experimental study of ilu preconditioners for indefinite matrices incomplete lu factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices however their failure rate is still too high for them to be useful as black box library software for general matrices besides fatal breakdowns due to zero pivots the major causes of failure are inaccuracy and instability of the triangular solves when there are small pivots both these problems can occur but these problems can also occur without small pivots through examples from actual problems this paper shows how these problems evince themselves how these problems can be detected and how these problems can sometimes be circumvented through pivoting reordering scaling perturbing diagonal elements and preserving symmetric structure the goal of this paper is to gain a better practical understanding of ilu preconditioners and help improve their reliability

dead beat control of linear periodic discrete time systems in this paper the dead beat control of linear periodic discrete time systems subject to periodic disturbances is studied necessary and sufficient conditions are given for the existence of a linear periodic controller such that both output dead beat control under periodic disturbances and state dead beat control under no disturbance are performed moreover synthesis procedures of the controller as well as of state dead beat observers are given which can be applied also to a class of bilinear systems

disturbance localization by measurement feedback for linear periodic discrete time systems abstract the recent notions of inner controllable outer controllable and outer reconstructible subspaces together with the extension of the notion of c a b pair of subspaces to the class of linear periodic discrete time systems are the main tools used in this paper in order to study the disturbance localization problem by measurement feedback for the same class of systems by means of these concepts the necessary and sufficient solvability conditions of such a problem and synthesis procedures of the solutions are found for three different cases i e with and without an additional requirement of output or state dead beat control

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

holder domains and poincare domains a domain d c rd of finite volume is said to be a p poincare domain if there is a constant mp d so that fu udii dx d then d is a p poincare domain this answers a question of axler and shields regarding the image of the unit disk under a holder continuous conformal mapping we also consider geometric conditions which imply that the imbedding of the sobolev space w p d lp d is compact and prove that this is the case for a holder domain d

block preconditioners for symmetric indefinite linear systems this paper presents a systematic theoretical and numerical evaluation of three common block preconditioners in a krylov subspace method for solving symmetric indefinite linear systems the focus is on large scale real world problems where block approximations are a practical necessity the main illustration is the performance of the block diagonal constrained and lower triangular preconditioners over a range of block approximations for the symmetric indefinite system arising from large scale finite element discretization of biot s consolidation equations this system of equations is of fundamental importance to geomechanics numerical studies show that simple diagonal approximations to the 1 1 block k and inexpensive approximations to the schur complement matrix s may not always produce the most spectacular time savings when k is explicitly available but is able to deliver reasonably good results on a consistent basis in addition the block diagonal preconditioner with a negative 2 2 block appears to be reasonably competitive when compared to the more complicated ones these observation are expected to remain valid for coefficient matrices whereby the 1 1 block is sparse diagonally significant a notion weaker than diagonal dominance moderately well conditioned and has a much larger block size than the 2 2 block copyright 2004 john wiley sons ltd

a modified ssor preconditioner for sparse symmetric indefinite linear systems of equations the standard ssor preconditioner is ineffective for the iterative solution of the symmetric indefinite linear systems arising from finite element discretization of the biot s consolidation equations in this paper a modified block ssor preconditioner combined with the eisenstat trick implementation is proposed for actual implementation a pointwise variant of this modified block ssor preconditioner is highly recommended to obtain a compromise between simplicity and effectiveness numerical experiments show that the proposed modified ssor preconditioned symmetric qmr solver can achieve faster convergence than several effective preconditioners published in the recent literature in terms of total runtime moreover the proposed modified ssor preconditioners can be generalized to non symmetric biot s systems copyright 2005 john wiley sons ltd

a note on quasihyperbolic boundary conditions and poincare domains we establish optimal poincare inequalities under a logarithmic growth condition on the quasihyperbolic metric

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

novel preconditioners for the iterative solution to fe discretized coupled consolidation equations a major computational issue in the finite element fe integration of coupled consolidation equations is the repeated solution in time of the resulting discretized indefinite system because of ill conditioning the iterative solution which is recommended in large size 3d settings requires the computation of a suitable preconditioner to guarantee convergence in this paper the coupled system is solved by a krylov subspace method preconditioned by an inexact constraint preconditioner icp preserving the same block structure as the native fe matrix the conditioning number of the preconditioned coupled problem depends on the quality of the approximation of the block corresponding to the structural stiffness matrix an efficient algorithm to implement icp into a krylov subspace method is developed numerical tests performed on realistic 3d problems reveal that icp typically outperforms standard ilut preconditioners and proves much more robust in severely ill conditioned problems

performance and robustness of block constraint preconditioners in finite element coupled consolidation problems block constraint preconditioners are a most recent development for the iterative solution to large scale often ill conditioned coupled consolidation problems a major limitation to their practical use however is the somewhat difficult selection of a number of user defined parameters at least 4 in a more or less optimal way the present paper investigates the robustness of three variants of the block constraint preconditioning in relation to the above parameters a theoretical analysis of the eigenspectrum of the preconditioned matrix provides relatively simple bounds of the eigenvalues as a function of these parameters a number of test problems used to validate the theoretical results show that both the mixed constraint preconditioner mcp combined with the symmetric quasi minimal residual sqmr solver and the mcp triangular variant t mcp combined with the bi conjugate gradient stabilized bi cgstab are efficient and robust tools for the solution to difficult finite element discretized coupled consolidation problems moreover the practical selection of the user defined parameters is relatively easy as a stable behavior is observed for a wide range of fill in degree values the theoretical bounds on the eigenspectrum of the preconditioned matrix may help to suggest the most appropriate parameter combination copyright 2009 john wiley sons ltd

parameterization of the class of deadbeat controllers this technical note deals with the problem of parameterizing the class of deadbeat controllers for a given system through the minimum of parameters it is shown that if n and m are dimensions of the state vector and control vector respectively and the i s are the controllability indexes then almost the whole class of deadbeat controllers is explicitly parameterizable through mn mu 1 3 mu 2 2m 1 mu m parameters

quasihyperbolic boundary conditions and poincare domains we prove that a domain in bbb r n whose quasihyperbolic metric satisfies a logarithmic growth condition with coefficient beta le 1 is a q p poincare domain for all p and q satisfying p in 1 infty cap n n beta n and q in p beta p where p np n p denotes the sobolev conjugate exponent an elementary example shows that the given ranges for p and q are sharp the proof makes use of estimates for a variational capacity when p 2 we give an application to the solvability of the neumann problem on domains with irregular boundaries we also discuss the relationship between this growth condition on the quasihyperbolic metric and the s john condition

minimal time minimal order deadbeat regulator with internal stability this paper is concerned with the output deadbeat finite settling time regulation with internal stability for linear discrete time multivariable systems the two cases are treated separately the one is the regulation by state feedback and the other is the regulation through function observers for both cases the basic solvability conditions the minimal settling time and the characterizations of the minimal time regulators are derived throughout the paper the well posedness condition plays a fundamental role a deterministic version of the so called separation theorem in lqg problem is derived namely the minimal time state feedback regulator combined with the minimal time minimal order function observer yields the overall minimal time minimal order regulator by output feedback

output dead beat controllers and function dead beat observers for linear periodic discrete time systems in this paper two problems are considered that of finding a possibly dynamic linear periodic controller such that both the regulated output and the control input of a linear periodic discrete time system become and remain zero and that of designing a dead beat observer of a linear function of the state of the system for both these problems necessary and sufficient conditions for the existence of a solution are given and synthesis procedures of both the controller and the observer are provided by the sufficiency proofs such conditions are very simple and concise when expressed in terms of controllable and or unreconstructible states

an efficient diagonal preconditioner for finite element solution of biot s consolidation equations finite element simulations of very large scale soil structure interaction problems e g excavations tunnelling pile rafts etc typically involve the solution of a very large ill conditioned and indefinite biot system of equations the traditional preconditioned conjugate gradient solver coupled with the standard jacobi sj preconditioner can be very inefficient for this class of problems this paper presents a robust generalized jacobi gj preconditioner that is extremely effective for solving very large scale biot s finite element equations using the symmetric quasi minimal residual method the gj preconditioner can be formed inverted and implemented within an element by element framework as readily as the sj preconditioner it was derived as a diagonal approximation to a theoretical form which can be proven mathematically to possess an attractive eigenvalue clustering property the effectiveness of the gj preconditioner over a wide range of soil stiffness and permeability was demonstrated numerically using a simple three dimensional footing problem this paper casts a new perspective on the potentialities of the simple diagonal preconditioner which has been commonly perceived as being useful only in situations where it can serve as an approximate inverse to a diagonally dominant coefficient matrix copyright 2002 john wiley sons ltd

robust tracking and regulation of linear periodic discrete time systems in this paper the robust asymptotic tracking and disturbance rejection problem is studied for linear periodic discrete time systems and for general classes of disturbance functions and reference signals consisting in free responses of periodic processes the necessary and sufficient conditions for the existence of a solution which involve the periodic notion of invariant zero are found a constructive design procedure of the periodic compensator is given it is based on the internal model principle

mixed constraint preconditioners for the iterative solution of fe coupled consolidation equations the finite element fe integration of the coupled consolidation equations requires the solution of linear symmetric systems with an indefinite saddle point coefficient matrix because of ill conditioning the repeated solution in time of the fe equations may be a major computational issue requiring ad hoc preconditioning strategies to guarantee the efficient convergence of krylov subspace methods in the present paper a mixed constraint preconditioner mcp is developed combining implicit and explicit approximations of the inverse of the structural sub matrix with the performance investigated in some representative examples an upper bound of the eigenvalue distance from unity is theoretically provided in order to give practical indications on how to improve the preconditioner the mcp is efficiently implemented into a krylov subspace method with the performance obtained in 2d and 3d examples compared to that of inexact constraint preconditioners and least square logarithm scaled ilut preconditioners two variants of mcp t mcp and d mcp developed with the aim at reducing the cost of the preconditioner application are also tested the results show that the mcp variants constitute a reliable and robust approach for the efficient solution of realistic coupled consolidation fe models and especially so in severely ill conditioned problems

the geometric approach for linear periodic discrete time systems abstract recently the disturbance localization problem and the problems of designing disturbance decoupled observers and of giving a geometric characterization of invariant zeros were solved for linear periodic discrete time systems through an extension of the geometric approach based on the notions of controlled invariant and conditionally invariant subspaces to periodic ones here the existing periodic geometric theory is supplemented with the notions of outer reachable subspace and controllability subspace and with some further results on the previously introduced notions of inner reachable controllable subspace and outer controllable subspace in addition it is shown that any such periodic geometric theory can be restated in an equivalent time invariant form which is just the same theory written for a suitable time invariant system having an extended state space

solving large scale semidefinite programs via an iterative solver on the augmented systems the search directions in an interior point method for large scale semidefinite programming sdp can be computed by applying a krylov iterative method to either the schur complement equation sce or the augmented equation both methods suffer from slow convergence as interior point iterates approach optimality numerical experiments have shown that a diagonally preconditioned conjugate residual method on the sce typically takes a huge number of steps to converge however it is difficult to incorporate cheap and effective preconditioners into the sce this paper proposes to apply the preconditioned symmetric quasi minimal residual psqmr method to a reduced augmented equation that is derived from the augmented equation by utilizing the eigenvalue structure of the interior point iterates numerical experiments on sdp problems arising from maximum clique and selected sdplib sdp library problems show that moderately accurate solutions can be obtained with a modest number of psqmr steps using the proposed preconditioned reduced augmented equation an sdp problem with 127600 constraints is solved in about 6 5 hours to an accuracy of 10 6 in relative duality gap

parametrized solution of the periodic dead beat control problem the dead beat control problem for linear periodic discrete time systems in presence of unmeasurable disturbances and external reference signals is considered using a polynomial method a parametrization of all causal output feedback controllers solving the problem is provided

a qmr based interior point algorithm for solving linear programs a new approach for the implementation of interior point methods for solving linear programs is proposed its main feature is the iterative solution of the symmetric but highly indefinite 2 2 block systems of linear equations that arise within the interior point algorithm these linear systems are solved by a symmetric variant of the quasi minimal residual qmr algorithm which is an iterative solver for general linear systems the symmetric qmr algorithm can be combined with indefinite preconditioners which is crucial for the efficient solution of highly indefinite linear systems yet it still fully exploits the symmetry of the linear systems to be solved to support the use of the symmetric qmr iteration a novel stable reduction of the original unsymmetric 3 3 block systems to symmetric 2 2 block systems is introduced and a measure for a low relative accuracy for the solution of these linear systems within the interior point algorithm is proposed some indefinite preconditioners are discussed finally we report results of a few preliminary numerical experiments to illustrate the features of the new approach

on the stabilization of a class of bilinear systems abstract in this paper the problem of the stabilization of a class of continuous time bilinear systems containing the class of the homogeneous ones is studied and simple stabiliza bility conditions are derived expressed in terms of the stability or stabilizability of some linear or homogeneous bilinear subsystems of the given system on this basis it is shown that this problem can be decomposed into subproblems and reduced to the stabilization of homogeneous subsystems of the system obtained through successive decompositions of it a sufficient condition for the stabilization of a homogeneous bilinear system is also given

a decomposition technique for integrable functions with applications to the divergence problem abstract let r n be a bounded domain that can be written as t t where t t is a countable collection of domains with certain properties in this work we develop a technique to decompose a function f l 1 with vanishing mean value into the sum of a collection of functions f t f t t subordinated to t t such that supp f t f t t and f t f t 0 as an application we use this decomposition to prove the existence of a solution in weighted sobolev spaces of the divergence problem div u f and the well posedness of the stokes equations on holder domains and some other domains with an external cusp arbitrarily narrow we also consider arbitrary bounded domains the weights used in each case depend on the type of domain

a lifting based approach to observer based fault detection of linear periodic systems in this note an approach based on lifting is introduced for the observer based fault detection fd of linear discrete time periodic systems the main contribution of this note is to give an analytic expression to transform the linear time invariant lti observer based residual generator designed based on the lifted lti reformulation of the periodic system into a periodic observer based residual generator to reduce detection delay the result enables a systematic transfer of the existing observer based fd methods for discrete lti systems to discrete time periodic systems

structural properties of a class of linear hybrid systems and output feedback stabilization in this paper we deal with the problem of output feedback stabilization for a class of linear hybrid systems this problem is addressed by characterizing the structural properties of such a class of systems namely reachability controllability stabilizability observability constructibility and detectability are framed in terms of algebraic and geometric conditions on the data of the system two canonical forms recalling the classical kalman decompositions with respect to reachability and observability are given by taking advantage of this characterization duality between control and observation structural properties is established and necessary and sufficient conditions for the existence of a linear time invariant output feedback compensator are stated compared with previous results no assumption is needed on the plant about minimum phaseness relative degree or squareness

periodic solutions of periodic riccati equations for periodically time varying matrix riccati equations controllability and observability in the usual sense are shown to be sufficient for the existence of a unique positive definite periodic solution

korn inequality on irregular domains abstract in this paper we study the weighted korn inequality on some irregular domains e g s john domains and domains satisfying quasihyperbolic boundary conditions examples regarding sharpness of the korn inequality on these domains are presented moreover we show that korn inequalities imply certain poincare inequality

a canonical decomposition of linear periodic discrete time systems in this paper a linear periodic discrete time system is studied on the basis of simple properties of the subspaces of controllable states and of unreconstructible ones a canonical structure theorem is derived this generalizes 1o such a system the classical kalman decomposition while preserving the constant dimensionality of the four subsystems which arise when a periodic continuous time system is decomposed the dynamic matrices of the non controllable and or non reconstructible subsystems are shown to be non singular at each time instant as those for a time invariant discrete time system are

deadbeat control and tracking of discrete time systems a new approach for forcing the state of a linear discrete time system to zero in a minimum number of steps is discussed the problem is formulated as the solution to a steady state optimal control problem with no cost on the control this problem in turn is set up as the solution to an associated eigenvalue problem no special assumption on the open loop system matrix and or the ratio of the number of states to controls is required stable numerical techniques are presented for solving for the feedback gain robust deadbeat tracking is also discussed