poles and zeros at infinity of linear time varying systems the notions of poles and zeros at infinity and their relations are extended to the case of linear continuous time varying systems this study is based on the notion of a newborn systems which is in a mathematical point of view a graded module extension over the noncommutative ring of differential operators it is proved to be a relevant generalization to the time varying case of the equivalence class for the so called restricted equivalence of rosenbrock s polynomial matrix descriptions the authors approach is intrinsic and unifies the definitions previously given in the literature in the time invariant case

the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

domain decomposition for least squares finite element methods for the stokes equations in this paper we consider the solutions of the discrete linear system resulting from least squares finite element discretization applied to the stokes equations additive and multiplicative domain decomposition methods are used to solve the linear system and the uniform convergence results are established for both methods

a parallel domain decomposition algorithm of mixed element equation for second order elliptic dirichlet boundary value problem a parallel domain decomposition algorithm of schwarz type is introduced to solve the mixed finite element equation for the dirichlet boundary value problem of a second order partial differential equation of elliptic type and convergence of this algorithm is analyzed

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

state feedback impulse elimination of linear time varying singular systems abstract linear time varying singular systems e t x a t x b t u are treated a necessary and sufficient condition for the existence of linear state feedback to eliminate impulsive behavior is derived the condition obtained is a natural extension of existing time invariant theory

an intrinsic algebraic setting for poles and zeros of linear time varying systems in this paper poles and zeros are defined for linear time varying systems using suitable ground field extensions the definitions of the system poles transmission poles invariant zeros hidden modes etc are given in an intrinsic module based framework and are consistent in the sense that the poles are connected to the stability of the system and the zeros to the zeroing of the output for non zero inputs in particular it is proved that the necessary and sufficient condition for a continuous time system to be exponentially stable is similar to the well known condition in the time invariant case

some basic structural properties of generalized linear systems abstract this paper shows how to give a kalman state variable realization to generalized linear systems simple characterizations of controllability observability of their duality and of a minimal realization are proposed our techniques based on elementary module theory are applicable to time varying systems as well as constant systems the note ends with a brief discussion of asymptotic observers in the constant case

new convergence estimates for multigrid algorithms in this paper new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems our theory relates the convergence of multigrid algorithms to a regularity and approximation parameter cap alpha epsilon 0 1 and the number of relaxations m we show that for the symmetric and nonsymmetric nu cycles the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1 cj sup 1 sup cap alpha sup sup cap alpha where j is the number of grid levels we then define a generalized nu cycle algorithm which involves exponentially increasing for example doubling the number of smoothings on successively coarser grids we show that the resulting symmetric and nonsymmetric multigrid iterations converge for any cap alpha with rates that are independent of the mesh size the theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods

convergence estimates for multigrid algorithms without regularity assumptions a new technique for proving rate of convergence estimates of multi grid algorithms for symmetric positive definite problems will be given in this paper the standard multigrid theory requires a regularity and approxima tion assumption in contrast the new theory requires only an easily verified approximation assumption this leads to convergence results for multigrid re finement applications problems with irregular coefficients and problems whose coefficients have large jumps in addition the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications

an application of the abstract multilevel theory to nonconforming finite element methods a multilevel product algorithm and its additive version or a multilevel preconditioner are proposed for the crouzeix raviart nonconforming finite element methods for second order elliptic equations the authors show that the preconditioned matrix has a condition number bounded uniformly from above if the domain can be partitioned into overlapping convex subdomains a uniform convergence for the multilevel product algorithm is also established the results are regularity free for the continuous problem and can be applied to problems with rough solutions and local grid refinement numerical results that illustrate the theory are presented

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

rate of convergence for some constraint decomposition methods for nonlinear variational inequalities some general subspace correction algorithms are proposed for a convex optimization problem over a convex constraint subset one of the nontrivial applications of the algorithms is the solving of some obstacle problems by multilevel domain decomposition and multigrid methods for domain decomposition and multigrid methods the rate of convergence for the algorithms for obstacle problems is of the same order as the rate of convergence for jump coefficient linear elliptic problems in order to analyse the convergence rate we need to decompose a finite element function into a sum of functions from the subspaces and also satisfying some constraints a special nonlinear interpolation operator is introduced for decomposing the functions

une interpretation algebrique de la transformation de laplace et des matrices de transfert abstract the tensor product of the module of a linear system with the quotient field of the ring of linear differential operators is a vector space where even in the time varying case a formal laplace transform and the transfer matrix are most naturally defined several classic problems are examined in this algebraic setting the relationship between left right coprime matrix decomposition and controllability observability the state variable canonical realization the transfer algebra with respect to parallel and series connections the input output inversion and model matching

index of an implicit time varying linear differential equation a noncommutative linear algebraic approach abstract an intrinsic definition of the index for implicit linear time varying differential systems is proposed this definition relies on linear algebraic techniques over the noncommutative principal ideal ring of nonconstant linear differential operators noncommutative transfer function or state variable representation accordance with the existing definition for constant coefficient linear systems is demonstrated

periodic polynomial interpretation for structural properties of linear periodic discrete time systems the structural properties of linear periodic discrete time systems are analyzed in the periodic polynomial representation it is shown that the classical polynomial approach for linear time invariant systems can be extended to periodic systems new definitions and properties are given in terms of skew polynomial rings and periodic difference algebra necessary and sufficient conditions for the characterization of reachability controllability observability and reconstructibility are given in this framework

convergence rate of a schwarz multilevel method for the constrained minimization of nonquadratic functionals in l badea convergence rate of a multiplicative schwarz method for strongly nonlinear inequalities v barbu i lasiecka d tiba and c varsan eds kluwer academic publishers boston 2003 the convergence of a subspace correction method applied to the constrained minimization of a functional in a general reflexive banach space has been proved provided that the convex set verifies a certain assumption this assumption is weaker than that in which the convex set is decomposed according to the space decomposition as a sum of subsets in the sobolev spaces the proposed method becomes a multiplicative schwarz method for the solution of the variational inequalities coming from the minimization of nonquadratic functionals we prove in this paper that this assumption holds for the one two and multilevel multiplicative schwarz methods in the finite element space and we explicitly write the constants in the error estimations depending on the overlapping and mesh parameters our error estimates are similar with those obtained for the minimization of quadratic functionals in l badea x c tai and j wang siam j numer anal 2003 pp 1052 1073 or with those obtained for the one obstacle problem in x c tai numer math 93 2003 pp 755 786

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

model matching and decoupling for continuous and discrete time linear time varying systems solutions to the exact model matching and block decoupling problems for both continuous and discrete time linear time varying systems are presented the parametrisation of the whole class of proper solutions is given for the decoupling the minimal delay problem is also considered in a time varying setting the approach is algebraic and based on the smith macmillan form at infinity of a transfer matrix of a time varying system which has been recently introduced in systems theory this avoids the difficulties related to the inversion of the transfer matrices with entries in non commutative fields over which the determinants of dieudonne or ore type are much more complicated the solutions presented here involve only standard matrix computations excluding direct matrix inversions and are thus easy to implement in practice examples are treated in detail to illustrate the theoretical results and the way in which the computations are done and a physical example is also shown

rosenbrock models and their homotopy equivalence as is known the notion of homotopy equivalence is a fundamental notion of mathematics and was introduced in order to formalize a relation that is weaker than isomorphism in this note we define a homotopy equivalence of rosenbrock systems and show that it coincides with the classical equivalences of rosenbrock and fuhrmann next we show that the homotopy equivalence does preserve the important properties of a system including the properties at infinity when these are properly understood finally we define in a simple manner the states and motions of a system and claim that they are homotopy invariants

the exact model matching problem for linear time varying systems an algebraic approach the exact model matching problem is formulated and solved for linear time varying systems the condition for the existence of a proper solution which is well known in the time invariant case is proven here to still be valid in the time varying case the properness is characterized using the smith macmillan form at infinity recently defined by the authors for the transfer matrices with time varying coefficients

an anisotropic h adaptive finite element method for compressible navier stokes equations abstract application of the finite element method with significantly stretched elements to solve compressible navier stokes equations is presented stretched elements are introduced in an adaptive process as a result of selective directional subsecting of elements an adaptive strategy of such anisotropic h type refinements was developed it is based on minimization of the interpolation errors reduction of the residual error indicators and an adaptive recovery of the viscous fluxes with the possibility to control their accuracy an iterative solver with the convergence characteristics independent of the aspect ratio of elements is presented the solver is the gmres algorithm with the domain decomposition schwarz type preconditioner the proposed techniques are used to solve a benchmark problem a hypersonic high reynolds number flow with strong shock boundary layer interaction

finite poles and zeros of linear systems an intrinsic approach several kinds of finite poles and zeros are defined and their relationships are studied in an intrinsic manner some of these definitions are more precise than the usual ones all kinds of multiplicities of an input output decoupling zero are defined here in addition we believe that our approach clarifies those notions and it makes it possible to very concisely establish fundamental relationships between various poles and zeros some new relationships are derived the notions of input output stability and transfer stability are also introduced several examples illustrate the theory

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

uniform convergence of the schwarz alternating method for solving singularly perturbed advection diffusion equations our purpose in this paper is to study the convergence rate of some variants of the schwarz alternating method for solving two types of singularly perturbed advection diffusion equations one in which the advection term first order term dominates and the other in which the lowest order term zeroth order term dominates for such problems boundary or other layer regions may be present for singular perturbation problems in which the advection dominates we choose two overlapping subregions one corresponding to an outer or elliptic region and the other corresponding to an inner or hyperbolic region our main result for this problem shows that if the subdomains can be chosen to follow the flow i e if the boundary interface of one of the subdomains corresponds to an outflow boundary for the streamlines of the flow then the schwarz iterates converge in the maximum norm with an error reduction factor per iteration that exponentially decays with increasing overlap or decreasing diffusion for the singular perturbation problem in which the lowest order term dominates we consider many overlapping subregions with no special requirement on outflow or inflow boundaries our main result here which is valid for two variants of the classical schwarz algorithm shows that the error reduction factor per iteration decays exponentially in the maximum norm with increasing overlap and increasing dominance of the zeroth order term for both problems our proofs rely on the maximum principle and the construction of barrier functions

structure and smith macmillan form of a rational matrix at infinity the use of special row and column operations in the reduction of a rational matrix to its smith macmillan form at infinity is investigated the connections between this reduction procedure and the valuation approach are established a graphical method for finding the smith macmillan form of a rational matrix at infinity from its bode magnitude array and some new results on realization theory for polynomial matrices are presented

time varying polynomial matrix systems abstract differential polynomial matrices with coefficients meromorphic on an open interval i r are studied first the subclass of full matrices p having the properties p is non singular antl every local solution of pf 0 extends analytically to i forms a sublattice with respect to right division of square matrices which is anti isomorphic to the lattice of the corresponding solution spaces this and further properties are then exploited in the study of time varying systems in differential operator representation results on equivalence state space models controllability observability and transfer functions are derived

global and uniform convergence of subspace correction methods for some convex optimization problems this paper gives some global and uniform convergence estimates for a class of subspace correction based on space decomposition iterative methods applied to some unconstrained convex optimization problems some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems

an additive analysis of multiplicative schwarz methods we present an analysis of multiplicative schwarz methods for symmetric positive definite problems that is based on the theory of additive schwarz preconditioners and discuss applications to multigrid methods and domain decomposition methods

convergence estimates for product iterative methods with applications to domain decomposition in this paper we consider iterative methods for the solution of symmetric positive definite problems on a space which are defined in terms of products of operators defined with respect to a number of subspaces the simplest algorithm of this sort has an error reducing operator which is the product of orthogonal projections onto the complement of the subspaces new normreduction estimates for these iterative techniques will be presented in an abstract setting applications are given for overlapping schwarz algorithms with many subregions for finite element approximation of second order elliptic problems

a new proof that for the contact process on homogeneous trees local survival implies complete convergence we provide a new proof substantially simpler than zhang s original one that for the contact process on homogeneous trees local survival implies complete convergence

parallel galerkin domain decomposition procedures for wave equation parallel galerkin domain decomposition procedures for wave equation are given these procedures use implicit method in the sub domains and simple explicit flux calculation on the inter boundaries of sub domains by integral mean method or extrapolation method thus the parallelism can be achieved by these procedures the explicit nature of the flux prediction induces a time step constraint that is necessary to preserve the stability l 2 norm error estimates are derived for these procedures experimental results are presented to confirm the theoretical results

multigrid convergence for second order elliptic problems with smooth complex coefficients the finite element method when applied to a second order partial differential equation in divergence form can generate operators that are neither hermitian nor definite when the coefficient function is complex valued for such problems under a uniqueness assumption we prove the continuous dependence of the exact solution and its finite element approximations on data provided that the coefficients are smooth and uniformly bounded away from zero then we show that a multigrid algorithm converges once the coarse mesh size is smaller than some fixed number providing an efficient solver for computing discrete approximations numerical experiments while confirming the theory also reveal pronounced sensitivity of gauss seidel iterations on the ordering of the unknowns for certain problems

convergence of algebraic multigrid based on smoothed aggregation we prove aconvergence estimate for the algebraic multigrid method with prolongations defined by aggregation using zero energy modes followed by a smoothing the method input is the problem matrix and a matrix of the zero energy modes the estimate depends only polylogarithmically on the mesh size and requires only a weak approximation property for the aggregates which can be a priori verified computationally construction of the prolongator in the case of a general second order system is described and the assumptions of the theorem are verified for a scalar problem discretized by linear conforming finite elements

schwarz methods of neumann neumann type for three dimensional elliptic finite element problems several domain decomposition methods of neumann neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements we will only consider problems in three dimensions several new variants of the basic algorithm are introduced in a schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multi level methods the neumann neumann algorithms have several advantages over other domain decomposition methods the subregions which define the subproblems only share the boundary degrees of freedom with their neighbors the subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries in its original form however the algorithm lacks a mechanism for global transportation of information and its performance therefore suffers when the number of subregions increases in the new variants of the algorithms considered in this paper the preconditioners include global components of low rank to overcome this difficulty bounds are established for the condition number of the iteration operator which are independent of the number of subregions and depend only polylogarithmically on the number of degrees of freedom of individual local subproblems results are also given for problems with arbitrarily large jumps in the coefficients across the interfaces separating the subregions 1995 john wiley sons inc

local multilevel methods for adaptive finite element methods for nonsymmetric and indefinite elliptic boundary value problems in this paper we propose some local multilevel algorithms for solving linear systems arising from adaptive finite element approximations of nonsymmetric and indefinite elliptic boundary value problems two types of local smoothers are constructed one is based on the original nonsymmetric problems and the other is defined in terms of the associated symmetric problems it is shown that the local multilevel methods for the nonsymmetric and indefinite elliptic boundary value problems are optimal which means that the convergence rates of the local multilevel methods are independent of mesh sizes and mesh levels provided that the coarsest grid is sufficiently fine numerical experiments are reported to confirm our theory

numerical analysis of vanka type solvers for steady stokes and navier stokes flows we consider vanka type smoothers for solving stokes and navier stokes problems in each iteration step this smoother requires the solution of several small local subproblems over finite element blocks it is shown that for particular choices for the blocks the algorithm always converges to the solution of the stokes problem and under suitable conditions to the solution of the navier stokes problem the convergence properties are analyzed and numerical examples are presented

domain decomposition for a least square finite element method for second order elliptic problem in this paper we consider the solutions of the discrete linear system resulting from least square mixed finite element discretization applied to a second order elliptic problems additive and multiplicative domain decomposition methods are used to solve the linear system and the uniform convergence results are established for both methods

analysis of a parallel schwarz algorithm for elliptic problems a parallel algorithm which is derived directly from the schwarz alternating algorithm with an overlapping domain decomposition for solving boundary value problems of elliptic partial differential equations is analyzed a new general analysis framework different from lions ones is established to describe the recursive relations of error functions between approximate solutions and the exact solution at the different iterative steps some abstract theories on convergence and convergent rate of the synchronously parallel algorithm are established under different conditions

realizations in generalized state space form for polynomial system matrices and the definitions of poles zeros and decoupling zeros at infinity this paper contains three main results firstly a realization procedure is presented that brings a polynomial system matrix of a non proper multivariable system to generalized state space form such that all relevant properties including phenomena of redundancy associated with finite and infinite decoupling zeros are retained secondly new definitions are proposed for poles zeros and decoupling zeros at infinity of a general polynomial system matrix as a third result a theorem of rosenbrock 1974 e on the redundancy of lcr multiports is generalized to include the decoupling zeros at infinity