exterior stable domain segmentation integral equation method for scattering problems this paper is concerned with the development of an exterior domain segmentation method for the solution of two or three dimensional time harmonic scattering problems in acoustic media this method based on a variational localized symmetric boundary integral equation formulation leads upon discretization to a sparse system of algebraic equations whose solution requires only o n multiplications where n is the number of unknown nodal pressures on the scatterer surface the new procedure is analogous to the one developed recently1 except that in the present formulation we avoid completely the use of the hypersingular operator thereby reducing the computational complexity numerical experiments for a rigid circular cylindrical scatterer subjected to a plane incident wave serve to assess its accuracy for normalized wave numbers ka ranging from 0 to 30 both directly on the scatterer and in the far field and to confirm that contrary to standard boundary integral equation formulations the present procedure is valid for critical frequencies

on the asymptotic stability ofθ methods for delay differential equations stability regions of methods for the linear delay differential test equations begin gathered y t py t qy t tau t 0 hfill y t varphi t t in tau 0 hfill end gathered where is a positive constant are presented in the case thatp andq are real constant coefficients necessary and sufficient conditions on the stepsize for the stability of a method are obtained furthermore whenp andq are complex coefficients sufficient conditions for the stability of the methods are also given

a stochastic lagrangian approach for geometrical uncertainties in electrostatics this work proposes a general framework to quantify uncertainty arising from geometrical variations in the electrostatic analysis the uncertainty associated with geometry is modeled as a random field which is first expanded using either polynomial chaos or karhunen loeve expansion in terms of independent random variables the random field is then treated as a random displacement applied to the conductors defined by the mean geometry to derive the stochastic lagrangian boundary integral equation the surface charge density is modeled as a random field and is discretized both in the random dimension and space using polynomial chaos and classical boundary element method respectively various numerical examples are presented to study the effect of uncertain geometry on relevant parameters such as capacitance and net electrostatic force the results obtained using the proposed method are verified using rigorous monte carlo simulations it has been shown that the proposed method accurately predicts the statistics and probability density functions of various relevant parameters

maximum likelihood estimation of stochastic chaos representations from experimental data this paper deals with the identification of probabilistic models of the random coefficients in stochastic boundary value problems sbvp the data used in the identification correspond to measurements of the displacement field along the boundary of domains subjected to specified external forcing starting with a particular mathematical model for the mechanical behaviour of the specimen the unknown field to be identified is projected on an adapted functional basis such as that provided by a finite element discretization for each set of measurements of the displacement field along the boundary an inverse problem is formulated to calculate the corresponding optimal realization of the coefficients of the unknown random field on the adapted basis realizations of these coefficients are then used in conjunction with the maximum likelihood principle to set up and solve an optimization problem for the estimation of the coefficients in a polynomial chaos representation of the parameters of the sbvp

stable localized symmetric integral equation method for acoustic scattering problems an energy based infinite boundary element integral equation method is developed for the solution of two or three dimensional time harmonic fluid scattering problems this method is essentially based on a domain decomposition that insures the validity for all frequencies and uses a hypersingular operator that can be integrated readily by standard procedures for single layers it leads to a set of sparse symmetric discretized equations numerical experiments for a rigid circular cylindrical scatterer subjected to a plane incident wave confirm the stability of the new procedure and serve to assess its accuracy for wave numbers ranging from 0 to 30 both directly on the scatterer and in the far field

scale space view of curve estimation scale space theory from computer vision leads to an interesting and novel approach to nonparametric curve estimation the family of smooth curve estimates indexed by the smoothing parameter can be represented as a surface called the scale space surface the smoothing parameter here plays the same role as that played by the scale of resolution in a visual system in this paper we study in detail various features of that surface from a statistical viewpoint weak convergence of the empirical scale space surface to its theoretical counterpart and some related asymptotic results have been established under appropriate regularity conditions our theoretical analysis provides new insights into nonparametric smoothing procedures and yields useful techniques for statistical exploration of features in the data in particular we have used the scale space approach for the development of an effective exploratory data analytic tool called sizer

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

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

exact mean integrated squared error an exact and easily computable expression for the mean integrated squared error mise for the kernel estimator of a general normal mixture density is given for gaussian kernels of arbitrary order this provides a powerful new way of understanding density estimation which complements the usual tools of simulation and asymptotic analysis the family of normal mixture densities is very flexible and the formulae derived allow simple exact analysis for a wide variety of density shapes a number of applications of this method giving important new insights into kernel density estimation are presented among these is the discovery that the usual asymptotic approximations to the mise can be quite inaccurate especially when the underlying density contains substantial fine structure and also strong evidence that the practical importance of higher order kernels is surprisingly small for moderate sample sizes 1 introduction substantial research has been devoted to kernel density estimation this is because it provides a simple yet appealing context in which to study problems and issues that arise in all types of nonparametric curve estimation this includes regression spectral density and hazard estimation and also a variety of other estimators including histograms splines and orthogonal series three important and useful tools for understanding the behavior of nonparametric curve estimators are asymptotic analysis simulation and numerical calculation of error criteria each of these methods provides many useful insights into the complicated structure present in the study of curve estimation however each has its limitations as well rhe strength of asymptotic analysis is that it frequently allows simultaneous study of many different specific examples through general results applying to entire classes of settings the weakness of asymptotics is that they only describe behavior in the limit this is still very useful in many situations because the asymptotics describe the actual situation quite well however it is less useful when the asymptotics have not yet kicked in that is in studying situations where the asymptotically dominant effect has not taken over yet perhaps the biggest drawback to asymptotics is that it is very difficult to determine in a given situation which of these possibilities is occurring

development and application of halanay type theory evolutionary differential and difference equations with time lag we consider extensions developments and modifications of a result due to halanay and the application of halanay type inequalities in the analysis and numerics of retarded functional differential equations difference equations and retarded functional difference equations our emphasis is on the variety structure and development and future development of halanay type results and their applications we classify and present novel results of halanay type linear and non linear discrete semi discrete and continuous and establish their relevance to delay differential equations discretized analogues we consider q methods and difference equations a role for such results in stability and contractivity analysis is made apparent

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

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

an alternative method of cross validation for the smoothing of density estimates cross validation with kullback leibler loss function has been applied to the choice of a smoothing parameter in the kernel method of density estimation a framework for this problem is constructed and used to derive an alternative method of cross validation based on integrated squared error recently also proposed by rudemo 1982 hall 1983 has established the consistency and asymptotic optimality of the new method for small and moderate sized samples the performances of the two methods of cross validation are compared on simulated data and specific examples

a coupled fe and boundary integral equation method based on exterior domain decomposition for fluid structure interface problems abstract a coupled finite element and exterior domain decomposition based boundary integral formulations for the solutions of two or three dimensional time harmonic fluid structure interaction problems is described in this paper it is known that the memory limitation of computers has been one of the major obstacles for solving large scale high frequency fluid structure interface systems using various existing nonlocal finite element and boundary integral equation coupling techniques due to the fully populated resultant matrix generated from the boundary integral equation representation the essence of this study is to decompose through domain decomposition of the exterior region the original exterior problem into arbitrary subproblems with data sharing only at the interfaces by decomposing the exterior infinite domain into an appropriate number of infinite subdomains this method not only ensures the validity of the formulation for all frequencies but also leads to a diagonalized blockwise banded system of discretized equations the size of an individual submatrix i e a block that is associated with an exterior subdomain may be decided by the user and may be selected such that the restriction due to the memory limitation of a given computer may be accommodated in addition the method is suited for parallel processing since the data associated with each subdomain impedance matrices load vectors etc may be generated in parallel and the communication needed will be only for the interface values most significantly unlike the existing coupled finite element and boundary integral equation techniques that are valid for all frequencies our method avoids the use of both the hypersingular operator and the double integrals therefore reducing the computational complexity numerical experiments have been performed for elastic cylindrical shells subjected to a plane incident wave the results have demonstrated the accuracy of the method for wavenumbers ranging from 0 to 30 both directly on the shell and in the far field and have confirmed that the procedure is valid for all frequencies

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

preservation of hopf bifurcation for neutral delay differential equations by θ methods this paper considers the preservation of hopf bifurcation of some neural delay differential equations by q method by analyzing the dynamics of the numerical discrete system derived by q method we show that q method could inherit the hopf bifurcation and the asymptotical stability for sufficiently small stepsize h 1 m where m is a positive integer in particular for q 1 2 the result holds for any stepsize h 1 m furthermore the stability of the closed invariant curve is established finally some numerical examples are illustrated to support the analytic results

stability properties of numerical methods for solving delay differential equations abstract stability properties of numerical methods for delay differential equations are considered some suitable definitions for the stability of the numerical methods are included and runge kutta type methods satisfying these properties are tested on a numerical example

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

kernel principal component analysis for stochastic input model generation stochastic analysis of random heterogeneous media provides useful information only if realistic input models of the material property variations are used these input models are often constructed from a set of experimental samples of the underlying random field to this end the karhunen loeve k l expansion also known as principal component analysis pca is the most popular model reduction method due to its uniform mean square convergence however it only projects the samples onto an optimal linear subspace which results in an unreasonable representation of the original data if they are non linearly related to each other in other words it only preserves the first order mean and second order statistics covariance of a random field which is insufficient for reproducing complex structures this paper applies kernel principal component analysis kpca to construct a reduced order stochastic input model for the material property variation in heterogeneous media kpca can be considered as a nonlinear version of pca through use of kernel functions kpca further enables the preservation of higher order statistics of the random field instead of just two point statistics as in the standard karhunen loeve k l expansion thus this method can model non gaussian non stationary random fields in this work we also propose a new approach to solve the pre image problem involved in kpca in addition polynomial chaos pc expansion is used to represent the random coefficients in kpca which provides a parametric stochastic input model thus realizations which are statistically consistent with the experimental data can be generated in an efficient way we showcase the methodology by constructing a low dimensional stochastic input model to represent channelized permeability in porous media

mapped infinite elements for exterior wave problems recently a new type of infinite elements which uses r 1 decay was proposed they were applied to exterior wave problems and good results were obtained in two dimensional problems however it was necessary to move the origin of the r 1 decay in order to model the outgoing wave more accurately because it decays roughly as r 1 2 in this paper the mapped infinite elements with r 1 2 decay and the necessary numerical integration procedure are presented these elements do not require any artificial movement of the origin several example problems are solved the results show that the infinite elements with r 1 2 decay here give much more accurate values than the infinite elements with exponential decay and any damper elements

the multi element probabilistic collocation method me pcm error analysis and applications stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters in this work we present and examine the multi element probabilistic collocation method me pcm which is a generalized form of the probabilistic collocation method in the me pcm the parametric space is discretized and a collocation cubature grid is prescribed on each element both full and sparse tensor product grids based on gauss and clenshaw curtis quadrature rules are considered we prove analytically and observe in numerical tests that as the parameter space mesh is refined the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness in addition the l 2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided numerical examples demonstrating adaptive me pcm are shown including low regularity problems and long time integration we test the me pcm on two dimensional navier stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions while the convergence rate of me pcm deteriorates in 50 dimensions the error in the mean and variance is two orders of magnitude lower than the error obtained with the monte carlo method using only a small number of samples e g 100 the computational cost of me pcm is found to be favorable when compared to the cost of other methods including stochastic galerkin monte carlo and quasi random sequence methods

stability assessment of a unified variational boundary integral method applicable to thin scatterers and scatterers with corners abstract the objective of this study is to assess the stability especially at and near critical frequencies of a new unified stable symmetric domain finite element and boundary integral methodology for solving time harmonic interface problems for scatterers of arbitrary shape the validity of this energy based variational procedure for rigid and penetrable smooth scatterers including the existence uniqueness and optimal convergence of the corresponding numerical approximations has been proved rigorously in the context of fluid structure interaction problems the emphasis here is on investigating the applicability of this procedure to the case of scatterers with corners using square and thin rectangular two dimensional rigid obstacles as prototypes we also establish that the apparently distinct direct boundary integral formulation due to burton and miller and the combined single and double layer indirect formulation due to leis brakhage and werner and panich introduced to insure that boundary integral equations for scattering problems are uniquely solvable for all wavenumbers are entirely equivalent within our variational setting by examining the condition number of the matrix of coefficients of the discretized equations as well as the resulting solutions both directly on the surface of the scatterers and in the far field it is demonstrated that the new methodology is robust and completely insensitive to critical frequencies even for thin objects most significantly the precise singular behaviour of the pressure field at corners is also predicted quite accurately by using standard finite elements along the boundary of the scatterer without having to introduce special singularity functions which for general interface problems may not be known in advance

stability analysis of numerical methods for delay differential equations this paper deals with the stability analysis of step by step methods for the numerical solution of delay differential equations we focus on the behaviour of such methods when they are applied to the linear testproblemu t u t u t with 0 and complex a general theorem is presented which can be used to obtain complete characterizations of the stability regions of these methods

the stability of difference formulas for delay differential equations a new simple geometric technique is presented for analyzing the stability of difference formulas for the model delay differential equation y t py t qy t 8 where p and q are complex constants and the delay 8 is a positive constant the technique is based on the argument principle and directly relates the region of absolute stability for ordinary differential equations corresponding to the py t term with the region corresponding to the delay term qy t 8 a sufficient condition for stability is that these regions be disjoint the technique is used to show that for each a stable a a stable or stiffly stable linear multistep formula for ordinary differential equations there is a correspond ing linear multistep formula for delay differential equations with analogous stability properties the analogy does not extend however to a stable one step formulas

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

numerical stability of the t methods for systems of differential equations with several delay terms abstract consider the following two dimensional delay differential equation dde u t a 1 u t b 1 v t d 2 v t a 2 v t b 2 u t d 1 t t 0 with the initial conditions u t u 0 t t 0 d 1 t t 0 v t v 0 t t 0 d 2 t t 0 where a j b j c d j 0 and u 0 t 0 t are continuous and complex valued the author give the sufficient conditions for the asymptotic stability of equation 0 1 the numerical stabilities of the linear method and the one leg method are investigated by analysing their stabilities behaviour in the solutions of 0 1 it is shown that the linear method is gp stable if and only if 1 2 1 and that the one leg method is gp stable if and only if 1

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

a symmetric galerkin bem variational framework for multi domain interface problems in this paper we discuss an energy based variational framework for the solution of interior problems in multiplyconnected domains comprising multiple piecewise homogeneous subdomains using exclusively boundary integral equations the primary goal is to provide a unified variational setting that lends itself naturally to symmetric galerkin boundary element formulations in terms of dirichlet type variables only the approach hinges on the explicit imposition of the normal derivative of the classical integral representation of the interior solution on each subdomain via lagrange multipliers in the augmented lagrangian of the system we use maue type identities to resolve the hypersingular kernels leading to a scheme that requires only standard singleand double layer evaluations in addition the usual difficulty with multi valued normals at subdomain corners is treated here within the same variational framework by incorporating into the variational formulation the constraint equation between the limiting normal derivatives at either side of the corner the resulting scheme remains fully symmetric the numerical implementation avoids the explicit presence of neumann type unknowns on the boundaries through condensation at the subdomain level in all integral evaluations three or four point gauss quadrature rules are sufficient for accurate results we describe the theory and present illustrative examples for thermal and acoustic problems governed by laplace and helmholtz equations respectively this technique however can be applied without essential modification to more general problems 2004 published by elsevier b v

improved statistical models for limited datasets in uncertainty quantification using stochastic collocation this paper presents a data driven framework for performing uncertainty quantification uq by choosing a stochastic model that accurately describes the sources of uncertainty in a system this model is propagated through an appropriate response surface function that approximates the behavior of this system using stochastic collocation given a sample of data describing the uncertainty in the inputs our goal is to estimate a probability density function pdf using the kernel moment matching kmm method so that this pdf can be used to accurately reproduce statistics like mean and variance of the response surface function instead of constraining the pdf to be optimal for a particular response function we show that we can use the properties of stochastic collocation to make the estimated pdf optimal for a wide variety of response functions we contrast this method with other traditional procedures that rely on the maximum likelihood approach like kernel density estimation kde and its adaptive modification akde we argue that this modified kmm method tries to preserve what is known from the given data and is the better approach when the available data is limited in quantity we test the performance of these methods for both univariate and multivariate density estimation by sampling random datasets from known pdfs and then measuring the accuracy of the estimated pdfs using the known pdf as a reference comparing the output mean and variance estimated with the empirical moments using the raw data sample as well as the actual moments using the known pdf we show that the kmm method performs better than kde and akde in predicting these moments with greater accuracy this improvement in accuracy is also demonstrated for the case of uq in electrostatic and electrothermomechanical microactuators we show how our framework results in the accurate computation of statistics in micromechanical systems

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

a brief survey of bandwidth selection for density estimation abstract there has been major progress in recent years in data based bandwidth selection for kernel density estimation some second generation methods including plug in and smoothed bootstrap techniques have been developed that are far superior to well known first generation methods such as rules of thumb least squares cross validation and biased cross validation we recommend a solve the equation plug in bandwidth selector as being most reliable in terms of overall performance this article is intended to provide easy accessibility to the main ideas for nonexperts

a domain adaptive stochastic collocation approach for analysis of mems under uncertainties this work proposes a domain adaptive stochastic collocation approach for uncertainty quantification suitable for effective handling of discontinuities or sharp variations in the random domain the basic idea of the proposed methodology is to adaptively decompose the random domain into subdomains within each subdomain a sparse grid interpolant is constructed using the classical smolyak construction s smolyak quadrature and interpolation formulas for tensor products of certain classes of functions soviet math dokl 4 1963 240 243 to approximate the stochastic solution locally the adaptive strategy is governed by the hierarchical surpluses which are computed as part of the interpolation procedure these hierarchical surpluses then serve as an error indicator for each subdomain and lead to subdivision whenever it becomes greater than a threshold value the hierarchical surpluses also provide information about the more important dimensions and accordingly the random elements can be split along those dimensions the proposed adaptive approach is employed to quantify the effect of uncertainty in input parameters on the performance of micro electromechanical systems mems specifically we study the effect of uncertain material properties and geometrical parameters on the pull in behavior and actuation properties of a mems switch using the adaptive approach we resolve the pull in instability in mems switches the results from the proposed approach are verified using monte carlo simulations and it is demonstrated that it computes the required statistics effectively

a method for computing acoustic fields based on the principle of wave superposition a method for computing the acoustic fields of arbitrarily shaped radiators is described that uses the principle of wave superposition the superposition integral which is shown to be equivalent to the helmholtz integral is based on the idea that the combined fields of an array of sources interior to a radiator can be made to reproduce a velocity prescribed on the surface of the radiator the strengths of the sources that produce this condition can in turn be used to compute the corresponding surface pressures the results of several numerical experiments are presented that demonstrate the simplicity of the method also the advantages that the superposition method has over the more commonly used boundary element methods are discussed these include simplicity of generating the matrix elements used in the numerical formulation and improved accuracy and speed the latter two being due to the avoidance of uniqueness and singularity problems inherent in the boundary element formulation

stochastic modeling of coupled electromechanical interaction for uncertainty quantification in electrostatically actuated mems abstract this work proposes a stochastic framework based on generalized polynomial chaos gpc to handle uncertain coupled electromechanical interaction arising from variations in material properties and geometrical parameters such as gap between the microstructures applicable to the static analysis of electrostatic mems the proposed framework comprises of two components a stochastic mechanical analysis which quantifies the uncertainty associated with the deformation of mem structures due to the variations in material properties and or applied traction and a stochastic electrostatic analysis to quantify the uncertainty in the electrostatic pressure due to variations in geometrical parameters or uncertain deformation of the conductors the stochastic analysis is based on a stochastic lagrangian approach where in addition to uncertain input parameters and unknown field variables the random deformed configuration is expanded in terms of gpc basis functions the spectral modes for the unknown field variables are finally obtained using galerkin projection in the space spanned by gpc basis functions the stochastic mechanical and electrostatic analyses are performed in a self consistent manner to obtain the random deformation of the mem structures various numerical examples are presented to study the effect of uncertain parameters on performance of various mems devices the results obtained using the proposed method are verified using rigorous monte carlo simulations it has been shown that the proposed method accurately predicts the statistics and probability density functions of various relevant parameters

stability in linear multistep methods for pure delay equations abstract the stability regions of linear multistep methods for pure delay equations are compared with the stability region of the delay equation itself a criterion is derived stating when the numerical stability region contains the analytical stability region this criterion yields an upper bound for the integration step conditional q stability these bounds are computed for the adams bashforth adams moulton and backward differentiation methods of orders 8 furthermore symmetric adams methods are considered which are shown to be unconditionally q stable finally the extended backward differentiation methods of cash are analysed

gpg stability of runge kutta methods for generalized delay differential systems abstract the gp g stability of runge kutta methods for the numerical solutions of the systems of delay differential equations is considered the stability behaviour of implicit runge kutta methods irk is analyzed for the solution of the system of linear test equations with multiple delay terms after an establishment of a sufficient condition for asymptotic stability of the solutions of the system a criterion of numerical stability of irk with the lagrange interpolation process is given for any stepsize of the method

p stability properties of runge kutta methods for delay differential equations recently the author and others have studied a class of runge kutta methods for ddes these methods make use of certain constrained meshes which allow to get optimal order results in this paper we study their asymptotic stability properties using the test equation begin gathered y t ay t by t tau t 0 hfill y t g t for tau leqq t leqq 0 hfill end gathered wherea b 0 andg t is continuous and complex valued it is known thaty t 0 ast if b re a in particular we show that any astable one step collocation method for odes inherits the same property when it is applied to ddes with a constrained mesh i e it is p stable

stability analysis for delay differential equations with multidelays and numerical examples in this paper we are concerned with the asymptotic stability of the delay differential equation x t a 0 x t a k x t k where a 0 a k c d d are constant complex matrices and x t k x 1 t k1 x 2 t k2 x d t kd t kl 0 stand for n x d constant delays k 1 n i 1 d we obtain two criteria for stability through the evaluation of a harmonic function on the boundary of a certain region we also get similar results for the neutral delay differential equation x t lx t m i x t i n j x t j where l m i and nj e c d d are constant complex matrices and i j 0 stands for constant delays i 1 m j 1 n numerical examples on various circumstances are shown to check our results which are more general than those already reported

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

asymptotic stability analysis of θ methods for functional differential equations stability analysis of methods for functional differential equations based on the test equation begin gathered y t ay t lambda by t t 0 hfill y t psi t t in left lambda 0 right hfill end gathered 0 is presented it is known thaty t 0 ast if and only if b a and we investigate whether this property is inherited by the numerical solution approximatingy

asymptotic sampling distribution for polynomial chaos representation from data a maximum entropy and fisher information approach a procedure is presented for characterizing the asymptotic sampling distribution of estimators of the polynomial chaos pc coefficients of a second order nonstationary and non gaussian random process by using a collection of observations the random process represents a physical quantity of interest and the observations made over a finite denumerable subset of the indexing set of the random process are considered to form a set of realizations of a random vector mathcal y representing a finite dimensional projection of the random process the karhunen loeve decomposition and a scaling transformation are employed to produce a reduced order model mathcal z of mathcal y the pc expansion of mathcal z is next determined by having recourse to the maximum entropy principle the metropolis hastings markov chain monte carlo algorithm and the rosenblatt transformation the resulting pc expansion has random coefficients where the random characteristics of the pc coefficients can be attributed to the limited data available from the experiment the estimators of the pc coefficients of mathcal y obtained from that of mathcal z are found to be maximum likelihood estimators as well as consistent and asymptotically efficient computation of the covariance matrix of the associated asymptotic normal distribution of estimators of the pc coefficients of mathcal y requires knowledge of the fisher information matrix fim the fim is evaluated here by using a numerical integration scheme as well as a sampling technique the resulting confidence interval on the pc coefficient estimators essentially reflects the effect of incomplete information due to data limitation on the characterization of the stochastic process this asymptotic distribution is significant as its characteristics can be propagated through predictive models for which the stochastic process in question describes uncertainty on some input parameters

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

mapped spheroidal wave envelope elements for unbounded wave problems this paper describes a family of axisymmetric spheroidal wave envelope elements for modelling exterior wave problems they are of variable radial order and can be used to represent steady and transient wave fields the formulation is presented for the axisymmetric case using elements which are based on oblate and prolate spheroidal geometries these offer the prospect of reduced dimensionality in comparison to conventional spherically formulated elements when used to represent wave fields in the vicinity of slender or flat objects conjugated weighting functions are used to give frequency independent acoustic mass stiffness and damping matrices this facilitates a simple extension of the method to transient problems the effectiveness and accuracy of the method is demonstrated by a comparison of computed and analytic solutions for sound fields generated by a rigid sphere in steady harmonic oscillation by a rigid sphere excited from rest and by a circular plate vibrating in a plane baffle

the optimal accuracy of difference schemes we consider difference approximations to the model hyperbolic equation ut ux which compute each new value u x t at as a combination of the known values u x ra x t u x s a x a t for such schemes we find the optimal order of accuracy stability is possible for small a t a x if and only if p min r s 2 r 2 2s a similar bound is established for implicit methods in this case the most accurate schemes are based on pade approximations p z q z to za near z 1 and we find an expression for the difference i q 12 p 12 this allows us to test the von neumann condition i p q i 1 we also determine the number of zeros of q in the unit circle which decides whether the implicit part is uniformly invertible

biased and unbiased cross validation in density estimation abstract nonparametric density estimation requires the specification of smoothing parameters the demands of statistical objectivity make it highly desirable to base the choice on properties of the data set in this article we introduce some biased cross validation criteria for selection of smoothing parameters for kernel and histogram density estimators closely related to one investigated in scott and factor 1981 these criteria are obtained by estimating l 2 norms of derivatives of the unknown density and provide slightly biased estimates of the average squared l 2 error or mean integrated squared error these criteria are roughly the analog of wahba s 1981 generalized cross validation procedure for orthogonal series density estimators we present the relationship of the biased cross validation procedure to the least squares cross validation procedure which provides unbiased estimates of the mean integrated squared error both methods are shown to be based on u statistics we compare the two metho