robust algorithms for solving stochastic partial differential equations a robust semi implicit central partial difference algorithm for the numerical solution of coupled stochastic parabolic partial differential equations pdes is described this can be used for calculating correlation functions of systems of interacting stochastic fields such field equations can arise in the description of hamiltonian and open systems in the physics of nonlinear processes and may include multiplicative noise sources the algorithm can be used for studying the properties of nonlinear quantum or classical field theories the general approach is outlined and applied to a specific example namely the quantum statistical fluctuations of ultra short optical pulses in 2 parametric waveguides this example uses a non diagonal coherent state representation and correctly predicts the sub shot noise level spectral fluctuations observed in homodyne detection measurements it is expected that the methods used will be applicable for higher order correlation functions and other physical problems as well a stochastic differencing technique for reducing sampling errors is also introduced this involves solving nonlinear stochastic parabolic pdes in combination with a reference process which uses the wigner representation in the example presented here a computer implementation on mimd parallel architectures is discussed

prediction of the mechanical response of the femur with uncertain elastic properties abstract a mandatory requirement for any reliable prediction of the mechanical response of bones based on quantitative computer tomography is an accurate relationship between material properties usually young s modulus e and bone density many such e relationships are available based on different experiments on femur specimens with a large spread due to uncertainties the first goal of this study is to pool and analyze the relevant available experimental data and develop a stochastic e relationship this analysis highlights that there is no experimental data available to cover the entire density range of the human femur and that some popular e relationships are based on data that contains extreme scatter while others are based on a very limited amount of information the second goal is to use the newly developed stochastic e relationship in high order finite element analyses feas for the computation of strains and displacements in two human proximal femurs mimicking in vitro experiments when compared with the experimental observations the fea predictions using the median of the stochastic e relationship follow the underlying distribution of the stochastic e relationship thus most deviations of the fea predictions from experimental observations can possibly be explained by uncertain elastic properties of the femur

recent results on multiplicative noise recent developments in the analysis of langevin equations with multiplicative noise mn are reported in particular we i present numerical simulations in three dimensions showing that the mn equation exhibits like the kardar parisi zhang kpz equation both a weak coupling fixed point and a strong coupling phase supporting the proposed relation between mn and kpz ii present a dimensional and mean field analysis of the mn equation to compute critical exponents iii show that the phenomenon of the noise induced ordering transition associated with the mn equation appears only in the stratonovich representation and not in the ito one and iv report the presence of a first order like phase transition at zero spatial coupling supporting the fact that this is the minimum model for noise induced ordering transitions

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

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

on type 2 fuzzy relations and interval valued type 2 fuzzy sets this paper introduces new operations on the algebra of fuzzy truth values extended supremum and extended infimum which are generalizations of the extended operations of maximum and minimum between fuzzy truth values for type 2 fuzzy sets respectively using these new operations the properties of type 2 fuzzy relations are discussed especially the compositions of type 2 fuzzy relations on this basis this paper introduces interval valued type 2 fuzzy sets and interval valued type 2 fuzzy relations and discusses their properties

uncertainty quantification for a 1d thermo hyperelastic coupled problem using polynomial chaos projection and p fems numerical solutions of non linear stochastic thermo hyperelastic problems at finite strains are addressed these belong to a category of non linear coupled problems that impose challenges on their numerical treatment both in the physical and stochastic spaces combining the high order finite element methods fems for discretizing the physical space and the polynomial chaos projection pcp method for discretizing the stochastic space a non intrusive scheme is obtained manifesting an exponential convergence rate the method is applied to a 1 d coupled stationary thermo hyperelastic system with stochastic material properties we derive exact stochastic solutions that serve for comparison to numerical results allowing their verification these demonstrate that stochastic coupled problems intractable by standard monte carlo mc methods may be easily computed by combining high order fems with the pcp method controlling discretization errors

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

stochastic dynamic analysis of marine risers considering gaussian system uncertainties abstract this paper performs the stochastic dynamic response analysis of marine risers with material uncertainties i e in the mass density and elastic modulus by using stochastic finite element method sfem and model reduction technique these uncertainties are assumed having gaussian distributions the random mass density and elastic modulus are represented by using the karhunen loeve kl expansion the polynomial chaos pc expansion is adopted to represent the vibration response because the covariance of the output is unknown model reduction based on the iterated improved reduced system iirs technique is applied to eliminate the pc coefficients of the slave degrees of freedom to reduce the dimension of the stochastic system monte carlo simulation mcs is conducted to obtain the reference response statistics two numerical examples are studied in this paper the response statistics from the proposed approach are compared with those from mcs it is noted that the computational time is significantly reduced while the accuracy is kept the results demonstrate the efficiency of the proposed approach for stochastic dynamic response analysis of marine risers

covariance kernel representations of multidimensional second order stochastic processes the dynamics of stationary stochastic processes in space is not exactly analogous to that of stationary stochastic processes in the time domain this is due to the unilateral nature of the time series that is only influenced by past values as opposed to the dependence in all directions of the spatial process in this work we unfold the connection that exits between the covariance kernel of a multi dimensional second order autoregressive random process and its underlying discrete random dynamical system starting from a discrete random dynamical system we show that the random process satisfying that system is governed by the modified helmholtz equation in the continuous limit we establish the dependence of the correlation constant on the grid size of the discretization we also show that the random forcing term in the continuous case turns out to be a white noise process a number of covariance functions are worked out for simple and more complex geometrical domains with various boundary conditions in multi dimensions we use both the discrete and the continuous systems in our computations

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

the mechanism of stochastic resonance it is shown that a dynamical system subject to both periodic forcing and random perturbation may show a resonance peak in the power spectrum which is absent when either the forcing or the perturbation is absent

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

spike frequency adapting neural ensembles beyond mean adaptation and renewal theories we propose a markov process model for spike frequency adapting neural ensembles that synthesizes existing mean adaptation approaches population density methods and inhomogeneous renewal theory resulting in a unified and tractable framework that goes beyond renewal and mean adaptation theories by accounting for correlations between subsequent interspike intervals a method for efficiently generating inhomogeneous realizations of the proposed markov process is given numerical methods for solving the population equation are presented and an expression for the first order interspike interval correlation is derived further we show that the full five dimensional master equation for a conductance based integrate and fire neuron with spike frequency adaptation and a relative refractory mechanism driven by poisson spike trains can be reduced to a two dimensional generalization of the proposed markov process by an adiabatic elimination of fast variables for static and dynamic stimulation negative serial interspike interval correlations and transient population responses respectively of monte carlo simulations of the full five dimensional system can be accurately described by the proposed two dimensional markov process

three dimensional stochastic finite element method for elasto plastic bodies a new stochastic finite element method sfem is formulated for three dimensional softening elasto plastic bodies with random material properties the method is based on the karhunen loeve and polynomial chaos expansions and able to efficiently estimate complete probabilistic characteristics of the response such as moments or pdfs to reduce the computational complexity in the three dimensional setting two alterations are made with respect to the two dimensional sfem proposed earlier by the authors first a variability preserving modification of the karhunen loeve expansion is rigorously derived and applied in the stochastic discretization of random fields representing material properties second an efficient algorithm for parallel processing is developed with time consumption being the same order as for an ordinary fem rendering the proposed sfem an effective alternative to monte carlo simulation the applicability of the proposed method to stochastic analysis of strain localization is examined using monte carlo simulation then it is applied to a fault formation problem which is a recent concern of earthquake engineering ground surface layers are modelled by a softening elasto plastic body and the evolution of probabilistic characteristics of the rupture process is analysed in detail some practical observations are made regarding the nature of the fault formation from the stochastic viewpoint copyright 2001 john wiley sons ltd

simplified type 2 fuzzy sliding controller for wing rock system wing rock is a highly nonlinear phenomenon in which aircrafts with slender delta wings undergo limit cycle roll oscillations at high angles of attack a simplified type 2 fuzzy sliding controller is designed for suppressing wing rock phenomena and tracking the desired trajectories to reduce the computational complexity of a type reducer the end points of a type reduced set are approximated by the outputs of two standard fuzzy sliding mechanisms in the proposed simplified type 2 fuzzy sliding controller furthermore the sliding modes of the fuzzy sliding control system are guaranteed simulation results are included to show the effectiveness of the proposed simplified type 2 fuzzy sliding controller

characterization of subthreshold voltage fluctuations in neuronal membranes synaptic noise due to intense network activity can have a significant impact on the electrophysiological properties of individual neurons this is the case for the cerebral cortex where ongoing activity leads to strong barrages of synaptic inputs which act as the main source of synaptic noise affecting on neuronal dynamics here we characterize the subthreshold behavior of neuronal models in which synaptic noise is represented by either additive or multiplicative noise described by ornstein uhlenbeck processes we derive and solve the fokker planck equation for this system which describes the time evolution of the probability density function for the membrane potential we obtain an analytic expression for the membrane potential distribution at steady state and compare this expression with the subthreshold activity obtained in hodgkin huxley type models with stochastic synaptic inputs the differences between multiplicative and additive noise models suggest that multiplicative noise is adequate to describe the high conductance states similar to in vivo conditions because the steady state membrane potential distribution is easily obtained experimentally this approach provides a possible method to estimate the mean and variance of synaptic conductances in real neurons

uncertainty bounds and their use in the design of interval type 2 fuzzy logic systems we derive inner and outer bound sets for the type reduced set of an interval type 2 fuzzy logic system fls based on a new mathematical interpretation of the karnik mendel iterative procedure for computing the type reduced set the bound sets can not only provide estimates about the uncertainty contained in the output of an interval type 2 fls but can also be used to design an interval type 2 fls we demonstrate by means of a simulation experiment that the resulting system can operate without type reduction and can achieve similar performance to one that uses type reduction therefore our new design method based on the bound sets can relieve the computation burden of an interval type 2 fls during its operation which makes an interval type 2 fls useful for real time applications

estimation of the input parameters in the feller neuronal model the stochastic feller neuronal model is studied and estimators of the model input parameters depending on the firing regime of the process are derived closed expressions for the first two moments of functionals of the first passage time ftp through a constant boundary in the suprathreshold regime are derived which are used to calculate moment estimators in the subthreshold regime the exponentiality of the ftp is utilized to characterize the input parameters the methods are illustrated on simulated data finally approximations of the first passage time moments are suggested and biological interpretations and comparisons of the parameters in the feller and the ornstein uhlenbeck models are discussed

high dimensional integration of smooth functions over cubes we construct a new algorithm for the numerical integration of functions that are defined on a d dimensional cube it is based on the clenshaw curtis rule for d 1 and on smolyak s construction this way we make the best use of the smoothness properties of any nonperiodic function we prove error bounds showing that our algorithm is almost optimal up to logarithmic factors for different classes of functions with bounded mixed derivative numerical results show that the new method is very competitive in particular for smooth integrands and d ge 8

synaptic shot noise and conductance fluctuations affect the membrane voltage with equal significance the subthreshold membrane voltage of a neuron in active cortical tissue is a fluctuating quantity with a distribution that reflects the firing statistics of the presynaptic population it was recently found that conductancebased synaptic drive can lead to distributions with a significant skew here it is demonstrated that the underlying shot noise caused by poissonian spike arrival also skews the membrane distribution but in the opposite sense using a perturbative method we analyze the effects of shot noise on the distribution of synaptic conductances and calculate the consequent voltage distribution to first order in the perturbation theory the voltage distribution is a gaussian modulated by a prefactor that captures the skew the gaussian component is identical to distributions derived using current based models with an effective membrane time constant the well known effective time constant approximation can therefore be identified as the leading order solution to the full conductance based model the higher order modulatory prefactor containing the skew comprises terms due to both shot noise and conductance fluctuations the diffusion approximation misses these shot noise effects implying that analytical approaches such as the fokker planck equation or simulation with filtered white noise cannot be used to improve on the gaussian approximation it is further demonstrated that quantities used for fitting theory to experiment such as the voltage mean and variance are robust against these non gaussian effects the effective time constant approximation is therefore relevant to experiment and provides a simple analytic base on which other pertinent biological details may be added

effects of synaptic conductance on the voltage distribution and firing rate of spiking neurons a neuron in an active cortical circuit is subject to a fluctuating synaptic drive mediated by conductance changes it was recently demonstrated that synaptic conductance effects in vivo significantly alter the integrative properties of neurons these effects are missed in models that approximate the synaptic drive as a fluctuating current here the membrane potential distribution and firing rate are derived for the integrate andfire neuron withd correlated conductance based synaptic input using the fokker planck formalism a number of different input scenarios are examined including balanced drive and fluctuation changes at constant conductance the latter of which corresponds to shifts in synchrony in the presynaptic population this minimal model captures many experimentally observed conductance related effects such as reduced membranepotential fluctuations in response to increasing synaptic noise the solvability of the model allows for a direct comparison with current based approaches providing a basis for assessing the validity of existing approximation schemes that have dealt with conductance change in particular a commonly used heuristic approach whereby the passive membrane time constant is replaced by a drive dependent effective time constant is examined it is demonstrated that this approximation is valid in the same limit that the underlying diffusion approximation holds both for d correlated as well as filtered synaptic drive

noise induced phase separation mean field results we present a study of a phase separation process induced by the presence of spatially correlated multiplicative noise we develop a mean field approach suitable for conserved order parameter systems and use it to obtain the phase diagram of the model mean field results are compared with numerical simulations of the complete model in two dimensions additionally a comparison between the noise driven dynamics of conserved and nonconserved systems is made at the level of the mean field approximation

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

numerical solution of spectral stochastic finite element systems abstract this paper addresses the issues involved in solving systems of linear equations which arise in the context of the spectral stochastic finite element ssfem formulation two efficient solution procedures are presented that dramatically reduce the amount of computations involved in numerically solving these problems a brief review is first provided of the underlying spectral approach which highlights the peculiar structure of the matrices generated and how their properties are related to both the level of approximation involved as well as to the convergence behavior of the proposed solution procedure the differences between these matrices from their deterministic finite element counterparts are illustrated an iterative solution scheme is proposed which utilizes their specific properties for efficient memory management and enhanced convergence behavior results from numerical tests are presented comparisons with standard algorithms illustrate the efficiency of the proposed algorithm the second solution procedure presented in this paper is based on hierarchical basis concepts results from numerical tests are again provided and the limitations of this approach are assessed the performance of both proposed algorithms indicates that the linear algebraic systems from the underlying ssfem formulation can be solved with considerably less effort in memory and computation time than their size suggests furthermore the data structures and the hierarchical concept introduced in this study are found to have great potential for the future development of adaptive procedures in stochastic fem

enhanced karnik mendel algorithms the karnik mendel km algorithms are iterative procedures widely used in fuzzy logic theory they are known to converge monotonically and superexponentially fast however several usually two to six iterations are still needed before convergence occurs methods to reduce their computational cost are proposed in this paper extensive simulations show that on average the enhanced km algorithms can save about two iterations which corresponds to more than a 39 reduction in computation time an additional at least 23 computational cost can be saved if no sorting of the inputs is needed

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

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

on the construction and analysis of stochastic models characterization and propagation of the errors associated with limited data this paper investigates the predictive accuracy of stochastic models in particular a formulation is presented for the impact of data limitations associated with the calibration of parameters for these models on their overall predictive accuracy in the course of this development a new method for the characterization of stochastic processes from corresponding experimental observations is obtained specifically polynomial chaos representations of these processes are estimated that are consistent in some useful sense with the data the estimated polynomial chaos coefficients are themselves characterized as random variables with known probability density function thus permitting the analysis of the dependence of their values on further experimental evidence moreover the error in these coefficients associated with limited data is propagated through a physical system characterized by a stochastic partial differential equation spde this formalism permits the rational allocation of resources in view of studying the possibility of validating a particular predictive model a bayesian inference scheme is relied upon as the logic for parameter estimation with its computational engine provided by a metropolis hastings markov chain monte carlo procedure

improving the speed of center of sets type reduction in interval type 2 fuzzy systems by eliminating the need for sorting in the deployment of interval type 2 fuzzy systems one of the most important steps is the type reduction the commonly used center of sets type reducer requires the solution of two nonlinear constrained optimization problems frequently used approaches to solve them are the karnik mendel algorithms and their variants however these algorithms suffer from the need for sorting which is known to be computationally very expensive using the reformulations proposed in this paper for center of sets type reducer it is possible to eliminate the need for sorting this makes interval type 2 fuzzy systems more appropriate for cost sensitive real time applications extensive simulations are presented to illustrate the faster convergence speed of the proposed method over six other enhanced variants of the karnik mendel algorithm as applied to center of sets type reduction of interval type 2 fuzzy systems

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

fuzzy weighted averages revisited the fuzzy weighted average operations are common operations in risk and decision analysis based on zadeh s extension principle dong and wang fuzzy sets and systems 21 1987 183 proposed a fwa method a method based on the cut representation of fuzzy sets and interval analysis the fwa method provides a discrete approximation of the cuts of fuzzy weighted average the fwa method has an exponential complexity subsequently liou and wang fuzzy sets and systems 49 1992 307 proved that indeed the fwa method can be simplified to be a nonconstrainted linear fractional programming problem with lower and upper bounds for each of the variables they proposed an improved algorithm called ifwa which is of o n2 complexity where n is the number of decision variables in ifwa guh et al comput math appl 32 8 1996 115 proposed a max min paired elimination method pfwa they claimed their pfwa was of order n in complexity yet the pfwa is a heuristic method lacking any proof of convergence in 1997 lee and park proposed an algorithm efwa which is of o nlogn complexity most recently kao and liu fuzzy sets and systems 120 2001 435 solved the fwa by transforming the fractional program into a linear program in this note we shall relate the ifwa framework with the nonconstrainted 0 1 integer linear fractional programming problem in the mathematical programming literature several o n time algorithms have been existing for instance robillard naval res logist quart 18 1971 47 and hansen et al math programming 52 1991 255 the algorithm of hansen et al will be presented for easy reference

formulation of geometrically non linear problems in the spatial reference frame this paper presents a new formulation for geometrically non linear problems and their numerical treatment by the p version of the finite element method the formulation is characterized by a weak form based on the spatial representation of the equilibrium equations a deformed geometry mapped by the displacement field a natural description for non conservative loads that keeps the symmetry of the equations and a direct non incremental iterative algorithm to solve the equations the paper is concerned only with problems for which the solution is unique i e the load displacement curves are single valued and the strains are small examples are presented 1997 by john wiley sons ltd

robust control of an lusm based x hbox y hbox theta motion control stage using an adaptive interval type 2 fuzzy neural network the robust control of a linear ultrasonic motor based x y thetas motion control stage to track various contours is achieved by using an adaptive interval type 2 fuzzy neural network ait2fnn control system in this study in the proposed ait2fnn control system an it2fnn which combines the merits of an interval type 2 fuzzy logic system and a neural network is developed to approximate an unknown dynamic function moreover adaptive learning algorithms are derived using the lyapunov stability theorem to train the parameters of the it2fnn online furthermore a robust compensator is proposed to confront the uncertainties including the approximation error optimal parameter vectors and higher order terms in taylor series to relax the requirement for the value of lumped uncertainty in the robust compensator an adaptive lumped uncertainty estimation law is also investigated in addition the circle and butterfly contours are planned using a nonuniform rational b spline curve interpolator the experimental results show that the contour tracking performance of the proposed ait2fnn is significantly improved compared with the adaptive type 1 fnn additionally the robustness to parameter variations external disturbances cross coupled interference and frictional force can also be obtained using the proposed ait2fnn

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

aggregation using the fuzzy weighted average as computed by the karnik mendel algorithms by connecting work from two different problems the fuzzy weighted average fwa and the generalized centroid of an interval type 2 fuzzy set a new alpha cut algorithm for solving the fwa problem has been obtained one that is monotonically and superexponentially convergent this new algorithm uses the karnik mendel km algorithms to compute the fwa cut end points it appears that the km cut algorithms approach for computing the fwa requires the fewest iterations to date and may therefore be the fastest available fwa algorithm to date

a multiscale probabilistic collocation method for subsurface flow in heterogeneous media 1 owing to the spatial variability of the media properties uncertainty quantification for subsurface flow and solute transport usually requires high resolution simulations in this work a multiscale probabilistic collocation method mspcm is developed for solving such problems in a computationally efficient manner the subsurface flow problem is cast in a stochastic framework and probabilistic collocation strategy is used to represent the original stochastic differential equation the resulting equations are a set of decoupled deterministic equations with respect to collocation points a multiscale finite element method is utilized to solve these deterministic problems on a coarse mesh coarse scale basis functions are constructed on a field in which the conductivity varies spatially at each set of stochastic collocation points the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the fine scale heterogeneity the fine scale solution is reconstructed after the coarse scale solution is available since the pcm and multiscale finite element method are implemented at different levels the mspcm inherits their respective advantages in which a stochastic problem is decomposed by fewer realizations and is solved on a coarser grid the performance of the proposed method is demonstrated with numerical examples the capability of mspcm in reproducing the probability density functions pdfs of head and velocity is investigated the numerical results show that the mspcm with proper coarsening level is able to capture small scale heterogeneity with a coarse mesh and to generate satisfactory probability density functions of head and velocity

design of a fuzzy logic controller as a suction controller abstract a fuzzy logic controller flc designed as a suction controller is proposed firstly we prove that there exists a switching line in the flc and the slope of the switching line is determined by scaling factors secondly a lyapunov function is selected and sufficient conditions for the fuzzy control system fcs to be globally asymptotically stable is determined based on the sufficient conditions a design procedure is suggested to guarantee that the fcs is globally asymptotically stable both a nonlinear oscillator and a cart pole system are simulated to demonstrate the proposed flc we show that not only is the flc similar to sliding mode control smc but also a realizable method for a suction controller is described in addition a lemma is introduced to interpret that the rise time of the fcs can be adjusted by the scaling factors

generalized extended fuzzy implications this paper is devoted to investigate generalized extended fuzzy implications in accordance with the generalized extension principle the algebraic properties of generalized extended fuzzy implications are discussed on different algebras of fuzzy truth values on this basis fuzzy valued fuzzy implications are constructed from arbitrary fuzzy implications and t norms moreover a fuzzy valued fuzzy implication is identical with a generalized extended fuzzy implication on the algebra of fuzzy values when they are induced by exactly the same continuous fuzzy implication and same upper semicontinuous t norm

stochastic elastic plastic finite elements abstract a computational framework has been developed for simulations of the behavior of solids and structures made of stochastic elastic plastic materials uncertain elastic plastic material properties are modeled as random fields which appear as the coefficient term in the governing partial differential equation of mechanics a spectral stochastic elastic plastic finite element method with fokker planck kolmogorov equation based probabilistic constitutive integrator is proposed for solution of this non linear elastic plastic partial differential equation with stochastic coefficient to this end the random field material properties are discretized in both spatial and stochastic dimension into finite numbers of independent basic random variables using karhunen loeve expansion those random variables are then propagated through the elastic plastic constitutive rate equation using fokker planck kolmogorov equation approach to obtain the evolutionary material properties as the material plastifies the unknown displacement solution random field is then assembled using polynomial chaos as a function of known basic random variables and unknown deterministic coefficients which are obtained by minimizing the error of finite representation by galerkin technique

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

stochastic simulation of riser sections with uncertain measured pressure loads and or uncertain material properties abstract we investigate three dimensional problems in solid mechanics with stochastic loading or material properties to solve these problems we use a spectral expansion of the solution and random inputs based on askey type orthogonal polynomials in terms of independent identically distributed i i d random variables a galerkin procedure using these types of expansions the generalized polynomial chaos gpc method is employed to solve linear elasticity problems an analagous spectral collocation formulation is used to study problems in nonlinear elasticity these methods both cast the stochastic problem as a coupled or decoupled high dimensional system of deterministic pdes which is then solved numerically using a deterministic p finite element solver we present algorithms for solving certain coupled systems arising from the stochastic galerkin projection without modifying the original deterministic solver three dimensional riser sections undergoing elastic deformations due to random pressure loads are considered we also model a riser section with stochastic young s modulus undergoing deterministic loads it is demonstrated that the gpc method provides accurate and efficient results at a speed up factor of two and three orders of magnitude compared to traditional monte carlo simulations for nonlinear problems the stochastic collocation method is also shown to be much faster than monte carlo simulation while still rivaling this method in simplicity of implementation

gradual numbers and their application to fuzzy interval analysis in this paper we introduce a new way of looking at fuzzy intervals instead of considering them as fuzzy sets we see them as crisp sets of entities we call gradual real numbers they are a gradual extension of real numbers not of intervals such a concept is apparently missing in fuzzy set theory gradual numbers basically have the same algebraic properties as real numbers but they are functions a fuzzy interval is then viewed as a pair of fuzzy thresholds which are monotonic gradual real numbers this view enables interval analysis to be directly extended to fuzzy intervals without resorting to alpha cuts in agreement with zadeh s extension principle several results show that interval analysis methods can be directly adapted to fuzzy interval computation where end points of intervals are changed into left and right fuzzy bounds our approach is illustrated on two known problems computing fuzzy weighted averages and determining fuzzy floats and latest starting times in activity network scheduling

comment on characterization of subthreshold voltage fluctuations in neuronal membranes by m rudolph and a destexhe in two recent articles rudolph and destexhe 2003 2005 studied a leaky integrator model an rc circuit driven by correlated colored gaussian conductance noise and gaussian current noise in the first article they derived an expression for the stationary probability density of the membrane voltage in the second they modified this expression to cover a larger parameter regime here we show by standard analysis of solvable limit cases white noise limit of additive and multiplicative noise sources only slow multiplicative noise only additive noise and by numerical simulations that their first result does not hold for the general colored noise case and uncover the errors made in the derivation of a fokker planck equation for the probability density furthermore we demonstrate analytically including an exact integral expression for the time dependent mean value of the voltage and by comparison to simulation results that the extended expression for the probability density works much better but still does not exactly solve the full colored noise problem we also show that at stronger synaptic input the stationary mean value of the linear voltage model may diverge and give an exact condition relating the system parameters for which this takes place