on the essential spectrum of the transfer operator for expanding markov maps the essential spectrum of the transfer operator for expanding markov maps of the interval is studied in detail to this end we construct explicityly an infinite set of eigenfunctions which allows us to prove that the essential spectrum inc k is a disk whose radius is related to the free energy of the liapunov exponent

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

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

spectral analysis and an area preserving extension of a piecewise linear intermittent map we investigate the spectral properties of a one dimensional piecewise linear intermittent map which has not only a marginal fixed point but also a singular structure suppressing injections of the orbits into neighborhoods of the marginal fixed point we explicitly derive generalized eigenvalues and eigenfunctions of the frobenius perron operator of the map for classes of observables and piecewise constant initial densities and it is found that the frobenius perron operator has two simple real eigenvalues 1 and ensuremath lambda d ensuremath 1 0 and a continuous spectrum on the real line 0 1 from these spectral properties we also found that this system exhibits a power law decay of correlations this analytical result is found to be in a good agreement with numerical simulations moreover the system can be extended to an area preserving invertible map defined on the unit square this extended system is similar to the baker transformation but does not satisfy hyperbolicity a relation between this area preserving map and a billiard system is also discussed

symmetric permutations for i matrices to delay and avoid small pivots during factorization in this article we present several new permutations for i matrices making these more suitable for incomplete lu factorization preconditioners used in solving linear systems by iterative methods a general matrix can be transformed by row permutation as well as row and columns scaling into an i matrix i e a matrix having elements of modulus 1 on the diagonal and elements of modulus of no more than 1 elsewhere reordering rows and columns by the same permutation clearly preserves i matrices in this article we consider such reordering techniques which make the permuted matrix more suitable for an incomplete lu factorization preconditioner than the original i matrix we use a multilevel iluc an incomplete lu factorization preconditioner using crout s implementation of gaussian elimination without pivoting to test these reorderings the combination of i matrix preprocessing with the various algorithms presented here and the multilevel incomplete lu factorizations forms a powerful preconditioning method for unsymmetric highly indefinite problems the c code has been made available in the software package ilu

a new pivoting strategy for gaussian elimination this paper discusses a method for determining a good pivoting sequence for gaussian elimination based on an algorithm for solving assignment problems the worst case complexity is o n3 in practice o n2 25 operations are sufficient

scale invariant motion in intermittent chaotic systems we investigate the dynamics generated from iterated maps and analyze the motion in terms of the probabilistic continuous time random walk ctrw approach two different ctrw models are considered i particles jump between sites turning points or ii particles move at a constant velocity between sites and choose a new direction at random for both models we study the mean squared displacement mathit r 2 t and the propagator p r t the probability to be at location r at time t having started at the origin at t 0 iterated maps are used to generate both dispersive and enhanced diffusion and the results are analyzed using the ctrw framework and scaling arguments for the case of dispersive motion we discuss the problem of the stationary state and point out its relevance

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

a robust and efficient ilu that incorporates the growth of the inverse triangular factors in this paper we present a new ilu decomposition which is based on an existing sparse direct solver in contrast to many incomplete lu decompositions this ilu incorporates information about the inverse factors l 1 and u 1 which have direct influence on the dropping strategy we demonstrate in several large scale examples that this implementation constructs a robust preconditioner

addressing the curse of dimensionality in stochastic dynamics a wiener path integral variational formulation with free boundaries a wiener path integral variational formulation with free boundaries is developed for determining the stochastic response of high dimensional nonlinear dynamical systems in a computationally efficie

preconditioning highly indefinite and nonsymmetric matrices standard preconditioners like incomplete factorizations perform well when the coefficient matrix is diagonally dominant but often fail on general sparse matrices we experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices the permutations and scalings are those developed by olschowka and neumaier linear algebra appl 240 1996 pp 131 151 and by duff and koster siam j matrix anal appl 20 1999 pp 889 901 tech report ral tr 99 030 rutherford appleton laboratory chilton uk 1999 we target highly indefinite nonsymmetric problems that cause difficulties for preconditioned iterative solvers our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing

on strong anomalous diffusion superdiffusive behavior i e x2 t t2 with 1 2 is in general not completely characterized by a unique exponent we study some systems exhibiting strong anomalous diffusion i e x t q tq q where 2 1 2 and q q is not a linear function of q this feature is different from the weak superdiffusive regime i e q const 1 2 occurring in random shear flows strong anomalous diffusion can be generated by nontrivial chaotic dynamics e g lagrangian motion in 2d time dependent incompressible velocity fields 2d symplectic maps and 1d intermittent maps typically the function q q is piecewise linear this is due to two mechanisms a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions in order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem this happens only in a very narrow region of the control parameters space in the presence of strong anomalous diffusion one does not have a unique exponent and therefore one has the failure of the usual scaling p x t t f x t of the probability density this implies that the effective equation at large scale and long time for p x t obeys neither the usual fick equation nor other linear equations involving temporal and or spatial fractional derivatives

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

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

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

beating the curse of dimension with accurate statistics for the fokker planck equation in complex turbulent systems abstract solving the fokker planck equation for high dimensional complex dynamical systems is an important issue recently the authors developed efficient statistically accurate algorithms for solving the fokker planck equations associated with high dimensional nonlinear turbulent dynamical systems with conditional gaussian structures which contain many strong non gaussian features such as intermittency and fat tailed probability density functions pdfs the algorithms involve a hybrid strategy with a small number of samples l l where a conditional gaussian mixture in a high dimensional subspace via an extremely efficient parametric method is combined with a judicious gaussian kernel density estimation in the remaining low dimensional subspace in this article two effective strategies are developed and incorporated into these algorithms the first strategy involves a judicious block decomposition of the conditional covariance matrix such that the evolutions of different blocks have no interactions which allows an extremely efficient parallel computation due to the small size of each individual block the second strategy exploits statistical symmetry for a further reduction of l l the resulting algorithms can efficiently solve the fokker planck equation with strongly non gaussian pdfs in much higher dimensions even with orders in the millions and thus beat the curse of dimension the algorithms are applied to a 1 000 1 000 dimensional stochastic coupled fitzhugh nagumo model for excitable media an accurate recovery of both the transient and equilibrium non gaussian pdfs requires only l 1 l 1 samples in addition the block decomposition facilitates the algorithms to efficiently capture the distinct non gaussian features at different locations in a 240 240 dimensional two layer inhomogeneous lorenz 96 model using only l 500 l 500 samples

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

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

statistical properties of maps with indifferent periodic points in this note we present an axiomatic approach to the decay of correlations for maps of arbitrary dimension with indifferent periodic points as applications we apply our results to the well known manneville pomeau equation and the inhomogeneous diophantine approximation algorithm

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

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

extensive numerical investigations on the ergodic properties of two coupled pomeau manneville maps abstract we present extensive numerical investigations on the ergodic properties of two identical pomeau manneville maps interacting on the unit square through a diffusive linear coupling the system exhibits anomalous statistics as expected but with strong deviations from the single intermittent map such differences are characterized by numerical experiments with densities which do not have singularities in the marginal fixed point escape and poincare recurrence time statistics that share a power law decay exponent modified by a clear dimensional scaling while the rate of phase space filling and the convergence of ensembles of lyapunov exponents show a stretched instead of pure exponential behavior in spite of the lack of rigorous results about this system the dependence on both the intermittency and the coupling parameters appears to be smooth paving the way for further analytical development we remark that dynamical exponents appear to be independent of the nonzero coupling strength

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

sporadicity between periodic and chaotic dynamical behaviors abstract we define the class of sporadic dynamical systems as the systems where the algorithmic complexity of kolmogorov kolmogorov a n 1983 russ math surv 38 29 40 and chaitin chaitin g j 1987 algorithmic information theory cambridge univ press cambridge u k as well as the logarithm of separation of initially nearby trajectories grow as nv0 log n v1 with 0 v0 1 or v0 1 and v1 0 as time n these systems present a behavior intermediate between the multiperiodic v0 0 v1 1 and the chaotic ones v0 1 v1 0 we show that intermittent systems of manneville manneville p 1980 j phys paris 41 1235 1243 as well as some countable markov chains may be sporadic and furthermore that the dynamical fluctuations of these systems may be of levy s type rather than gaussian

multilevel ilu with reorderings for diagonal dominance this paper presents a preconditioning method based on combining two sided permutations with a multilevel approach the nonsymmetric permutation exploits a greedy strategy to put large entries of the matrix in the diagonal of the upper leading submatrix the method can be regarded as a complete pivoting version of the incomplete lu factorization this leads to an effective incomplete factorization preconditioner for general nonsymmetric irregularly structured sparse linear systems

multilevel preconditioners constructed from inverse based ilus this paper analyzes dropping strategies in a multilevel incomplete lu decomposition context and presents a few strategies for obtaining related ilus with enhanced robustness the analysis shows that the incomplete lu factorization resulting from dropping small entries in gaussian elimination produces a good preconditioner when the inverses of these factors have norms that are not too large as a consequence a few strategies are developed whose goal is to achieve this feature a number of templates for enabling implementations of these factorizations are presented numerical experiments show that the resulting ilus offer a good compromise between robustness and efficiency

spectral properties of a piecewise linear intermittent map for a piecewise linear intermittent map the evolution of statistical averages of a class of observables with respect to piecewise constant initial densities is investigated and generalized eigenfunctions of the frobenius perron operator p are explicitly derived the evolution of the averages are shown to be a superposition of the contributions from two simple eigenvalues 1 and d 1 0 and a continuous spectrum on the unit interval 0 1 of p power law decay of correlations are controlled by the continuous spectrum also the non normalizable invariant measure in the non stationary regime is shown to determine the strength of the power law decay

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

escape from intermittent repellers periodic orbit theory for crossover from exponential to algebraic decay we apply periodic orbit theory to study the asymptotic distribution of escape times from an intermittent map the dynamical zeta function exhibits a branch point which is associated with an asymptotic power law escape by an analytic continuation technique we compute a pair of complex conjugate zeroes beyond the branch point associated with a preasymptotic exponential decay the crossover time from an exponential to a power law is also predicted the theoretical predictions are confirmed by numerical simulation applications to conductance fluctuations in quantum dots are discussed

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

a 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 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

convergence to equilibrium for intermittent symplectic maps we investigate a class of area preserving non uniformly hyperbolic maps of the two torus first we establish some results on the regularity of the invariant foliations then we use this knowledge to estimate the rate of mixing

towards a complete determination of the spectrum of a transfer operator associated with intermittency the physical phenomenon of intermittency can be investigated via the spectral analysis of a transfer operator associated with the dynamics of an interval map with indifferent fixed point for an example of such an intermittent map the farey map we give a simple proof that the transfer operator is self adjoint on a suitably defined hilbert space and characterize its spectrum using a suitable first return map we present a highly efficient numerical method for the determination of all the eigenvalues including those embedded in the continuous spectrum

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

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

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

dynamical and transport properties in a family of intermittent area preserving maps we introduce a family of area preserving maps representing a nontrivial two dimensional extension of the pomeau manneville family in one dimension we analyze the long time behavior of recurrence time distributions and correlations providing analytical and numerical estimates we study the transport properties of a suitable lift and use a probabilistic argument to derive the full spectrum of transport moments finally the dynamical effects of a stochastic perturbation are considered

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

numerical approximation of incremental infinitesimal gradient plasticity we investigate a representative model of continuum infinitesimal gradient plasticity the formulation is an extension of classical rate independent infinitesimal plasticity based on the additive decomposition of the symmetric strain tensor into elastic and plastic parts it is assumed that dislocation processes contribute to the storage of energy in the material whereby the curl of the plastic distortion appears in the thermodynamic potential and leads to an additional nonlocal backstress tensor the formulation is cast into a numerical framework by a saddle point approximation of the corresponding minimization problem in each incremental loading step this allows one to reformulate the nonlocal dissipation inequality to a point wise flow rule and yields a solution scheme which is a direct extension of the standard approach in classical plasticity our numerical results show the regularizing effects of the additional physically motivated terms copyright 2008 john wiley sons ltd

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

combined compact difference scheme for linear second order partial differential equations with mixed derivative a combined compact difference scheme is proposed for linear second order partial differential equations with mixed derivative the scheme is based on a nine point stencil at the interior with sixth order accurate local truncation error fourier analysis is used to analyze the spectral resolution of the proposed scheme numerical tests demonstrate at least sixth order convergence rate with dirichlet boundary condition and fifth order with robin boundary condition a bonus is that high reynolds numbers do not interfere with the order of accuracy

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

wiener path integral based response determination of nonlinear systems subject to non white non gaussian and non stationary stochastic excitation abstract the recently developed wiener path integral wpi technique for determining the joint response probability density function of nonlinear systems subject to gaussian white noise excitation is generalized herein to account for non white non gaussian and non stationary excitation processes specifically modeling the excitation process as the output of a filter equation with gaussian white noise as its input it is possible to define an augmented response vector process to be considered in the wpi solution technique a significant advantage relates to the fact that the technique is still applicable even for arbitrary excitation power spectrum forms in such cases it is shown that the use of a filter approximation facilitates the implementation of the wpi technique in a straightforward manner without compromising its accuracy necessarily further in addition to dynamical systems subject to stochastic excitation the technique can also account for a special class of engineering mechanics problems where the media properties are modeled as stochastic fields several numerical examples pertaining to both single and multi degree of freedom systems are considered including a marine structural system exposed to flow induced non white excitation as well as a bending beam with a non gaussian and non homogeneous young s modulus comparisons with monte carlo simulation data demonstrate the accuracy of the technique

levy flights from a continuous time process l evy flight dynamics can stem from simple random walks in a system whose operational time number of steps n typically grows superlinearly with physical time t thus this process is a kind of continuous time random walk ctrw dual to the typical scher montroll model in which n grows sublinearly with t models in which l evy flights emerge due to a temporal subordination allow one easily to discuss the response of a random walker to a weak outer force which is shown to be nonlinear on the other hand the relaxation of an ensemble of such walkers in a harmonic potential follows a simple exponential pattern and leads to a normal boltzmann distribution mixed models describing normal ctrw s in superlinear operational time and l evy flights under the operational time of subdiffusive ctrw s lead to a paradoxical diffusive behavior similar to the one found in transport on polymer chains the relaxation to the boltzmann distribution in such models is slow and asymptotically follows a power law

ilu preconditioners for nonsymmetric saddle point matrices with application to the incompressible navier stokes equations motivated by the numerical solution of the linearized incompressible navier stokes equations we study threshold incomplete lu factorizations for nonsymmetric saddle point matrices the resulting preconditioners are used to accelerate the convergence of a krylov subspace method applied to finite element discretizations of fluid dynamics problems in three space dimensions the paper presents and examines an extension for nonsymmetric matrices of the tismenetsky kaporin incomplete factorization it is shown that in numerically challenging cases of higher reynolds number flows one benefits from using this two parameter modification of a standard threshold ilu preconditioner the performance of the ilu preconditioners is studied numerically for a wide range of flow and discretization parameters and the efficiency of the approach is shown if threshold parameters are chosen suitably the practical utility of the method is further demonstrated for the haemodynamic problem of simulating blood flow in a right c

crout versions of ilu for general sparse matrices this paper presents an efficient implementation of the incomplete lu ilu factorization derived from the crout version of gaussian elimination at step k of the elimination the kth row of u and the kth column of l are computed using previously computed rows of u and columns of l the data structure and implementation borrow from already known techniques used in developing both sparse direct solution codes and incomplete cholesky factorizations this version of ilu can be computed much faster than standard threshold based ilu factorizations computed rowwise or columnwise in addition the data structure allows efficient implementations of more rigorous and effective dropping strategies