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

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

ranking fuzzy numbers with integral value abstract ranking fuzzy numbers is important in decision making since very often the alternatives are evaluated by fuzzy numbers in a vague environment a comparison between these fuzzy numbers is indeed a comparison between alternatives this paper proposes a method of ranking fuzzy numbers with integral value the method which is independent of the type of membership functions used and the normality of the functions can rank more than two fuzzy numbers simultaneously it is relatively simple in computation especially in ranking triangular and trapezoidal fuzzy numbers further an index of optimism is used to reflect the decision maker s optimistic attitude discussion on comparative advantages is included

fuzzy shortest paths abstract shortest paths are one of the simplest and most widely used concepts in nonfuzzy networks fuzzy shortest paths also have a variety of applications but are hampered by the possibility that the fuzzy shortest length may not correspond to an actual path in the network in this paper we present new models based on fuzzy shortest paths that circumvent this problem and we also give a general algorithm based on dynamic programming to solve the new models in addition we analyze the fuzzy shortest path algorithms in terms of submodular functions the analysis concepts developed for fuzzy shortest paths are also discussed in terms of general fuzzy mathematical programming

constructing least squares polynomial approximations polynomial approximations constructed using a least squares approach form a ubiquitous technique in numerical computation one of the simplest ways to generate data for least squares problems is wi

on comparing interval numbers abstract this paper first presents a brief survey of the existing works on comparing and ranking any two interval numbers on the real line and then on the basis of this gives two approaches to compare any two interval numbers the first one describes a value judgement index along with a discussion on its strength and weakness over the other approaches this index has also been called an optimistic decision maker s preference index the other approach defines strict and fuzzy preference ordering between any two interval numbers from a pessimistic decision maker s point of view choice of different decision makers with different degrees of pessimism has also been considered

fuzzy shortest path problems incorporating interactivity among paths abstract this paper deals with a shortest path problem on a network in which a fuzzy number instead of a real number is assigned to each arc length such a problem is ill posed because each arc cannot be identified as being either on the shortest path or not therefore based on the possibility theory we introduce the concept of degree of possibility that an arc is on the shortest path every pair of distinct paths from the source node to any other node is implicitly assumed to be noninteractive in the conventional approaches this assumption is unrealistic and involve inconsistencies to overcome this drawback we define a new comparison index between the sum of fuzzy numbers by considering interactivity among fuzzy numbers an algorithm is presented to determine the degree of possibility for each arc on a network finally this algorithm is evaluated by means of large scale numerical examples consequently we can find this approach is efficient even for real world practical networks

fuzzy location problems on networks location problems concern a wide set of fields where it is usually assumed that exact data are known however in real applications the location of the facility considered can be full of linguistic vagueness that can be appropriately modelled using networks with fuzzy values thus fuzzy location problems on networks arise this paper deals with their general formulation and the description of the ways to solve them namely we show the variety of problems that can be considered in this context and for some of them we propose the most operative approaches for their solution

a new approach for ranking fuzzy numbers by distance method abstract many ranking methods have been proposed so far however there is yet no method that can always give a satisfactory solution to every situation some are counterintuitive not discriminating some use only the local information of fuzzy values some produce different rankings for the same situation for overcoming the above problems we propose a new method for ranking fuzzy numbers by distance method our method is based on calculating the centroid point where the distance means from original point to the centroid point x 0 y 0 and the x 0 index is the same as murakami et al s x 0 however the y 0 index is integrated from the inverse functions of an lr type fuzzy number thus we use ranking function r a x 2 y 2 distance index as the order quantities in a vague environment our method can rank more than two fuzzy numbers simultaneously and the fuzzy numbers need not be normal furthermore we also propose the coefficient of variation cv index to improve lee and li s method comput math appl 15 1988 887 896 lee and li rank fuzzy numbers based on two different criteria namely the fuzzy mean and the fuzzy spread of the fuzzy numbers and they pointed out that human intuition would favor a fuzzy number with the following characteristics higher mean value and at the same time lower spread however when higher mean value and at the same time higher spread or lower mean value and at the same time lower spread exists it is not easy to compare its orderings clearly our cv index is defined as cv standard error mean which can overcome lee and li s problem efficiently in this way our proposed method can also be easily calculated by the mathematica package to solve problems of ranking fuzzy numbers at last we present three numerical examples to illustrate our proposed method and compare with other ranking methods

computing multivariate fekete and leja points by numerical linear algebra we discuss and compare two greedy algorithms that compute discrete versions of fekete like points for multivariate compact sets by basic tools of numerical linear algebra the first gives the so called approximate fekete points by qr factorization with column pivoting of vandermonde like matrices the second computes discrete leja points by lu factorization with row pivoting moreover we study the asymptotic distribution of such points when they are extracted from weakly admissible meshes

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

nonlocal transform domain filter for volumetric data denoising and reconstruction we present an extension of the bm3d filter to volumetric data the proposed algorithm bm4d implements the grouping and collaborative filtering paradigm where mutually similar d dimensional patches are stacked together in a d 1 dimensional array and jointly filtered in transform domain while in bm3d the basic data patches are blocks of pixels in bm4d we utilize cubes of voxels which are stacked into a 4 d group the 4 d transform applied on the group simultaneously exploits the local correlation present among voxels in each cube and the nonlocal correlation between the corresponding voxels of different cubes thus the spectrum of the group is highly sparse leading to very effective separation of signal and noise through coefficient shrinkage after inverse transformation we obtain estimates of each grouped cube which are then adaptively aggregated at their original locations we evaluate the algorithm on denoising of volumetric data corrupted by gaussian and rician noise as well as on reconstruction of volumetric phantom data with non zero phase from noisy and incomplete fourier domain k space measurements experimental results demonstrate the state of the art denoising performance of bm4d and its effectiveness when exploited as a regularizer in volumetric data reconstruction

constructing nested nodal sets for multivariate polynomial interpolation we present a robust method for choosing multivariate polynomial interpolation nodes our algorithm is an optimization method to greedily minimize a measure of interpolant sensitivity a variant of a weighted lebesgue function nodes are therefore chosen that tend to control oscillations in the resulting interpolant this method can produce an arbitrary number of nodes and is not constrained by the dimension of a complete polynomial space our method is therefore flexible nested nodal sets are produced in spaces of arbitrary dimensions and the number of nodes added at each stage can be arbitrary the algorithm produces a nodal set given a probability measure on the input space thus parameterizing interpolants with respect to finite measures we present examples to show that the method yields nodal sets that behave well with respect to standard interpolation diagnostics the lebesgue constant the vandermonde determinant and the vandermonde condition number we also show that a nongreedy version of the

a near optimal sampling strategy for sparse recovery of polynomial chaos expansions abstract compressive sampling has become a widely used approach to construct polynomial chaos surrogates when the number of available simulation samples is limited originally these expensive simulation samples would be obtained at random locations in the parameter space it was later shown that the choice of sample locations could significantly impact the accuracy of resulting surrogates this motivated new sampling strategies or design of experiment approaches such as coherence optimal sampling which aim at improving the coherence property in this paper we propose a sampling strategy that can identify near optimal sample locations that lead to improvement in local coherence property and also enhancement of cross correlation properties of measurement matrices we provide theoretical motivations for the proposed sampling strategy along with several numerical examples that show that our near optimal sampling strategy produces substantially more accurate results compared to other sampling strategies

unified approach fuzzy graph problems we present a taxonomy of fuzzy graphs that treats fuzziness in vertex existence edge existence edge connectivity and edge weight within that framework we formulate some standard graph theoretic problems shortest paths and minimum cut for fuzzy graphs using a unified approach distinguished by its uniform application of guiding principles such as the construction of membership grades via the ranking of fuzzy numbers the preservation of membership grade normalization and the collapsing of fuzzy sets of graphs into fuzzy graphs finally we provide algorithmic solutions to these problems with examples

the shortest path problem on networks with fuzzy parameters shortest path problem where the costs have vague values is one of the most studied problems in fuzzy sets and systems area however due to its high computational complexity previously published algorithms present peculiarities and problems that need to be addressed e g they find costs without an existing path they determine a fuzzy solution set but do not give any guidelines to help the decision maker choose the best path they can only be applied in graphs with fuzzy non negative parameters etc in this paper one proposes an iterative algorithm that assumes a generic ranking index for comparing the fuzzy numbers involved in the problem in such a way that each time in which the decision maker wants to solve a concrete problem s he can choose or propose the ranking index that best suits that problem this algorithm that solves the above remarked drawbacks is based on the ford moore bellman algorithm for classical graphs and in concrete it can be applied in graphs with negative parameters and it can detect whether there are negative circuits for the sake of illustrating the performance of the algorithm in the paper it has been here developed using only certain order relations but it is not restricted at all to use these comparison relations exclusively the proposed algorithm is easy of understanding as the theoretical base of a decision support system oriented to solving this kind of problems

reweighted l1 minimization method for stochastic elliptic differential equations abstract we consider elliptic stochastic partial differential equations spdes with random coefficients and solve them by expanding the solution using generalized polynomial chaos gpc under some mild conditions on the coefficients the solution is sparse in the random space i e only a small number of gpc basis makes considerable contribution to the solution to exploit this sparsity we employ reweighted l 1 minimization to recover the coefficients of the gpc expansion we also combine this method with random sampling points based on the chebyshev probability measure to further increase the accuracy of the recovery of the gpc coefficients we first present a one dimensional test to demonstrate the main idea and then we consider 14 and 40 dimensional elliptic spdes to demonstrate the significant improvement of this method over the standard l 1 minimization method for moderately high dimensional 10 problems the combination of chebyshev measure with reweighted l 1 minimization performs well while for higher dimensional problems reweighted l 1 only is sufficient the proposed approach is especially suitable for problems for which the deterministic solver is very expensive since it reuses the sampling results and exploits all the information available from limited sources

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

a procedure for ordering fuzzy subsets of the unit interval abstract we introduce a function to help in the ordering of fuzzy subsets of the unit interval this function is the integral of the mean of the level sets associated with the fuzzy subsets various properties of this function are studied

a least squares approximation of partial differential equations with high dimensional random inputs uncertainty quantification schemes based on stochastic galerkin projections with global or local basis functions and also stochastic collocation methods in their conventional form suffer from the so called curse of dimensionality the associated computational cost grows exponentially as a function of the number of random variables defining the underlying probability space of the problem in this paper to overcome the curse of dimensionality a low rank separated approximation of the solution of a stochastic partial differential spde with high dimensional random input data is obtained using an alternating least squares als scheme it will be shown that in theory the computational cost of the proposed algorithm grows linearly with respect to the dimension of the underlying probability space of the system for the case of an elliptic spde an a priori error analysis of the algorithm is derived finally different aspects of the proposed methodology are explored through its application to some numerical experiments

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

nonadaptive quasi optimal points selection for least squares linear regression in this paper we present a quasi optimal sample set for ordinary least squares ols regression the quasi optimal set is designed in such a way that for a given number of samples it can deliver the regression result as close as possible to the result obtained by a much larger set of candidate samples the quasi optimal set is determined by maximizing a quantity measuring the mutual column orthogonality and the determinant of the model matrix this procedure is nonadaptive in the sense that it does not depend on the sample data this is useful in practice as it allows one to determine prior to the potentially expensive data collection procedure where to sample the underlying system in addition to presenting the theoretical motivation of the quasi optimal set we also present its efficient implementation via a greedy algorithm along with several numerical examples to demonstrate its efficacy since the quasi optimal set allows one to obtain a near optimal regression result under any affordable nu

on a near optimal sampling strategy for least squares polynomial regression abstract we present a sampling strategy of least squares polynomial regression the strategy combines two recently developed methods for least squares method christoffel least squares algorithm and quasi optimal sampling more specifically our new strategy first choose samples from the pluripotential equilibrium measure and then re order the samples by the quasi optimal algorithm a weighted least squares problem is solved on a much smaller sample set to obtain the regression result it is then demonstrated that the new strategy results in a polynomial least squares method with high accuracy and robust stability at almost minimal number of samples

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

point collocation nonintrusive polynomial chaos method for stochastic computational fluid dynamics this paper describes a point collocation nonintrusive polynomial chaos technique used for uncertainty propagation in computational fluid dynamics simulations the application of point collocation nonintrusive polynomial chaos to stochastic computational fluid dynamics is demonstrated with two examples 1 a stochastic expansion wave problem with an uncertain deflection angle geometric uncertainty and 2 a stochastic transonic wing case with uncertain freestream mach number and angle of attack for each problem input uncertainties are propagated with both the nonintrusive polynomial chaos method and monte carlo techniques to obtain the statistics of various output quantities confidence intervals for monte carlo statistics are calculated using the bootstrap method for the expansion wave problem a fourth degree polynomial chaos expansion which requires five deterministic computational fluid dynamics evaluations has been sufficient to predict the statistics within the confidence interval of 10 000 crude monte carlo simulations in the transonic wing case for various output quantities of interest it has been shown that a fifth degree point collocation nonintrusive polynomial chaos expansion obtained with hammersley sampling was capable of estimating the statistics at an accuracy level of 1000 latin hypercube monte carlo simulations with a significantly lower computational cost overall the examples demonstrate that the point collocation nonintrusive polynomial chaos has a promising potential as an effective and computationally efficient uncertainty propagation technique for stochastic computational fluid dynamics simulations

image denoising by sparse 3 d transform domain collaborative filtering we propose a novel image denoising strategy based on an enhanced sparse representation in transform domain the enhancement of the sparsity is achieved by grouping similar 2d image fragments e g blocks into 3d data arrays which we call groups collaborative altering is a special procedure developed to deal with these 3d groups we realize it using the three successive steps 3d transformation of a group shrinkage of the transform spectrum and inverse 3d transformation the result is a 3d estimate that consists of the jointly filtered grouped image blocks by attenuating the noise the collaborative filtering reveals even the finest details shared by grouped blocks and at the same time it preserves the essential unique features of each individual block the filtered blocks are then returned to their original positions because these blocks are overlapping for each pixel we obtain many different estimates which need to be combined aggregation is a particular averaging procedure which is exploited to take advantage of this redundancy a significant improvement is obtained by a specially developed collaborative wiener filtering an algorithm based on this novel denoising strategy and its efficient implementation are presented in full detail an extension to color image denoising is also developed the experimental results demonstrate that this computationally scalable algorithm achieves state of the art denoising performance in terms of both peak signal to noise ratio and subjective visual quality

comments on image denoising by sparse 3 d transform domain collaborative filtering in order to resolve the problem that the denoising performance has a sharp drop when noise standard deviation reaches 40 proposed to replace the wavelet transform by the dct in this comment we argue that this replacement is unnecessary and that the problem can be solved by adjusting some numerical parameters we also present this parameter modification approach here experimental results demonstrate that the proposed modification achieves better results in terms of both peak signal to noise ratio and subjective visual quality than the original method for strong noise

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

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

the fuzzy shortest path problem and its most vital arcs abstract the shortest path problem is to find the shortest distance between two specified nodes in a network an arc is called a single most vital arc in the network if its removal from the network results in the greatest increase in the shortest distance the availability of arcs plays a key role in network problems the most vital arcs problems provide a means by which the importance of arc s availability can be measured in the traditional shortest path problem the arc lengths are assumed to be crisp numbers in this paper we consider the case that the arc lengths are fuzzy numbers in particular we derive the membership function of the shortest distance by using a fuzzy linear programming approach based on this result we then propose an algorithm for finding the single most vital arc in a network

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

a procedure for ranking fuzzy numbers using fuzzy relations abstract this paper presents a method to give fuzzy order relations between fuzzy numbers these are founded on the concept of comparison function defined in the paper and the use of fuzzy measures related with the same numbers the link of such relations with fuzzy operations is investigated too

a new algorithm for the discrete fuzzy shortest path problem in a network in a network the arc lengths may represent time or cost in practical situations it is reasonable to assume that each arc length is a discrete fuzzy set we called it the discrete fuzzy shortest path problem there were several methods reported to solve this kind of problem in the open literature in these methods they can obtain either the fuzzy shortest length or the shortest path in this paper we propose a new algorithm that can obtain both of them the discrete fuzzy shortest length method is proposed to find the fuzzy shortest length and the fuzzy similarity measure is utilized to get the shortest path an illustrative example is given to demonstrate our proposed algorithm

multiobjective programming in optimization of the interval objective function abstract in this paper the mathematical programming problem whose objective function has interval coefficients is investigated to maximize the interval objective function the order relations which represent the decision maker s preference between interval profits are defined by the right limit the left limit the center and the width of an interval the maximization problem with the interval objective function is converted into a multiobjective problem using the order relations the minimization problem is also converted into a multiobjective problem using the order relations which represent the decision maker s preference between interval costs