detrended fluctuation analysis of intertrade durations abstract the intraday pattern long memory and multifractal nature of the intertrade durations which are defined as the waiting times between two consecutive transactions are investigated based upon the limit order book data and order flows of 23 liquid chinese stocks listed on the shenzhen stock exchange in 2003 an inverse u shaped intraday pattern in the intertrade durations with an abrupt drop in the first minute of the afternoon trading is observed based on a detrended fluctuation analysis we find a crossover of power law scaling behaviors for small box sizes trade numbers in boxes and large box sizes and strong evidence in favor of long memory in both regimes in addition the multifractal nature of intertrade durations in both regimes is confirmed by a multifractal detrended fluctuation analysis for individual stocks with a few exceptions in the small duration regime the intraday pattern has little influence on the long memory and multifractality

on periodic boundary conditions and ergodicity in computational homogenization of heterogeneous materials with random microstructure abstract due to high computational costs associated with stochastic computational homogenization a highly complex random material microstructure is often replaced by simplified parametric ergodic and sometimes periodic models this replacement is often criticized in the literature due to unclear error resulting from the periodicity and ergodicity assumptions in the current contribution we perform a validation of both assumptions through various numerical examples to this end we compare large scale non simplified and non ergodic models with simplified ergodic and periodic solutions in addition we analyze the hill mandel condition for stochastic homogenization problems and demonstrate that for a stochastic problem there are more than three classical types of boundary conditions as an example we propose two novel stochastic periodic boundary conditions which possess a clear physical meaning the effect of these novel periodic boundary conditions is also analyzed by comparing with non ergodic simulation results

convex extensions and envelopes of lower semi continuous functions we define a convex extension of a lower semi continuous function to be a convex function that is identical to the given function over a pre specified subset of its domain convex extensions are not necessarily constructible or unique we identify conditions under which a convex extension can be constructed when multiple convex extensions exist we characterize the tightest convex extension in a well defined sense using the notion of a generating set we establish conditions under which the tightest convex extension is the convex envelope then we employ convex extensions to develop a constructive technique for deriving convex envelopes of nonlinear functions finally using the theory of convex extensions we characterize the precise gaps exhibited by various underestimators of x y over a rectangle and prove that the extensions theory provides convex relaxations that are much tighter than the relaxation provided by the classical outer linearization of bilinear terms

fuzzy versions of the covering circle problem abstract in this paper we consider two fuzzy versions of the well known problem of determining the smallest circle center and radius that would cover a given finite set of points in the plane when the locations of points are not precise but fuzzy the first is modeled as a possibility constrained mathematical program while the second is modeled as a necessity constrained one polynomial algorithms are presented for both the versions also a numerical example is included for one case these models and solutions are of interest in both theoretical and practical contexts

identification of random shapes from images through polynomial chaos expansion of random level set functions in this paper an efficient method is proposed for the identification of random shapes in a form suitable for numerical simulation within the extended stochastic finite element method x sfem the method starts from a collection of images representing different outcomes of the random shape to identify the key point of the method is to represent the random geometry in an implicit manner using the level set technique in this context the problem of random geometry identification is equivalent to the identification of a random level set function which is a random field this random field is represented on a polynomial chaos pc basis and various efficient numerical strategies are proposed in order to identify the coefficients of its pc decomposition the performance of these strategies is evaluated through some manufactured problems and useful conclusions are provided the propagation of geometrical uncertainties in structural analysis using the x sfem is finally examined copyright 2009 john wiley sons ltd

activity spectrum from waiting time distribution in high frequency financial data not only returns but also waiting times between trades are random variables in this work we analyze the spectra of the waiting time processes for tick by tick trades the numerical problem strictly related with the real inversion of laplace transforms is analyzed by using tikhonov s regularization method we also analyze these spectra by a rough method using a comb of dirac s delta functions

fuzzy stochastic fem based homogenization framework for materials with polymorphic uncertainties in the microstructure uncertainties in the macroscopic response of heterogeneous materials result from two sources the natural variability in the microstructure s geometry and the lack of sufficient knowledge regarding the microstructure the first type of uncertainty is denoted aleatoric uncertainty and may be characterized by a known probability density function the second type of uncertainty is denoted epistemic uncertainty this kind of uncertainty cannot be described using probabilistic methods models considering both sources of uncertainties are called polymorphic in the case of polymorphic uncertainties some combination of stochastic methods and fuzzy arithmetic should be used thus in the current work we examine a fuzzy stochastic finite element method based homogenization framework for materials with random inclusion sizes we analyze an experimental radii distribution of inclusions and develop a stochastic representative volume element the stochastic finite element method is used to obtain the material response in the case of random inclusion radii due to unavoidable noise in experimental data an insufficient number of samples and limited accuracy of the fitting procedure the radii distribution density cannot be obtained exactly thus it is described in terms of fuzzy location and scale parameters the influence of fuzzy input on the homogenized stress measures is analyzed

determination of rve size for random composites with local volume fraction variation abstract in this paper a novel computational procedure is proposed for the determination of the representative volume element rve size for random composites which is the basis of homogenization methods the proposed approach takes into account the local volume fraction variation by processing computer simulated images of composites with randomly scattered inclusions a variable number of microstructure models are derived directly from the images using the moving window technique each microstructure model contains different amount of inclusions measured by image analysis tools the proposed computational procedure is based on the solution of multiple boundary value problems under kinematic and static uniform boundary conditions using the extended finite element method xfem coupled with monte carlo simulation mcs in this way the statistical characteristics of the upper and lower bounds of the apparent material properties are computed the rve is attained within a prescribed tolerance by examining the convergence of these two bounds with respect to the mesoscale size the numerical results clearly demonstrate the significant effect of matrix inclusion stiffness ratio and volume fraction on the convergence rate the proposed computational procedure leads to rves which can give more realistic results when used in homogenization problems mesoscale random fields describing the spatial variation of the components of the apparent elasticity tensor are also determined using the proposed approach

solving elliptic boundary value problems with uncertain coefficients by the finite element method the stochastic formulation this work studies a linear elliptic problem with uncertainty the introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations the major emphasis

global optimization method for finding dense packings of equal circles in a circle abstract this paper considers the problem of finding the densest packing of n n 1 2 equal circles in a circle this problem is perhaps the most classical packing problem it is also a natural and challenging test system for evaluating various global optimization methods we propose a quasi physical global optimization method by simulating two kinds of movements of n elastic disks smooth movement driven by elastic pressures and abrupt movement driven by strong repulsive forces and attractive forces the algorithm is tested on the instances of n 1 2 200 using the best known record of the radius of the container as an upper bound we find 63 new packings better than the best known ones reported in literature

a multiscale stochastic finite element method on elliptic problems involving uncertainties in this study a multiscale stochastic finite element method mssfem is developed to resolve scale coupling stochastic elliptic problems based on formulations of a stochastic variational approach and scale bridging multiscale shape functions by employing polynomial chaos expansions and lagrange polynomials stochastic galerkin method and stochastic collocation method are introduced and combined with deterministic multiscale methods i e variational multiscale method and multiscale finite element method the resulting spectral and pseudo spectral stochastic finite element methods are incorporated into the fine and coarse mesh computation of the mssfem a benchmark multiscale model and the numerical experiment based on the formulated mssfem are provided for illustration it is expected that the proposed mssfem can act as a paradigm for solving of general stochastic partial differential equations involving multiscale stochastic data

damped arrow hurwicz algorithm for sphere packing we consider algorithms that from an arbitrarily sampling of n spheres possibly overlapping find a close packed configuration without overlapping these problems can be formulated as minimization problems with non convex constraints for such packing problems we observe that the classical iterative arrow hurwicz algorithm does not converge we derive a novel algorithm from a multi step variant of the arrow hurwicz scheme with damping we compare this algorithm with classical algorithms belonging to the class of linearly constrained lagrangian methods and show that it performs better we provide an analysis of the convergence of these algorithms in the simple case of two spheres in one spatial dimension finally we investigate the behaviour of our algorithm when the number of spheres is large in two and three spatial dimensions

deterministic and stochastic global optimization techniques for planar covering with ellipses problems abstract problems of planar covering with ellipses are tackled in this work ellipses can have a fixed angle or each of them can be freely rotated deterministic global optimization methods are developed for both cases while a stochastic version of the method is also proposed for large instances of the latter case numerical results show the effectiveness and efficiency of the proposed methods

an adaptive multi element generalized polynomial chaos method for stochastic differential equations we formulate a multi element generalized polynomial chaos me gpc method to deal with long term integration and discontinuities in stochastic differential equations we first present this method for legendre chaos corresponding to uniform random inputs and subsequently we generalize it to other random inputs the main idea of me gpc is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value in each subdomain or random element we then employ a generalized polynomial chaos expansion we develop a criterion to perform such a decomposition adaptively and demonstrate its effectiveness for odes including the kraichnan orszag three mode problem as well as advection diffusion problems the new method is similar to spectral element method for deterministic problems but with h p discretization of the random space

multi element stochastic reduced basis methods this paper presents multi element stochastic reduced basis methods me srbms for solving linear stochastic partial differential equations in me srbms the domain of definition of the random inputs is decomposed into smaller subdomains or random elements stochastic reduced basis methods srbms are employed in each random element to evaluate the response statistics these elemental statistics are assimilated to compute the overall statistics the effectiveness of the method is demonstrated by solving the stochastic steady state heat transfer equation on two geometries involving different types of boundary conditions numerical studies are conducted to investigate the h convergence rates of global and local preconditioning strategies

an outer approximation algorithm for a class of mixed integer nonlinear programs an outer approximation algorithm is presented for solving mixed integer nonlinear programming problems of a particular class linearity of the integer or discrete variables and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure based on principles of decomposition outer approximation and relaxation the proposed algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed integer linear master program convergence and optimality properties of the algorithm are presented as well as a general discussion on its implementation numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper finally a theoretical comparison with generalized benders decomposition is presented on the lower bounds predicted by the relaxed master programs

a branch and cut algorithm for nonconvex quadratically constrained quadratic programming we present a branch and cut algorithm that yields in finite time a globally e optimal solution with respect to feasibility and optimality of the nonconvex quadratically constrained quadratic programming problem the idea is to estimate all quadratic terms by successive linearizations within a branching tree using reformulation linearization techniques rlt to do so four classes of linearizations cuts depending on one to three parameters are detailed for each class we show how to select the best member with respect to a precise criterion the cuts introduced at any node of the tree are valid in the whole tree and not only within the subtree rooted at that node in order to enhance the computational speed the structure created at any node of the tree is flexible enough to be used at other nodes computational results are reported that include standard test problems taken from the literature some of these problems are solved for the first time with a proof of global optimality

mixtures of exponential distributions arbitrary nonparametric mixtures of exponential and weibull fixed shape distributions are considered as possible models for a lifetime distribution a characterization of such distributions is given by the well known characterization of laplace transforms the maximum likelihood estimate of the mixing distribution is investigated and found to be supported on a finite number of points it is shown to be unique and weakly convergent to the true mixing measure with probability one a practical algorithm for computing the maximum likelihood estimate is described its performance is briefly discussed and some illustrative examples given

a reduced spectral function approach for the stochastic finite element analysis the stochastic finite element analysis of elliptic type partial differential equations with non gaussian random fields are considered a novel approach by projecting the solution of the discretized equation into a reduced finite dimensional orthonormal vector basis is investigated it is shown that the solution can be obtained using a finite series comprising functions of random variables and orthonormal vectors these functions called as the spectral functions can be expressed in terms of the spectral properties of the deterministic coefficient matrices arising due to the discretization of the governing partial differential equation based on the projection in a reduced orthonormal vector basis a galerkin error minimization approach is proposed the constants appearing in the galerkin method are solved from a system of linear equations which has much smaller dimension compared to the original discretized equation a hybrid analytical and simulation based computational approach is proposed to obtain the moments and probability density function of the solution the method is illustrated using the stochastic nanomechanics of a zinc oxide zno nanowire deflected under the atomic force microscope afm tip the results are compared with the results obtained using direct monte carlo simulation classical neumann expansion and polynomial chaos approach for different correlation lengths and strengths of randomness

iterated tabu search and variable neighborhood descent for packing unequal circles into a circular container this paper presents an iterated tabu search and variable neighborhood descent its vnd algorithm for packing unequal circles into a circular container pucc the algorithm adapts the tabu search procedure of iterated tabu search algorithms and proposes a tabu search and variable neighborhood descent ts vnd procedure we observe there are strong complementarities between the small circles and the large vacant places and propose the insert neighborhood to match up the small circles with the large vacant places although the insert neighborhood is inefficient and time consuming it is an important supplement to the classic swap neighborhood as it could arrange the small circles properly predicated on these features we employ the insert neighborhood only at chosen local minima of the swap neighborhood that shows promise for an improvement the traditional tabu search procedure is then transformed into a hybrid procedure composed of two alternative parts namely variable neighborhood descent and tabu search respectively besides this reformed procedure its vnd also incorporates other new features such as an adaptive evaluation function a novel method for accelerating the neighborhood exploration and the collision accidents criterion for evaluating how intensively the area near the current solution has been explored the computational results on three well established benchmark sets show that the proposed algorithm not only has a good discovery capability but also can provide good results within a reasonable time for a total of 84 benchmark instances the proposed algorithm improves the best known results on 23 instances matches 60 and only misses one

a global minlp approach to symbolic regression symbolic regression methods generate expression trees that simultaneously define the functional form of a regression model and the regression parameter values as a result the regression problem can search many nonlinear functional forms using only the specification of simple mathematical operators such as addition subtraction multiplication and division among others currently state of the art symbolic regression methods leverage genetic algorithms and adaptive programming techniques genetic algorithms lack optimality certifications and are typically stochastic in nature in contrast we propose an optimization formulation for the rigorous deterministic optimization of the symbolic regression problem we present a mixed integer nonlinear programming minlp formulation to solve the symbolic regression problem as well as several alternative models to eliminate redundancies and symmetries we demonstrate this symbolic regression technique using an array of experiments based upon literature instances we then use a set of 24 minlps from symbolic regression to compare the performance of five local and five global minlp solvers finally we use larger instances to demonstrate that a portfolio of models provides an effective solution mechanism for problems of the size typically addressed in the symbolic regression literature

on convex envelopes for bivariate functions over polytopes in this paper we discuss convex envelopes for bivariate functions satisfying suitable assumptions over polytopes we first propose a technique to compute the value and a supporting hyperplane of the convex envelope over a general two dimensional polytope through the solution of a three dimensional convex subproblem with continuously differentiable constraint functions then for quadratic functions as well as for some polynomial and rational ones again satisfying suitable assumptions we show how the same computations can be carried out through the solution of a single semidefinite problem

augmented lagrangian methods under the constant positive linear dependence constraint qualification two augmented lagrangian algorithms for solving kkt systems are introduced the algorithms differ in the way in which penalty parameters are updated possibly infeasible accumulation points are characterized it is proved that feasible limit points that satisfy the constant positive linear dependence constraint qualification are kkt solutions boundedness of the penalty parameters is proved under suitable assumptions numerical experiments are presented

optimization as an analysis tool for human complex problem solving we present a problem class of mixed integer nonlinear programs minlps with nonconvex continuous relaxations which stem from economic test scenarios that are used in the analysis of human complex problem solving in a round based scenario participants hold an executive function a posteriori a performance indicator is calculated and correlated to personal measures such as intelligence working memory or emotion regulation altogether we investigate 2088 optimization problems that differ in size and initial conditions based on real world experimental data from 12 rounds of 174 participants the goals are twofold first from the optimal solutions we gain additional insight into a complex system which facilitates the analysis of a participant s performance in the test second we propose a methodology to automatize this process by providing a new criterion based on the solution of a series of optimization problems by providing a mathematical optimization model and this methodology we disprove the ass

a fuzzy stochastic multiscale model for fiber composites a one dimensional study abstract we study mathematical and computational models for computing the deformation of fiber reinforced polymers due to external forces a thorough study requires an understanding of both micro structural effects and uncertainty variability in the manufacturing process such as uncertainty in the size and distribution of fibers and variability in material properties and fracture parameters we first show that the present uncertainties variabilities which are of both random and non random types cannot be accurately characterized by current stochastic multiscale models based on precise probability theory such as stationary random fields next we present a new hybrid fuzzy stochastic model which can more accurately describe uncertainties variabilities in fiber composites the new model which is referred to as a fuzzy stationary random field consists of a random field with fuzzy moments we then construct a global local multiscale algorithm in a fuzzy stochastic framework for efficiently computing output quantities of interest such as displacements and stresses in regions of relatively small size e g hot spots the algorithm utilizes the concept of representative volume elements and homogenization and constructs a global solution to compute a local approximation that captures the microscale features of the problem the results are based on and backed by real experimental data through a calibration validation approach

theory of earthquakes interevent times applied to financial markets abstract we analyze the probability density function pdf of waiting times between financial loss exceedances the empirical pdfs are fitted with the self excited hawkes conditional poisson process with a long power law memory kernel the hawkes process is the simplest extension of the poisson process that takes into account how past events influence the occurrence of future events by analyzing the empirical data for 15 different financial assets we show that the formalism of the hawkes process used for earthquakes can successfully model the pdf of interevent times between successive market losses

higher order extended fem for weak discontinuities level set representation quadrature and application to magneto mechanical problems in this article we present the application of bilinear and biquadratic extended fem xfem formulations to model weak discontinuities in magnetic and coupled magneto mechanical boundary value problems for properly resolving the location of curved interfaces and the discontinuous physical behaviour the major part of the contribution is devoted to review and develop methods for level set representation of curved interfaces and numerical integration of the weak form in higher order xfem formulations in order to reduce the complexity of the representation of curved interfaces an element local approach that allows for an automated computation of the level set values and also improves the compatibility between the level set representation and the integration subdomains is proposed integration rules for polygons and strain smoothing are applied in conjunction with biquadratic elements and compared with curved integration subdomains eventually a coupled magneto mechanical demonstration problem is modelled and solved by xfem for demonstration purposes a magneto mechanical coupling due to magnetic stresses is considered errors and convergence rates are analysed for the different level set representations and numerical integration procedures as well as their dependence on the ratio of material parameters at an interface copyright c 2012 john wiley sons ltd

a computational approach to handle complex microstructure geometries in multiscale analysis of components there is usually a need to solve microstructures with complex geometries in this paper we use the extended finite element method x fem to solve scales involving complex geometries the x fem allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach for material interfaces this is achieved by introducing a new enrichment strategy although the mesh does not need to conform to the physical surfaces it needs to be fine enough to capture the geometry of these surfaces a simple algorithm is described to adaptively refine the mesh to meet this geometrical requirement numerical experiments on the periodic homogenization of two phase complex cells demonstrate the accuracy and simplicity of the x fem c 2003 elsevier b v all rights reserved

uncoupled continuous time random walk finite jump length probability density function in this work the uncoupled continuous time random walk model with the exponential jump length probability density function pdf is considered exact solutions for the pdf and second moment are obtained and investigated for different kinds of waiting time pdf

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

generalized spectral decomposition for stochastic nonlinear problems we present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems the method consists in the construction of a reduced basis approximation of the galerkin solution and is independent of the stochastic discretization selected polynomial chaos stochastic multi element or multi wavelets two algorithms are proposed for the sequential construction of the successive generalized spectral modes they involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems compared to the classical galerkin method the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes the methodology is detailed and tested on two model problems the one dimensional steady viscous burgers equation and a two dimensional nonlinear diffusion problem these examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

exact quadratic convex reformulations of mixed integer quadratically constrained problems we propose a solution approach for the general problem qp of minimizing a quadratic function of bounded integer variables subject to a set of quadratic constraints the resolution is based on the reformulation of the original problem qp into an equivalent quadratic problem whose continuous relaxation is convex so that it can be effectively solved by a branch and bound algorithm based on quadratic convex relaxation we concentrate our efforts on finding a reformulation such that the continuous relaxation bound of the reformulated problem is as tight as possible furthermore we extend our method to the case of mixed integer quadratic problems with the following restriction all quadratic sub functions of purely continuous variables are already convex finally we illustrate the different results of the article by small examples and we present some computational experiments on pure integer and mixed integer instances of qp most of the considered instances with up to 53 variables can be solved by our approach combined with the use of cplex

estimation of the optical constants and the thickness of thin films using unconstrained optimization the problem of estimating the thickness and the optical constants of thin films using transmission data only is very challenging from the mathematical point of view and has a technological and an economic importance in many cases it represents a very ill conditioned inverse problem with many local nonglobal solutions in a recent publication we proposed nonlinear programming models for solving this problem well known software for linearly constrained optimization was used with success for this purpose in this paper we introduce an unconstrained formulation of the nonlinear programming model and we solve the estimation problem using a method based on repeated calls to a recently introduced unconstrained minimization algorithm numerical experiments on computer generated films show that the new procedure is reliable

an adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations in recent years there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary partial differential equations to this end the spectral stochastic finite element method ssfem is the most popular method due to its fast convergence rate recently the stochastic sparse grid collocation method has emerged as an attractive alternative to ssfem it approximates the solution in the stochastic space using lagrange polynomial interpolation the collocation method requires only repetitive calls to an existing deterministic solver similar to the monte carlo method however both the ssfem and current sparse grid collocation methods utilize global polynomials in the stochastic space thus when there are steep gradients or finite discontinuities in the stochastic space these methods converge very slowly or even fail to converge in this paper we develop an adaptive sparse grid collocation strategy using piecewise multi linear hierarchical basis functions hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region numerical examples especially for problems related to long term integration and stochastic discontinuity are presented comparisons with monte carlo and multi element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method

examples of computational approaches for elliptic possibly multiscale pdes with random inputs we overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness the specific equations manipulated are linear elliptic and arise in the context of multiscale problems but the purpose is more general on a set of prototypical situations we investigate two critical issues present in many settings variance reduction techniques to obtain sufficiently accurate results at a limited computational cost when solving pdes with random coefficients and finite element techniques that are sufficiently flexible to carry over to geometries with random fluctuations some elements of theoretical analysis and numerical analysis are briefly mentioned numerical experiments although simple provide convincing evidence of the efficiency of the approaches

on solving elliptic stochastic partial differential equations abstract a model elliptic boundary value problem of second order with stochastic coefficients described by the karhunen loeve expansion is addressed this problem is transformed into an equivalent deterministic one the perturbation method and the method of successive approximations is analyzed rigorous error estimates in the framework of sobolev spaces are given

autoregressive conditional duration a new model for irregularly spaced transaction data this paper proposes a new statistical model for the analysis of data which arrive at irregular intervals the model treats the time between events as a stochastic process and proposes a new class of point processes with dependent arrival rates the conditional intensity is developed and compared with other self exciting processes because the model focuses on the expected duration between events it is called the autoregressive conditional duration acd model asymptotic properties of the quasi maximum likelihood estimator are developed as a corollary to arch model results strong evidence is provided for duration clustering for the financial transaction data analyzed both deterministic time of day effects and stochastic effects are important the model is applied to the arrival times of trades and therefore is a model of transaction volume and also to the arrival of other events such as price changes models for the volatility of prices are estimated with price based durations and examined from a market microstructure point of view

mathematical programming techniques in water network optimization abstract in this article we survey mathematical programming approaches to problems in the field of drinking water distribution network optimization among the predominant topics treated in the literature we focus on two different but related problem classes one can be described by the notion of network design while the other is more aptly termed by network operation the basic underlying model in both cases is a nonlinear network flow model and we give an overview on the more specific modeling aspects in each case the overall mathematical model is a mixed integer nonlinear program having a common structure with respect to how water dynamics in pipes are described finally we survey the algorithmic approaches to solve the proposed problems and we discuss computation on various types of water networks

global minimization using an augmented lagrangian method with variable lower level constraints a novel global optimization method based on an augmented lagrangian framework is introduced for continuous constrained nonlinear optimization problems at each outer iteration k the method requires the varepsilon k global minimization of the augmented lagrangian with simple constraints where varepsilon k to varepsilon global convergence to an varepsilon global minimizer of the original problem is proved the subproblems are solved using the bb method numerical experiments are presented

the stochastic finite element method past present and future a powerful tool in computational stochastic mechanics is the stochastic finite element method sfem sfem is an extension of the classical deterministic fe approach to the stochastic framework i e to the solution of static and dynamic problems with stochastic mechanical geometric and or loading properties the considerable attention that sfem received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large scale problems this article aims at providing a state of the art review of past and recent developments in the sfem area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future

stochastic reduced basis methods stochastic reduced basis methods for solving large scale linear random algebraic systems of equations such as those obtained by discretizing linear stochastic partial differential equations in space time and the random dimension are introduced the fundamental idea employed is to represent the system response using a linear combination of stochastic basis vectors with undetermined deterministic coefficients or random functions we present a theoretical justification for employing basis vectors spanning the preconditioned stochastic krylov subspace to approximate the response process subsequently variants of the bubnov galerkin scheme are employed to compute the undetermined coefficients which allow explicit expressions for the response quantities to be derived we also examine some theoretical properties of the projection scheme and procedures for computing the response statistics numerical studies are presented for static and dynamic analysis of stochastic structural systems we demonstrate that significant improvements over the neumann expansion scheme as well as other relevant techniques in the literature can be achieved

a polyhedral branch and cut approach to global optimization a variety of nonlinear including semidefinite relaxations have been developed in recent years for nonconvex optimization problems their potential can be realized only if they can be solved with sufficient speed and reliability unfortunately state of the art nonlinear programming codes are significantly slower and numerically unstable compared to linear programming software in this paper we facilitate the reliable use of nonlinear convex relaxations in global optimization via a polyhedral branch and cut approach our algorithm exploits convexity either identified automatically or supplied through a suitable modeling language construct in order to generate polyhedral cutting planes and relaxations for multivariate nonconvex problems we prove that if the convexity of a univariate or multivariate function is apparent by decomposing it into convex subexpressions our relaxation constructor automatically exploits this convexity in a manner that is much superior to developing polyhedral outer approximators for the original function the convexity of functional expressions that are composed to form nonconvex expressions is also automatically exploited root node relaxations are computed for 87 problems from globallib and minlplib and detailed computational results are presented for globally solving 26 of these problems with baron 7 2 which implements the proposed techniques the use of cutting planes for these problems reduces root node relaxation gaps by up to 100 and expedites the solution process often by several orders of magnitude

partitioned treatment of uncertainty in coupled domain problems a separated representation approach abstract this work is concerned with the propagation of uncertainty across coupled domain problems with high dimensional random inputs a stochastic model reduction approach based on low rank separated representations is proposed for the partitioned treatment of the uncertainty space the construction of the coupled domain solution is achieved though a sequence of approximations with respect to the dimensionality of the random inputs associated with each individual sub domain and not the combined dimensionality hence drastically reducing the overall computational cost the coupling between the sub domain solutions is done via the classical finite element tearing and interconnecting feti method thus providing a well suited framework for parallel computing two high dimensional stochastic problems a 2d elliptic pde with random diffusion coefficient and a stochastic linear elasticity problem have been considered to study the performance and accuracy of the proposed stochastic coupling approach

scaling in the distribution of intertrade durations of chinese stocks abstract the distribution of intertrade durations defined as the waiting times between two consecutive transactions is investigated based upon the limit order book data of 23 liquid chinese stocks listed on the shenzhen stock exchange in the whole year 2003 a scaling pattern is observed in the distributions of intertrade durations where the empirical density functions of the normalized intertrade durations of all 23 stocks collapse onto a single curve the scaling pattern is also observed in the intertrade duration distributions for filled and partially filled trades and in the conditional distributions the ensemble distributions for all stocks are modeled by the weibull and the tsallis q exponential distributions maximum likelihood estimation shows that the weibull distribution outperforms the q exponential for not too large intertrade durations which account for more than 98 5 of the data alternatively nonlinear least squares estimation selects the q exponential as a better model in which the optimization is conducted on the distance between empirical and theoretical values of the logarithmic probability densities the distribution of intertrade durations is weibull followed by a power law tail with an asymptotic tail exponent close to 3

integer programming subject to monomial constraints we investigate integer programs containing constraints of the type prod i in i x i alpha i b due to the number theoretic nature of these constraints standard methods based on linear algebra cannot be applied directly instead we present a reformulation resulting in integer programs with linear constraints and polynomial objective functions using prime decompositions of the right hand sides b moreover we show that minimizing a linear objective function with nonnegative coefficients over bivariate constraints is possible in polynomial time

stochastic finite element analysis of shells with combined random material and geometric properties abstract in this work an extended stochastic formulation of the triangular composite facet shell element tric is presented for the case of combined uncertain material young s modulus poisson s ratio and geometric thickness properties these properties are assumed to be described by uncorrelated two dimensional homogeneous stochastic fields the stochastic finite element analysis of shell structures is performed using the spectral representation method for the description of the random fields in conjunction with monte carlo simulation mcs for the computation of the response variability useful conclusions regarding the influence of each one of the structural parameters on the response variability are derived from the numerical tests examined

efficient iterative solvers for stochastic galerkin discretizations of log transformed random diffusion problems we consider the numerical solution of a steady state diffusion problem where the diffusion coefficient is the exponent of a random field the standard stochastic galerkin formulation of this problem is computationally demanding because of the nonlinear structure of the uncertain component of it we consider a reformulated version of this problem as a stochastic convection diffusion problem with random convective velocity that depends linearly on a fixed number of independent truncated gaussian random variables the associated galerkin matrix is nonsymmetric but sparse and allows for fast matrix vector multiplications with optimal complexity we construct and analyze two block diagonal preconditioners for this galerkin matrix for use with krylov subspace methods such as the generalized minimal residual method we test the efficiency of the proposed preconditioning approaches and compare the iterative solver performance for a model problem posed in both diffusion and convection diffusion formulations

solving circle packing problems by global optimization numerical results and industrial applications abstract a general circle packing is an optimized arrangement of n arbitrary sized circles inside a container e g a rectangle or a circle such that no two circles overlap in this paper we present several circle packing problems review their industrial applications and some exact and heuristic strategies for their solution we also present illustrative numerical results using generic global optimization software packages our work highlights the relevance of global optimization in solving circle packing problems and points towards the necessary advancements in both theory and numerical practice

iterated tabu search for the circular open dimension problem abstract this paper investigates the circular open dimension problem codp which consists of packing a set of circles of known radii into a strip of fixed width and unlimited length without overlapping the objective is to minimize the length of the strip in this paper codp is solved by a series of sub problems each corresponding to a fixed strip length for each sub problem an iterated tabu search approach named its is proposed its starts from a randomly generated solution and attempts to gain improvements by a tabu search procedure after that if the obtained solution is not feasible a perturbation operator is subsequently employed to reconstruct the incumbent solution and an acceptance criterion is implemented to determine whether or not accept the perturbed solution as a supplementary method the length of the strip is determined in monotonously decreasing way with the aid of some post processing techniques the search terminates and returns the best found solution after the allowed computation time has been elapsed computational experiments based on numerous well known benchmark instances show that its produces quite competitive results with respect to the best known results while the computational time remains reasonable for each instance

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

uncertainty quantification in homogenization of heterogeneous microstructures modeled by xfem an extended finite element method xfem coupled with a monte carlo approach is proposed to quantify the uncertainty in the homogenized effective elastic properties of multiphase materials the methodology allows for an arbitrary number aspect ratio location and orientation of elliptic inclusions within a matrix without the need for fine meshes in the vicinity of tightly packed inclusions and especially without the need to remesh for every different generated realization of the microstructure moreover the number of degrees of freedom in the enriched elements is dynamically reallocated for each monte carlo sample run based on the given volume fraction the main advantage of the proposed xfem based methodology is a major reduction in the computational effort in extensive monte carlo simulations compared with the standard fem approach monte carlo and xfem appear to work extremely efficiently together the monte carlo approach allows for the modeling of the size aspect ratios orientations and spatial distribution of the elliptical inclusions as random variables with any prescribed probability distributions numerical results are presented and the uncertainty of the homogenized elastic properties is discussed copyright 2011 john wiley sons ltd

a stochastic computational multiscale approach application to mems resonators abstract the aim of this work is to develop a stochastic multiscale model for polycrystalline materials which accounts for the uncertainties in the micro structure at the finest scale we model the micro structure using a random voronoi tessellation each grain being assigned a random orientation then we apply a computational homogenization procedure on statistical volume elements to obtain a stochastic characterization of the elasticity tensor at the meso scale a random field of the meso scale elasticity tensor can thus be generated based on the information obtained from the sve simulations finally using a stochastic finite element method these meso scale uncertainties are propagated to the coarser scale as an illustration we study the resonance frequencies of mems micro beams made of poly silicon materials and we show that the stochastic multiscale approach predicts results in agreement with a monte carlo analysis applied directly on the fine finite element model i e with an explicit discretization of the grains

an extended stochastic finite element method for solving stochastic partial differential equations on random domains abstract recently a new strategy was proposed to solve stochastic partial differential equations on random domains it is based on the extension to the stochastic framework of the extended finite element method x fem this method leads by a direct calculus to an explicit solution in terms of the variables describing the randomness on the geometry it relies on two major points the implicit representation of complex geometries using random level set functions and the use of a galerkin approximation at both stochastic and deterministic levels in this article we detail the basis of this technique from theoretical and technical points of view several numerical examples illustrate the efficiency of this method and compare it to other approaches

linear models for the approximate solution of the problem of packing equal circles into a given domain abstract the linear models for the approximate solution of the problem of packing the maximum number of equal circles of the given radius into a given closed bounded domain g are proposed we construct a grid in g the nodes of this grid form a finite set of points t and it is assumed that the centers of circles to be packed can be placed only at the points of t the packing problems of equal circles with the centers at the points of t are reduced to 0 1 linear programming problems a heuristic algorithm for solving the packing problems based on linear models is proposed this algorithm makes it possible to solve packing problems for arbitrary connected closed bounded domains independently of their shape in a unified manner numerical results demonstrating the effectiveness of this approach are presented

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

a milp formulation for generalized geometric programming using piecewise linear approximations abstract generalized geometric programming ggp problems are converted to mixed integer linear programming milp problems using piecewise linear approximations our approach is to approximate a multiple term log sum function of the form log x 1 x 2 x n in terms of a set of linear equalities or inequalities of log x 1 log x 2 and log x n where x 1 x n are strictly positive the advantage of this approach is its simplicity and readiness to implement and solve using commercial milp solvers while milp problems in general are no easier than ggp problems this approach is justified by the phenomenal progress of computing power of both personal computers and commercial milp solvers the limitation of this approach is discussed along with numerical tests

polyhedral approximation in mixed integer convex optimization generalizing both mixed integer linear optimization and convex optimization mixed integer convex optimization possesses broad modeling power but has seen relatively few advances in general purpose solvers in recent years in this paper we intend to provide a broadly accessible introduction to our recent work in developing algorithms and software for this problem class our approach is based on constructing polyhedral outer approximations of the convex constraints resulting in a global solution by solving a finite number of mixed integer linear and continuous convex subproblems the key advance we present is to strengthen the polyhedral approximations by constructing them in a higher dimensional space in order to automate this extended formulation we rely on the algebraic modeling technique of disciplined convex programming dcp and for generality and ease of implementation we use conic representations of the convex constraints although our framework requires a manual translation of existing models into dcp form after performing this transformation on the minlplib2 benchmark library we were able to solve a number of unsolved instances and on many other instances achieve superior performance compared with state of the art solvers like bonmin scip and artelys knitro

a multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties abstract we here propose a multiscale numerical method for the solution of stochastic parametric partial differential equations with localized uncertainties described with a finite number of random variables it is based on a multiscale domain decomposition method that exploits the localized side of uncertainties and incidentally improves the conditioning of the problem by operating a separation of scales an efficient iterative algorithm is proposed that requires the solution of a sequence of simple global problems at a macro scale involving a deterministic operator and local problems at a micro scale for which we have the possibility to use fine approximation spaces global and local problems are solved using tensor approximation methods allowing the representation of high dimensional stochastic parametric solutions convergence properties of these tensor based methods which are closely related to spectral decompositions benefit from the separation of scales different types of uncertainties are considered at the micro level they may be associated with some variability in the operator or source terms or even with some geometrical variability in the latter case specific reformulations of local problems using fictitious domain methods are introduced

packing different sized circles into a rectangular container abstract this paper is motivated by the problem of fitting pipes of different diameters into a shipping container here we study the subproblem of fitting circles of different sizes into a rectangle since that problem is a central part of the larger problem we formulate this situation as a nonlinear mixed integer programming problem and develop a number of heuristic procedures for approximately solving this problem the heuristics are based on a variety of solution building rules that emulate the process of packing a container some of these methods including a genetic algorithm were based on a more structured design intended to provide solutions which are stable from a stowage viewpoint these heuristics are described in detail and their relative performances are compared for a sample set of 66 randomly generated problems based on this sample the best performing heuristic methods were a quasi random technique and a genetic algorithm of the stable solution structure

global optimization in hilbert space we propose a complete search algorithm for solving a class of non convex possibly infinite dimensional optimization problems to global optimality we assume that the optimization variables are in a bounded subset of a hilbert space and we determine worst case run time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set because these run time bounds are independent of the number of optimization variables and in particular are valid for optimization problems with infinitely many optimization variables we prove that the algorithm converges to an varepsilon suboptimal global solution within finite run time for any given termination tolerance varepsilon 0 finally we illustrate these results for a problem of calculus of variations

global optimization of mixed integer nonlinear programs a theoretical and computational study this work addresses the development of an efficient solution strategy for obtaining global optima of continuous integer and mixed integer nonlinear programs towards this end we develop novel relaxation schemes range reduction tests and branching strategies which we incorporate into the prototypical branch and bound algorithm in the theoretical algorithmic part of the paper we begin by developing novel strategies for constructing linear relaxations of mixed integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties we then use lagrangian linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming this theory leads to new range reduction schemes including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch and bound tree finally we incorporate these relaxation and reduction strategies in a branch and bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems in the computational part of the paper we describe our implementation discussing wherever appropriate the use of suitable data structures and associated algorithms we present computational experience with benchmark separable concave quadratic programs fractional 0 1 programs and mixed integer nonlinear programs from applications in synthesis of chemical processes engineering design just in time manufacturing and molecular design

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

acceleration of the spectral stochastic fem using pod and element based discrete empirical approximation for a micromechanical model of heterogeneous materials with random geometry abstract the spectral stochastic fem with local basis functions in the stochastic domain sl fem is one of the most flexible and accurate stochastic methods however also the most computationally expensive these expenses are traditionally associated with the extra large tangent stiffness matrix and a huge number of elements which need to be re integrated in every iteration in this work we incorporate the proper orthogonal decomposition pod into the sl fem thus performing a drastic reduction of the stiffness matrix in order to reduce the integration costs by hyperreduction a novel element based modification of the discrete empirical interpolation the so called element based empirical approximation method edeam is developed and combined with the pod particular advantages of the sl fem for order reduction and hyperreduction compared to other stochastic techniques are discussed the new reduced order sl fem is applied to the computational homogenization of materials with random geometry of the microstructure i e to a general class of problems exhibiting strongly nonlinear non smooth and sometimes discontinuous dependency of the solution on some random parameters the reduced order sl fem demonstrates a high accuracy and a high solution speed whereby the solution time for the reduced order sl fem is comparable to the solution time of only one single monte carlo sample

long term correlations and multifractal nature in the intertrade durations of a liquid chinese stock and its warrant abstract the intertrade duration of equities is an important financial measure characterizing trading activities it is defined as the waiting time between successive trades of an equity using the ultrahigh frequency data of a liquid chinese stock and its associated warrant we perform a comparative investigation of the statistical properties of their intertrade duration time series the distributions of the two equities can be better described by the shifted power law form than the weibull form and their scaled distributions do not collapse onto a single curve although the intertrade durations of the two equities have very different magnitude their intraday patterns exhibit very similar shapes both detrended fluctuation analysis dfa and detrending moving average analysis dma show that the 1 min intertrade duration time series of the two equities are strongly correlated in addition both multifractal detrended fluctuation analysis mfdfa and multifractal detrending moving average analysis mfdma unveil that the 1 min intertrade durations possess multifractal nature however the difference between the two singularity spectra of the two equities obtained from the mfdma is much smaller than that from the mfdfa

optimizing the packing of cylinders into a rectangular container a nonlinear approach abstract the container loading problem has important industrial and commercial applications an increase in the number of items in a container leads to a decrease in cost for this reason the related optimization problem is of economic importance in this work a procedure based on a nonlinear decision problem to solve the cylinder packing problem with identical diameters is presented this formulation is based on the fact that the centers of the cylinders have to be inside the rectangular box defined by the base of the container a radius far from the frontier and far from each other at least one diameter with this basic premise the procedure tries to find the maximum number of cylinder centers that satisfy these restrictions the continuous nature of the problem is one of the reasons that motivated this study a comparative study with other methods of the literature is presented and better results are achieved

a new approach for the stochastic analysis of finite element modelled structures with uncertain parameters a method for evaluating the static response of uncertain finite element fe discretised structures is presented the method is comparable with the perturbation procedures from a computational point of view but it overcomes the drawbacks related to these procedures in fact the present method gives excellent level of accuracy even for high amount of uncertainties moreover it produces the exact solution for statically determinate structures the accuracy is remarkable even in the evaluation of high order statistics and probability density function of the response so that reliability consideration can be drawn

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

efficient solvers for a linear stochastic galerkin mixed formulation of diffusion problems with random data we introduce a stochastic galerkin mixed formulation of the steady state diffusion equation and focus on the efficient iterative solution of the saddle point systems obtained by combining standard finite element discretizations with two distinct types of stochastic basis functions so called mean based preconditioners based on fast solvers for scalar diffusion problems are introduced for use with the minimum residual method we derive eigenvalue bounds for the preconditioned system matrices and report on the efficiency of the chosen preconditioning schemes with respect to all the discretization parameters

the gradual covering decay location problem on a network abstract in covering problems it is assumed that there is a critical distance within which the demand point is fully covered while beyond this distance it is not covered at all in this paper we define two distances within the lower distance a demand point is fully covered and beyond the larger distance it is not covered at all for a distance between these two values we assume a gradual coverage decreasing from full coverage at the lower distance to no coverage at the larger distance

acceleration of uncertainty updating in the description of transport processes in heterogeneous materials the prediction of thermo mechanical behaviour of heterogeneous materials such as heat and moisture transport is strongly influenced by the uncertainty in parameters such materials occur e g in historic buildings and the durability assessment of these therefore needs a reliable and probabilistic simulation of transport processes which is related to the suitable identification of material parameters in order to include expert knowledge as well as experimental results one can employ an updating procedure such as bayesian inference the classical probabilistic setting of the identification process in bayes form requires the solution of a stochastic forward problem via computationally expensive sampling techniques which makes the method almost impractical in this paper novel stochastic computational techniques such as the stochastic galerkin method are applied in order to accelerate the updating procedure the idea is to replace the computationally expensive forward simulation via the conventional finite element fe method by the evaluation of a polynomial chaos expansion pce such an approximation of the fe model for the forward simulation perfectly suits the bayesian updating the presented uncertainty updating techniques are applied to the numerical model of coupled heat and moisture transport in heterogeneous materials with spatially varying coefficients defined by random fields

extended stochastic finite element method for the numerical simulation of heterogeneous materials with random material interfaces an extended stochastic finite element method has been recently proposed for the numerical solution of partial differential equations defined on random domains this method is based on a marriage between the extended finite element method and spectral stochastic methods in this article we propose an extension of this method for the numerical simulation of random multi phased materials the random geometry of material interfaces is described implicitly by using random level set functions a fixed deterministic finite element mesh which is not conforming to the random interfaces is then introduced in order to approximate the geometry and the solution classical spectral stochastic finite element approximation spaces are not able to capture the irregularities of the solution field with respect to spatial and stochastic variables which leads to a deterioration of the accuracy and convergence properties of the approximate solution in order to recover optimal convergence properties of the approximation we propose an extension of the partition of unity method to the spectral stochastic framework this technique allows the enrichment of approximation spaces with suitable functions based on an a priori knowledge of the irregularities in the solution numerical examples illustrate the efficiency of the proposed method and demonstrate the relevance of the enrichment procedure copyright 2010 john wiley sons ltd

domain decomposition of stochastic pdes theoretical formulations we present a novel theoretical framework for the domain decomposition of uncertain systems defined by stochastic partial differential equations the methodology involves a domain decomposition method in the geometric space and a functional decomposition in the probabilistic space the probabilistic decomposition is based on a version of stochastic finite elements based on orthogonal decompositions and projections of stochastic processes the spatial decomposition is achieved through a schur complement based domain decomposition the methodology aims to exploit the full potential of high performance computing platforms by reducing discretization errors with high resolution numerical model in conjunction to giving due regards to uncertainty in the system the mathematical formulation is numerically validated with an example of waves in random media copyright 2008 john wiley sons ltd

generalized spectral decomposition method for solving stochastic finite element equations invariant subspace problem and dedicated algorithms stochastic galerkin methods have become a significant tool for the resolution of stochastic partial differential equations spde however they suffer from prohibitive computational times and memory requirements when dealing with large scale applications and high stochastic dimensionality some alternative techniques based on the construction of suitable reduced deterministic or stochastic bases have been proposed in order to reduce these computational costs recently a new approach based on the concept of generalized spectral decomposition gsd has been introduced for the definition and the automatic construction of reduced bases in this paper the concept of gsd initially introduced for a class of linear elliptic spde is extended to a wider class of stochastic problems the proposed definition of the gsd leads to the resolution of an invariant subspace problem which is interpreted as an eigen like problem this interpretation allows the construction of efficient numerical algorithms for building optimal reduced bases which are associated with dominant generalized eigenspaces the proposed algorithms by separating the resolution of reduced stochastic and deterministic problems lead to drastic computational savings their efficiency is illustrated on several examples where they are compared to classical resolution techniques

the transformation method for the simulation and analysis of systems with uncertain parameters in this paper the transformation method is introduced as a powerful approach for both the simulation and the analysis of systems with uncertain model parameters based on the concept of cuts the method represents a special implementation of fuzzy arithmetic that avoids the well known effect of overestimation which usually arises when fuzzy arithmetic is reduced to interval computation systems with uncertain model parameters can thus be simulated without any artificial widening of the simulation results as a by product of the implementation scheme the transformation method also provides a measure of influence to quantitatively analyze the uncertain system with respect to the effect of each uncertain model parameter on the overall uncertainty of the model output by this a special kind of sensitivity analysis can be defined on the basis of fuzzy arithmetic finally to show the efficiency of the transformation method the method is applied to the simulation and analysis of a model for the friction interface between the sliding surfaces of a bolted joint connection