bayesian guided pattern search for robust local optimization optimization for complex systems in engineering often involves the use of expensive computer simulation by combining statistical emulation using treed gaussian processes with pattern search optimization we are able to perform robust local optimization more efficiently and effectively than when using either method alone our approach is based on the augmentation of local search patterns with location sets generated through improvement prediction over the input space we further develop a computational framework for asynchronous parallel implementation of the optimization algorithm we demonstrate our methods on two standard test problems and our motivating example of calibrating a circuit device simulator

on discrete least squares projection in unbounded domain with random evaluations and its application to parametric uncertainty quantification this work is concerned with approximating multivariate functions in an unbounded domain by using a discrete least squares projection with random point evaluations particular attention is given to functions with random gaussian or gamma parameters we first demonstrate that the traditional hermite laguerre polynomials chaos expansion suffers from the instability in the sense that an unfeasible number of points which is relevant to the dimension of the approximation space is needed to guarantee the stability in the least squares framework we then propose to use the hermite laguerre functions rather than polynomials as bases in the expansion the corresponding design points are obtained by mapping the uniformly distributed random points in bounded intervals to the unbounded domain which involved a mapping parameter l by using the hermite laguerre functions and a proper mapping parameter the stability can be significantly improved even if the number of design points scales linearly up to a logar

a stochastic projection method for fluid flow ii random process an uncertainty quantification scheme is developed for the simulation of stochastic thermofluid processes the scheme relies on spectral representation of uncertainty using the polynomial chaos pc system the solver combines a galerkin procedure for the determination of pc coefficients with a projection method for efficiently simulating the resulting system of coupled transport equations implementation of the numerical scheme is illustrated through simulations of natural convection in a 2d square cavity with stochastic temperature distribution at the cold wall the properties of the uncertainty representation scheme are analyzed and the predictions are contrasted with results obtained using a monte carlo approach

multivariate discrete least squares approximations with a new type of collocation grid in this work we discuss the problem of approximating a multivariate function by discrete least squares projection onto a polynomial space using a specially designed deterministic point set the independent variables of the function are assumed to be random variables stemming from the motivating application of uncertainty quantification our deterministic points are inspired by a theorem due to andre weil we first work with the chebyshev measure and consider the approximation in chebyshev polynomial spaces we prove the stability and an optimal convergence estimate provided the number of points scales quadratically with the dimension of the polynomial space a possible application for quantifying epistemic uncertainties is then discussed we show that the point set asymptotically equidistributes to the product chebyshev measure allowing us to propose a weighted least squares framework and extending our method to more general polynomial approximations numerical examples are given to confirm the theore

fuzzy s irresolute mappings in this paper the concepts of fuzzy s irresolute semi s irresolute and s precontinuous mappings in fuzzy topological spaces in sostak sense are introduced and studied some of their characteristic properties are considered also fuzzy s connectedness is introduced and studied with the help of r fuzzy strongly semi open sets

novel computational methods for high dimensional stochastic sensitivity analysis summary this paper presents three new computational methods for calculating design sensitivities of statistical moments and reliability of high dimensional complex systems subject to random input the first method represents a novel integration of the polynomial dimensional decomposition pdd of a multivariate stochastic response function and score functions applied to the statistical moments the method provides mean square convergent analytical expressions of design sensitivities of the first two moments of a stochastic response the second and third methods relevant to probability distribution or reliability analysis exploit two distinct combinations built on pdd the pdd saddlepoint approximation spa or pdd spa method entailing spa and score functions and the pdd monte carlo simulation mcs or pdd mcs method utilizing the embedded mcs of the pdd approximation and score functions for all three methods developed the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation numerical examples including a 100 dimensional mathematical problem indicate that the new methods developed provide not only theoretically convergent or accurate design sensitivities but also computationally efficient solutions a practical example involving robust design optimization of a three hole bracket illustrates the usefulness of the proposed methods copyright 2014 john wiley sons ltd

on characterizations of some covering properties in l fuzzy topological spaces in sostak s sense in this work various characterizations of some weaker and stronger forms of l fuzzy compactness are obtained in l fuzzy topological spaces in sostak s sense where l is a fuzzy lattice

compressive sampling of polynomial chaos expansions sampling orthogonal polynomial bases via monte carlo is of interest for uncertainty quantification of models with random inputs using polynomial chaos pc expansions it is known that bounding a probabilistic parameter referred to as coherence yields a bound on the number of samples necessary to identify coefficients in a sparse pc expansion via solution to an 1 minimization problem utilizing results for orthogonal polynomials we bound the coherence parameter for polynomials of hermite and legendre type under their respective natural sampling distribution in both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation for more general orthonormal bases we propose the coherence optimal sampling a markov chain monte carlo sampling which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support we demonstrate these different sampling strategies numerically in both high order and high dimensional manufactured pc expansions in addition the quality of each sampling method is compared in the identification of solutions to two differential equations one with a high dimensional random input and the other with a high order pc expansion in both cases the coherence optimal sampling scheme leads to similar or considerably improved accuracy

k fan s theorem in fuzzifying topology in this paper we offer an application of ukasiewicz logic in topology by ukasiewicz logic we introduce a new definition of connectivity in fuzzifying topology and prove k fan s theorem of connectivity in fuzzifying topology

a data driven stochastic collocation approach for uncertainty quantification in mems this work presents a data driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi physics simulation of micro electromechanical systems mems the proposed framework comprises of two key steps first probabilistic characterization of the input uncertain parameters based on available experimental information and second propagation of these uncertainties through the predictive model to relevant quantities of interest the uncertain input parameters are modeled as independent random variables for which the distributions are estimated based on available experimental observations using a nonparametric diffusion mixing based estimator botev nonparametric density estimation via diffusion mixing technical report 2007 the diffusion based estimator derives from the analogy between the kernel density estimation kde procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent the diffusion model allows for the incorporation of the prior density and leads to an improved density estimate in comparison with the standard kde approach as demonstrated through several numerical examples following the characterization step the uncertainties are propagated to the output variables using the stochastic collocation approach based on sparse grid interpolation smolyak soviet math dokl 1963 4 240 243 the developed framework is used to study the effect of variations in young s modulus induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a mems switch copyright 2010 john wiley sons ltd

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

review and unification of methods for computing derivatives of multidisciplinary computational models this paper presents a review of all existing discrete methods for computing the derivatives of computational models within a unified mathematical framework this framework hinges on a new equation the unifying chain rule from which all the methods can be derived the computation of derivatives is described as a two step process the evaluation of the partial derivatives and the computation of the total derivatives which are dependent on the partial derivatives finite differences the complex step method and symbolic differentiation are discussed as options for computing the partial derivatives it is shown that these are building blocks with which the total derivatives can be assembled using algorithmic differentiation the direct and adjoint methods and coupled analytic methods for multidisciplinary systems each of these methods is presented and applied to a common numerical example to demonstrate and compare the various approaches the paper ends with a discussion of current challenges and possib

on i fuzzy topologies derived from a decreasing family of i topologies the aim of this paper is to study i fuzzy topologies fuzzy topologies in sostak s sense derived from decreasing families of i topologies fuzzy topologies in chang s sense defined on a set x

three topological structures of smooth topological spaces abstract various concepts related to smooth topological spaces have been introduced and relations among them studied by several authors 2 6 in this paper we introduce the notions of smooth quasi smooth and weak smooth structure showing that various properties of a smooth topological space can be expressed in terms of these structures in particular the definitions and the results of 2 6 may be expressed in terms of the smooth and quasi smooth structures then we will set out the fundamental concepts of these three structures and the relations between them

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

gradations of openness and chang s fuzzy topologies abstract we study some properties of gradations of openness defined on a set x and prove that each gradation of openness is the supremum infimum of a strictly increasing decreasing sequence of gradations of openness which are equivalent to also we characterize those fuzzy topological spaces x t with the property that there exists a gradation of openness from i x onto i such that g t iff g 0

gradations of uniformity and gradations of proximity abstract in 1992 a new definition of a fuzzy topological space was given by introducing the concept of a gradation of openness in this paper a new definition of a fuzzy uniform resp proximity space is given by introducing the concept of a gradation of uniformity resp proximity on a non empty set x how to construct a gradation of openness induced by a gradation of uniformity resp proximity is explained the connexions between gradations of proximity and gradations of uniformity are investigated

uncertainty quantification of limit cycle oscillations different computational methodologies have been developed to quantify the uncertain response of a relatively simple aeroelastic system in limit cycle oscillation subject to parametric variability the aeroelastic system is that of a rigid airfoil supported by pitch and plunge structural coupling with nonlinearities in the component in pitch the nonlinearities are adjusted to permit the formation of a either a subcritical or supercritical branch of limit cycle oscillations uncertainties are specified in the cubic coefficient of the torsional spring and in the initial pitch angle of the airfoil stochastic projections of the time domain and cyclic equations governing system response are carried out leading to both intrusive and non intrusive computational formulations non intrusive formulations are examined using stochastic projections derived from wiener expansions involving haar wavelet and b spline bases while wiener hermite expansions of the cyclic equations are employed intrusively and non intrusively application of the b spline stochastic projection is extended to the treatment of aerodynamic nonlinearities as modeled through the discrete euler equations the methodologies are compared in terms of computational cost convergence properties ease of implementation and potential for application to complex aeroelastic systems

sensitivity analysis with finite changes an application to modified eoq models abstract in this work we introduce a new method for the sensitivity analysis of model output in the presence of finite changes in one or more of the exogenous variables we define sensitivity measures that do not rest on differentiability we relate the sensitivity measures to classical differential and comparative statics indicators we prove a result that allows us to obtain the sensitivity measures at the same cost of one variable at a time methods thus making their estimation feasible also for computationally intensive models we discuss in detail the derivation of managerial insights formulating a procedure based on the concept of settings the method is applied to the sensitivity analysis of a discrete change in optimal order quantity following a jump in the exogenous variables of a non linear programming inventory model

fuzzy proximities and fuzzy uniformities abstract a new definition of fuzzy proximity is given and some of its properties are studied we also give a definition of fuzzy uniformity whose underlying fuzzy proximity is a fuzzy proximity in our sense

the rost nilpotence theorem for geometrically rational surfaces we show that the rost nilpotence theorem is true for geometrically rational surfaces over fields of characteristic zero

a posteriori error estimation and adaptivity in stochastic galerkin fem for parametric elliptic pdes beyond the affine case abstract we consider a linear elliptic partial differential equation pde with a generic uniformly bounded parametric coefficient the solution to this pde problem is approximated in the framework of stochastic galerkin finite element methods we perform a posteriori error analysis of galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations practical versions of this error estimate are discussed and tested numerically for a model problem with non affine parametric representation of the coefficient furthermore we use the error reduction indicators derived from spatial and parametric error estimators to guide an adaptive solution algorithm for the given parametric pde problem the performance of the adaptive algorithm is tested numerically for model problems with two different non affine parametric representations of the coefficient

tolerance sensitivity and optimality bounds in linear programming traditional sensitivity analysis in linear programming usually focuses on variations of one coefficient or term at a time the tolerance approach was proposed to provide a decision maker with an effective and easy to use method to summarize the effects of simultaneous and independent changes in selected parameters in particular for variations of the objective function coefficients the approach gives a maximum tolerance percentage within which selected coefficients may vary from their estimated values within a priori limits while still retaining the same optimal basic feasible solution although an optimal solution may cease being optimal for variations beyond the maximum tolerance percentage it may still be close to optimal herein we characterize the potential loss of optimality for variations beyond the maximum tolerance percentage as a maximum regret function we consider theoretical properties of this function and propose a method to compute a relevant portion of it

analysis of regression confidence intervals and bayesian credible intervals for uncertainty quantification 1 confidence intervals based on classical regression theories augmented to include prior information and credible intervals based on bayesian theories are conceptually different ways to quantify parametric and predictive uncertainties because both confidence and credible intervals are used in environmental modeling we seek to understand their differences and similarities this is of interest in part because calculating confidence intervals typically requires tens to thousands of model runs while bayesian credible intervals typically require tens of thousands to millions of model runs given multi gaussian distributed observation errors our theoretical analysis shows that for linear or linearized nonlinear models confidence and credible intervals are always numerically identical when consistent prior information is used for nonlinear models nonlinear confidence and credible intervals can be numerically identical if parameter confidence regions defined using the approximate likelihood method and parameter credible regions estimated using markov chain monte carlo realizations are numerically identical and predictions are a smooth monotonic function of the parameters both occur if intrinsic model nonlinearity is small while the conditions of gaussian errors and small intrinsic model nonlinearity are violated by many environmental models heuristic tests using analytical and numerical models suggest that linear and nonlinear confidence intervals can be useful approximations of uncertainty even under significantly nonideal conditions in the context of epistemic model error for a complex synthetic nonlinear groundwater problem the linear and nonlinear confidence and credible intervals for individual models performed similarly enough to indicate that the computationally frugal confidence intervals can be useful in many circumstances experiences with these groundwater models are expected to be broadly applicable to many environmental models we suggest that for environmental problems with lengthy execution times that make credible intervals inconvenient or prohibitive confidence intervals can provide important insight during model development when frequent calculation of uncertainty intervals is important to understanding the consequences of various model construction alternatives and data collection strategies strategic use of both confidence and credible intervals can be critical

probabilistic collocation for period 1 limit cycle oscillations in this paper probabilistic collocation for limit cycle oscillations pclco is proposed probabilistic collocation pc is a non intrusive approach to compute the polynomial chaos description of uncertainty numerically polynomial chaos can require impractical high orders to approximate long term time integration problems due to the fast increase of required polynomial chaos order with time pclco is a pc formulation for modeling the long term stochastic behavior of dynamical systems exhibiting a periodic response i e a limit cycle oscillation lco in the pc method deterministic time series are computed at collocation points in probability space in pclco pc is applied to a time independent parametrization of the periodic response of the deterministic solves instead of to the time dependent functions themselves due to the time independent parametrization the accuracy of pclco is independent of time the approach is applied to period 1 oscillations with one main frequency subject to a random parameter numerical results are presented for the harmonic oscillator a two dof airfoil flutter model and the fluid structure interaction of an elastically mounted cylinder

fuzzy proximities structures and fuzzy grills abstract in this paper definitions of basic proximities of fuzzy sets and basic fuzzy proximities are given and a classification scheme for proximities of fuzzy sets as well as fuzzy proximities are introduced

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

convergence in i fuzzy topological spaces the purpose of this work is to combine the fuzzy q neighborhood system of fuzzy points introduced by sostak with the convergence structures of lee et al using the l filter approach of gahler and eklund

uncertainty quantification in chemical systems summary we demonstrate the use of multiwavelet spectral polynomial chaos techniques for uncertainty quantification in non isothermal ignition of a methane air system we employ bayesian inference for identifying the probabilistic representation of the uncertain parameters and propagate this uncertainty through the ignition process we analyze the time evolution of moments and probability density functions of the solution we also examine the role and significance of dependence among the uncertain parameters we finish with a discussion of the role of non linearity and the performance of the algorithm copyright q 2009 john wiley sons ltd

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

on several types of degree of fuzzy compactness abstract sostak 1985 defined a fuzzy topology on a set x as a fuzzy subset of the family i x of fuzzy subsets of x satisfying some axioms the concept of the degree of compactness of a fuzzy set was first introduced by sostak 1985 in this paper we discuss the degrees of almost compactness near compactness countable compactness light compactness and strong compactness of a fuzzy set also we investigate the behavior of the degrees of compactness under several types of fuzzy continuous functions

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

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

provably stable overset grid methods for computational aeroacoustics the simulation of sound generating flows in complex geometries requires accurate numerical methods that are non dissipative and stable and well posed boundary conditions a structured mesh approach is often desired for a higher order discretization that better uses the provided grids but at the expense of complex geometry capabilities relative to techniques for unstructured grids one solution is to use an overset mesh based discretization where locally structured meshes are globally assembled in an unstructured manner this article discusses recent advancements in overset methods also called chimera methods concerning boundary conditions parallel methods for overset grid management and stable and accurate interpolation between the grids several examples are given some of which include moving grids

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

coherence motivated sampling and convergence analysis of least squares polynomial chaos regression independent sampling of orthogonal polynomial bases via monte carlo is of interest for uncertainty quantification of models using polynomial chaos pc expansions it is known that bounding the spectral radius of a random matrix consisting of pc samples yields a bound on the number of samples necessary to identify coefficients in the pc expansion via solution to a least squares regression problem we present a related analysis which guarantees a mean square convergence using a coherence parameter of the sampled pc basis that may be both analytically bounded and computationally estimated utilizing asymptotic results for orthogonal polynomials we bound the coherence parameter for polynomials of hermite and legendre type under each respective natural sampling distribution in both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the pc basis for more general orthonormal bases we propose the coherence optimal sampling a markov chain monte carlo sampling which directly uses the basis functions under consideration to achieve a statistical optimality among all such sampling schemes with identical support and which guarantees recovery with a number of samples that is up to a logarithmic factor linear in the number of basis functions considered we demonstrate these different sampling strategies numerically in both high order and high dimensional manufactured pc expansions in addition the quality of each sampling method is compared in the identification of solutions to two differential equations one with a high dimensional random input and the other with a high order pc expansion in all observed cases the coherence optimal sampling leads to similar or considerably improved accuracy over the other considered sampling distributions

on several types of compactness in smooth topological spaces abstract the smooth closure and smooth interior of a fuzzy set w r t a smooth topology were defined by gayyar et al 1994 and some relations between a few types of compactness were established in the presence of strong restrictions in this paper by constructing new definitions of smooth closure and smooth interior which have more desirable properties than those of gayyar et al 1994 we prove that several hypothesis in the results of gayyar et al 1994 can be weakened and show that the relations which hold between various types of compactness in fuzzy topological spaces in chang s sense cfts for short di concilio and gerla 1984 haydar es 1987 can be extended to smooth topological spaces

approximation of quantities of interest in stochastic pdes by the random discrete l 2 projection on polynomial spaces in this work we consider the random discrete l 2 projection on polynomial spaces hereafter rdp for the approximation of scalar quantities of interest qois related to the solution of a partial differential equation model with random input parameters in the rdp technique the qoi is first computed for independent samples of the random input parameters as in a standard monte carlo approach and then the qoi is approximated by a multivariate polynomial function of the input parameters using a discrete least squares approach we consider several examples including the darcy equations with random permeability the linear elasticity equations with random elastic coefficient and the navier stokes equations in random geometries and with random fluid viscosity we show that the rdp technique is well suited to qois that depend smoothly on a moderate number of random parameters our numerical tests confirm the theoretical findings in g migliorati f nobile e von schwerin and r tempone analysis of the

adaptive sparse polynomial dimensional decomposition methods for high dimensional stochastic computing this article presents two novel adaptive sparse polynomial dimensional decomposition pdd methods for solving high dimensional uncertainty quantification problems in computational science and engineering the methods entail global sensitivity analysis for retaining important pdd component functions and a full or sparse grid dimension reduction integration or quasi monte carlo simulation for estimating the pdd expansion coefficients a unified algorithm endowed with two distinct ranking schemes for grading component functions was created for their numerical implementation the fully adaptive sparse pdd method is comprehensive and rigorous leading to the second moment statistics of a stochastic response that converges to the exact solution when the tolerances vanish a partially adaptive sparse pdd method obtained through regulated adaptivity and sparsity is economical and is therefore expected to solve practical problems with numerous variables compared with past developments the adaptive sparse pdd methods do not require their truncation parameter s to be assigned a priori or arbitrarily the numerical results reveal that an adaptive sparse pdd method achieves a desired level of accuracy with considerably fewer coefficients compared with existing pdd approximations for a required accuracy in calculating the probabilistic response characteristics the new bivariate adaptive sparse pdd method is more efficient than the existing bivariately truncated pdd method by almost an order of magnitude finally stochastic dynamic analysis of a disk brake system was performed demonstrating the ability of the new methods to tackle practical engineering problems

fuzzy topology redefined abstract in this paper we give a new definition of fuzzy topology by introducing a concept of gradation of openness of fuzzy subsets and then study fuzzy continuity

global sensitivity analysis in applications of operations research models decision makers must assess the sensitivity of outputs to imprecise values for some of the model s parameters existing analytic approaches for classic optimization models rely heavily on duality properties for assessing the impact of local parameter variations parametric programming for examining systematic variations in model coefficients or stochastic programming for ascertaining a robust solution this paper accommodates extensive simultaneous variations in any of an operations research model s parameters for constrained optimization models the paper demonstrates practical approaches for determining relative parameter sensitivity with respect to a model s optimal objective function value decision variables and other analytic functions of a solution relative sensitivity is assessed by assigning a portion of variation in an output value to each parameter that is imprecisely specified the computing steps encompass optimization monte carlo sampling

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

a polynomial dimensional decomposition for stochastic computing this article presents a new polynomial dimensional decomposition method for solving stochastic problems commonly encountered in engineering disciplines and applied sciences the method involves a hierarchical decomposition of a multivariate response function in terms of variables with increasing dimensions a broad range of orthonormal polynomial bases consistent with the probability measure for fourier polynomial expansion of component functions and an innovative dimension reduction integration for calculating the coefficients of the expansion the new decomposition method does not require sample points as in the previous version yet it generates a convergent sequence of lower variate estimates of the probabilistic characteristics of a generic stochastic response the results of five numerical examples indicate that the proposed decomposition method provides accurate convergent and computationally efficient estimates of the tail probability of random mathematical functions or the reliability of mechanical systems

time varying sensitivity analysis clarifies the effects of watershed model formulation on model behavior 1 lumped rainfall runoff models are widely used for flow prediction but a long recognized need exists for diagnostic tools to determine whether the process level behavior of a model aligns with the expectations inherent in its formulation to this end we develop a comprehensive exploration of dominant parameters in the hymod hbv and sacramento soil moisture accounting sac sma model structures model controls are isolated using time varying sobol sensitivity analysis for twelve mopex watersheds in the eastern united states over a 10 year period sensitivity indices are visualized along gradients of observed precipitation and streamflow to identify key behavioral differences between the three models and to connect these back to the models underlying assumptions results indicate that the models dominant parameters strongly depend on time varying hydroclimatic conditions parameters associated with surface processes such as evapotranspiration and runoff generally dominate under dry conditions when high evaporative fluxes are required for accurate simulation parameters associated with routing processes typically dominate under high flow conditions when performance depends on the timing of flow events the results highlight significant inter model differences in performance controls even in cases where the models share similar process formulations the dominant parameters identified can be counterintuitive even these simple models represent complex nonlinear systems and the links between formulation and behavior are difficult to discern a priori as complexity increases scrutinizing the links between model formulation and behavior becomes an important diagnostic approach particularly in applications such as predictions under change where dominant model controls will shift under hydrologic extremes

using bounds on the data in linear programming the tolerance approach to sensitivity analysis in contrast to traditional sensitivity analysis in linear programming the tolerance approach considers simultaneous and independent variations in a number of parameters a primary focus of this approach is to determine a maximum tolerance percentage for selected right hand side terms in which the same basis is optimal as long as each term is accurate to within that percentage of its estimated value similarly the approach yields a maximum tolerance percentage for selected objective function coefficients this paper shows how the tolerance approach can exploit information on the range of possible values over which terms and coefficients can vary to yield larger maximum tolerance percentages

towards essential improvement for the parareal tr and parareal gauss4 algorithms parareal is an iterative algorithm and is characterized by two propagators g and f which are respectively associated with large step size t and small step size t where t j t and j 2 is an integer for symmetric positive definite spd system u t a u t g t arising from semi discretizing time dependent pdes if we fix the g propagator to the backward euler method and choose for f some l stable time integrator it can be proven that the convergence factors of the corresponding parareal algorithms satisfy 1 3 j 2 and a 0 where a is the spectrum of the matrix a however this result does not hold when time integrators that lack l stability such as the trapezoidal rule and the 4th order gauss rk method are chosen as the f propagator the parareal algorithms using these two methods for the f propagator are denoted by parareal tr and parareal gauss4 in this paper we propose a strategy to let these two parareal algorithms possess such a uniform convergence property the idea is to choose an l stable propagator f and on each coarse time interval t n t n 1 we perform first two steps of f then followed by j 2 steps of f precisely for the trapezoidal rule we select the 2nd order sdirk method as the f propagator and for the 4th order gauss rk method we select the 4th order lobatto iii c method as the f propagator numerical results are given to support our theoretical conclusions

basis adaptive sample efficient polynomial chaos base pc abstract for a large class of orthogonal basis functions there has been a recent identification of expansion methods for computing accurate stable approximations of a quantity of interest this paper presents within the context of uncertainty quantification a practical implementation using basis adaptation and coherence motivated sampling which under assumptions has satisfying guarantees this implementation is referred to as basis adaptive sample efficient polynomial chaos base pc a key component of this is the use of anisotropic polynomial order which admits evolving global bases for approximation in an efficient manner leading to consistently stable approximation for a practical class of smooth functionals this fully adaptive non intrusive method requires no a priori information of the solution and has satisfying theoretical guarantees of recovery a key contribution to stability is the use of a presented correction sampling for coherence optimal sampling in order to improve stability and accuracy within the adaptive basis scheme theoretically the method may dramatically reduce the impact of dimensionality in function approximation and numerically the method is demonstrated to perform well on problems with dimension up to 1000

probabilistic sensitivity analysis of complex models a bayesian approach summary in many areas of science and technology mathematical models are built to simulate complex real world phenomena such models are typically implemented in large computer programs and are also very complex such that the way that the model responds to changes in its inputs is not transparent sensitivity analysis is concerned with understanding how changes in the model inputs influence the outputs this may be motivated simply by a wish to understand the implications of a complex model but often arises because there is uncertainty about the true values of the inputs that should be used for a particular application a broad range of measures have been advocated in the literature to quantify and describe the sensitivity of a model s output to variation in its inputs in practice the most commonly used measures are those that are based on formulating uncertainty in the model inputs by a joint probability distribution and then analysing the induced uncertainty in outputs an approach which is known as probabilistic sensitivity analysis we present a bayesian framework which unifies the various tools of prob abilistic sensitivity analysis the bayesian approach is computationally highly efficient it allows effective sensitivity analysis to be achieved by using far smaller numbers of model runs than standard monte carlo methods furthermore all measures of interest may be computed from a single set of runs

a derivative based sensitivity measure of failure probability in the presence of epistemic and aleatory uncertainties in order to analyze the effect of the epistemic uncertainty of random variables distribution parameters on the safety of the structure system a novel sensitivity measure of the failure probability is constructed by integrating the derivative of failure probability in the parameter space compared with the variance based sensitivity index the new derivative based sensitivity measure can be evaluated with less computational cost it is noticed that the ranking of the new derivative based sensitivity measure is the same as that of the variance based one for the problem of the variance based sensitivity index with large computational cost the standard sobol s method is employed and the quasi monte carlo method and double loop point estimate method are then utilized to compute the derivative based sensitivity measure for comparison four examples are employed to demonstrate the reasonability of the proposed sensitivity measure and the efficiency of the proposed method

permutation modules and chow motives of geometrically rational surfaces abstract we prove that the chow motive with integral coefficient of a geometrically rational surfaces s over a perfect field k is zero dimensional if and only if the picard group of k k s where k is an algebraic closure of k is a direct summand of a gal k k permutation module and s possesses a zero cycle of degree one as shown by colliot thelene in a letter to the author which we have reproduced in the appendix this is in turn equivalent to s having a zero cycle of degree 1 and ch 0 k s k s being torsion free

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

sensitivity analysis and parameter identification of a time dependent constitutive model for rock salt the tendency to shift from fossil and nuclear energy sources to renewable energy carriers has increased during the past couple of decades subsequently development of effective energy storage systems has become more attractive nowadays caverns excavated in rock salt formations are recognized as the appropriate places for underground storage of energy in the form of compressed air or hydrogen accurate design of these underground cavities requires suitable numerical simulations employing appropriate constitutive models to describe the material behavior of rock salt under various geological conditions it is obvious that to have a realistic numerical simulation it is essential to have a comprehensive knowledge concerning the unknown material parameters and their influence on the calculation results in this paper a time dependent model is selected to describe the mechanical response of the rock salt around the cavern this model is implemented in a finite element code and its application in numerical modeling of salt caverns is illustrated in addition global sensitivity analysis is used to investigate the influence of material parameters on the mechanical behavior of the salt cavern finally inverse analysis of the synthetic data is performed to identify the material parameters of the selected model the applied global sensitivity and inverse analysis algorithms employ metamodeling technique in order to reduce the time which is needed for these computationally expensive calculations

on separation axioms in a newly defined fuzzy topology abstract in fuzzy sets and systems 45 1992 79 82 hazra et al defined a fuzzy topology on a non empty set in a new fashion here we introduce separation axioms and some allied notions in this new set up and prove several results related to these notions

fuzzy topology fuzzy closure operator fuzzy compactness and fuzzy connectedness abstract in this paper definitions of fuzzy closure operator fuzzy compactness and fuzzy connectedness are given in a redefined fuzzy topological space and then some characteristic properties of fuzzy closure operator product theorems of fuzzy compactness invariance of fuzzy compactness and fuzzy connectedness under gradation preserving maps and invariance of fuzzy connectedness under the fuzzy closure operator are studied

polynomial chaos expansion with latin hypercube sampling for estimating response variability a computationally efficient procedure for quantifying uncertainty and finding significant parameters of uncertainty models is presented to deal with the random nature of input parameters of structural models several efficient probabilistic methods are investigated specifically the polynomial chaos expansion with latin hypercube sampling is used to represent the response of an uncertain system latin hypercube sampling is employed for evaluating the generalized fourier coefficients of the polynomial chaos expansion because the key challenge in uncertainty analysis is to find the most significant components that drive response variability analysis of variance is employed to find the significant parameters of the approximation model several analytical examples and a large finite element model of a joined wing are used to verify the effectiveness of this procedure

gradation of openness fuzzy topology abstract in an earlier paper we have given a new definition of fuzzy topology by introducing a concept of gradation of openness of fuzzy subsets in order to make the concept more appropriate in this paper we modify the definition of gradation function and then study subspace of fuzzy topological spaces gradation preserving maps and category of fuzzy topological spaces and gradation preserving maps

invariant probabilistic sensitivity analysis in evaluating opportunities investors wish to identify key sources of uncertainty we propose a new way to measure how sensitive model outputs are to each probabilistic input e g revenues growth idiosyncratic risk parameters we base our approach on measuring the distance between cumulative distributions risk profiles using a metric that is invariant to monotonic transformations thus the sensitivity measure will not vary by alternative specifications of the utility function over the output to measure separation we propose using either kuiper s metric or kolmogorov smirnov s metric we illustrate the advantages of our proposed sensitivity measure by comparing it with others most notably the contribution to variance measures our measure can be obtained as a by product of a monte carlo simulation we illustrate our approach in several examples focusing on investment analysis situations this paper was accepted by peter wakker decision analysis

weighted discrete least squares polynomial approximation using randomized quadratures we discuss the problem of polynomial approximation of multivariate functions using discrete least squares collocation the problem stems from uncertainty quantification uq where the independent variables of the functions are random variables with specified probability measure we propose to construct the least squares approximation on points randomly and uniformly sampled from tensor product gaussian quadrature points we analyze the stability properties of this method and prove that the method is asymptotically stable provided that the number of points scales linearly up to a logarithmic factor with the cardinality of the polynomial space specific results in both bounded and unbounded domains are obtained along with a convergence result for chebyshev measure numerical examples are provided to verify the theoretical results

comparative study of projection schemes for stochastic finite element analysis we present a comparison of subspace projection schemes for stochastic finite element analysis in terms of accuracy and computational efficiency more specifically we compare the polynomial chaos projection scheme with reduced basis projection schemes based on the preconditioned stochastic krylov subspace numerical studies are presented for two problems 1 static analysis of a plate with random young s modulus and 2 settlement of a foundation supported on a randomly heterogeneous soil monte carlo simulation results based on exact structural analysis are used to generate benchmark results against which the projection schemes are compared we show that stochastic reduced basis methods require significantly less computer memory and execution time compared to the polynomial chaos approach particularly for large scale problems with many random variables for the class of problems considered we find that stochastic reduced basis methods can be up to orders of magnitude faster while providing results of comparable or better accuracy

the tolerance approach to sensitivity analysis in linear programming in constrast to ordinary sensitivity analysis in linear programming the tolerance approach considers simultaneous and independent changes in the objective function coefficients and in the right hand side terms this approach yields a maximum tolerance percentage such that as long as selected coefficients or terms are accurate to within that percentage of their estimated values the same basis is optimal in particular if the objective function coefficients are accurate to within the maximum tolerance percentage of their specified values then the same solution is optimal

on polynomial chaos expansion via gradient enhanced l 1 minimization gradient enhanced uncertainty quantification uq has received recent attention in which the derivatives of a quantity of interest qoi with respect to the uncertain parameters are utilized to improve the surrogate approximation polynomial chaos expansions pces are often employed in uq and when the qoi can be represented by a sparse pce 1 minimization can identify the pce coefficients with a relatively small number of samples in this work we investigate a gradient enhanced 1 minimization where derivative information is computed to accelerate the identification of the pce coefficients for this approach stability and convergence analysis are lacking and thus we address these here with a probabilistic result in particular with an appropriate normalization we show the inclusion of derivative information will almost surely lead to improved conditions e g related to the null space and coherence of the measurement matrix for a successful solution recovery further we demonstrate our analysis empirically via three numerical examples a manufactured pce an elliptic partial differential equation with random inputs and a plane poiseuille flow with random boundaries these examples all suggest that including derivative information admits solution recovery at reduced computational cost an approach for sparse polynomial chaos pc expansions including derivative information is presented method relies on 1 minimization to solve for the pc coefficients formal analysis is presented to guarantee the stability and convergence of gradient enhanced 1 minimization three examples presented to illustrate cost accuracy improvements achieved by gradient enhanced 1 minimization

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

least squares polynomial chaos expansion a review of sampling strategies abstract as non intrusive polynomial chaos expansion pce techniques have gained growing popularity among researchers we here provide a comprehensive review of major sampling strategies for the least squares based pce traditional sampling methods such as monte carlo latin hypercube quasi monte carlo optimal design of experiments ode gaussian quadratures as well as more recent techniques such as coherence optimal and randomized quadratures are discussed we also propose a hybrid sampling method dubbed alphabetic coherence optimal that employs the so called alphabetic optimality criteria used in the context of ode in conjunction with coherence optimal samples a comparison between the empirical performance of the selected sampling methods applied to three numerical examples including high order pce s high dimensional problems and low oversampling ratios is presented to provide a road map for practitioners seeking the most suitable sampling technique for a problem at hand we observed that the alphabetic coherence optimal technique outperforms other sampling methods specially when high order ode are employed and or the oversampling ratio is low

on fuzzy bitopological spaces abstract in this paper we have used the fuzzy topologies 1 and 2 to generate a family s which is a supra fuzzy topology on x using the family s we introduce and study the concepts of separation axioms continuity resp openness closedness of a mapping and compactness for a fuzzy bitopological space x 1 2 our definitions preserve much of the correspondence between concepts of fuzzy bitopological spaces and the associated fuzzy topological spaces we then investigate the relationship between these concepts and their correspondence with the fuzzy bitopological spaces kandil and el shafee 1991

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

functional anova ultramodularity and monotonicity applications in multiattribute utility theory abstract utility function properties as monotonicity and concavity play a fundamental role in reflecting a decision maker s preference structure these properties are usually characterized via partial derivatives however elicitation methods do not necessarily lead to twice differentiable utility functions furthermore while in a single attribute context concavity fully reflects risk aversion in multiattribute problems such correspondence is not one to one we show that tsetlin and winkler s multivariate risk attitudes imply ultramodularity of the utility function we demonstrate that geometric properties of a multivariate utility function can be successfully studied by utilizing an integral function expansion functional anova the necessary and sufficient conditions under which monotonicity and or ultramodularity of single attribute functions imply the monotonicity and or ultramodularity of the corresponding multiattribute function under additive preferential and mutual utility independence are then established without reliance on the utility function differentiability we also investigate the relationship between the presence of interactions among the attributes of a multiattribute utility function and the decision maker s multivariate risk attitudes

on the relative importance of input factors in mathematical models safety assessment for nuclear waste disposal this article deals with global quantitative sensitivity analysis of the level e model a computer code used in safety assessment for nuclear waste disposal the level e code has been the subject of two international benchmarks of risk assessment codes and monte carlo methods and is well known in the literature we discuss the level e model with reference to two different settings in the first setting the objective is to find the input factor that drives most of the output variance in the second setting we strive to achieve a preestablished reduction in the variance of the model output by fixing the smallest number of factors the emphasis of this work is on how to define the concept of importance in an unambiguous way and how to assess it in the simultaneous occurrence of correlated input factors and non additive models

neighborhood structures of smooth topological spaces abstract smooth topological spaces and their fundamental concepts have been discussed in the literature 2 4 7 in this paper we present the notions of base and two sorts of neighborhood structure of smooth topological spaces and give some of their properties which are results in 3 extended to smooth topological spaces

semiorthogonal decompositions and birational geometry of del pezzo surfaces over arbitrary fields we study the birational properties of geometrically rational surfaces from a derived categorical perspective in particular we give a criterion for the rationality of a del pezzo surface s over an arbitrary field namely that its derived category decomposes into zero dimensional components when s has degree at least 5 we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of s from brauer classes and chern classes associated to these vector bundles

on l fuzzy topological spaces in this paper we study l fuzzy topological spaces where l represents a completely distributive lattice we shall investigate the level decomposition of l fuzzy topologies on x and the corresponding l fuzzy continuous maps in addition we shall establish the representation theorems of l fuzzy topologies on x