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

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

simplex stochastic collocation with eno type stencil selection for robust uncertainty quantification multi element uncertainty quantification approaches can robustly resolve the high sensitivities caused by discontinuities in parametric space by reducing the polynomial degree locally to a piecewise linear approximation it is important to extend the higher degree interpolation in the smooth regions up to a thin layer of linear elements that contain the discontinuity to maintain a highly accurate solution this is achieved here by introducing essentially non oscillatory eno type stencil selection into the simplex stochastic collocation ssc method for each simplex in the discretization of the parametric space the stencil with the highest polynomial degree is selected from the set of candidate stencils to construct the local response surface approximation the application of the resulting ssc eno method to a discontinuous test function shows a sharper resolution of the jumps and a higher order approximation of the percentiles near the singularity ssc eno is also applied to a chemical model problem and a shock tube problem to study the impact of uncertainty both on the formation of discontinuities in time and on the location of discontinuities in space

convergence of unsymmetric kernel based meshless collocation methods this paper proves convergence of variations of the unsymmetric kernel based collocation method introduced by kansa in 1986 since then this method has been very successfully used in many applications though it may theoretically fail in special situations and though it had no error bound or convergence proof up to now thus it is necessary to add assumptions or to make modifications our modifications prevent numerical failure by dropping strict collocation and allow a rigorous mathematical analysis proving error bounds and convergence rates these rates improve with the smoothness of the solution the domain and the kernel providing the trial spaces but they are currently not yet optimal and deserve refinement they are based on rates of approximation to the residuals by nonstationary meshless kernel based trial spaces and they are independent of the type of differential operator the results are applicable to large classes of linear problems in strong form provided that there is a smooth solution and the test and trial discretizations are chosen with some care our analysis does not require assumptions like ellipticity and it can be extended to ill posed problems

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

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

further analysis of multilevel monte carlo methods for elliptic pdes with random coefficients we consider the application of multilevel monte carlo methods to elliptic pdes with random coefficients we focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter and that only have limited spatial regularity we extend the finite element error analysis for this type of equation carried out in charrier et al siam j numer anal 2013 to more difficult problems posed on non smooth domains and with discontinuities in the coefficient for this wider class of model problem we prove convergence of the multilevel monte carlo algorithm for estimating any bounded linear functional and any continuously frechet differentiable non linear functional of the solution we further improve the performance of the multilevel estimator by introducing level dependent truncations of the karhunen loeve expansion of the random coefficient numerical results complete the paper

unsymmetric meshless methods for operator equations a general framework for proving error bounds and convergence of a large class of unsymmetric meshless numerical methods for solving well posed linear operator equations is presented the results provide optimal convergence rates if the test and trial spaces satisfy a stability condition operators need not be elliptic and the problems can be posed in weak or strong form without changing the theory non stationary kernel based trial and test spaces are shown to fit into the framework disregarding the operator equation as a special case unsymmetric meshless kernel based methods solving weakly posed problems with distributional data are treated in some detail this provides a foundation of certain variations of the meshless local petrov galerkin technique of s n atluri and collaborators

aggregating preference rankings using owa operator weights one important issue of aggregating preference rankings is to determine the weights of different ranking places this paper proposes the use of ordered weighted averaging owa operator weights to aggregate preference rankings which allows the weights associated with different ranking places to be determined in terms of a decision maker dm s optimism level characterized by an orness degree by adjusting the dm s optimism level ties can be avoided and winner can be selected two numerical examples are examined using owa operator weights to show their applications simplicity and flexibility in aggregating preference rankings

torsional springs for two dimensional dynamic unstructured fluid meshes abstract dynamic fluid grids are commonly used for the solution of flow problems with moving boundaries they are often represented by a network of fictitious lineal springs that can become unreliable when the fluid mesh undergoes large displacements and or deformations in this paper we propose to control the arbitrary motion of two dimensional dynamic unstructured fluid grids with additional torsional springs we show that such springs can be designed to prohibit the interpenetration of neighboring triangles and therefore to provide the method of spring analogy with the robustness needed for enlarging its range of applications we illustrate our new dynamic mesh motion algorithm with several examples that highlight its advantages in terms of robustness quality and performance

uncertainty quantification for systems of conservation laws uncertainty quantification through stochastic spectral methods has been recently applied to several kinds of non linear stochastic pdes in this paper we introduce a formalism based on kinetic theory to tackle uncertain hyperbolic systems of conservation laws with polynomial chaos pc methods the idea is to introduce a new variable the entropic variable in bijection with our vector of unknowns which we develop on the polynomial basis by performing a galerkin projection we obtain a deterministic system of conservation laws we state several properties of this deterministic system in the case of a general uncertain system of conservation laws we then apply the method to the case of the inviscid burgers equation with random initial conditions and we present some preliminary results for the euler system we systematically compare results from our new approach to results from the stochastic galerkin method in the vicinity of discontinuities the new method bounds the oscillations due to gibbs phenomenon to a certain range through the entropy of the system without the use of any adaptative random space discretizations it is found to be more precise than the stochastic galerkin method for smooth cases but above all for discontinuous cases

multi level monte carlo finite element method for elliptic pdes with stochastic coefficients in monte carlo methods quadrupling the sample size halves the error in simulations of stochastic partial differential equations spdes the total work is the sample size times the solution cost of an instance of the partial differential equation a multi level monte carlo method is introduced which allows in certain cases to reduce the overall work to that of the discretization of one instance of the deterministic pde the model problem is an elliptic equation with stochastic coefficients multi level monte carlo errors and work estimates are given both for the mean of the solutions and for higher moments the overall complexity of computing mean fields as well as k point correlations of the random solution is proved to be of log linear complexity in the number of unknowns of a single multi level solve of the deterministic elliptic problem numerical examples complete the theoretical analysis

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

a continuation multi level monte carlo c mlmc method for uncertainty quantification in compressible inviscid aerodynamics in this work we apply the continuation multi level monte carlo c mlmc algorithm proposed in collier et al 2014 to efficiently propagate operating and geometric uncertainties in inviscid compressible aerodynamics numerical simulations the key idea of mlmc is that one can draw mc samples simultaneously and independently on several approximations of the problem under investigations on a hierarchy of nested computational grids levels the expectation of an output quantity is computed as a sample average using the coarsest solutions and corrected by averages of the differences of the solutions of two consecutive grids in the hierarchy by this way most of the computational effort is transported from the finest level as in a standard monte carlo approach to the coarsest one the continuation algorithm c mlmc is a robust and self tuning version that estimates on the fly the optimal number of level and realizations per level in this work we describe in detail how c mlmc can be adapted to perform uncertainty quantification analysis in compressible aerodynamics and we apply it to the relevant test cases of a quasi 1d convergent divergent laval nozzle and the 2d transonic rae 2822 airfoil

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

all well posed problems have uniformly stable and convergent discretizations this paper considers a large class of linear operator equations including linear boundary value problems for partial differential equations and treats them as linear recovery problems for functions from their data well posedness of the problem means that this recovery is continuous discretization recovers restricted trial functions from restricted test data and it is well posed or stable if this restricted recovery is continuous after defining a general framework for these notions this paper proves that all well posed linear problems have stable and refinable computational discretizations with a stability that is determined by the well posedness of the problem and independent of the computational discretization provided that sufficiently many test data are used the solutions of discretized problems converge when enlarging the trial spaces and the convergence rate is determined by how well the data of the function solving the analytic problem can be approximated by the data of the trial functions this allows new and very simple proofs of convergence rates for generalized finite elements symmetric and unsymmetric kansa type collocation and other meshfree methods like meshless local petrov galerkin techniques it is also shown that for a fixed trial space weak formulations have a slightly better convergence rate than strong formulations but at the expense of numerical integration since convergence rates are reduced to those coming from approximation theory and since trial spaces are arbitrary this also covers various spectral and pseudospectral methods all of this is illustrated by examples

convergence analysis of general spectral methods abstract if a spectral numerical method for solving ordinary or partial differential equations is written as a biinfinite linear system b z a with a map z l 2 l 2 that has a continuous inverse this paper shows that one can discretize the biinfinite system in such a way that the resulting finite linear system b z a is uniquely solvable and is unconditionally stable i e the stability can be made to depend on z only not on the discretization convergence rates of finite approximations b of b then carry over to convergence rates of finite approximations a of a spectral convergence is a special case some examples are added for illustration

finite element error analysis of elliptic pdes with random coefficients and its application to multilevel monte carlo methods we consider a finite element approximation of elliptic partial differential equations with random coefficients such equations arise for example in uncertainty quantification in subsurface flow modeling models for random coefficients frequently used in these applications such as log normal random fields with exponential covariance have only very limited spatial regularity and lead to variational problems that lack uniform coercivity and boundedness with respect to the random parameter in our analysis we overcome these challenges by a careful treatment of the model problem almost surely in the random parameter which then enables us to prove uniform bounds on the finite element error in standard bochner spaces these new bounds can then be used to perform a rigorous analysis of the multilevel monte carlo method for these elliptic problems that lack full regularity and uniform coercivity and boundedness to conclude we give some numerical results that confirm the new bounds