multi resolution analysis scheme for uncertainty quantification in chemical systems this paper presents a multi resolution approach for the propagation of parametric uncertainty in chemical systems it is motivated by previous studies where galerkin formulations of wiener hermite expansions were found to fail in the presence of steep dependences of the species concentrations with regard to the reaction rates the multi resolution scheme is based on representation of the uncertain concentration in terms of compact polynomial multi wavelets allowing for the control of the convergence in terms of polynomial order and resolution level the resulting representation is shown to greatly improve the robustness of the galerkin procedure in presence of steep dependences however this improvement comes with a higher computational cost which drastically increases with the number of uncertain reaction rates to overcome this drawback an adaptive strategy is proposed to control locally in the parameter space and in time the resolution level the efficiency of the method is demonstrated for an uncertain chemical system having eight random parameters

global existence for energy critical waves in 3 d domains we prove that the defocusing quintic wave equation with dirichlet boundary conditions is globally well posed on for any smooth compact domain the main ingredient in the proof is an spectral projector estimate obtained recently by smith and sogge combined with a precise study of the boundary value problem

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

an optimal adaptive wavelet method without coarsening of the iterands in this paper an adaptive wavelet method for solving linear operator equations is constructed that is a modification of the method from math comp 70 2001 pp 27 75 by cohen dahmen and devore in the sense that there is no recurrent coarsening of the iterands despite this it will be shown that the method has optimal computational complexity numerical results for a simple model problem indicate that the new method is more efficient than an existing alternative adaptive wavelet method

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

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

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

continuous fields of kirchberg c algebras abstract in this paper we study the c algebras associated to continuous fields over locally compact metrizable zero dimensional spaces whose fibers are kirchberg c algebras satisfying the uct we show that these algebras are inductive limits of finite direct sums of kirchberg algebras and they are classified up to isomorphism by topological invariants

analysis of the hum control operator and exact controllability for semilinear waves in uniform time we present a regularity result for the hum optimal control associated with the interior control of linear waves we use this analysis together with strichartz inequalities to get results on the exact controllability for subcritical nonlinear waves in a bounded domain of mathbb r 3

approximations of c algebras and the ideal property we introduce several classes of c algebras using for this local approximations by nice c algebras that generalize the ah algebras among others we initiate their study proving mainly results about the ideal property but also about the ideals generated by their projections the real rank zero the weak projection property minimal tensor products extensions quasidiagonal extensions ideal structure the largest ideal with the ideal property and short exact sequences some of the previous results of the second named author are generalized 2007 elsevier inc all rights reserved

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

purely infinite corona algebras of simple c algebras in this paper we study the problem of when the corona algebra of a non unital c algebra is purely infinite a complete answer is obtained for stabilisations of simple and unital algebras that have enough comparison of positive elements our result relates the pure infiniteness condition from its strongest to weakest forms to the geometry of the tracial simplex of the algebra and to the behaviour of corona projections despite the fact that there is no real rank zero condition

on stabilization and control for the critical klein gordon equation on a 3 d compact manifold abstract in this article we study the internal stabilization and control of the critical nonlinear klein gordon equation on 3 d compact manifolds under a geometric assumption slightly stronger than the classical geometric control condition we prove exponential decay for some solutions bounded in the energy space but small in a lower norm the proof combines profile decomposition and microlocal arguments this profile decomposition analogous to the one of bahouri and gerard 1999 2 on r 3 is performed by taking care of possible geometric effects it uses some results of s ibrahim 2004 21 on the behavior of concentrating waves on manifolds

finite group actions on c algebras with the rohlin property i basic properties of finite group actions with the rohlin property on unital c algebras are investigated a characterization of finite group actions with the rohlin property on the cuntz algebra 02 is given in terms of central sequences which may be considered as an equivariant version of e kirchberg and n c phillips s characterization of 02 a large class of symmetries on 02 are classified in terms of the fixed point algebras for conjugacy and the crossed products for cocycle conjugacy model actions of symmetries of 02 are constructed for given k theoretical invariants

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

convergence rates of multilevel and sparse tensor approximations for a random elliptic pde partial differential equations with random coefficients can be cast as parametric problems with a potentially infinite dimensional parameter domain for a class of elliptic equations we derive convergence rates of approximations based on a tensorized polynomial basis on the parameter domain and a sequence of spatial discretizations we prove that a sparse tensor product construction achieves essentially the same convergence rate as a multilevel approximation in which the spatial discretization level is chosen separately for each coefficient of the solution with respect to the polynomial basis in some cases the same rate is attained also by a single level approximation using just one spatial discretization for all coefficients we suggest an adaptive algorithm that reaches the optimal convergence rate with respect to the total number of degrees of freedom in a multilevel setting without prior knowledge of this rate numerical computations confirm theoretical results

a generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations we propose a new robust technique for solving stochastic partial differential equations the solution is approximated by a series of terms each of which being the product of a scalar stochastic function by a deterministic function none of these functions are fixed a priori but determined by solving a problem which can be interpreted as an extended eigenvalue problem this technique generalizes the classical spectral decomposition namely the karhunen loeve expansion ad hoc iterative techniques to build the approximation inspired by the power method for classical eigenproblems then transform the problem into the resolution of a few uncoupled deterministic problems and stochastic equations this method drastically reduces the calculation costs and memory requirements of classical resolution techniques used in the context of galerkin stochastic finite element methods finally this technique is particularly suitable to non linear and evolution problems since it enables the construction of a relevant reduced basis of deterministic functions which can be efficiently reused for subsequent resolutions

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

solution verification goal oriented adaptive methods for stochastic advection diffusion problems abstract a goal oriented analysis of linear stochastic advection diffusion models is presented which provides both a method for solution verification as well as a basis for improving results through adaptation of both the mesh and the way random variables are approximated a class of model problems with random coefficients and source terms is cast in a variational setting specific quantities of interest are specified which are also random variables a stochastic adjoint problem associated with the quantities of interest is formulated and a posteriori error estimates are derived these are used to guide an adaptive algorithm which adjusts the sparse probabilistic grid so as to control the approximation error numerical examples are given to demonstrate the methodology for a specific model problem

galerkin methods for linear and nonlinear elliptic stochastic partial differential equations abstract stationary systems modelled by elliptic partial differential equations linear as well as nonlinear with stochastic coefficients random fields are considered the mathematical setting as a variational problem existence theorems and possible discretisations in particular with respect to the stochastic part are given and investigated with regard to stability different and increasingly sophisticated computational approaches involving both wiener s polynomial chaos as well as the karhunen loeve expansion are addressed in conjunction with stochastic galerkin procedures and stability within the galerkin framework is established new and effective algorithms to compute the mean and covariance of the solution are proposed the similarities and differences with better known monte carlo methods are exhibited as well as alternatives to integration in high dimensional spaces hints are given regarding the numerical implementation and parallelisation numerical examples serve as illustration

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

high order methods as an alternative to using sparse tensor products for stochastic galerkin fem solutions of random elliptic boundary value problems admit efficient approximations by polynomials on the parameter domain each coefficient in such an expansion is a spatially dependent function and can be approximated within a hierarchy of finite element spaces if the finite elements are of sufficiently high order using just a single spatial mesh is predicted to achieve the same convergence rate with respect to the total number of degrees of freedom as sparse tensor product constructions and other multilevel stochastic galerkin approximations numerical computations for an elliptic two point boundary value problem confirm this and indicate no loss of accuracy for a single level method compared to using a sparse tensor product with the same total number of degrees of freedom

crossed products by finite group actions with the rokhlin property we prove that a number of classes of separable unital c algebras are closed under crossed products by finite group actions with the rokhlin property including a ai algebras at algebras and related classes characterized by direct limit decompositions using semiprojective building blocks b simple unital ah algebras with slow dimension growth and real rank zero c c algebras with real rank zero or stable rank one d simple c algebras for which the order on projections is determined by traces e c algebras whose quotients all satisfy the universal coefficient theorem f c algebras with a unique tracial state along the way we give a systematic treatment of the derivation of direct limit decompositions from local approximation conditions by homomorphic images which are not necessarily injective

the tracial rokhlin property for actions of finite groups on c algebras we define tracial analogs of the rokhlin property for actions of finite groups approximate representability of actions of finite abelian groups and approximate innerness we prove the following four analogs of related nontracial results bullet the crossed product of an infinite dimensional simple separable unital c algebra with tracial rank zero by an action of a finite group with the tracial rokhlin property again has tracial rank zero bullet an outer action of a finite abelian group on an infinite dimensional simple separable unital c algebra has the tracial rokhlin property if and only if its dual is tracially approximately representable and is tracially approximately representable if and only if its dual has the tracial rokhlin property bullet if a strongly tracially approximately inner action of a finite cyclic group on an infinite dimensional simple separable unital c algebra has the tracial rokhlin property then it is tracially approximately representable bullet an automorphism of an infinite dimensional simple separable unital c algebra a with tracial rank zero is tracially approximately inner if and only if it is the identity on k 0 a mod infinitesimals

non simple purely infinite c algebras a c algebra a is defined to be purely infiniteif there are no characters on a a nd if for every pair of positive elements a b in a such that b lies in the closed two sided ideal generated by a there exists a sequence rn in a such that rn arn b this definition agrees with the usual definition by j cuntz when a is simple it is shown that the property of being purely infinite is preserved under extensions morita equivalence inductive limits and it passes to quotients and to hereditary sub c algebras it is shown that a is purely infinite for every c algebra a purely infinite c algebras admit no traces and conversely an approximately divisible exact c algebra is purely infinite if it admits no nonzero trace

crossed products by spectrally free actions abstract we define spectral freeness for actions of discrete groups on c algebras we relate spectral freeness to other freeness conditions an example result is that for an action of a finite group g spectral freeness is equivalent to strong pointwise outerness and also to the condition that g 1 for every g g 1 we then prove permanence results for reduced crossed products by exact spectrally free actions for crossed products by arbitrary actions of z 2 z and for extensions direct limits stable isomorphism and several related constructions for the following properties the combination of pure infiniteness and the ideal property residual hereditary infiniteness closely related to pure infiniteness residual sp a strengthening of property sp suitable for nonsimple c algebras the weak ideal property closely related to the ideal property for the weak ideal property we can allow arbitrary crossed products by any finite abelian group these properties of c algebras are shown to have formulations of the same general type allowing them all to be handled using a common set of theorems

a multilevel sparse kernel based stochastic collocation finite element method for elliptic problems with random coefficients abstract a new stochastic collocation finite element method is proposed for the numerical solution of elliptic boundary value problems bvp with random coefficients assuming that the randomness is well approximated by a finite number of random variables with given probability distributions the proposed method consists of a finite element approximation in physical space along with a stochastic collocation quadrature approach utilizing the recent multilevel sparse kernel based interpolation musik technique georgoulis et al 2013 musik is based on a multilevel sparse grid type algorithm with the basis functions consisting of directionally anisotropic gaussian radial basis functions kernels placed at directionally uniform grid points we prove that musik is interpolatory at these nodes and therefore can be naturally used to define a quadrature scheme numerical examples are also presented assessing the performance of the new algorithm in the context of high dimensional stochastic collocation finite element methods

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 wavelet methods for elliptic operator equations convergence rates this paper is concerned with the construction and analysis of wavelet based adaptive algorithms for the numerical solution of elliptic equations these algorithms approximate the solution u of the equation by a linear combination of n wavelets therefore a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of n wavelets so called n term approximation which would be obtained by keeping the n largest wavelet coefficients of the real solution which of course is unknown the main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error o n s in the energy norm whenever such a rate is possible by n term approximation the range of s 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator moreover it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to n the adaptive algorithm applies to a wide class of elliptic problems and wavelet bases the analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization