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

accurate multiscale finite element method for numerical simulation of two phase flow in fractured media using discrete fracture model numerical simulation in naturally fractured media is challenging because of the coexistence of porous media and fractures on multiple scales that need to be coupled we present a new approach to reservoir simulation that gives accurate resolution of both large scale and fine scale flow patterns multiscale methods are suitable for this type of modeling because it enables capturing the large scale behavior of the solution without solving all the small features dual porosity models in view of their strength and simplicity can be mainly used for sugar cube representation of fractured media in such a representation the transfer function between the fracture and the matrix block can be readily calculated for water wet media for a mixed wet system the evaluation of the transfer function becomes complicated due to the effect of gravity in this work we use a multiscale finite element method msfem for two phase flow in fractured media using the discrete fracture model by combining msfem with the discrete fracture model we aim towards a numerical scheme that facilitates fractured reservoir simulation without upscaling msfem uses a standard darcy model to approximate the pressure and saturation on a coarse grid whereas fine scale effects are captured through basis functions constructed by solving local flow problems using the discrete fracture model the accuracy and the robustness of msfem are shown through several examples in the first example we consider several small fractures in a matrix and then compare the results solved by the finite element method then we use the msfem in more complex models the results indicate that the msfem is a promising path toward direct simulation of highly resolution geomodels

upper bounds of limit cycles in abel differential equations with invariant curves abstract new criteria are established for upper bounds on the number of limit cycles of periodic abel differential equations having two periodic invariant curves one of them bounded the criteria are applied to obtain upper bounds of either zero or one limit cycle for planar differential systems

kriging sparse polynomial dimensional decomposition surrogate model with adaptive refinement uncertainty quantification and global sensitivity analysis problems are made more difficult in the case of applications which involve expensive computer simulations this because a limited amount of simulations is available to build a sufficiently accurate metamodel of the quantities of interest in this work a numerical technique for the construction of a low cost and accurate metamodel is proposed having in mind applications with expensive computer codes two main points are intro duced firstly a technique which couples universal kriging with sparse polynomial dimensional decomposition pdd to build a metamodel with improved accuracy the polynomials selected by the adaptive pdd representation are used as a sparse basis to build an universal kriging surrogate model the second is a strategy derived from anisotropic mesh adaptation to adaptively add a fixed number of new training points to an existing design of experiments the convergence of the proposed algorithm is analyzed and assessed on different test functions with an increasing size of the input space finally the algorithm is used to propagate uncertainties in two high dimensional real problems related to the atmospheric reentry

simplex stochastic collocation method with improved scalability the simplex stochastic collocation ssc method is a robust tool used to propagate uncertain input distributions through a computer code however it becomes prohibitively expensive for problems with dimensions higher than 5 the main purpose of this paper is to identify bottlenecks and to improve upon this bad scalability in order to do so we propose an alternative interpolation stencil technique based upon the set covering problem and we integrate the ssc method in the high dimensional model reduction framework in addition we address the issue of ill conditioned sample matrices and we present an analytical map to facilitate uniformly distributed simplex sampling

a new algorithm for high dimensional uncertainty quantification based on dimension adaptive sparse grid approximation and reduced basis methods in this work we develop an adaptive and reduced computational algorithm based on dimension adaptive sparse grid approximation and reduced basis methods for solving high dimensional uncertainty quantification uq problems in order to tackle the computational challenge of curse of dimensionality commonly faced by these problems we employ a dimension adaptive tensor product algorithm 16 and propose a verified version to enable effective removal of the stagnation phenomenon besides automatically detecting the importance and interaction of different dimensions to reduce the heavy computational cost of uq problems modelled by partial differential equations pde we adopt a weighted reduced basis method 7 and develop an adaptive greedy algorithm in combination with the previous verified algorithm for efficient construction of an accurate reduced basis approximation the efficiency and accuracy of the proposed algorithm are demonstrated by several numerical experiments

a geometric criterion for equation x i 0mai t xi having at most m isolated periodic solutions abstract this paper is devoted to the investigation of generalized abel equation x s x t i 0 m a i t x i where a i c 0 1 a solution x t is called a periodic solution if x 0 x 1 in order to estimate the number of isolated periodic solutions of the equation we propose a hypothesis h which is only concerned with s x t on m straight lines there exist m real numbers 1 m such that either 1 i s i t 0 for i 1 m or 1 i s i t 0 for i 1 m by means of lagrange interpolation formula we prove that the equation has at most m isolated periodic solutions counted with multiplicities if hypothesis h holds and the upper bound is sharp furthermore this conclusion is also valid under some weaker geometric hypotheses applying our main result for the trigonometrical generalized abel equation with coefficients of degree one we give a criterion to obtain the upper bound for the number of isolated periodic solutions this criterion is almost equivalent to hypothesis h and can be much more effectively checked

on anova expansions and strategies for choosing the anchor point the classic lebesgue anova expansion offers an elegant way to represent functions that depend on a high dimensional set of parameters and it often enables a substantial reduction in the evaluation cost of such functions once the anova representation is constructed unfortunately the construction of the expansion itself is expensive due to the need to evaluate high dimensional integrals a way around this is to consider an alternative formulation known as the anchored anova expansion this formulation requires no integrals but has an accuracy that depends sensitively on the choice of a special parameter known as the anchor point we present a comparative study of several strategies for the choice of this anchor point and argue that the optimal choice of this anchor point is the center point of a sparse grid quadrature this choice induces no additional cost and as we shall show results in a natural truncation of the anova expansion the efficiency and accuracy is illustrated through several standard benchmarks and this choice is shown to outperform the alternatives over a range of applications c 2010 elsevier inc all rights reserved

chebyshev legendre super spectral viscosity method for nonlinear conservation laws in this paper a super spectral viscosity method using the chebyshev differential operator of high order d s sqrt 1 x 2 opx s is developed for nonlinear conservation laws the boundary conditions are treated by a penalty method compared with the second order spectral viscosity method the super one is much weaker while still guaranteeing the convergence of the bounded solution of the chebyshev galerkin chebyshev collocation or legendre galerkin approximations to nonlinear conservation laws which is proved by compensated compactness arguments

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

strong tractability of multivariate integration using quasi monte carlo algorithms we study quasi monte carlo algorithms based on low discrepancy sequences for multivariate integration we consider the problem of how the minimal number of function evaluations needed to reduce the worst case error from its initial error by a factor of e depends on e 1 and the dimension s strong tractability means that it does not depend on s and is bounded by a polynomial in e 1 the least possible value of the power of e 1 is called the e exponent of strong tractability sloan and wozniakowski established a necessary and sufficient condition of strong tractability in weighted sobolev spaces and showed that the e exponent of strong tractability is between 1 and 2 however their proof is not constructive in this paper we prove in a constructive way that multivariate integration in some weighted sobolev spaces is strongly tractable with e exponent equal to 1 which is the best possible value under a stronger assumption than sloan and wozniakowski s assumption we show that quasi monte carlo algorithms using niederreiter s t s sequences and sobol sequences achieve the optimal convergence order o n 1 for any 0 independent of the dimension with a worst case deterministic guarantee where n is the number of function evaluations this implies that strong tractability with the best e exponent can be achieved in appropriate weighted sobolev spaces by using niederreiter s t s sequences and sobol sequences

the multiscale restriction smoothed basis method for fractured porous media f msrsb a novel multiscale method for multiphase flow in heterogeneous fractured porous media is devised the discrete fine scale system is described using an embedded fracture modeling approach in which the heterogeneous rock matrix and highly conductive fractures are represented on independent grids given this fine scale discrete system the method first partitions the fine scale volumetric grid representing the matrix and the lower dimensional grids representing fractures into independent coarse grids then basis functions for matrix and fractures are constructed by restricted smoothing which gives a flexible and robust treatment of complex geometrical features and heterogeneous coefficients from the basis functions one constructs a prolongation operator that maps between the coarse and fine scale systems the resulting method allows for general coupling of matrix and fracture basis functions giving efficient treatment of a large variety of fracture conductivities in addition basis functions can be adaptively updated using efficient global smoothing strategies to account for multiphase flow effects the method is conservative and because it is described and implemented in algebraic form it is straightforward to employ it to both rectilinear and unstructured grids through a series of challenging test cases for single and multiphase flow in which synthetic and realistic fracture maps are combined with heterogeneous petrophysical matrix properties we validate the method and conclude that it is an efficient and accurate approach for simulating flow in complex large scale fractured media

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

numerical methods for high dimensional probability density function equations in this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables these types of equations arise naturally in many different areas of mathematical physics e g in particle systems liouville and boltzmann equations stochastic dynamical systems fokker planck and dostupov pugachev equations random wave theory malakhov saichev equations and coarse grained stochastic systems mori zwanzig equations we propose three different classes of new algorithms addressing high dimensionality the first one is based on separated series expansions resulting in a sequence of low dimensional problems that can be solved recursively and in parallel by using alternating direction methods the second class of algorithms relies on truncation of interaction in low orders that resembles the bogoliubov born green kirkwood yvon bbgky framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations the third class of algorithms is based on high dimensional model representations e g the anova method and probabilistic collocation methods a common feature of all these approaches is that they are reducible to the problem of computing the solution to high dimensional equations via a sequence of low dimensional problems the effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations with up to 120 variables

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

minimal multi element stochastic collocation for uncertainty quantification of discontinuous functions we propose a multi element stochastic collocation method that can be applied in high dimensional parameter space for functions with discontinuities lying along manifolds of general geometries the key feature of the method is that the parameter space is decomposed into multiple elements defined by the discontinuities and thus only the minimal number of elements are utilized on each of the resulting elements the function is smooth and can be approximated using high order methods with fast convergence properties the decomposition strategy is in direct contrast to the traditional multi element approaches which define the sub domains by repeated splitting of the axes in the parameter space such methods are more prone to the curse of dimensionality because of the fast growth of the number of elements caused by the axis based splitting the present method is a two step approach firstly a discontinuity detector is used to partition parameter space into disjoint elements in each of which the function is smooth the detector uses an efficient combination of the high order polynomial annihilation technique along with adaptive sparse grids and this allows resolution of general discontinuities with a smaller number of points when the discontinuity manifold is low dimensional after partitioning an adaptive technique based on the least orthogonal interpolant is used to construct a generalized polynomial chaos surrogate on each element the adaptive technique reuses all information from the partitioning and is variance suppressing we present numerous numerical examples that illustrate the accuracy efficiency and generality of the method when compared against standard locally adaptive sparse grid methods the present method uses many fewer number of collocation samples and is more accurate

an adaptive multi element generalized polynomial chaos method for stochastic differential equations we formulate a multi element generalized polynomial chaos me gpc method to deal with long term integration and discontinuities in stochastic differential equations we first present this method for legendre chaos corresponding to uniform random inputs and subsequently we generalize it to other random inputs the main idea of me gpc is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value in each subdomain or random element we then employ a generalized polynomial chaos expansion we develop a criterion to perform such a decomposition adaptively and demonstrate its effectiveness for odes including the kraichnan orszag three mode problem as well as advection diffusion problems the new method is similar to spectral element method for deterministic problems but with h p discretization of the random space

abel like differential equations with no periodic solutions we present various criteria for the non existence of positive periodic solutions of generalized abel differential equations with periodic coefficients that can change sign as an application we obtain some families of planar vector fields without limit cycles

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

periodic solutions for equation x a t x m b t x n c t x l with a t and b t changing signs in this paper we investigate the differential equation x s x t a t xm b t xn c t xl where a b c c 0 1 m n l and m n l z a solution x t with x 1 x 0 is called a periodic solution under some hypotheses which admit a t and b t without fixed sign we obtain the upper bound sometimes sharp for the number of isolated periodic solutions of the equation applying these results for the abel equation i e m 3 n 2 l 1 we get that if there exists 0 such that s t c t 0 resp s t a t b t 0 then the equation has at most 2 resp 4 non zero isolated periodic solutions furthermore suppose that a t t is a smooth curve which lies in r 0 0 1 with a 0 a 1 we obtain that if vector fields s x t 1 resp and c t x 1 are transverse to resp s x t 1 on in opposite directions then the number of non zero isolated periodic solutions of this abel equation is still no more than 2 resp 4 these conclusions generalize the known criteria about the abel equation which only refer to the cases with either a t or b t keeping sign finally as an application we study a kind of trigonometrical abel equation

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

a generalized dimension reduction method for multidimensional integration in stochastic mechanics summary a new generalized multivariate dimension reduction method is presented for calculating statistical moments of the response of mechanical systems subject to uncertainties in loads material properties and geometry the method involves an additive decomposition of an n dimensional response function into at most s dimensional functions where s n an approximation of response moments by moments of input random variables and a moment based quadrature rule for numerical integration a new theorem is presented which provides a convenient means to represent the taylor series up to a specific dimension without involving any partial derivatives a complete proof of the theorem is given using two lemmas also proved in this paper the proposed method requires neither the calculation of partial derivatives of response as in commonly used taylor expansion perturbation methods nor the inversion of random matrices as in the neumann expansion method eight numerical examples involving elementary mathematical functions and solid mechanics problems illustrate the proposed method results indicate that the multivariate dimension reduction method generates convergent solutions and provides more accurate estimates of statistical moments or multidimensional integration than existing methods such as first and second order taylor expansion methods statistically equivalent solutions quasi monte carlo simulation and the fully symmetric interpolatory rule while the accuracy of the dimension reduction method is comparable to that of the fourth order neumann expansion method a comparison of cpu time suggests that the former is computationally far more efficient than the latter copyright 2004 john wiley sons ltd

convergence of a nonconforming multiscale finite element method the multiscale finite element method msfem t y hou x h wu and z cai math comp 1998 to appear t y hou and x h wu j comput phys 134 1997 pp 169 189 has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients this is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator our previous study reveals that the leading order error in this approach is caused by the resonant sampling which leads to large error when the mesh size is close to the small scale of the continuous problem similar difficulty also arises in numerical upscaling methods an oversampling technique has been introduced to alleviate this difficulty t y hou and x h wu j comput phys 134 1997 pp 169 189 a consequence of the oversampling method is that the resulting finite element method is no longer conforming here we give a detailed analysis of the nonconforming error our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale we show that the cell resonance error is of lower order our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice

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

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

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

generalized multiscale finite element methods gmsfem in this paper we propose a general approach called generalized multiscale finite element method gmsfem for performing multiscale simulations for problems without scale separation over a complex input space as in multiscale finite element methods msfems the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space in the proposed approach we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage the enrichment in the online stage is performed based on a spectral decomposition of the offline space in the online stage for any input parameter a multiscale space is constructed to solve the global problem on a coarse grid the online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues the computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re used for solving the forward problem with any forcing and boundary condition compared with the other approaches where global snapshots are used the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse grid level we present various examples in the paper and some numerical results to demonstrate the effectiveness of our method

stochastic collocation and mixed finite elements for flow in porous media the aim of this paper is to quantify uncertainty of flow in porous media through stochastic modeling and computation of statistical moments the governing equations are based on darcy s law with stochastic permeability starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion mixed finite element approximations are used in the spatial domain and collocation at the zeros of tensor product hermite polynomials is used in the stochastic dimensions error analysis is performed and experimentally verified with numerical simulations computational results include incompressible and slightly compressible single and two phase flow

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

locally conservative galerkin and finite volume methods for two phase flow in porous media in this paper we propose a locally conservative galerkin lcg finite element method for two phase flow simulations in heterogeneous porous media the main idea of the method is to use the property of local conservation at steady state conditions in order to define a numerical flux at element boundaries it provides a way to apply standard galerkin finite element method in two phase flow simulations in porous media the lcg method has all the advantages of the standard finite element method while explicitly conserving fluxes over each element all the examples presented show that the formulation employed is accurate and robust while using less cpu time than finite volume method

a new chebyshev family with applications to abel equations we prove that a family of functions defined through some definite integrals forms an extended complete chebyshev system the key point of our proof consists of reducing the study of certain wronskians to the gram determinants of a suitable set of new functions our result is then applied to give upper bounds for the number of isolated periodic solutions of some perturbed abel equations

adaptive surrogate modeling by anova and sparse polynomial dimensional decomposition for global sensitivity analysis in fluid simulation the polynomial dimensional decomposition pdd is employed in this work for the global sensitivity analysis and uncertainty quantification uq of stochastic systems subject to a moderate to large number of input random variables due to the intimate connection between the pdd and the analysis of variance anova approaches pdd is able to provide a simpler and more direct evaluation of the sobol sensitivity indices when compared to the polynomial chaos expansion pc unfortunately the number of pdd terms grows exponentially with respect to the size of the input random vector which makes the computational cost of standard methods unaffordable for real engineering applications in order to address the problem of the curse of dimensionality this work proposes essentially variance based adaptive strategies aiming to build a cheap meta model i e surrogate model by employing the sparse pdd approach with its coefficients computed by regression three levels of adaptivity are carried out in this paper 1 the truncated dimensionality for anova component functions 2 the active dimension technique especially for second and higher order parameter interactions and 3 the stepwise regression approach designed to retain only the most influential polynomials in the pdd expansion during this adaptive procedure featuring stepwise regressions the surrogate model representation keeps containing few terms so that the cost to resolve repeatedly the linear systems of the least squares regression problem is negligible the size of the finally obtained sparse pdd representation is much smaller than the one of the full expansion since only significant terms are eventually retained consequently a much smaller number of calls to the deterministic model is required to compute the final pdd coefficients

the multi element probabilistic collocation method me pcm error analysis and applications stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters in this work we present and examine the multi element probabilistic collocation method me pcm which is a generalized form of the probabilistic collocation method in the me pcm the parametric space is discretized and a collocation cubature grid is prescribed on each element both full and sparse tensor product grids based on gauss and clenshaw curtis quadrature rules are considered we prove analytically and observe in numerical tests that as the parameter space mesh is refined the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness in addition the l 2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided numerical examples demonstrating adaptive me pcm are shown including low regularity problems and long time integration we test the me pcm on two dimensional navier stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions while the convergence rate of me pcm deteriorates in 50 dimensions the error in the mean and variance is two orders of magnitude lower than the error obtained with the monte carlo method using only a small number of samples e g 100 the computational cost of me pcm is found to be favorable when compared to the cost of other methods including stochastic galerkin monte carlo and quasi random sequence methods

iterative approximation of limit cycles for a class of abel equations this article considers the analytical approximation of limit cycles that may appear in abel equations written in the normal form the procedure uses an iterative approach that takes advantage of the contraction mapping theorem thus the obtained sequence exhibits uniform convergence to the target periodic solution the effectiveness of the technique is illustrated through the approximation of an unstable limit cycle that appears in an abel equation arising in a tracking control problem that affects an elementary second order bilinear power converter

a non linear dimension reduction methodology for generating data driven stochastic input models stochastic analysis of random heterogeneous media polycrystalline materials porous media functionally graded materials provides information of significance only if realistic input models of the topology and property variations are used this paper proposes a framework to construct such input stochastic models for the topology and thermal diffusivity variations in heterogeneous media using a data driven strategy given a set of microstructure realizations input samples generated from given statistical information about the medium topology the framework constructs a reduced order stochastic representation of the thermal diffusivity this problem of constructing a low dimensional stochastic representation of property variations is analogous to the problem of manifold learning and parametric fitting of hyper surfaces encountered in image processing and psychology denote by m the set of microstructures that satisfy the given experimental statistics a non linear dimension reduction strategy is utilized to map m to a low dimensional region a we first show that m is a compact manifold embedded in a high dimensional input space r n an isometric mapping f from m to a low dimensional compact connected set email protected r d email protected n is constructed given only a finite set of samples of the data the methodology uses arguments from graph theory and differential geometry to construct the isometric transformation f m a asymptotic convergence of the representation of m by a is shown this mapping f serves as an accurate low dimensional data driven representation of the property variations the reduced order model of the material topology and thermal diffusivity variations is subsequently used as an input in the solution of stochastic partial differential equations that describe the evolution of dependant variables a sparse grid collocation strategy smolyak algorithm is utilized to solve these stochastic equations efficiently we showcase the methodology by constructing low dimensional input stochastic models to represent thermal diffusivity in two phase microstructures this model is used in analyzing the effect of topological variations of two phase microstructures on the evolution of temperature in heat conduction processes

an adaptive least squares global sensitivity method and application to a plasma coupled combustion prediction with parametric correlation abstract we introduce an efficient non intrusive surrogate based methodology for global sensitivity analysis and uncertainty quantification modified covariance based sensitivity indices mcov si are defined for outputs that reflect correlated effects the overall approach is applied to simulations of a complex plasma coupled combustion system with disparate uncertain parameters in sub models for chemical kinetics and a laser induced breakdown ignition seed the surrogate is based on an analysis of variance anova expansion such as widely used in statistics with orthogonal polynomials representing the anova subspaces and a polynomial dimensional decomposition pdd representing its multi dimensional components the coefficients of the pdd expansion are obtained using a least squares regression which both avoids the direct computation of high dimensional integrals and affords an attractive flexibility in choosing sampling points this facilitates importance sampling using a bayesian calibrated posterior distribution which is fast and thus particularly advantageous in common practical cases such as our large scale demonstration for which the asymptotic convergence properties of polynomial expansions cannot be realized due to computation expense effort instead is focused on efficient finite resolution sampling standard covariance based sensitivity indices cov si are employed to account for correlation of the uncertain parameters magnitude of cov si is unfortunately unbounded which can produce extremely large indices that limit their utility alternatively mcov si are then proposed in order to bound this magnitude 0 1 the polynomial expansion is coupled with an adaptive anova strategy to provide an accurate surrogate as the union of several low dimensional spaces avoiding the typical computational cost of a high dimensional expansion it is also adaptively simplified according to the relative contribution of the different polynomials to the total variance the approach is demonstrated for a laser induced turbulent combustion simulation model which includes parameters with correlated effects

cubic systems and abel equations abstract we investigate cubic systems which can be transformed to an equation of abel form the conditions for the origin to be a centre and in particular an isochronous centre are obtained the maximum number of limit cycles which can bifurcate from a fine focus is determined and some information is obtained about the global phase portrait

a locally conservative multiscale finite element method for multiphase flow simulation through heterogeneous and fractured porous media abstract a multiscale locally conservative galerkin mslcg method is proposed to accurately simulate multiphase flow in heterogeneous and fractured porous media mslcg employs a coarse partition of the fine grids and multiscale basis function for mapping the fine scale information to the coarse scale unknowns different from standard multiscale finite element method msfem the main improvement of our mslcg is to use the property of local conservation at steady state conditions to define a numerical flux at element boundaries mslcg provides a way to extend standard msfem to handle challenging multiphase flow problems in heterogeneous and fractured porous media mslcg preserves all the advantages of the standard msfem while it provides explicitly conservative fluxes through each element we present a number of representative numerical examples to demonstrate that our method is efficient and accurate for simulating multiphase flow through heterogeneous and fractured porous media

a multigrid method for adaptive sparse grids sparse grids have become an important tool to reduce the number of degrees of freedom of discretizations of moderately high dimensional partial differential equations however the reduction in degrees of freedom comes at the cost of an almost dense and unconventionally structured system of linear equations to guarantee overall efficiency of the sparse grid approach special linear solvers are required we present a multigrid method that exploits the sparse grid structure to achieve an optimal runtime that scales linearly with the number of sparse grid points our approach is based on a novel decomposition of the right hand sides of the coarse grid equations that leads to a reformulation in so called auxiliary coefficients with these auxiliary coefficients the right hand sides can be represented in a nodal point basis on low dimensional full grids our proposed multigrid method directly operates in this auxiliary coefficient representation circumventing most of the computationally cumbersome sparse gr

anchored anova petrov galerkin projection schemes for parabolic stochastic partial differential equations abstract we present a numerical scheme based on the combination of hoeffding s functional analysis of variance anova decomposition with stochastic galerkin projection for solving a class of high dimensional parabolic stochastic partial differential equations spdes the central idea underpinning the proposed approach is to approximate the spde solution using a hoeffding anova decomposition whose component functions are constrained to be orthogonal with respect to an appropriate measure we show that when a dirac product measure is chosen to enforce the orthogonality constraints and the test functions are chosen appropriately the original stochastic weak formulation can be decoupled into low dimensional subproblems that can be efficiently solved in parallel using polynomial chaos based stochastic projection schemes as a result the proposed approach scales very well to spdes with large number of random variables we theoretically analyze the proposed formulation and provide a priori error estimates as a function of the spatial stochastic and temporal discretization parameters and the anova expansion order that hold under appropriate stochastic regularity assumptions numerical studies are presented for a set of time dependent stochastic diffusion problems with up to 50 random variables to demonstrate the effectiveness of the proposed approach and comparisons are made against classical polynomial chaos stochastic galerkin projection scheme gpc and the generalized spectral decomposition scheme these studies show that the proposed approach provides accuracy comparable to classical gpc galerkin projection schemes while incurring significantly lower computational cost

anchored analysis of variance petrov galerkin projection schemes for linear stochastic structural dynamics in this paper we propose anchored functional analysis of variance petrovgalerkin aapg projection schemes originally developed in the context of parabolic stochastic partial differential equatio

sequential experimental design based generalised anova over the last decade surrogate modelling technique has gained wide popularity in the field of uncertainty quantification optimization model exploration and sensitivity analysis this approach relies on experimental design to generate training points and regression interpolation for generating the surrogate in this work it is argued that conventional experimental design may render a surrogate model inefficient in order to address this issue this paper presents a novel distribution adaptive sequential experimental design da sed the proposed da sed has been coupled with a variant of generalised analysis of variance g anova developed by representing the component function using the generalised polynomial chaos expansion moreover generalised analytical expressions for calculating the first two statistical moments of the response which are utilized in predicting the probability of failure have also been developed the proposed approach has been utilized in predicting probability of failure of three structural mechanics problems it is observed that the proposed approach yields accurate and computationally efficient estimate of the failure probability

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

a new uniqueness criterion for the number of periodic orbits of abel equations abstract a solution of the abel equation x a t x 3 b t x 2 such that x 0 x 1 is called a periodic orbit of the equation our main result proves that if there exist two real numbers a and b such that the function a a t b b t is not identically zero and does not change sign in 0 1 then the abel differential equation has at most one non zero periodic orbit furthermore when this periodic orbit exists it is hyperbolic this result extends the known criteria about the abel equation that only refer to the cases where either a t 0 or b t 0 does not change sign we apply this new criterion to study the number of periodic solutions of two simple cases of abel equations the one where the functions a t and b t are 1 periodic trigonometric polynomials of degree one and the case where these two functions are polynomials with three monomials finally we give an upper bound for the number of isolated periodic orbits of the general abel equation x a t x 3 b t x 2 c t x when a t b t and c t satisfy adequate conditions

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

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

sparse high order fem for elliptic spdes abstract we describe the analysis and the implementation of two finite element fe algorithms for the deterministic numerical solution of elliptic boundary value problems with stochastic coefficients they are based on separation of deterministic and stochastic parts of the input data by a karhunen loeve expansion truncated after m terms with a change of measure we convert the problem to a sequence of m dimensional parametric deterministic problems two sparse high order polynomial approximations of the random solution s joint pdf s parametrized in the input data s karhunen loeve expansion coordinates are analyzed a sparse stochastic galerkin fem sparse sgfem and a sparse stochastic collocation fem sparse scfem a priori and a posteriori error analysis is used to tailor the sparse polynomial approximations of the random solution s joint pdf s to the stochastic regularity of the input data scfem and sgfem yield deterministic approximations of the random solutions joint pdf s that converge spectrally in the number of deterministic problems to be solved numerical examples with random inputs of small correlation length in diffusion problems are presented high order gpc approximations of solutions with stochastic parameter spaces of dimension up to m 80 are computed on workstations

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

existence of non trivial limit cycles in abel equations with symmetries abstract we study the periodic solutions of the generalized abel equation x a 1 a 1 t x n 1 a 2 a 2 t x n 2 a 3 a 3 t x n 3 where n 1 n 2 n 3 1 are distinct integers a 1 a 2 a 3 r and a 1 a 2 a 3 are 2 periodic analytic functions such that a 1 t sin t a 2 t cos t a 3 t sin t cos t are periodic positive even functions when n 3 n 1 n 3 n 2 0 we prove that the equation has no non trivial different from zero limit cycle for any value of the parameters a 1 a 2 a 3 when n 3 n 1 n 3 n 2 0 we obtain under additional conditions the existence of non trivial limit cycles in particular we obtain limit cycles not detected by abelian integrals

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

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

locally conservative stabilized finite element methods for variably saturated flow standard galerkin finite element methods for variably saturated groundwater flow have several deficiencies for instance local oscillations can appear around sharp infiltration fronts without the use of mass lumping and velocity fields obtained from differentiation of pressure fields are discontinuous at element boundaries here we consider conforming finite element discretizations based on a multiscale formulation along with recently developed local postprocessing schemes the resulting approach maintains the basic flexibility and appeal of traditional finite element methods while controlling nonphysical oscillations and producing element wise mass conservative velocity fields accuracy and efficiency of the proposed schemes are evaluated through a series of steady state and transient variably saturated groundwater flow problems in homogeneous as well as heterogeneous domains the schemes are formulated for a generic nonlinear advection diffusion equation and are thus applicable to many other flow models

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

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

adaptive multiscale finite volume method for nonlinear multiphase transport in heterogeneous formations in the previous multiscale finite volume msfv method an efficient and accurate multiscale approach was proposed to solve the elliptic flow equation the reconstructed fine scale velocity field was then used to solve the nonlinear hyperbolic transport equation for the fine scale saturations using an overlapping schwarz scheme a coarse scale system for the transport equations was not derived because of the hyperbolic character of the governing equations and intricate nonlinear interactions between the saturation field and the underlying heterogeneous permeability distribution in this paper we describe a sequential implicit multiscale finite volume framework for coupled flow and transport with general prolongation and restriction operations for both pressure and saturation in which three adaptive prolongation operators for the saturation are used in regions with rapid pressure and saturation changes the original approach with full reconstruction of the velocity field and overlapping schwarz is used to compute the saturations in regions where the temporal changes in velocity or saturation can be represented by asymptotic linear approximations two additional approximate prolongation operators are proposed the efficiency and accuracy are evaluated for two phase incompressible flow in two and three dimensional domains the new adaptive algorithm is tested using various models with homogeneous and heterogeneous permeabilities it is demonstrated that the multiscale results with the adaptive transport calculation are in excellent agreement with the fine scale solutions furthermore the adaptive multiscale scheme of flow and transport is much more computationally efficient compared with the previous msfv method and conventional fine scale reservoir simulation methods

control volume method for numerical simulation of two phase immiscible flow in two and three dimensional discrete fractured media 1 we provide a numerical procedure for the simulation of two phase immiscible and incompressible flow in two and three dimensional discrete fractured media the concept of cross flow equilibrium is used to reduce the fracture dimension from n to n 1 in the calculation of flow in the fractures this concept which is often referred to as the discrete fracture model has a significant effect on the reduction of computational time the spatial discretization is performed with the control volume method this method is locally conservative and allows the use of unstructured grids to represent complex geometries such as discrete fracture configurations the relative permeability is upwinded with a criterion based on the evaluation of the flux direction at the boundaries of the control volumes which is consistent with the physics of fluid flow the system of partial differential equations is decoupled and solved using the implicit pressure explicit saturation impes approach the algorithm has been successfully tested in two and three dimensional numerical simulations of wetting phase fluid injection such as water in discrete fractured media saturated by a nonwetting phase such as nonaqueous phase liquid or oil with mild to high nonlinearity in relative permeability and capillary pressure to the best of our knowledge results for simulations of two phase immiscible and incompressible flow in three dimensional discrete fractured media including capillary and gravity effects are the first to appear in the literature

multi element probabilistic collocation method in high dimensions we combine multi element polynomial chaos with analysis of variance anova functional decomposition to enhance the convergence rate of polynomial chaos in high dimensions and in problems with low stochastic regularity specifically we employ the multi element probabilistic collocation method mepcm 1 and so we refer to the new method as mepcm a we investigate the dependence of the convergence of mepcm a on two decomposition parameters the polynomial order m and the effective dimension n with n n and n the nominal dimension numerical tests for multi dimensional integration and for stochastic elliptic problems suggest that n m for monotonic convergence of the method we also employ mepcm a to obtain error bars for the piezometric head at the hanford nuclear waste site under stochastic hydraulic conductivity conditions finally we compare the cost of mepcm a against monte carlo in several hundred dimensions and we find mepcm a to be more efficient for up to 600 dimensions for a specific multi dimensional integration problem involving a discontinuous function

a stochastic collocation method for elliptic partial differential equations with random input data this work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms these input data are assumed to depend on a finite number of random variables the method consists of a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it treats easily a wide range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction of the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method finally we include a section with developments posterior to the original publication of this work there we review sparse grid stochastic collocation methods which are effective collocation strategies for problems that depend on a moderately large number of random variables

periodic orbits in complex abel equations abstract this paper is devoted to prove two unexpected properties of the abel equation d z d t z 3 b t z 2 c t z where b and c are smooth 2 periodic complex valuated functions t r and z c the first one is that there is no upper bound for its number of isolated 2 periodic solutions in contrast recall that if the functions b and c are real valuated then the number of complex 2 periodic solutions is at most three the second property is that there are examples of the above equation with b and c being low degree trigonometric polynomials such that the center variety is formed by infinitely many connected components in the space of coefficients of b and c this result is also in contrast with the characterization of the center variety for the examples of abel equations d z d t a t z 3 b t z 2 studied in the literature where the center variety is located in a finite number of connected components

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

a probabilistic graphical model approach to stochastic multiscale partial differential equations we develop a probabilistic graphical model based methodology to efficiently perform uncertainty quantification in the presence of both stochastic input and multiple scales both the stochastic input and model responses are treated as random variables in this framework their relationships are modeled by graphical models which give explicit factorization of a high dimensional joint probability distribution the hyperparameters in the probabilistic model are learned using sequential monte carlo smc method which is superior to standard markov chain monte carlo mcmc methods for multi modal distributions finally we make predictions from the probabilistic graphical model using the belief propagation algorithm numerical examples are presented to show the accuracy and efficiency of the predictive capability of the developed graphical model

wick malliavin approximation to nonlinear stochastic partial differential equations analysis and simulations approximating nonlinearities in stochastic partial differential equations spdes via the wick product has often been used in quantum field theory and stochastic analysis the main benefit is simplification of the equations but at the expense of introducing modelling errors in this paper we study the accuracy and computational efficiency of wick type approximations to spdes and demonstrate that the wick propagator i e the system of equations for the coefficients of the polynomial chaos expansion of the solution has a sparse lower triangular structure that is seemingly universal i e independent of the type of noise we also introduce new higher order stochastic approximations via wick malliavin series expansions for gaussian and uniformly distributed noises and demonstrate convergence as the number of expansion terms increases our results are for diffusion burgers and navier stokes equations but the same approach can be readily adopted for other nonlinear spdes and more general noises

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

limit cycles of abel equations of the first kind abstract consider the scalar differential equation x i 0 m a i t x n i where a i t are t periodic analytic functions and 1 n i n for any polynomial q x x n 0 i 1 m i x n i the equation can be written as x a 0 q x r t x let w be the wronskian of q and r with respect to x and q w the previous polynomials after removing multiplicity of roots and solutions of the differential equation we prove that if the vector field defined by the differential equation is transversal at every point of q x 0 or w t x 0 then the number of limit cycles isolated periodic solutions in the set of periodic solutions of the differential equation is at most 3 n 1

sensitivity analysis using anchored anova expansion and high order moments computation an anchored analysis of variance anova method is proposed in this paper to decompose the statistical moments compared to the standard anova with mutually orthogonal component functions the anchored anova with an arbitrary choice of the anchor point loses the orthogonality if employing the same measure however an advantage of the anchored anova consists in the considerably reduced number of deterministic solver s computations which renders the uncertainty quantification of real engineering problems much easier different from existing methods the covariance decomposition of the output variance is used in this work to take account of the interactions between non orthogonal components yielding an exact variance expansion and thus with a suitable numerical integration method provides a strategy that converges this convergence is verified by studying academic tests in particular the sensitivity problem of existing methods to the choice of anchor point is analyzed via the ishigami case and we point out that covariance decomposition survives from this issue also with a truncated anchored anova expansion numerical results prove that the proposed approach is less sensitive to the anchor point the covariance based sensitivity indices si are also used compared to the variance based si furthermore we emphasize that the covariance decomposition can be generalized in a straightforward way to decompose higher order moments for academic problems results show the method converges to exact solution regarding both the skewness and kurtosis finally the proposed method is applied on a realistic case that is estimating the chemical reactions uncertainties in a hypersonic flow around a space vehicle during an atmospheric reentry

a stochastic variational multiscale method for diffusion in heterogeneous random media a stochastic variational multiscale method with explicit subgrid modelling is provided for numerical solution of stochastic elliptic equations that arise while modelling diffusion in heterogeneous random media the exact solution of the governing equations is split into two components a coarse scale solution that can be captured on a coarse mesh and a subgrid solution a localized computational model for the subgrid solution is derived for a generalized trapezoidal time integration rule for the coarse scale solution the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the subgrid model the generalized polynomial chaos method combined with the finite element technique is used for the solution of equations resulting from the coarse formulation and subgrid models finally various numerical examples are considered for evaluating the method

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

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

locally conservative fluxes for the continuous galerkin method the standard continuous galerkin cg finite element method for second order elliptic problems suffers from its inability to provide conservative flux approximations a much needed quantity in many applications we show how to overcome this shortcoming by using a two step postprocessing the first step is the computation of a numerical flux trace defined on element interfaces and is motivated by the structure of the numerical traces of discontinuous galerkin methods this computation is nonlocal in that it requires the solution of a symmetric positive definite system but the system is well conditioned independently of mesh size so it can be solved at asymptotically optimal cost the second step is a local element by element postprocessing of the cg solution incorporating the result of the first step this leads to a conservative flux approximation with continuous normal components this postprocessing applies for the cg method in its standard form or for a hybridized version of it we present the hybridized version since it allows easy handling of variable degree polynomials and hanging nodes furthermore we provide an a priori analysis of the error in the postprocessed flux approximation and display numerical evidence suggesting that the approximation is competitive with the approximation provided by the raviart thomas mixed method of corresponding degree

adaptive leja sparse grid constructions for stochastic collocation and high dimensional approximation we propose an adaptive sparse grid stochastic collocation approach based upon leja interpolation sequences for approximation of parameterized functions with high dimensional parameters leja sequences are arbitrarily granular any number of nodes may be added to a current sequence producing a new sequence and thus are a good choice for the univariate composite rule used to construct adaptive sparse grids in high dimensions when undertaking stochastic collocation one is often interested in constructing weighted approximation where the weights are determined by the probability densities of the random variables this paper establishes that a certain weighted formulation of one dimensional leja sequences produces a sequence of nodes whose empirical distribution converges to the corresponding limiting distribution of the gauss quadrature nodes associated with the weight function this property is true even for unbounded domains we apply the leja sparse grid approach to several high dimensional problems and

uncertainty quantification for complex systems with very high dimensional response using grassmann manifold variations abstract this paper addresses uncertainty quantification uq for problems where scalar or low dimensional vector response quantities are insufficient and instead full field very high dimensional responses are of interest to do so an adaptive stochastic simulation based methodology is introduced that refines the probability space based on grassmann manifold variations the proposed method has a multi element character discretizing the probability space into simplex elements using a delaunay triangulation for every simplex the high dimensional solutions corresponding to its vertices sample points are projected onto the grassmann manifold the pairwise distances between these points are calculated using appropriately defined metrics and the elements with large total distance are sub sampled and refined as a result regions of the probability space that produce significant changes in the full field solution are accurately resolved an added benefit is that an approximation of the solution within each element can be obtained by interpolation on the grassmann manifold the method is applied to study the probability of shear band formation in a bulk metallic glass using the shear transformation zone theory

reweighted l1 minimization method for stochastic elliptic differential equations abstract we consider elliptic stochastic partial differential equations spdes with random coefficients and solve them by expanding the solution using generalized polynomial chaos gpc under some mild conditions on the coefficients the solution is sparse in the random space i e only a small number of gpc basis makes considerable contribution to the solution to exploit this sparsity we employ reweighted l 1 minimization to recover the coefficients of the gpc expansion we also combine this method with random sampling points based on the chebyshev probability measure to further increase the accuracy of the recovery of the gpc coefficients we first present a one dimensional test to demonstrate the main idea and then we consider 14 and 40 dimensional elliptic spdes to demonstrate the significant improvement of this method over the standard l 1 minimization method for moderately high dimensional 10 problems the combination of chebyshev measure with reweighted l 1 minimization performs well while for higher dimensional problems reweighted l 1 only is sufficient the proposed approach is especially suitable for problems for which the deterministic solver is very expensive since it reuses the sampling results and exploits all the information available from limited sources