generalized entropic uncertainty relations a new class of uncertainty relations is derived for pairs of observables in a finite dimensional hilbert space which do not have any common eigenvector this class contains an entropic uncertainty relation which improves a previous result of deutsch and confirms a recent conjecture by kraus some comments are made on the extension of these relations to the case where the hilbert space is infinite dimensional

characterization of discontinuities in high dimensional stochastic problems on adaptive sparse grids in this paper we present a set of efficient algorithms for detection and identification of discontinuities in high dimensional space the method is based on extension of polynomial annihilation for discontinuity detection in low dimensions compared to the earlier work the present method poses significant improvements for high dimensional problems the core of the algorithms relies on adaptive refinement of sparse grids it is demonstrated that in the commonly encountered cases where a discontinuity resides on a small subset of the dimensions the present method becomes optimal in the sense that the total number of points required for function evaluations depends linearly on the dimensionality of the space the details of the algorithms will be presented and various numerical examples are utilized to demonstrate the efficacy of the method

a dynamically adaptive sparse grids method for quasi optimal interpolation of multidimensional functions in this work we develop a dynamically adaptive sparse grids sg method for quasi optimal interpolation of multidimensional analytic functions defined over a product of one dimensional bounded domains the goal of such approach is to construct an interpolant in space that corresponds to the best m terms based on sharp a priori estimate of polynomial coefficients in the past sg methods have been successful in achieving this with a traditional construction that relies on the solution to a knapsack problem only the most profitable hierarchical surpluses are added to the sg however this approach requires additional sharp estimates related to the size of the analytic region and the norm of the interpolation operator i e the lebesgue constant instead we present an iterative sg procedure that adaptively refines an estimate of the region and accounts for the effects of the lebesgue constant our approach does not require any a priori knowledge of the analyticity or operator norm is easily generalized to both affine and non affine analytic functions and can be applied to sparse grids built from one dimensional rules with arbitrary growth of the number of nodes in several numerical examples we utilize our dynamically adaptive sg to interpolate quantities of interest related to the solutions of parametrized elliptic and hyperbolic pdes and compare the performance of our quasi optimal interpolant to several alternative sg schemes

renormalization group and asymptotics of solutions of nonlinear parabolic equations we present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations the method does not rely on a priori estimates such as the maximum principle it applies to systems of coupled equations to boundary conditions at infinity creating a front and to higher possibly fractional differential linear terms we present in detail the analysis for nonlinear diffusion type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity in an accompanying paper 5 the method is applied to systems of equations where some variables are slaved such as the complex ginzburg landau equation c 1994 john wiley sons inc

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

stochastic galerkin framework with locally reduced bases for nonlinear two phase transport in heterogeneous formations abstract the generalized polynomial chaos method with multiwavelet basis functions is applied to the buckley leverett equation we consider a spatially homogeneous domain modeled as a random field the problem is projected onto stochastic basis functions which yields an extended system of partial differential equations analysis and numerical methods leading to reduced computational cost are presented for the extended system of equations the accurate representation of the evolution of a discontinuous stochastic solution over time requires a large number of stochastic basis functions adaptivity of the stochastic basis to reduce computational cost is challenging in the stochastic galerkin setting since the change of basis affects the system matrix itself to achieve adaptivity without adding overhead by rewriting the entire system of equations for every grid cell we devise a basis reduction method that distinguishes between locally significant and insignificant modes without changing the actual system matrices results are presented for problems in one and two spatial dimensions with varying number of stochastic dimensions we show how to obtain stochastic velocity fields from realistic permeability fields and demonstrate the performance of the stochastic galerkin method with local basis reduction the system of conservation laws is discretized with a finite volume method and we demonstrate numerical convergence to the reference solution obtained through monte carlo sampling

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

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

preparation uncertainty implies measurement uncertainty in a class of generalized probabilistic theories in quantum theory it is known for a pair of noncommutative observables that there is no state on which they take simultaneously definite values and that there is no joint measurement of them they are called preparation uncertainty and measurement uncertainty respectively and research has unveiled that they are not independent from but related with each other in a quantitative way this study aims to reveal whether similar relations to quantum ones hold also in generalized probabilistic theories gpts in particular a certain class of gpts is considered which can be characterized by transitivity and self duality and regarded as extensions of quantum theory it is proved that there are close connections expressed quantitatively between two types of uncertainty on a pair observables also in those theories if preparation uncertainty exists then measurement uncertainty also exists and they are described by similar inequalities our results manifest that their correspondences are not specific to quantum theory but more universal ones

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

field theoretic renormalization group for a nonlinear diffusion equation this paper is an attempt to relate two vast areas of the applicability of the renormalization group rg field theoretic models and partial differential equations it is shown that the green function of a nonlinear diffusion equation can be viewed as a correlation function in a field theoretic model with an ultralocal term concentrated at a space time point this field theory is shown to be multiplicatively renormalizable so that the rg equations can be derived in a standard fashion and the rg functions the beta function and anomalous dimensions can be calculated within a controlled approximation a direct calculation carried out in the two loop approximation for the nonlinearity of the form phi alpha where alpha 1 is not necessarily integer confirms the validity and self consistency of the approach the explicit self similar solution is obtained for the infrared asymptotic region with exactly known exponents its range of validity and relationship to previous treatments are briefly discussed

the kronecker product and stochastic automata networks this paper can be thought of as a companion paper to van loan s the ubiquitous kronecker product paper j comput appl math 123 2000 85 we collect and catalog the most useful properties of the kronecker product and present them in one place we prove several new properties that we discovered in our search for a stochastic automata network preconditioner we conclude by describing one application of the kronecker product omitted from van loan s list of applications namely stochastic automata networks

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

random graph model with power law distributed triangle subgraphs clustering is well known to play a prominent role in the description and understanding of complex networks and a large spectrum of tools and ideas have been introduced to this end in particular it has been recognized that the abundance of small subgraphs is important here we study the arrangement of triangles in a model for scale free random graphs and determine the asymptotic behavior of the clustering coefficient the average number of triangles as well as the number of triangles attached to the vertex of maximum degree we prove that triangles are power law distributed among vertices and characterized by both vertex and edge coagulation when the degree exponent satisfies 2l ensuremath beta l2 5 furthermore a finite density of triangles appears as ensuremath beta 2 1 3

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

intrusive galerkin methods with upwinding for uncertain nonlinear hyperbolic systems this paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws we consider problems with parametric uncertainty in initial conditions and model coefficients whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables the stochastic spectral method relies on multi resolution schemes where the stochastic domain is discretized using tensor product stochastic elements supporting local polynomial bases a galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution hyperbolicity of the resulting galerkin system is analyzed a finite volume scheme with a roe type solver is used for discretization of the spatial and time variables an original technique is introduced for the fast evaluation of approximate upwind matrices which is particularly well adapted to local polynomial bases efficiency and robustness of the overall method are assessed on the burgers and euler equations with shocks

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

sparse tensor discretization of elliptic spdes we propose and analyze sparse deterministic stochastic tensor galerkin finite element methods sparse sgfems for the numerical solution of elliptic partial differential equations pdes with random coefficients in a physical domain d subset mathbb r d in tensor product sgfems the variational solution to the boundary value problem is approximated in tensor product finite element spaces v gamma otimes v d where v gamma and v d denote suitable finite dimensional subspaces of the stochastic and deterministic function spaces respectively these approaches lead to sgfem algorithms of complexity o n gamma n d where n gamma dim v gamma and n d dim v d in this work we use hierarchic sequences v gamma 1 subset v gamma 2 subset ldots and v d 1 subset v d 2 subset ldots of finite dimensional spaces to approximate the law of the random solution the hierarchies of approximation spaces allow us to define sparse tensor product spaces v gamma ell hat otimes v d ell ell 1 2 dots yielding algorithms of o n gamma log n d n d log n gamma work and memory we estimate the convergence rate of sgfem for algebraic decay of the input random field karhunen loeve coefficients we give an algorithm for an input adapted a priori selection of deterministic and stochastic discretization spaces the convergence rate in terms of the total number of degrees of freedom of the proposed method is superior to monte carlo approximations numerical examples illustrate the theoretical results and demonstrate superiority of the sparse tensor product discretization proposed here versus the full tensor product approach

a stochastic particle mesh scheme for uncertainty propagation in vortical flows abstract a new mesh particle scheme is constructed for uncertainty propagation in vortical flow the scheme is based on the incorporation of polynomial chaos pc expansions into a lagrangian particle approximation of the navier stokes equations the main idea of the method is to use a unique set of particles to transport the stochastic modes of the solution the particles are transported by the mean velocity field while their stochastic strengths are updated to account for diffusive and convective effects induced by the coupling between stochastic modes an integral treatment is used for the evaluation of the coupled stochastic terms following the framework of the particle strength exchange pse methods which yields a conservative algorithm it is also shown that it is possible to apply solution algorithms used in deterministic setting including particle mesh techniques and particle remeshing thus the method combines the advantages of particles discretizations with the efficiency of pc representations validation of the method on uncertain diffusion and convection problems is first performed an example is then presented of natural convection of a hot patch of fluid in infinite domain and the computations are used to illustrate the effectiveness of the approach for both large number of particles and high order pc expansions

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

random construction of riemann surfaces we develop a new approach for the study of typical riemann surfaces with high genus the method that we use is the construction of random riemann surfaces from oriented cubic graphs this construction enables us to get a control over the global geometry properties of compact riemann surfaces we use the theory of random regular graphs to show that almost all such surfaces have large first eigenvalues and large cheeger constants moreover a closer analysis of the probability space of oriented cubic graphs shows that on a typical surface there is a large embedded hyperbolic ball

hyperspherical sparse approximation techniques for high dimensional discontinuity detection this work proposes a hyperspherical sparse approximation framework for detecting jump discontinuities in functions in high dimensional spaces the need for a novel approach results from the theoretical and computational inefficiencies of well known approaches such as adaptive sparse grids for discontinuity detection our approach constructs the hyperspherical coordinate representation of the discontinuity surface of a function then sparse approximations of the transformed function are built in the hyperspherical coordinate system with values at each point estimated by solving a one dimensional discontinuity detection problem due to the smoothness of the hypersurface the new technique can identify jump discontinuities with significantly reduced computational cost compared to existing methods several approaches are used to approximate the transformed discontinuity surface in the hyperspherical system including adaptive sparse grid and radial basis function interpolation discrete least squares proj

probabilistic collocation method for flow in porous media comparisons with other stochastic methods 1 an efficient method for uncertainty analysis of flow in random porous media is explored in this study on the basis of combination of karhunen loeve expansion and probabilistic collocation method pcm the random log transformed hydraulic conductivity field is represented by the karhunen loeve expansion and the hydraulic head is expressed by the polynomial chaos expansion probabilistic collocation method is used to determine the coefficients of the polynomial chaos expansion by solving for the hydraulic head fields for different sets of collocation points the procedure is straightforward and analogous to the monte carlo method but the number of simulations required in pcm is significantly reduced steady state flows in saturated random porous media are simulated with the probabilistic collocation method and comparisons are made with other stochastic methods monte carlo method the traditional polynomial chaos expansion pce approach based on galerkin scheme and the moment equation approach based on karhunen loeve expansion klme this study reveals that pcm and klme are more efficient than the galerkin pce approach while the computational efforts are greatly reduced compared to the direct sampling monte carlo method the pcm and klme approaches are able to accurately estimate the statistical moments and probability density function of the hydraulic head

uncertainty quantification given discontinuous model response and a limited number of model runs we outline a methodology for forward uncertainty quantification in systems with uncertain parameters discontinuous model response and a limited number of model runs our approach involves two stages first we detect the discontinuity with bayesian inference thus obtaining a probabilistic representation of the discontinuity curve for arbitrarily distributed input parameters then employing the rosenblatt transform we construct spectral representations of the uncertain model output using polynomial chaos pc expansions on either side of the discontinuity curve leading to an averaged pc representation of the forward model response that allows efficient uncertainty quantification we obtain pc modes by either orthogonal projection or bayesian inference and argue for a hybrid approach that targets a balance between the accuracy provided by the orthogonal projection and the flexibility provided by the bayesian inference the uncertain model output is then computed by taking an ensemble average over pc expansions corresponding to sampled realizations of the discontinuity curve the methodology is demonstrated on synthetic examples of discontinuous model response with adjustable sharpness and structure

spectra of real world graphs beyond the semicircle law results on the spectra of adjacency matrices corresponding to models of real world graphs we find that when the number of links grows as the number of nodes the spectral density of uncorrelated random matrices does not converge to the semicircle law furthermore the spectra of real world graphs have specific features depending on the details of the corresponding models in particular scale free graphs develop a trianglelike spectral density with a power law tail while small world graphs have a complex spectral density consisting of several sharp peaks these and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks

inverse modeling of geochemical and mechanical compaction in sedimentary basins through polynomial chaos expansion we present an inverse modeling procedure for the estimation of model parameters of sedimentary basins subject to compaction driven by mechanical and geochemical processes we consider a sandstone basin whose dynamics are governed by a set of unknown key quantities these include geophysical and geochemical system attributes as well as pressure and temperature boundary conditions we derive a reduced or surrogate model of the system behavior based on generalized polynomial chaos expansion gpce approximations which are directly linked to the variance based sobol indices associated with the selected uncertain model parameters parameter estimation is then performed within a maximum likelihood ml framework we then study the way the ml inversion procedure can benefit from the adoption of anisotropic polynomial approximations a gpce in which the surrogate model is refined only with respect to selected parameters according to an analysis of the nonlinearity of the input output mapping as quantified through the sobol sensitivity indices results are illustrated for a one dimensional setting involving quartz cementation and mechanical compaction in sandstones the reliability of gpce and a gpce approximations in the context of the inverse modeling framework is assessed the effects of a the strategy employed to build the surrogate model leading either to a gpce or a gpce representation and b the type and quality of calibration data on the goodness of the parameter estimates is then explored

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

uncertainty in quantum measurements the object of this letter is to show that except in the case of canonically conjugate observables the generalized heisenberg inequality does not properly express the quantum uncertainty principle it is in general too weak an inequality is obtained which does express the principle

renormalization group and the ginzburg landau equation we use renormalization group methods to prove detailed long time asymptotics for the solutions of the ginzburg landau equations with initial data approaching asx different spiraling stationary solutions a universal pattern is formed depending only on this asymptotics at spatial infinity

universal joint measurement uncertainty relation for error bars we formulate and prove a new universally valid uncertainty relation for the necessary error bar widths in any approximate joint measurement of position and momentum

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

nonlinear diffusion through large complex networks containing regular subgraphs transport through generalized trees is considered trees contain the simple nodes and supernodes either well structured regular subgraphs or those with many triangles we observe a superdiffusion for the highly connected nodes while it is brownian for the rest of the nodes transport within a supernode is affected by the finite size effects vanishing as n for the even dimensions of space d 2 4 6 the finite size effects break down the perturbation theory at small scales and can be regularized by using the heat kernel expansion

sparse pseudospectral approximation method multivariate global polynomial approximations such as polynomial chaos or stochastic collocation methods are now in widespread use for sensitivity analysis and uncertainty quantification the pseudospectral variety of these methods uses a numerical integration rule to approximate the fourier type coefficients of a truncated expansion in orthogonal polynomials for problems in more than two or three dimensions a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation however when using a sparse rule to approximately integrate these coefficients one often finds unacceptable errors in the coefficients associated with higher degree polynomials by reexamining smolyak s algorithm and exploiting the connections between interpolation and projection in tensor product spaces we construct a sparse pseudospectral approximation method that accurately reproduces the coefficients for basis functions that naturally correspond to the sparse grid integration rule the compelling numerical results show that this is the proper way to use sparse grid integration rules for pseudospectral approximation

convergence of quasi optimal stochastic galerkin methods for a class of pdes with random coefficients in this work we consider quasi optimal versions of the stochastic galerkin method for solving linear elliptic pdes with stochastic coefficients in particular we consider the case of a finite number n of random inputs and an analytic dependence of the solution of the pde with respect to the parameters in a polydisc of the complex plane c n we show that a quasi optimal approximation is given by a galerkin projection on a weighted anisotropic total degree space and prove a sub exponential convergence rate as a specific application we consider a thermal conduction problem with non overlapping inclusions of random conductivity numerical results show the sharpness of our estimates

pade legendre approximants for uncertainty analysis with discontinuous response surfaces a novel uncertainty propagation method for problems characterized by highly non linear or discontinuous system responses is presented the approach is based on a pade legendre pl formalism which does not require modifications to existing computational tools non intrusive approach and it is a global method the paper presents a novel pl method for problems in multiple dimensions which is non trivial in the pade literature in addition a filtering procedure is developed in order to minimize the errors introduced in the approximation close to the discontinuities the numerical examples include fluid dynamic problems characterized by shock waves a simple dual throat nozzle problem with uncertain initial state and the turbulent transonic flow over a transonic airfoil where the flight conditions are assumed to be uncertain results are presented in terms of statistics of both shock position and strength and are compared to monte carlo simulations

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

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 grid reduced basis bayesian inversion we extend the reduced basis rb accelerated bayesian inversion methods for affine parametric linear operator equations which are considered in 16 17 to non affine nonlinear parametric operator equations we generalize the analysis of sparsity of parametric forward solution maps in 20 and of bayesian inversion in 48 49 to the fully discrete setting including petrov galerkin high fidelity hifi discretization of the forward maps we develop adaptive stochastic collocation based reduction methods for the efficient computation of reduced bases on the parametric solution manifold the nonaffinity and nonlinearity with respect to w r t the distributed uncertain parameters and the unknown solution is collocated specifically by the so called empirical interpolation method eim for the corresponding bayesian inversion problems computational efficiency is enhanced in two ways first expectations w r t the posterior are computed by adaptive quadratures with dimension independent convergence rates proposed in 49 the present work generalizes 49 to account for the impact of the pg discretization in the forward maps on the convergence rates of the quantities of interest qoi for short second we propose to perform the bayesian estimation only w r t a parsimonious rb approximation of the posterior density based on the approximation results in 49 the infinite dimensional parametric deterministic forward map and operator admit n term rb and eim approximations which converge at rates which depend only on the sparsity of the parametric forward map in several numerical experiments the proposed algorithms exhibit dimension independent convergence rates which equal at least the currently known rate estimates for n term approximation we propose to accelerate bayesian estimation by first offline construction of reduced basis surrogates of the bayesian posterior density the parsimonious surrogates can then be employed for online data assimilation and for bayesian estimation they also open a perspective for optimal experimental design

random walks on bundled structures bundled structures bs are discrete structures obtained joining to each point of a base graph a copy of a fiber graph in condensed matter physics bs are used as realistic models for the geometry and dynamics of nontranslationally invariant systems polymers inhomogeneous systems etc we present an analytical solution for the random walk problem on these structures which is possible when we know the solution for base and fiber separately we obtain an expression for the spectral dimension of the bs as a function of the spectral dimensions of its components moreover we discuss some applications of these results concerning anomalous diffusion laws proving the existence of nondisordered structures with logarithmic and sublogarithmic diffusion laws due only to geometric features

global large time self similarity of a thermal diffusive combustion system with critical nonlinearity we study the initial value problem of the thermal diffusive combustion system u 1 t u 1 x x u 1 u 2 2 u 2 t du 2 xx u 1 u 2 2 x is an element of r 1 for non negative spatially decaying initial data of arbitrary size and for any positive constant d we show that if the initial data decay to zero sufficiently fast at infinity then the solution u 1 u 2 converges to a self similar solution of the reduced system u 1 t u 1 xx u 1 u 2 2 u 2 t du 2 xx in the large time limit in particular u 1 decays to zero like o t 1 2 delta where delta 0 is an anomalous exponent depending on the initial data and u 2 decays to zero with normal rate o t 1 2 the idea of the proof is to combine the a priori estimates for the decay of global solutions with the renormalization group method for establishing the self similarity of the solutions in the large time limit c 1996 academic press inc

the number of large graphs with a positive density of triangles we give upper and lower bounds on the number of graphs of fixed degree which have a positive density of triangles in particular we show that there are very few such graphs when compared to the number of graphs without this restriction we also show that in this case the triangles seem to cluster even at low density

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

the h p and h p versions of the finite element method in 1 dimension this paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting the main emphasis is placed on the analysis when the exact solution has singularity of x type the first part analyzes thep version the second theh version and generalh p version and the final third part addresses the problems of the adaptiveh p version

probabilistic collocation method for strongly nonlinear problems 1 transform by location 1 in this work we propose a new collocation method for uncertainty quantification in strongly nonlinear problems based on polynomial construction the traditional probabilistic collocation method pcm approximates the model output response which is a function of the random input parameter from the eulerian point of view in specific locations in some cases especially when the advection dominates the model response has a strongly nonlinear profile with a discontinuous shock or large gradient this nonlinearity in the space domain is then translated to nonlinearity in the random parametric domain which causes nonphysical oscillation and inaccurate estimation using the traditional pcm to address this issue a new location based transformed probabilistic collocation method xtpcm is developed in this study inspired by the lagrangian point of view in which model response is represented by an alternative variable i e the location of a particular response value which is relatively linear to the random parameter with a smooth profile the location is then approximated by polynomial construction from which a sufficient number of location samples are randomly generated and transformed back to obtain the response samples and to estimate the statistical properties the advantage of the xtpcm is demonstrated through applications to multiphase flow and solute transport in porous media which shows that the xtpcm achieves higher statistical accuracy than does the pcm and produces more reasonable realizations without oscillation while computational effort is greatly reduced compared to the direct sampling monte carlo method

adaptive anisotropic spectral stochastic methods for uncertain scalar conservation laws this paper deals with the design of adaptive anisotropic discretization schemes for conservation laws with stochastic parameters a finite volume scheme is used for the deterministic discretization while a piecewise polynomial representation is used at the stochastic level the methodology is designed in the context of intrusive galerkin projection methods with a roe type solver the adaptation aims at selecting the stochastic resolution level based on the local smoothness of the solution in the stochastic domain in addition the stochastic features of the solution greatly vary in space and time so that the constructed stochastic approximation space depends on space and time the dynamically evolving stochastic discretization uses a tree structure representation that allows for the efficient implementation of the various operators needed to perform anisotropic multiresolution analysis efficiency of the overall adaptive scheme is assessed on a stochastic nonlinear conservation law with uncertain initial

a multiscale probabilistic collocation method for subsurface flow in heterogeneous media 1 owing to the spatial variability of the media properties uncertainty quantification for subsurface flow and solute transport usually requires high resolution simulations in this work a multiscale probabilistic collocation method mspcm is developed for solving such problems in a computationally efficient manner the subsurface flow problem is cast in a stochastic framework and probabilistic collocation strategy is used to represent the original stochastic differential equation the resulting equations are a set of decoupled deterministic equations with respect to collocation points a multiscale finite element method is utilized to solve these deterministic problems on a coarse mesh coarse scale basis functions are constructed on a field in which the conductivity varies spatially at each set of stochastic collocation points the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the fine scale heterogeneity the fine scale solution is reconstructed after the coarse scale solution is available since the pcm and multiscale finite element method are implemented at different levels the mspcm inherits their respective advantages in which a stochastic problem is decomposed by fewer realizations and is solved on a coarser grid the performance of the proposed method is demonstrated with numerical examples the capability of mspcm in reproducing the probability density functions pdfs of head and velocity is investigated the numerical results show that the mspcm with proper coarsening level is able to capture small scale heterogeneity with a coarse mesh and to generate satisfactory probability density functions of head and velocity

the h p and h p versions of the finite element method in 1 dimension part ii the error analysis of the h and h p versions the paper is the second in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimensional setting the first part 1 analyzed thep version the present one concentrates on theh andh p versions

efficient localization of discontinuities in complex computational simulations surrogate models for computational simulations are input output approximations that allow computationally intensive analyses such as uncertainty propagation and inference to be performed efficiently when a simulation output does not depend smoothly on its inputs the error and convergence rate of many approximation methods deteriorate substantially this paper details a method for efficiently localizing discontinuities in the input parameter domain so that the model output can be approximated as a piecewise smooth function the approach comprises an initialization phase which uses polynomial annihilation to assign function values to different regions and thus seed an automated labeling procedure followed by a refinement phase that adaptively updates a kernel support vector machine representation of the separating surface via active learning the overall approach avoids structured grids and exploits any available simplicity in the geometry of the separating surface thus reducing the number of model

uncertainty relations for joint localizability and joint measurability in finite dimensional systems two quantities quantifying uncertainty relations are examined busch and pearson j math phys 48 082103 2007 investigated the limitation on joint localizability and joint measurement of position and momentum by introducing overall width and error bar width in this paper we show a simple relationship between these quantities for finite dimensional systems our result indicates that if there is a bound on joint localizability it is possible to obtain a similar bound on joint measurability for finite dimensional systems uncertainty relations for a pair of general projection valued measures are obtained as by products

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

uncertainty quantification of geochemical and mechanical compaction in layered sedimentary basins abstract in this work we propose an uncertainty quantification methodology for sedimentary basins evolution under mechanical and geochemical compaction processes which we model as a coupled time dependent non linear monodimensional depth only system of pdes with uncertain parameters while in previous works formaggia et al 2013 porta et al 2014 we assumed a simplified depositional history with only one material in this work we consider multi layered basins in which each layer is characterized by a different material and hence by different properties this setting requires several improvements with respect to our earlier works both concerning the deterministic solver and the stochastic discretization on the deterministic side we replace the previous fixed point iterative solver with a more efficient newton solver at each step of the time discretization on the stochastic side the multi layered structure gives rise to discontinuities in the dependence of the state variables on the uncertain parameters that need an appropriate treatment for surrogate modeling techniques such as sparse grids to be effective we propose an innovative methodology to this end which relies on a change of coordinate system to align the discontinuities of the target function within the random parameter space the reference coordinate system is built upon exploiting physical features of the problem at hand we employ the fact that the locations of the material interfaces display a smooth dependence on the random parameters and are therefore amenable to sparse grid polynomial approximations we showcase the capabilities of our numerical methodologies through two synthetic test cases in particular we show that our methodology reproduces with high accuracy multi modal probability density functions displayed by target state variables e g porosity

random walks on disordered networks random walks are studied on disordered cellular networks in 2 and 3dimensional spaces with arbitrary curvature the coefficients of the evolution equation are calculated in term of the structural properties of the cellular system the effects of disorder and space curvature on the diffusion phenomena are investigated in disordered systems the mean square displacement displays an enhancement at short time and a lowering at long ones with respect to the ordered case the asymptotic expression for the diffusion equation on hyperbolic cellular systems relates random walk on curved lattices to hyperbolic brownian motion

convergence of quasi optimal sparse grid approximation of hilbert space valued functions application to random elliptic pdes in this work we provide a convergence analysis for the quasi optimal version of the sparse grids stochastic collocation method we presented in a previous work on the optimal polynomial approximation of stochastic pdes by galerkin and collocation methods beck et al math models methods appl sci 22 09 2012 the construction of a sparse grid is recast into a knapsack problem a profit is assigned to each hierarchical surplus and only the most profitable ones are added to the sparse grid the convergence rate of the sparse grid approximation error with respect to the number of points in the grid is then shown to depend on weighted summability properties of the sequence of profits this is a very general argument that can be applied to sparse grids built with any uni variate family of points both nested and non nested as an example we apply such quasi optimal sparse grids to the solution of a particular elliptic pde with stochastic diffusion coefficients namely the inclusions problem we detail the convergence estimates obtained in this case using polynomial interpolation on either nested clenshaw curtis or non nested gauss legendre abscissas verify their sharpness numerically and compare the performance of the resulting quasi optimal grids with a few alternative sparse grid construction schemes recently proposed in the literature

scales of fluctuation and the propagation of uncertainty in random porous media 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 heterogeneity of the medium is modeled as a superposition of scales of heterogeneity that are statistically uncorrelated the stochastic process making up a hydraulic property is thus represented as a linear combination of normalized deterministic shapes multiplied by a set of uncorrelated random variables the karhunen loeve expansion is used to formalize this representation the uncertainty contained in each of these scales is then propagated through the system using the governing differential equation as a constraint the solution process will thus consist of the original scales used to represent the porous medium as well as various interaction between these scales the objective of this paper is to develop a procedure for quantifying the uncertainty in the field variables using the scales of fluctuation of the porous medium as the measuring stick

epidemic spreading in a variety of scale free networks we have shown that the epidemic spreading in scale free networks is very sensitive to the statistics of degree distribution characterized by the index gamma the effective spreading rate lambda the social strategy used by individuals to choose a partner and the policy of administrating a cure to an infected node depending on the interplay of these four factors the stationary fractions of infected population f gamma as well as the epidemic threshold properties can be essentially different we have given an example of the evolutionary scale free network which is disposed to the spreading and the persistence of infections at any spreading rate lambda 0 for any gamma probably it is impossible to obtain a simple immunization program that can be simultaneously effective for all types of scale free networks we have also studied the dynamical solutions for the evolution equation governed by the epidemic spreading in scale free networks and found that for the case of vanishingly small cure rate delta 1 the initial configuration of infected nodes would feature the solution for very long times

sparse spectral approximations of high dimensional problems based on hyperbolic cross hyperbolic cross approximations by some classical orthogonal polynomials functions in both bounded and unbounded domains are considered in this paper optimal error estimates in proper anisotropic weighted korobov spaces for both regular hyperbolic cross approximations and optimized hyperbolic cross approximations are established these fundamental approximation results indicate that spectral methods based on hyperbolic cross approximations can be effective for treating certain high dimensional problems and will serve as basic tools for analyzing sparse spectral methods in high dimensions

mesh refinement for uncertainty quantification through model reduction we present a novel way of deciding when and where to refine a mesh in probability space in order to facilitate uncertainty quantification in the presence of discontinuities in random space a discontinuity in random space makes the application of generalized polynomial chaos expansion techniques prohibitively expensive the reason is that for discontinuous problems the expansion converges very slowly an alternative to using higher terms in the expansion is to divide the random space in smaller elements where a lower degree polynomial is adequate to describe the randomness in general the partition of the random space is a dynamic process since some areas of the random space particularly around the discontinuity need more refinement than others as time evolves in the current work we propose a way to decide when and where to refine the random space mesh based on the use of a reduced model the idea is that a good reduced model can monitor accurately within a random space element the cascade of activity to higher degree terms in the chaos expansion in turn this facilitates the efficient allocation of computational sources to the areas of random space where they are more needed for the kraichnan orszag system the prototypical system to study discontinuities in random space we present theoretical results which show why the proposed method is sound and numerical results which corroborate the theory

eulerian lagrangian approach for modeling of flow and transport in heterogeneous geological formations this paper presents a new eulerian lagrangian method for modeling flow and transport of passive solutes in heterogeneous porous formations the physically plausible random velocity fields are generated by a geostatistical based model the ability of the method to correctly reproduce the velocity field statistics and to satisfy mass balance is tested and demonstrated for the case of two dimensional flow the method is found particularly useful in dealing with large fields where numerical fixed grid methods such as finite difference or finite elements become very demanding from a computational point of view the transport problem is solved in a lagrangian framework by the particle tracking approach which uses a suitable grid refinement to correctly handle the velocity fluctuation at any a priori defined resolution the new method is employed to compute the probability distribution functions of the concentration and travel times and to investigate the limitations of existing methods for predicting concentrations

multi index stochastic collocation for random pdes in this work we introduce the multi index stochastic collocation method misc for computing statistics of the solution of a pde with random data misc is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data we propose an optimization procedure to select the most effective mixed differences to include in the misc estimator such optimization is a crucial step and allows us to build a method that provided with sufficient solution regularity is potentially more effective than other multi level collocation methods already available in literature we then provide a complexity analysis that assumes decay rates of product type for such mixed differences showing that in the optimal case the convergence rate of misc is only dictated by the convergence of the deterministic solver applied to a one dimensional problem we show the effectiveness of misc with some computational tests comparing it with other related methods available in the literature such as the multi index and multilevel monte carlo multilevel stochastic collocation quasi optimal stochastic collocation and sparse composite collocation methods

use of global sensitivity analysis and polynomial chaos expansion for interpretation of nonreactive transport experiments in laboratory scale porous media in this work we show how the use of global sensitivity analysis gsa in conjunction with the polynomial chaos expansion pce methodology can provide relevant information for the interpretation of transport experiments in laboratory scale heterogeneous porous media we perform gsa by calculating the sobol indices which provide a variance based importance measure of the effects of uncertain parameters on the output of a chosen interpretive transport model the choice of pce has the following two benefits 1 it provides the global sensitivity indices in a straightforward manner and 2 pce can serve as a surrogate model for the calibration of parameters the coefficients of the pce are computed by probabilistic collocation the methodology is applied to two nonreactive transport experiments available in the literature while considering both transient and pseudo steady state transport regimes this method allows a rigorous investigation of the relative effects and importance of different uncertain quantities which include boundary conditions as well as porous medium hydraulic and dispersive parameters the parameters that are most relevant to depicting the system s behavior can then be evaluated in addition one can assess the space time distribution of measurement points which is the most influential factor for the identifiability of parameters our work indicates that these methods can be valuable tools in the proper design of model based transport experiments

multi index monte carlo when sparsity meets sampling we propose and analyze a novel multi index monte carlo mimc method for weak approximation of stochastic models that are described in terms of differential equations either driven by random measures or with random coefficients the mimc method is both a stochastic version of the combination technique introduced by zenger griebel and collaborators and an extension of the multilevel monte carlo mlmc method first described by heinrich and giles inspired by giles s seminal work we use in mimc high order mixed differences instead of using first order differences as in mlmc to reduce the variance of the hierarchical differences dramatically this in turn yields new and improved complexity results which are natural generalizations of giles s mlmc analysis and which increase the domain of the problem parameters for which we achieve the optimal convergence mathcal o mathrm tol 2 o tol 2 moreover in mimc the rate of increase of required memory with respect to mathrm tol tol is independent of the number of directions up to a logarithmic term which allows far more accurate solutions to be calculated for higher dimensions than what is possible when using mlmc we motivate the setting of mimc by first focusing on a simple full tensor index set we then propose a systematic construction of optimal sets of indices for mimc based on properly defined profits that in turn depend on the average cost per sample and the corresponding weak error and variance under standard assumptions on the convergence rates of the weak error variance and work per sample the optimal index set turns out to be the total degree type in some cases using optimal index sets mimc achieves a better rate for the computational complexity than the corresponding rate when using full tensor index sets we also show the asymptotic normality of the statistical error in the resulting mimc estimator and justify in this way our error estimate which allows both the required accuracy and the confidence level in our computational results to be prescribed finally we include numerical experiments involving a partial differential equation posed in three spatial dimensions and with random coefficients to substantiate the analysis and illustrate the corresponding computational savings of mimc

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

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

optimized general sparse grid approximation spaces for operator equations this paper is concerned with the construction of optimized sparse grid approximation spaces for elliptic pseudodifferential operators of arbitrary order based on the framework of tensor product biorthogonal wavelet bases and stable subspace splittings we construct operator adapted subspaces with a dimension smaller than that of the standard full grid spaces but which have the same approximation order as the standard full grid spaces provided that certain additional regularity assumptions on the solution are fulfilled specifically for operators of positive order their dimension is o 2 j independent of the dimension n of the problem compared to o 2 jn for the full grid space also for operators of negative order the overall cost is significantly in favor of the new approximation spaces we give cost estimates for the case of continuous linear information we show these results in a constructive manner by proposing a galerkin method together with optimal preconditioning the theory covers elliptic boundary value problems as well as boundary integral equations

stochastic uncertainty analysis for solute transport in randomly heterogeneous media using a karhunen loeve based moment equation approach 1 a new approach has been developed for solving solute transport problems in randomly heterogeneous media using the karhunen loeve based moment equation klme technique proposed by zhang and lu 2004 the klme approach combines the karhunen loeve decomposition of the underlying random conductivity field and the perturbative and polynomial expansions of dependent variables including the hydraulic head flow velocity dispersion coefficient and solute concentration the equations obtained in this approach are sequential and their structure is formulated in the same form as the original governing equations such that any existing simulator such as modular three dimensional multispecies transport model for simulation of advection dispersion and chemical reactions of contaminants in groundwater systems mt3dms can be directly applied as the solver through a series of two dimensional examples the validity of the klme approach is evaluated against the classical monte carlo simulations results indicate that under the flow and transport conditions examined in this work the klme approach provides an accurate representation of the mean concentration for the concentration variance the accuracy of the klme approach is good when the conductivity variance is 0 5 as the conductivity variance increases up to 1 0 the mismatch on the concentration variance becomes large although the mean concentration can still be accurately reproduced by the klme approach our results also indicate that when the conductivity variance is relatively large neglecting the effects of the cross terms between velocity fluctuations and local dispersivities as done in some previous studies can produce noticeable errors and a rigorous treatment of the dispersion terms becomes more appropriate