multi element least square hdmr methods and their applications for stochastic multiscale model reduction stochastic multiscale modeling has become a necessary approach to quantify uncertainty and characterize multiscale phenomena for many practical problems such as flows in stochastic porous media the numerical treatment of the stochastic multiscale models can be very challengeable as the existence of complex uncertainty and multiple physical scales in the models to efficiently take care of the difficulty we construct a computational reduced model to this end we propose a multi element least square high dimensional model representation hdmr method through which the random domain is adaptively decomposed into a few subdomains and a local least square hdmr is constructed in each subdomain these local hdmrs are represented by a finite number of orthogonal basis functions defined in low dimensional random spaces the coefficients in the local hdmrs are determined using least square methods we paste all the local hdmr approximations together to form a global hdmr approximation to further reduce computational cost we present a multi element reduced least square hdmr which improves both efficiency and approximation accuracy in certain conditions to effectively treat heterogeneity properties and multiscale features in the models we integrate multiscale finite element methods with multi element least square hdmr for stochastic multiscale model reduction this approach significantly reduces the original model s complexity in both the resolution of the physical space and the high dimensional stochastic space we analyze the proposed approach and provide a set of numerical experiments to demonstrate the performance of the presented model reduction techniques multi element least square hdmr is proposed to treat stochastic models random domain is adaptively decomposed into some subdomains to obtain adaptive multi element hdmr least square reduced hdmr is proposed to enhance computation efficiency and approximation accuracy in certain conditions integrating msfem and multi element least square hdmr can significantly reduce computation complexity

multi element generalized polynomial chaos for arbitrary probability measures we develop a multi element generalized polynomial chaos me gpc method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs given a stochastic input with an arbitrary probability measure its random space is decomposed into smaller elements subsequently in each element a new random variable with respect to a conditional probability density function pdf is defined and a set of orthogonal polynomials in terms of this random variable is constructed numerically then the generalized polynomial chaos gpc method is implemented element by element numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost efficiency and convergence of me gpc are studied numerically by considering some commonly used random variables me gpc provides an efficient and flexible approach to solving differential equations with random inputs especially for problems related to long term integration large perturbation and stochastic discontinuities

a mixed multiscale finite element method for elliptic problems with oscillating coefficients the recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large scale structure of the solutions without resolving all the fine scale structures motivated by the numerical simulation of flow transport in highly heterogeneous porous media we propose a mixed multiscale finite element method with an over sampling technique for solving second order elliptic equations with rapidly oscillating coefficients the multiscale finite element bases are constructed by locally solving neumann boundary value problems we provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain the asymptotic structure of the solutions numerical experiments are carried out for flow transport in a porous medium with a random log normal relative permeability to demonstrate the efficiency and accuracy of the proposed method

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

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 multiscale reduced basis method for parametrized elliptic partial differential equations with multiple scales we present a technique for solving parametrized elliptic partial differential equations with multiple scales the technique is based on the combination of the reduced basis method c prud homme d rovas k veroy y maday a t patera g turinici reliable real time solution of parametrized partial differential equations reduced basis output bound methods journal of fluids engineering 124 1 2002 70 80 and the multiscale finite element method t y hou x h wu a multiscale finite element method for elliptic problems in composite materials and porous media journal of computational physics 134 1 1997 169 189 to treat problems in which the differential coefficient is characterized by a large number of independent parameters for the multiscale finite element method a large number of cell problems has to be solved at the fine local mesh for each new configuration of the differential coefficient in order to improve the computational efficiency of this method we construct reduced basis spaces that are adapted to the local parameter dependence of the differential operator the approximate solutions of the cell problems are computed accurately and efficiently via performing galekin projection onto the reduced basis spaces and implementing the offline online computational procedure therefore a large number of similar computations at the fine local mesh can be carried out with lower computational cost for each new configuration of the differential coefficient numerical results are provided to demonstrate the accuracy and efficiency of the proposed approach

error estimates for the anova method with polynomial chaos interpolation tensor product functions we focus on the analysis of variance anova method for high dimensional function approximation using jacobi polynomial chaos to represent the terms of the expansion first we develop a weight theory inspired by quasi monte carlo theory to identify which functions have low effective dimension using the anova expansion in different norms we then present estimates for the truncation error in the anova expansion and for the interpolation error using multielement polynomial chaos in the weighted korobov spaces over the unit hypercube we consider both the standard anova expansion using the lebesgue measure and the anchored anova expansion using the dirac measure the optimality of different sets of anchor points is also examined through numerical examples

linear systems with commensurate time delays stability and stabilization independent of delay notions of exponential stability independent of delay and stabilizability independent of delay are developed for the class of delay differential systems of the retarded type with commensurate time delays various criteria for exponential stability independent of delay with a given order are specified in terms of matrices whose entries are functions of a single real parameter and polynomials in one variable whose coefficients are functions of a single real parameter sufficient conditions and a necessary condition based on local stabilizability are given for stabilizability independent of delay using state feedback with commensurate time delays constructive methods for determining a stabilizing feedback are also presented the last part of the paper deals with a standard type of observer and regulator with the requirement that the closed loop system be stable independent of delay

new approach to constrained shape optimization using genetic algorithms a robust genetic algorithm for constrained functional optimization is described the function being sought is represented both in a piecewise linear fashion and in two different types of orthogonal series representations satisfying in each case specified end conditions of both dirichlet and neumann types the search for the optimal function is translated to one of determining the coefficients of a series expansion and a genetic algorithm is developed for this purpose the method is validated in terms of test problems for which the global optimum solutions are known the results indicate that if the population size of the chromosome pool is held constant the performance of the piecewise linear representation approach deteriorates considerably as the number of degrees of freedom increases in contrast the orthogonal series representations do not suffer from this drawback and a significant reduction in the population size can be achieved therefore the latter methodology offers a far more efficient approach to functional optimization than previously attempted the developed methodology was applied to the determination of an optimal micropump shape the genetic algorithm uncovered shapes that were nonintuitive but yielded vastly superior pump performance

least squares residuals and minimal residual methods we study krylov subspace methods for solving unsymmetric linear algebraic systems that minimize the norm of the residual at each step minimal residual mr methods mr methods are often formulated in terms of a sequence of least squares ls problems of increasing dimension we present several basic identities and bounds for the ls residual these results are interesting in the general context of solving ls problems when applied to mr methods they show that the size of the mr residual is strongly related to the conditioning of different bases of the same krylov subspace using different bases is useful in theory because relating convergence to the characteristics of different bases offers new insight into the behavior of mr methods different bases also lead to different implementations which are mathematically equivalent but can differ numerically our theoretical results are used for a finite precision analysis of implementations of the gmres method y saad and m h schultz siam j sci statist comput 7 1986 pp 856 869 we explain that the choice of the basis is fundamental for the numerical stability of the implementation as demonstrated in the case of simpler gmres h f walker and l zhou numer linear algebra appl 1 1994 pp 571 581 the best orthogonalization technique used for computing the basis does not compensate for the loss of accuracy due to an inappropriate choice of the basis in particular we prove that simpler gmres is inherently less numerically stable than the classical gmres implementation due to saad and schultz siam j sci statist comput 7 1986 pp 856 869

a stochastic mortar mixed finite element method for flow in porous media with multiple rock types this paper presents an efficient multiscale stochastic framework for uncertainty quantification in modeling of flow through porous media with multiple rock types the governing equations are based on darcy s law with nonstationary stochastic permeability represented as a sum of local karhunen loeve expansions the approximation uses stochastic collocation on either a tensor product or a sparse grid coupled with a domain decomposition algorithm known as the multiscale mortar mixed finite element method the latter method requires solving a coarse scale mortar interface problem via an iterative procedure the traditional implementation requires the solution of local fine scale linear systems on each iteration we employ a recently developed modification of this method that precomputes a multiscale flux basis to avoid the need for subdomain solves on each iteration in the stochastic setting the basis is further reused over multiple realizations leading to collocation algorithms that are more efficient than the traditional implementation by orders of magnitude error analysis and numerical experiments are presented

to be or not to be intrusive the solution of parametric and stochastic equations the plain vanilla galerkin case in parametric equations stochastic equations are a special case one may want to approximate the solution such that it is easy to evaluate its dependence on the parameters interpolation in the parameters is an obvious possibility in this context often labeled as a collocation method in the frequent situation where one has a solver for a given fixed parameter value this may be used nonintrusively as a black box component to compute the solution at all the interpolation points independently of each other by extension all other methods and especially simple galerkin methods which produce some kind of coupled system are often classed as intrusive we show how for such plain vanilla galerkin formulations one may solve the coupled system in a nonintrusive way and even the simplest form of block solver has comparable efficiency this opens at least two avenues for possible speed up first to benefit from the coupling in the iteration by using more sophisticated block solvers and second

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

stochastic computation based on orthogonal expansion of random fields abstract in solving stochastic differential equations recently a random variable based polynomial chaos rv pc method has been developed as a major numerical solver for many realistic random media problems the rv pc method however confronts a critical challenge of curse of dimensionality since a random field is represented by random variables the use of various optimal sampling techniques and conventional high dimensional methods still faces the curse of dimensionality distinguished from all the random variable based methods in this study a novel random field based orthogonal expansion rf oe method is proposed in aim to circumvent the curse of dimensionality for many physical systems whereas the input information is represented as random fields or stochastic processes e g seismic ocean wave wind load shock wave and geophysical media multiscale modeling of random media problems is selected as the benchmark problem to test the rf oe method especially the rf oe method provides a perfect matching with the higher order mehler s formula by replacing high dimensional random variable representations with a series of orthogonal expansion terms about an underlying random field process the rf oe method reduces the number of dimensions of a stochastic differential equation exponentially in the first example the rf oe method is verified with monte carlo simulation on a lognormal random media flow transport problem in the second example the rf oe method is applied to a time domain problem involving orthogonal expansion of random excitations in the conclusion the items for further development of the rf oe method are identified

on decompositions of multivariate functions we present formulas that allow us to decompose a function f of d variables into a sum of 2 d terms fu indexed by subsets u of f1 dg where each term fu depends only on the variables with indices in u the decomposition depends on the choice of d commuting projections fpjg d 1 where pj f does not depend on the variable xj we present an explicit formula for fu which is new even for the anova and anchored decompositions both are special cases of the general decomposition we show that the decomposition is minimal in the following sense if f is expressible as a sum in which there is no term that depends on all of the variables indexed by the subset z then for every choice of fpjg d 1 the terms fu 0 for all subsets u containing z furthermore in a reproducing kernel hilbert space setting we give su cient conditions for the terms fu to be mutually orthogonal

multi scale finite volume method for elliptic problems in subsurface flow simulation in this paper we present a multi scale finite volume msfv method to solve elliptic problems with many spatial scales arising from flow in porous media the method efficiently captures the effects of small scales on a coarse grid is conservative and treats tensor permeabilities correctly the underlying idea is to construct transmissibilities that capture the local properties of the differential operator this leads to a multi point discretization scheme for the finite volume solution algorithm transmissibilities for the msfv have to be constructed only once as a preprocessing step and can be computed locally therefore this step is perfectly suited for massively parallel computers furthermore a conservative fine scale velocity field can be constructed from the coarse scale pressure solution two sets of locally computed basis functions are employed the first set of basis functions captures the small scale heterogeneity of the underlying permeability field and it is computed in order to construct the effective coarse scale transmissibilities a second set of basis functions is required to construct a conservative fine scale velocity field the accuracy and efficiency of our method is demonstrated by various numerical experiments

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

stochastic analysis of unsaturated flow with probabilistic collocation method 1 in this study we present an efficient approach called the probabilistic collocation method pcm for uncertainty analysis of flow in unsaturated zones in which the constitutive relationship between the pressure head and the unsaturated conductivity is assumed to follow the van genuchten mualem model spatial variability of soil parameters leads to uncertainty in predicting flow behaviors the aim is to quantify the uncertainty associated with flow quantities such as the pressure head and the effective saturation in the proposed approach input random fields i e the soil parameters are represented via the karhunen loeve expansion and the flow quantities are expressed by polynomial chaos expansions pces the coefficients in the pces are determined by solving the equations for a set of carefully selected collocation points in the probability space to illustrate this approach we use two dimensional examples with different input variances and correlation scales and under steady state and transient conditions we also demonstrate how to deal with multiple input random parameters to validate the pcm we compare the resulting mean and variance of the flow quantities with those from monte carlo mc simulations the comparison reveals that the pcm can accurately estimate the flow statistics with a much smaller computational effort than the mc

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 multiscale stochastic finite element method on elliptic problems involving uncertainties in this study a multiscale stochastic finite element method mssfem is developed to resolve scale coupling stochastic elliptic problems based on formulations of a stochastic variational approach and scale bridging multiscale shape functions by employing polynomial chaos expansions and lagrange polynomials stochastic galerkin method and stochastic collocation method are introduced and combined with deterministic multiscale methods i e variational multiscale method and multiscale finite element method the resulting spectral and pseudo spectral stochastic finite element methods are incorporated into the fine and coarse mesh computation of the mssfem a benchmark multiscale model and the numerical experiment based on the formulated mssfem are provided for illustration it is expected that the proposed mssfem can act as a paradigm for solving of general stochastic partial differential equations involving multiscale stochastic data