection dominated linear quadratic elliptic optimal control problems abstract we present an optimization level domain decomposition dd preconditioner for the solution of advection dominated elliptic linear quadratic optimal control problems which arise in many science and engineering applications the dd preconditioner is based on a decomposition of the optimality conditions for the elliptic linear quadratic optimal control problem into smaller subdomain optimality conditions with dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces these subdomain optimality conditions are coupled through robin transmission conditions for the states and the adjoints the parameters in the robin transmission condition depend on the advection this decomposition leads to a schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces the schur complement operator is the sum of subdomain schur complement operators the application of which is shown to correspond to the solution of subdomain optimal control problems which are essentially smaller copies of the original optimal control problem we show that under suitable conditions the application of the inverse of the subdomain schur complement operators requires the solution of a subdomain elliptic linear quadratic optimal control problem with robin boundary conditions for the state numerical tests for problems with distributed and with boundary control show that the dependence of the preconditioners on mesh size and subdomain size is comparable to its counterpart applied to a single advection dominated equation these tests also show that the preconditioners are insensitive to the size of the control regularization parameter

transient effective hydraulic conductivities under slowly and rapidly varying mean gradients in bounded three dimensional random media we have shown elsewhere tartakovsky and neuman this issue a that in randomly heterogeneous media the ensemble mean transient flux is generally nonlocal in space time and therefore non darcian we have also shown tartakovsky and neuman this issue b that there are special situations in which this flux can be localized so as to render it darcian in real or transformed domains each such situation gives rise to an effective hydraulic conductivity which relates mean gradient to mean flux at any point in real or transformed space time in this paper we develop first order analytical expressions for effective hydraulic conductivity under three dimensional transient flow through a boxshaped domain due to a mean hydraulic gradient that varies slowly in space and time when the mean gradient varies rapidly in time the laplace transform of the mean flux is local but its real time equivalent includes a temporal convolution integral we develop analytical expressions for the real time kernel of this convolution integral the box is embedded within a statistically homogeneous natural log hydraulic conductivity field that is gaussian and exhibits an anisotropic exponential spatial correlation structure by the effective hydraulic conductivity of a finite box in such a field we imply the ensemble mean expected value of all random equivalent conductivities that one could associate with the box under these conditions we explore the influence of domain size time and statistical anisotropy on effective conductivity and include a simple new formula for its variation with statistical anisotropy ratio in an infinite domain under steady state appendixes c e are available with entire article on microfiche order by mail from american geophysical union 2000 florida avenue n w washington dc 20009 or by phone at 800 966 2481 2 50 document w97 002 payment must accompany order

reduced basis anova methods for partial differential equations with high dimensional random inputs in this paper we present a reduced basis anova approach for partial deferential equations pdes with random inputs the anova method combined with stochastic collocation methods provides model reduction in high dimensional parameter space through decomposing high dimensional inputs into unions of low dimensional inputs in this work to further reduce the computational cost we investigate spatial low rank structures in the anova collocation method and develop efficient spatial model reduction techniques using hierarchically generated reduced bases we present a general mathematical framework of the methodology validate its accuracy and demonstrate its efficiency with numerical experiments

an algorithm for the approximate solution of integral equations of mellin type the cruciform crack problem of elasticity gives rise to an integral equation of the second kind on 0 1 whose kernel has a fixed singularity at 0 0 we introduce a transformation of 0 1 onto itself such that an arbitrary number of derivatives vanish at the end points 0 and 1 if the transformed kernel is dominated near the origin by a mellin kernel then we have given conditions under which the use of a modified euler maclaurin quadrature rule and the nystrom method gives an approximate solution which converges to the exact solution of the original equation the method is illustrated with a numerical example

approximate partitioned method of snapshots for pod the proper orthogonal decomposition pod method has been widely used to construct efficient numerical surrogate models for computationally intensive applications in control and optimization an inherent challenge with this method is that pod basis generation can be computationally expensive due to the huge size of the input snapshot data obtained from typical high fidelity large scale dynamic system simulations however if the process can be distributed into much smaller tasks over multiple processors in parallel computational time can be drastically reduced in this paper we put forth a novel partitioned method for generating the pod basis from snapshot data this method preserves the distributed nature of the data and takes advantage of parallelism for computation additionally it greatly reduces subtask communication volume two numerical examples are presented that demonstrate the effectiveness of the new method

hitting matrix and domino tiling with diagonal impurities as a continuation to our previous work nakano and sadahiro in fundam inform 117 249 264 2012 nakano and sadahiro in j stat phys 139 4 565 597 2010 we consider the domino tiling problem with impurities 1 if we have more than two impurities on the boundary we can compute the number of corresponding perfect matchings by using the hitting matrix method fomin in trans am math soc 353 9 3563 3583 2001 2 we have an alternative proof of the main result in nakano and sadahiro fundam inform 117 249 264 2012 and result in 1 above using the formula by kenyon and wilson trans am math soc 363 3 1325 1364 2011 electron j comb 16 1 112 2009 of counting the number of groves on circular planar graphs 3 we study the behavior of the probability of finding the impurity at a given site when the size of the graph tends to infinity as well as the scaling limit of those

a domain decomposition approach for uncertainty analysis this paper proposes a decomposition approach for uncertainty analysis of systems governed by partial differential equations pdes the system is split into local components using domain decomposition our domain decomposed uncertainty quantification dduq approach performs uncertainty analysis independently on each local component in an offline phase and then assembles global uncertainty analysis results using precomputed local information in an online phase at the heart of the dduq approach is importance sampling which weights the precomputed local pde solutions appropriately so as to satisfy the domain decomposition coupling conditions to avoid global pde solves in the online phase a proper orthogonal decomposition reduced model provides an efficient approximate representation of the coupling functions

a nonstationary spectral method for solving stochastic groundwater problems unconditional analysis stochastic analyses of groundwater flow and transport are frequently based on partial differential equations which have random coefficients or forcing terms analytical methods for solving these equations rely on restrictive assumptions which may not hold in some practical applications numerically oriented alternatives are computationally demanding and generally not able to deal with large three dimensional problems in this paper we describe a hybrid solution approach which combines classical fourier transform concepts with numerical solution techniques our approach is based on a nonstationary generalization of the spectral representation theorem commonly used in time series analysis the generalized spectral representation is expressed in terms of an unknown transfer function which depends on space time and wave number the transfer function is found by solving a linearized deterministic partial differential equation which has the same form as the original stochastic flow or transport equation this approach can accomodate boundary conditions spatially variable mean gradients measurement conditioning and other sources of nonstationarity which cannot be included in classical spectral methods here we introduce the nonstationary spectral method and show how it can be used to derive unconditional statistics of interest in groundwater flow and transport applications

a nystrom method for boundary integral equations in domains with corners we give a convergence and error analysis for a nystrom method on a graded mesh for the numerical solution of boundary integral equations for the harmonic dirichlet problem in plane domains with corners

algorithm refinement for stochastic partial differential equations i linear diffusion a hybrid particle continuum algorithm is formulated for fickian diffusion in the fluctuating hydrodynamic limit the particles are taken as independent random walkers the fluctuating diffusion equation is solved by finite differences with deterministic and white noise fluxes at the interface between the particle and continuum computations the coupling is by flux matching giving exact mass conservation this methodology is an extension of adaptive mesh and algorithm refinement to stochastic partial differential equations results from a variety of numerical experiments are presented for both steady and time dependent scenarios in all cases the mean and variance of density are captured correctly by the stochastic hybrid algorithm for a nonstochastic version i e using only deterministic continuum fluxes the mean density is correct but the variance is reduced except in particle regions away from the interface extensions of the methodology to fluid mechanics applications are discussed

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 1 theory and computational approach we consider the effect of measuring randomly varying hydraulic conductivitiesk x on one s ability to predict numerically without resorting to either monte carlo simulation or upscaling steady state flow in bounded domains driven by random source and boundary terms our aim is to allow optimum unbiased prediction of hydraulic heads h x and fluxes q x by means of their ensemble moments h x c and q x c respectively conditioned on measurements of k x these predictors have been shown by neuman and orr 1993a to satisfy exactly an integrodifferential conditional mean flow equation in which q x c is nonlocal and non darcian here we develop complementary integrodifferential equations for second conditional moments of head and flux which serve as measures of predictive uncertainty obtain recursive closure approximations for both the first and second conditional moment equations through expansion in powers of a small parameter y which represents the standard estimation error of ln k x and show how to solve these equations to first order in y2 by finite elements on a rectangular grid in two dimensions in the special case where one treats k x as if it was locally homogeneous and mean flow as if it was locally uniform one obtains a localized darcian approximation q x c kc x h x c in which kc x is a space dependent conditional hydraulic conductivity tensor this leads to the traditional deterministic darcian steady state flow equation which however acquires a nontraditional meaning in that its parameters and state variables are data dependent and therefore inherently nonunique it further explains why parameter estimates obtained by traditional inverse methods tend to vary as one modifies the database localized equations yield no information about predictive uncertainty our stochastic derivation of these otherwise standard deterministic flow equations makes clear that uncertainty measures associated with estimates of head and flux obtained by traditional inverse methods are generally smaller often considerably so than measures of corresponding predictive uncertainty which can be assessed only by means of stochastic models such as ours we present a detailed comparison between finite element solutions of nonlocal and localized moment equations and monte carlo simulations under superimposed mean uniform and convergent flow regimes in two dimensions paper 1 presents the theory and computational approach and paper 2 guadagnini and neuman this issue describes unconditional and conditional computational results

a nystrom method for a boundary integral equation related to the dirichlet problem on domains with corners the authors consider the interior dirichlet problem for laplace s equation on planar domains with corners they provide a complete analysis of a natural method of nystrom type based on the global gauss lobatto quadrature rule in order to approximate the solution of the corresponding double layer boundary integral equation mellin type integral operators are involved and as usual a modification of the method close to the corners is needed a new modification is proposed and the convergence and stability of the modified quadrature method are proved some numerical tests are also included

sequential imputations and bayesian missing data problems abstract for missing data problems tanner and wong have described a data augmentation procedure that approximates the actual posterior distribution of the parameter vector by a mixture of complete data posteriors their method of constructing the complete data sets is closely related to the gibbs sampler both required iterations and similar to the em algorithm convergence can be slow we introduce in this article an alternative procedure that involves imputing the missing data sequentially and computing appropriate importance sampling weights in many applications this new procedure works very well without the need for iterations sensitivity analysis influence analysis and updating with new data can be performed cheaply bayesian prediction and model selection can also be incorporated examples taken from a wide range of applications are used for illustration

a polynomial collocation method for the numerical solution of weakly singular and singular integral equations on non smooth boundaries the aim of this paper is to show the efficiency of the use of smoothing changes of variable in the numerical treatment of 1d and 2d weakly singular and singular integral equations the introduction of a smoothing transformation besides smoothing the solution allows also the use of a very simple and efficient collocation method based on chebyshev polynomials of the first kind and their zeros further we propose proper smoothing changes of variable also for the numerical approximation of those collocation matrix elements which are given by weakly singular singular or nearly singular integrals several numerical tests are given to point out the efficiency of the numerical approach we propose copyright 2003 john wiley sons ltd

partitioned treatment of uncertainty in coupled domain problems a separated representation approach abstract this work is concerned with the propagation of uncertainty across coupled domain problems with high dimensional random inputs a stochastic model reduction approach based on low rank separated representations is proposed for the partitioned treatment of the uncertainty space the construction of the coupled domain solution is achieved though a sequence of approximations with respect to the dimensionality of the random inputs associated with each individual sub domain and not the combined dimensionality hence drastically reducing the overall computational cost the coupling between the sub domain solutions is done via the classical finite element tearing and interconnecting feti method thus providing a well suited framework for parallel computing two high dimensional stochastic problems a 2d elliptic pde with random diffusion coefficient and a stochastic linear elasticity problem have been considered to study the performance and accuracy of the proposed stochastic coupling approach

flow in heterogeneous media displaying a linear trend in the log conductivity the purpose of this study is to derive a general solution for the problem of flow in nonstationary geological formations where the nonstationary manifests itself in the form of a spatial trend in the mean log conductivity a stochastic frame of reference is adopted to account for the spatial variability of the hydraulic conductivity for a complete stochastic description we derive the expected values and spatial covariances of the hydraulic head and the fluid flux vector as well as a relation between the expected values of the head and the fluxes these expressions are obtained using a perturbation expansion of the log conductivity about its nonstationary mean and they are correct to the first order in the variance of the log conductivity the expressions we derive are applicable for any space dimensionality and for arbitrary orientation of the trend in space a general methodology is outlined for derivation of these expressions for any type of spatial covariance and for demonstration explicit results are obtained for a gaussian isotropic covariance 16 refs 6 figs

numerical solutions of moment equations for flow in heterogeneous composite aquifers 1 we analyze flow in heterogeneous media composed of multiple materials whose hydraulic properties and geometries are uncertain our analysis relies on the composite media theory of winter and tartakovksy 2000 2002 which allows one to derive and solve moment equations even when the medium is highly heterogeneous we use numerical solutions of darcy flows through a representative composite medium to investigate the robustness of perturbation approximations in porous medium with total log conductivity variances as high as 20 we also investigate the relative importance of the two sources of uncertainty in composite media material properties and geometry in our examples the uncertain geometry by itself captures the main features of the mean head estimated by the full composite model even when the within material conductivities are deterministic however neglecting randomness within materials leads to head variance estimates that are qualitatively and quantitatively wrong we compare the composite media approach to approximations that replace statistically inhomogeneous conductivity fields with pseudohomogeneous random fields with deterministic trends we demonstrate that models with a deterministic trend can be expected to give a poor estimate of the statistics of head

an indirect assessment on the impact of connectivity of conductivity classes upon longitudinal asymptotic macrodispersivity solute transport takes place in heterogeneous porous formations with the log conductivity y ln k modeled as a stationary random space function of given univariate normal probability density fu

port reduction in parametrized component static condensation approximation and a posteriori error estimation summary we introduce a port interface approximation and a posteriori error bound framework for a general component based static condensation method in the context of parameter dependent linear elliptic partial differential equations the key ingredients are as follows i efficient empirical port approximation spaces the dimensions of these spaces may be chosen small to reduce the computational cost associated with formation and solution of the static condensation system and ii a computationally tractable a posteriori error bound realized through a non conforming approximation and associated conditioner the error in the global system approximation or in a scalar output quantity may be bounded relatively sharply with respect to the underlying finite element discretization our approximation and a posteriori error bound framework is of particular computational relevance for the static condensation reduced basis element scrbe method we provide several numerical examples within the scrbe context which serve to demonstrate the convergence rate of our port approximation procedure as well as the efficacy of our port reduction error bounds copyright 2013 john wiley sons ltd

modeling multiscale diffusion processes in random heterogeneous media a simple stochastic up scaling strategy is presented for incorporating the effects of micro scale variations of thermal conductivity in the thermal analysis of macro scale systems incomplete characterization of the fine scale variation necessitates posing the thermal conductivity as a stochastic multiscale random field a data driven approach coupled with a non linear dimension reduction algorithm is used to construct a low dimensional stochastic descriptor of the fine scale variation of the stochastic conductivity a representative volume element rve argument is utilized to construct a simple model for up scaling the stochastic thermal conductivity in this model the effective stochastic thermal conductivity is estimated by virtually interrogating the stochastic rv a realistic example of the proposed stochastic multiscale diffusion strategy is illustrated utilizing limited experimental data

bilevel integrated system synthesis for concurrent and distributed processing a new version is introduced of the bilevel integrated system synthesis method intended for optimization of engineering systems conducted by distributed specialty groups working concurrently in a multiprocessor computing environment the method decomposes the overall optimization task into subtasks associated with disciplines or subsystems where the local design variables are numerous and a single system level optimization whose design variables are relatively few the subtasks are fully autonomous as to their inner operations and decision making their purpose is to eliminate the local design variables and generate a wide spectrum of feasible designs whose behavior is represented by response surfaces to be accessed by a system level optimization it is shown that if the problem is convex the solution of the decomposed problem is the same as that obtained without decomposition a simplie ed example of an aircraft design shows the method working as intended a discussion of the method merits and demerits as well as recommendations for further research is included

groundwater flow in heterogeneous composite aquifers 1 we introduce a stochastic model of flow through highly heterogeneous composite porous media that greatly improves estimates of pressure head statistics composite porous media consist of disjoint blocks of permeable materials each block comprising a single material type within a composite medium hydraulic conductivity can be represented through a pair of random processes 1 a boundary process that determines block arrangement and extent and 2 a stationary process that defines conductivity within a given block we obtain second order statistics for hydraulic conductivity in the composite model and then contrast them with statistics obtained from a standard univariate model that ignores the boundary process and treats a composite medium as if it were statistically homogeneous next we develop perturbation expansions for the first two moments of head and contrast them with expansions based on the homogeneous approximation in most cases the bivariate model leads to much sharper perturbation approximations than does the usual model of flow through an undifferentiated material when both are applied to highly heterogeneous media we make this statement precise we illustrate the composite model with examples of one dimensional flows which are interesting in their own right and which allow us to compare the accuracy of perturbation approximations of head statistics to exact analytical solutions we also show the boundary process of our bivariate model is equivalent to the indicator functions often used to represent composite media in monte carlo simulations

on polynomial collocation for second kind integral equations with fixed singularities of mellin type we consider a polynomial collocation for the numerical solution of a second kind integral equation with an integral kernel of mellin convolution type using a stability result by junghanns and one of the authors we prove that the error of the approximate solution is less than a logarithmic factor times the best approximation and using the asymptotics of the solution we derive the rates of convergence finally we describe an algorithm to compute the stiffness matrix based on simple gaus quadratures and an alternative algorithm based on a recursion in the spirit of monegato and palamara orsi all together an almost best approximation to the solution of the integral equation can be computed with n2 log n 2 resp n2 operations where n is the dimension of the polynomial trial space

uncertainty quantification via random domain decomposition and probabilistic collocation on sparse grids quantitative predictions of the behavior of many deterministic systems are uncertain due to ubiquitous heterogeneity and insufficient characterization by data we present a computational approach to quantify predictive uncertainty in complex phenomena which is modeled by partial differential equations with uncertain parameters exhibiting multi scale variability the approach is motivated by flow in random composites whose internal architecture spatial arrangement of constitutive materials and spatial variability of properties of each material are both uncertain the proposed two scale framework combines a random domain decomposition rdd and a probabilistic collocation method pcm on sparse grids to quantify these two sources of uncertainty respectively the use of sparse grid points significantly reduces the overall computational cost especially for random processes with small correlation lengths a series of one two and three dimensional computational examples demonstrate that the combined rdd pcm approach yields efficient robust and non intrusive approximations for the statistics of diffusion in random composites

reduced order modeling of time dependent pdes with multiple parameters in the boundary data abstract the computational approximation of solutions to nonlinear partial differential equations pdes such as the navier stokes equations is often a formidable task for this reason there is significant interest in the development of very low dimensional models that can be used to determine reasonably accurate approximations in simulation and control problems for pdes many concrete tests have been reported on in the literature that show that several classes of reduced order models roms are effective i e an accurate approximate solution can be obtained using very low dimensional roms most of these tests involved problems for which the solution depends on only a single parameter appearing in the boundary data the extension of the techniques used in those tests to the case of multiple parameters is not a straightforward matter here we present test and compare two methods for treating within the context of a class of roms inhomogeneous dirichlet type boundary conditions that contain multiple parameters in this study we focus on the proper orthogonal decomposition pod approach to rom however the issues and results discussed here in the pod context apply equally well to other rom approaches

algorithm refinement for stochastic partial differential equations ii correlated systems we analyze a hybrid particle continuum algorithm for a hydrodynamic system with long ranged correlations specifically we consider the so called train model for viscous transport in gases which is based on a generalization of the random walk process for the diffusion of momentum this discrete model is coupled with its continuous counterpart given by a pair of stochastic partial differential equations at the interface between the particle and continuum computations the coupling is by flux matching giving exact mass and momentum conservation this methodology is an extension of our stochastic algorithm refinement ar hybrid for simple diffusion f alexander a garcia d tartakovsky algorithm refinement for stochastic partial differential equations i linear diffusion j comput phys 182 2002 47 66 results from a variety of numerical experiments are presented for steady state scenarios in all cases the mean and variance of density and velocity are captured correctly by the stochastic hybrid algorithm for a non stochastic version i e using only deterministic continuum fluxes the long range correlations of velocity fluctuations are qualitatively preserved but at reduced magnitude

goal oriented model constrained optimization for reduction of large scale systems optimization oriented reduced order models should target a particular output functional span an applicable range of dynamic and parametric inputs and respect the underlying governing equations of the system to achieve this goal we present an approach for determining a projection basis that uses a goal oriented model constrained optimization framework the mathematical framework permits consideration of general dynamical systems with general parametric variations and is applicable to both linear and nonlinear systems results for a simple linear model problem of the two dimensional heat equation demonstrate the ability of the goal oriented approach to target a particular output functional of interest application of the methodology to a more challenging example of a subsonic blade row governed by the unsteady euler flow equations shows a significant advantage of the new method over the proper orthogonal decomposition

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

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

loop erased walks and total positivity we consider matrices whose elements enumerate weights of walks in planar directed weighted graphs not necessarily acyclic these matrices are totally nonnegative more precisely all their minors are formal power series in edge weights with nonnegative coefficients a combinatorial explanation of this phenomenon involves loop erased walks applications include total positivity of hitting matrices of brownian motion in planar domains

a bijection theorem for domino tilings with diagonal impurities we consider the dimer problem on a planar non bipartite graph g where there are two types of dimers one of which we regard as impurities computer simulations reveal a reminiscence of the cheerios effect that is impurities are attracted to the boundary which is the motivation to study this particular graph our main theorem is a variant of the temperley bijection a bijection between the set of dimer coverings and the set of spanning forests with certain conditions we further discuss some implications of this theorem 1 the local move connectedness yielding an ergodic markov chain on the set of all possible dimer coverings and 2 a rough bound for the number of dimer coverings and that for the probability of finding an impurity at a given edge which is an extension of a result in nakano and sadahiro in arxiv 0901 4824

domain decomposition of stochastic pdes theoretical formulations we present a novel theoretical framework for the domain decomposition of uncertain systems defined by stochastic partial differential equations the methodology involves a domain decomposition method in the geometric space and a functional decomposition in the probabilistic space the probabilistic decomposition is based on a version of stochastic finite elements based on orthogonal decompositions and projections of stochastic processes the spatial decomposition is achieved through a schur complement based domain decomposition the methodology aims to exploit the full potential of high performance computing platforms by reducing discretization errors with high resolution numerical model in conjunction to giving due regards to uncertainty in the system the mathematical formulation is numerically validated with an example of waves in random media copyright 2008 john wiley sons ltd

a modified nystrom method for integral equations with mellin type kernels the aim of this paper is to propose a new modified nystrom method for the approximation of the solutions of second kind integral equations with fixed singularities of mellin convolution type the stability and the convergence are proved in l 2 spaces and error estimates in l 2 norm are given finally numerical tests showing the effectiveness of the method are presented

reduced chaos expansions with random coefficients in reduced dimensional stochastic modeling of coupled problems summary we address the curse of dimensionality in methods for solving stochastic coupled problems with an emphasis on stochastic expansion methods such as those involving polynomial chaos expansions the proposed method entails a partitioned iterative solution algorithm that relies on a reduced dimensional representation of information exchanged between subproblems to allow each subproblem to be solved within its own stochastic dimension while interacting with a reduced projection of the other subproblems the proposed method extends previous work by the authors by introducing a reduced chaos expansion with random coefficients the representation of the exchanged information by using this reduced chaos expansion with random coefficients enables an expeditious construction of doubly stochastic polynomial chaos expansions that separate the effect of uncertainty local to a subproblem from the effect of statistically independent uncertainty coming from other subproblems through the coupling after laying out the theoretical framework we apply the proposed method to a multiphysics problem from nuclear engineering copyright 2013 john wiley sons ltd