nilpotent and high order approximations of vector field systems this paper is concerned with the study of structurally stable properties associated with systems described by one or several real analytic vector fields via approximating systems which retain inf

efficient numerical strategies for spectral stochastic finite element models the use of spectral stochastic finite element models results in large systems of equations requiring specialized solution strategies this paper discusses three different numerical algorithms for solving these large systems of equations it presents a trade off of these algorithms in terms of memory usage and computation time it also shows that the structure of the spectral stochastic stiffness matrix can be exploited to accelerate the solution process while keeping the memory usage to a minimum copyright 2005 john wiley sons ltd

a multigrid solver for two dimensional stochastic diffusion equations steady and unsteady diffusion equations with stochastic diffusivity coefficient and forcing term are modeled in two dimensions by means of stochastic spectral representations problem data and solution variables are expanded using the polynomial chaos system the approach leads to a set of coupled problems for the stochastic modes spatial finite difference discretization of these coupled problems results in a large system of equations whose dimension necessitates the use of iterative approaches in order to obtain the solution within a reasonable computational time to accelerate the convergence of the iterative technique a multigrid method based on spatial coarsening is implemented numerical experiments show good scaling properties of the method both with respect to the number of spatial grid points and the stochastic resolution level

a new perspective on the solution of uncertainty quantification and reliability analysis of large scale problems abstract this work revisits the computational performance of non intrusive monte carlo versus intrusive galerkin methods of large scale stochastic systems in the framework of high performance computing environments the purpose of this work is to perform an assessment of the range of the relative superiority of these approaches with regard to a variety of stochastic parameters in both approaches the solution of the resulting algebraic equations is performed with a combination of primal and dual domain decomposition methods implementing specifically tailored preconditioners the solution of repeated simulations of the monte carlo method is accelerated with an a orthogonalization procedure aiming at reducing the iterations of subsequent simulations while the solution of the augmented equations of the stochastic galerkin method is enhanced with preconditioners which combine the block diagonal features of the resulting matrices as well as the sparsity pattern of the off block diagonal terms numerical results are presented demonstrating the efficiency of the proposed implementations on a large scale 3d problem with different stochastic characteristics and useful conclusions are derived regarding the ranges of stochastic parameters in which non intrusive solvers have a superior performance compared to intrusive ones and vice versa

preasymptotic stability and homogeneous approximations of hybrid dynamical systems hybrid dynamical systems are systems that combine features of continuous time dynamical systems and discrete time dynamical systems and can be modeled by a combination of differential equations or inclusions difference equations or inclusions and constraints preasymptotic stability is a concept that results from separating the conditions that asymptotic stability places on the behavior of solutions from issues related to existence of solutions in this paper techniques for approximating hybrid dynamical systems that generalize classical linearization techniques are proposed the approximation techniques involve linearization tangent cones homogeneous approximations of functions and set valued mappings and tangent homogeneous cones where homogeneity is considered with respect to general dilations the main results deduce preasymptotic stability of an equilibrium point for a hybrid dynamical system from preasymptotic stability of the equilibrium point for an approximate system further results relate the degree of homogeneity of a hybrid system to the zeno phenomenon that can appear in the solutions of the system

on the regularization of zeno hybrid automata fundamental properties of hybrid automata such as existence and uniqueness of executions are studied particular attention is devoted to zeno hybrid automata which are hybrid automata that take infinitely many discrete transitions in finite time it is shown that regularization techniques can be used to extend the zeno executions of these automata to times beyond the zeno time different types of regularization may however lead to different extensions a water tank control problem and a bouncing ball system are used to illustrate the results

relaxed persistent flow jump conditions for uniform global asymptotic stability for hybrid systems sufficient conditions are derived for the uniform global asymptotic stability of a given closed set these conditions are written in terms of a lyapunov function candidate and assume a semiglobal practical persistent flow resp persistent jump property of the solutions to the hybrid system the use of the new conditions is illustrated via the stability analysis of a physically inspired example and of an event triggered control algorithm

stochastic finite element analysis of structures in the presence of multiple imprecise random field parameters the research work presents the study on non deterministic problems in the presence of the multiple imprecise random field uncertainties by extending the spectral stochastic finite element framework the standard random field is expanded to characterize the behaviour of physical model with imprecise randomness more appropriately for real engineering problems based on available sources of uncertainty the young s modulus and body force of structures are considered as imprecise random fields with bounded statistical moments mathematical expressions and solution procedure are developed to produce the uncertain but bounded statistical characteristics of responses namely interval mean value interval standard deviation and bounding distribution functions the bounding probability density and cumulative distribution of the imprecise random response are then effectively visualized by means of polynomial chaos expansion the feasibility and effectiveness of the presented method are illustrated by three numerical examples

solutions to hybrid inclusions via set and graphical convergence with stability theory applications motivated by questions in stability theory for hybrid dynamical systems we establish some fundamental properties of the set of solutions to such systems using the notion of a hybrid time domain and general results on set and graphical convergence we establish under weak regularity and local boundedness assumptions that the set of solutions is sequentially compact and upper semicontinuous with respect to initial conditions and system perturbations the general facts are then used to establish several results for the behavior of hybrid systems that have asymptotically stable compact sets these results parallel what is already known for differential inclusions and difference inclusions for example the basin of attraction for a compact attractor is relatively open the attractivity is uniform from compact subsets of the basin of attraction and asymptotic stability is robust with respect to small perturbations

assessment of spectral representation and karhunen loeve expansion methods for the simulation of gaussian stochastic fields abstract from the wide variety of methods developed for the simulation of gaussian stochastic processes and fields two are most often used in applications the spectral representation method and the karhunen loeve k l expansion in this paper an in depth assessment on the capabilities of the two methods is presented the spectral representation method expands the stochastic field as a sum of trigonometric functions with random phase angles and or amplitudes the version having only random phase angles is used in this work a wavelet galerkin scheme is adopted for the efficient numerical solution of the fredholm integral equation appearing in the k l expansion a one dimensional homogeneous gaussian random field with two types of autocovariance function exponential and square exponential is used as the benchmark test the accuracy achieved and the computational effort required by the k l expansion and the spectral representation for the simulation of the stochastic field are investigated the accuracy obtained by the two approaches is examined by comparing their ability to produce sample functions that match the target correlation structure and the gaussian probability distribution or alternatively its low order statistical moments mean variance and skewness

homogeneous lyapunov function for homogeneous continuous vector field the goal of this article is to provide a construction of a homogeneous lyapunov function p associated with a system of differential equations j f x x r n 1 under the hypotheses 1 f c r n vanishes at x 0 and is homogeneous 2 the zero solution of this system is locally asymptotically stable moreover the lyapunov function v x tends to infinity with 1 x i and belongs to c r 0 r n cp with p e as large as wanted as application to the theory of homogeneous systems we present two well known results of robustness in a slightly extended form and with simpler proofs

dynamical properties of hybrid automata hybrid automata provide a language for modeling and analyzing digital and analogue computations in real time systems hybrid automata are studied here from a dynamical systems perspective necessary and sufficient conditions for existence and uniqueness of solutions are derived and a class of hybrid automata whose solutions depend continuously on the initial state is characterized the results on existence uniqueness and continuity serve as a starting point for stability analysis lyapunov s theorem on stability via linearization and lasalle s invariance principle are generalized to hybrid automata

finite gain lp stability for hybrid dynamical systems we characterize the finite gain l p stability properties for hybrid dynamical systems by defining a suitable concept of the hybrid l p norm we introduce hybrid storage functions and provide sufficient lyapunov conditions for the l p stability of hybrid systems which cover the well known continuous time and discrete time l p stability notions as special cases we then focus on homogeneous hybrid systems and prove a result stating the equivalence among local asymptotic stability of the origin global exponential stability existence of a homogeneous lyapunov function with suitable properties for the hybrid system with no inputs and input to state stability and we show how these properties all imply l p stability finally we characterize systems with direct and reverse average dwell time properties and establish parallel results for this class of systems we also make several connections to the existing results on dissipativity properties of hybrid dynamical systems

basis random variables in finite element analysis galerkin s method is applied to random operator equations appropriate hilbert spaces are defined for random functions and solutions are projected into these spaces allowing the first and second moment properties of the solution to be calculated an equivalent energy based approach similar to the rayleigh ritz method is developed from which a stochastic finite element technique is derived several one and two dimensional example problems are solved and the results discussed

stabilization by constrained controls a stabilization problem for a general nonlinear control system is considered in particular the control corresponding to the equilibrium position may belong to the boundary of the control set a linear control system is considered as a first approximation for the original problem the right hand side of the linear system generates a set valued map of a special type known as a convex process this set valued map has a number of properties similar to those of a linear operator they allow one to establish necessary and sufficient conditions for solvability of the regulator design problem for the first approximation and to construct a lyapunov function based on these results the nonlinear stabilization problem is investigated different statements of the regulator design problem are studied stabilization problems for some mechanical systems are considered to illustrate the regulator design techniques the properties of transient characteristics the peak effect are discussed for a linear stabilization p

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

lyapunov functions and duality for convex processes the paper studies convex lyapunov functions for differential and difference inclusions with right hand sides given by convex processes that is by set valued mappings the graphs of which are convex cones convex conjugacy between weak lyapunov functions for such inclusions and lyapunov functions for adjoint inclusions is established asymptotic stability concepts are compared and the existence of convex lyapunov functions for classes of convex processes is shown the relevance of the results for the study of asymptotic controllability or stabilizability and of detectability for linear control systems with conical control and state constraints is underlined

homogeneous state feedback stabilization of homogenous systems we show that for any asymptotically controllable homogeneous system in euclidean space not necessarily lipschitz at the origin there exists a homogeneous control lyapunov function and a homogeneous possibly discontinuous state feedback law stabilizing the corresponding sampled closed loop system if the system satisfies the usual local lipschitz condition on the whole space we obtain semiglobal stability of the sampled closed loop system for each sufficiently small fixed sampling rate if the system satisfies a global lipschitz condition we obtain global exponential stability for each sufficiently small fixed sampling rate the control lyapunov function and the feedback are based on the lyapunov exponents of a suitable auxiliary system and admit a numerical approximation

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

linear complementarity systems zeno states a linear complementarity system lcs is a hybrid dynamical system defined by a linear time invariant ordinary differential equation coupled with a finite dimensional linear complementarity problem lcp the present paper is the first of several papers whose goal is to study some fundamental issues associated with an lcs specifically this paper addresses the issue of zeno states and the related issue of finite number of mode switches in such a system the cornerstone of our study is an expansion of a solution trajectory to the lcs near a given state in terms of an observability degree of the state on the basis of this expansion and an inductive argument we establish that an lcs satisfying the p property has no strongly zeno states we next extend the analysis for such an lcs to a broader class of problems and provide sufficient conditions for a given state to be weakly non zeno while related mode switch results have been proved by brunovsky and sussmann for more general hybrid systems our analysis exploits the special structure of the lcs and yields new results for the latter that are of independent interest and complement those by these two and other authors

a domain decomposition method of stochastic pdes recent advances in high performance computing systems and sensing technologies motivate computational simulations with extremely high resolution models with capabilities to quantify uncertainties for credible numerical predictions a two level domain decomposition method is reported in this investigation to devise a linear solver for the large scale system in the galerkin spectral stochastic finite element method ssfem in particular a two level scalable preconditioner is introduced in order to iteratively solve the large scale linear system in the intrusive ssfem using an iterative substructuring based domain decomposition solver the implementation of the algorithm involves solving a local problem on each subdomain that constructs the local part of the preconditioner and a coarse problem that propagates information globally among the subdomains the numerical and parallel scalabilities of the two level preconditioner are contrasted with the previously developed one level preconditioner for two dimensional flow through porous media and elasticity problems with spatially varying non gaussian material properties a distributed implementation of the parallel algorithm is carried out using mpi and petsc parallel libraries the scalabilities of the algorithm are investigated in a linux cluster

smooth lyapunov functions for hybrid systems part ii pre asymptotically stable compact sets it is shown that pre asymptotic stability which generalizes asymptotic stability of a compact set for a hybrid system satisfying mild regularity assumptions is equivalent to the existence of a smooth lyapunov function this result is achieved with the intermediate result that asymptotic stability of a compact set for a hybrid system is generically robust to small state dependent perturbations as a special case we state a converse lyapunov theorem for systems with logic variables and use this result to establish input to state stabilization using hybrid feedback control the converse lyapunov theorems are also used to establish semiglobal practical robustness to slowly varying weakly jumping parameters to temporal regularization to the insertion of jumps according to an ldquoaverage dwell timerdquo rule and to the insertion of flow according to a ldquoreverse average dwell timerdquo rule

impulse differential inclusions a viability approach to hybrid systems impulse differential inclusions are introduced as a framework for modeling hybrid phenomena connections to standard problems in the area of hybrid systems are discussed conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion for sets that violate these conditions methods are developed for approximating their viability and invariance kernels that is the largest subset that is viable or invariant under the action of the impulse differential inclusion the results are demonstrated on examples

a feti preconditioned conjugate gradient method for large scale stochastic finite element problems in the spectral stochastic finite element method for analyzing an uncertain system the uncertainty is represented by a set of random variables and a quantity of interest such as the system response is considered as a function of these random variables consequently the underlying galerkin projection yields a block system of deterministic equations where the blocks are sparse but coupled the solution of this algebraic system of equations becomes rapidly challenging when the size of the physical system and or the level of uncertainty is increased this paper addresses this challenge by presenting a preconditioned conjugate gradient method for such block systems where the preconditioning step is based on the dual primal finite element tearing and interconnecting method equipped with a krylov subspace reusage technique for accelerating the iterative solution of systems with multiple and repeated right hand sides preliminary performance results on a linux cluster suggest that the proposed solution method is numerically scalable and demonstrate its potential for making the uncertainty quantification of realistic systems tractable

spectral stochastic finite element analysis for laminated composite plates abstract a spectral stochastic finite element formulation with consideration of multi layer effect and spatial variability of material properties is developed for probabilistic analysis of laminated composite plates the material properties of each lamina are modeled as a set of random fields and represented by the karhunen loeve expansion the expansion is incorporated into a spatial discretization in accordance with a standard finite element procedure based on the first order shear deformation theory a spectral expansion with use of polynomial chaos is employed to represent the stochastic nodal displacements in terms of standard normal random variables a preconditioning matrix is then proposed for the solution of spectral stochastic finite element equations with use of a preconditioning conjugate gradient technique the various statistics of interest of the system responses are finally obtained by means of the coefficients of the spectral expansion the numerical accuracy and the computational efficiency of the method are demonstrated by comparison with monte carlo simulation

spectral stochastic isogeometric analysis of linear elasticity abstract the stochastic isogeometric analysis of linear elasticity problem is investigated in this study the spectral stochastic analysis is introduced into isogeometric analysis iga and a novel yet robust stochastic analysis framework namely the spectral stochastic isogeometric analysis ssiga is freshly proposed unlike traditional numerical solutions of the karhunen loeve k l expansion the non uniform rational b spline nurbs and t spline basis functions are employed within the proposed framework of ssiga so the random fields acting on a continuous physical medium with complex geometry can be handled in an appropriate physically feasible and efficient fashion the polynomials chaos expansion pce is implemented to represent the stochastic structural response e g displacement strain and stress such that all corresponding statistical characteristics e g mean and standard deviation can be robustly acquired furthermore by utilizing the nonparametric statistical analysis both probability density functions pdfs and cumulative distribution functions cdfs of concerned structural displacements and stresses can be effectively established within the framework of iga by meticulously implementing the concept of the higher order k refinement the proposed ssiga provides a more legitimate and efficient stochastic computational approach for modern engineering structures which are complicated by both spatially dependent uncertainties and complex geometries

efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering in this study we describe the algebraic computations required to implement the stochastic finite element method for solving problems in which uncertainty is restricted to right hand side data coming from forcing functions or boundary conditions we show that the solution can be represented in a compact outer product form which leads to efficiencies in both work and storage and we demonstrate that block iterative methods for algebraic systems with multiple right hand sides can be used to advantage to compute this solution we also show how to generate a variety of statistical quantities from the computed solution finally we examine the behavior of these statistical quantities in one setting derived from a model of acoustic scattering

measure driven differential inclusions measure driven differential inclusions arise when we attempt to derive necessary conditions of optimality for optimal impulsive control problems with nonsmooth data we introduce the concept of a robust solution to a measure driven inclusion which extends to a multifunction setting interpretations of solutions to measure driven differential equations provided by dal maso and rampazzo and others closure properties of sets of robust solutions are established and notions of relaxation investigated implications for optimality conditions for impulsive control problems are pursued in a companion paper

local controllability and infinitesimal generators of semigroups of set valued maps we study the semigroup properties of reachable sets of a differential inclusion x in f x using the derivative of the set valued map which associates to each initial state the set of solutions the results are applied to the local controllability problem

iterative solution of the random eigenvalue problem with application to spectral stochastic finite element systems a new algorithm for the computation of the spectral expansion of the eigenvalues and eigenvectors of a random non symmetric matrix is proposed the algorithm extends the deterministic inverse power method using a spectral discretization approach the convergence and accuracy of the algorithm is studied for both symmetric and non symmetric matrices the method turns out to be efficient and robust compared to existing methods for the computation of the spectral expansion of random eigenvalues and eigenvectors

improved mixing condition on the grid for counting and sampling independent sets the hard core model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics a multicast model of calls in communication networks and as a weighted independent set problem in combinatorics probability and theoretical computer science in this model each independent set i in a graph g is weighted proportionally to i for a positive real parameter for large computing the partition function namely the normalizing constant which makes the weighting a probability distribution on a finite graph on graphs of maximum degree 3 is a well known computationally challenging problem more concretely let lambda c mathbb t delta denote the critical value for the so called uniqueness threshold of the hard core model on the infinite regular tree recent breakthrough results of weitz proceedings of the 38th annual acm symposium on theory of computing stoc pp 140 149 2006 and sly proceedings of the 51st annual ieee symposium on foundations of computer science focs pp 287 296 2010 have identified lambda c mathbb t delta as a threshold where the hardness of estimating the above partition function undergoes a computational transition we focus on the well studied particular case of the square lattice mathbb z 2 and provide a new lower bound for the uniqueness threshold in particular taking it well above lambda c mathbb t 4 our technique refines and builds on the tree of self avoiding walks approach of weitz resulting in a new technical sufficient criterion of wider applicability for establishing strong spatial mixing and hence uniqueness for the hard core model our new criterion achieves better bounds on strong spatial mixing when the graph has extra structure improving upon what can be achieved by just using the maximum degree applying our technique to mathbb z 2 we prove that strong spatial mixing holds for all 2 3882 improving upon the work of weitz that held for 27 16 1 6875 our results imply a fully polynomial deterministic approximation algorithm for estimating the partition function as well as rapid mixing of the associated glauber dynamics to sample from the hard core distribution