restarted gmres preconditioned by deflation this paper presents a new preconditioning technique for the restarted gmres algorithm it is based on an invariant subspace approximation which is updated at each cycle numerical examples show that this deflation technique gives a more robust scheme than the restarted algorithm at a low cost of operations and memory

the numerical range of products of normal matrices abstract in an earlier paper the author developed a formula for the trace class multiplier norm of a matrix of rank at most 2 in this article applications of this formula are given in the main result we suppose that f 1 f n and g 1 g n are given sets of complex numbers a description is given of the union of the numerical ranges of the product fg as f and g run over all n n normal matrices with the given sets as eigenvalues

hitting forbidden subgraphs in graphs of bounded treewidth we study the complexity of a generic hitting problem h subgraph hitting where given a fixed pattern graph h and an input graph g we seek for the minimum size of a set x v g that hits all subgraphs of g isomorphic to h in the colorful variant of the problem each vertex of g is precolored with some color from v h and we require to hit only h subgraphs with matching colors standard techniques e g courcelle s theorem show that for every fixed h and the problem is fixed parameter tractable parameterized by the treewidth of g however it is not clear how exactly the running time should depend on treewidth for the colorful variant we demonstrate matching upper and lower bounds showing that the dependence of the running time on treewidth of g is tightly governed by h the maximum size of a minimal vertex separator in h that is we show for every fixed h that on a graph of treewidth t the colorful problem can be solved in time 2 mathcal o t mu h cdot v g but cannot be solved in time 2 o t mu h cdot v g o 1 assuming the exponential time hypothesis eth furthermore we give some preliminary results showing that in the absence of colors the parameterized complexity landscape of h subgraph hitting is much richer

finite element method for conserved phase fields stress mediated diffusional phase transformation phase field models with conserved phase field variables result in a 4th order evolution partial differential equation pde when coupled with the usual 2nd order thermo mechanics equations such problems require special treatment in the past the finite element method fem has been successfully applied to non conserved phase fields governed by a 2nd order pde for higher order equations the convergence of the standard galerkin fem requires that the interpolation functions belong to a higher continuity class we consider the cahn hilliard phase field model for diffusion controlled solid state phase transformation in binary alloys coupled with elasticity of the solid phases a galerkin finite element formulation is developed with mixed order interpolation c 0 interpolation functions for displacements and c 1 interpolation functions for the phase field variable to demonstrate convergence of the mixed interpolation scheme we first study a one dimensional problem nucleation and growth of the intermediate phase in a thin film diffusion couple with elasticity effects then we study the effects of completeness of c 1 interpolation on parabolic problems in two space dimensions by considering the growth of the intermediate phase in a binary system quadratic convergence expected for conforming elements is achieved for both one and two dimensional systems

on the numerical analysis of the von karman equations mixed finite element approximation and continuation techniques the purpose of this paper is to study the approximation of the von karman equations by the mixed finite element scheme of miyoshi and to follow the solutions arcs at a neighbourhood of the first eigenvalue of the linearized problem this last problem is solved by a continuation method

mode jumping in the von karman equations we study numerical approximations of bifurcating solution curves of the von karman equations with simply supported and clamped boundary conditions respectively of special interest here is the splitting of a double bifurcation point into two simple bifurcation points and the tracing of the associated secondary solution branches which corresponds to the phenomenon of mode jumping in the buckling of a rectangular plate a continuation unsymmetric lanczos algorithm is utilized for curve tracking our numerical results verify the theoretical prediction of schaeffer and golubitsky namely mode jumping occurs when the boundary is partially clamped but not if it is merely simply supported

hitting minors on bounded treewidth graphs ii single exponential algorithms abstract for a finite collection of graphs f the f m deletion resp f tm deletion problem consists in given a graph g and an integer k decide whether there exists s v g with s k such that g s does not contain any of the graphs in f as a minor resp topological minor we are interested in the parameterized complexity of both problems when the parameter is the treewidth of g denoted by tw and specifically in the cases where f contains a single connected planar graph h we present algorithms running in time 2 o tw n o 1 called single exponential when h is either p 3 p 4 c 4 the paw the chair and the banner for both h m deletion and h tm deletion and when h k 1 i with i 1 for h tm deletion some of these algorithms use the rank based approach introduced by bodlaender et al 2015 7 this is the second of a series of articles on this topic and the results given here together with other ones allow us in particular to provide a tight dichotomy on the complexity of h m deletion in terms of h

maximal spectral distance abstract the exact upper bound for the norm distance between two normal matrices is given as a function of their eigenvalues among the applications we obtain an inequality related to the cartesian decomposition of a matrix and the evaluation of the hadamard multiplier norms for matrices in a certain broad class

multiple bifurcation in the von karman equations we numerically investigate nontrivial solutions for some thin plate problems we find that at a corank 2 bifurcation point of the von karman equations with simply supported boundary conditions four nontrivial solution branches bifurcate thus the results of holder schaeffer and golubitsky are numerically verified where two symmetry breaking solution branches are obtained we also show that the result of allgower bohmer and mei in second order semilinear elliptic problems can be extended to certain fourth order elastic plate problems the local bifurcation behaviors of the former and the latter are different the practical implementations are given sample numerical results are reported

finite dimensional approximation of non linear problems 3 simple bifurcation points note univ pavia cnr ist analisi numer i 27100 pavia italy ecole polytech f 91128 palaiseau france ecole polytech ctr math appl cnrs era 747 f 91128 palaiseau france brezzi f univ pavia ist mat appl i 27100 pavia italy isi document delivery no mn015times cited 75cited reference count 20 reference asn article 1981 001 record created on 2006 08 24 modified on 2017 05 12

a newton gmres approach for the analysis of the postbuckling behavior of the solutions of the von karman equations we propose a newton gmres type algorithm to solve the discrete von karman problem we show that this algorithm is efficient both in memory and computation time and robust in the neighborhood of the singular points of the bifurcation diagrams placing ourselves in the context of the schaeffer and golubitsky theory we use this algorithm to study the postbuckling behavior of a rectangular plate clamped and compressed along its four sides

equivalences and transformations of regular systems applications to recursive program schemes and grammars abstract this work presents a unified theory of recursive program schemes context free grammars grammars on arbitrary algebraic structures and in fact recursive definitions of all kind by means of regular systems the equivalences of regular systems associated with either all their solutions or their least solutions in all domains of appropriate type satisfying a set of algebraic laws expressed by equations are systematically investigated and characterized in some cases in terms of system transformations by folding unfolding and rewriting according to the equational algebraic laws grammars are better characterized in terms of polynomial systems which are regular systems involving the operation of set union and the same questions are raised for them we also examine conditions insuring the uniqueness of the solution of a regular or of a polynomial system this theory applies to grammars of many kinds which generate trees graphs etc we formulate some classical transformations of context free grammars in terms of correct transformations which only use folding unfolding and algebraic laws and we immediately obtain their correctness

buckling and postbuckling analysis of rectangular plates resting on elastic foundations with the use of the spectral method abstract buckling and contact effects for a von karman s plate unilaterally connected with a nonlinear elastic tensionless foundation are investigated the foundation is modeled as nonlinear elastic winkler type and shear deformable pasternak type medium compressive and tensile loading forces along the edges of the plate are considered the spectral method is developed for the discretization of the boundary value problems a parametric analysis of compressive and tensile loadings is given bifurcation phenomena of the solution of the equations are explored the iterative scheme with the path following algorithm is employed to solve the resulting system of nonlinear algebraic equations numerical results are presented

multigrid solution of automatically generated high order discretizations for the biharmonic equation in this work we use a symbolic algebra package to derive a family of finite difference approximations for the biharmonic equation on a 9 point compact stencil the solution and its first derivatives are carried as unknowns at the grid points dirichlet boundary conditions are thus incorporated naturally since the approximations use the 9 point compact stencil no special formulas are needed near the boundaries both second order and fourth order discretizations are derived the fourth order approximation produce more accurate results than the 13 point classical stencil or the commonly used system of two second order equations coupled with the boundary condition the method suffers from slow convergence when classical iteration methods such as gauss seidel or sor are employed in order to alleviate this problem we propose several multigrid techniques that exhibit grid independent convergence and solve the biharmonic equation in a small amount of computer time test results from three different problems including stokes flow in a driven cavity are reported

criteria for combining inverse and rayleigh quotient iteration a method is presented to find selected eigenvalue eigenvector pairs of the generalized problem ax lambda bx here a and b are real symmetric matrices and b is positive definite it is further assumed that the matrices are large and sparse and that factorization of either of them is impractical our method finds an eigenvalue in a given interval j gamma eta gamma eta or determines that j is free of eigenvalues while computing an approximation to the eigenvalue closest to this interval the corresponding eigenvector is also computed the method consists of inverse and rayleigh quotient iteration steps the convergence is studied and it is shown how an inclusion theorem gives one of the criteria for switching from inverse to rayleigh quotient iteration the existence of an eigenvalue in the desired interval is guaranteed when this criterion is fulfilled some numerical experiments are reported which suggest that the method is optimal in the sense that when the mesh is refined the computa

tracing the buckling of a rectangular plate with the block gmres method we investigate bifurcation scenario of the von karman equations with partially clamped boundary conditions defined in rectangular domains first we study how the preconditioned block gmres method can be used in the context of continuation methods for tracing solution curves of large systems of nonlinear equations next we discuss linear stabilities of the von karman equations with partially clamped boundary conditions in particular we calculate the first seven eigenvalues and its associated eigenfunctions of the linearized von karman equations via computer algebra the block gmres method is used to solve linear systems and to detect singularity along solution paths of the discrete problem sample numerical results are reported

a linear time algorithm for finding tree decompositions of small treewidth in this paper we give for constant k a linear time algorithm that given a graph g v e determines whether the treewidth of g is at most k and if so finds a tree decomposition of g with treewidth at most k a consequence is that every minor closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm another consequence is that a similar result holds when we look instead for path decompositions with pathwidth at most some constant k

simultaneous solution of large scale linear systems and eigenvalue problems with a parallel gmres method a method for simultaneous solution of large and sparse linearized equation sets and the corresponding eigenvalue problems is presented such problems arise from the discretization and the solution of nonlinear problems with the finite element method and newton iteration the method is based on a parallel version of the preconditioned gmres m by deflation the parallel code exploits the architecture of the computational clusters using the mpi message passing interface the convergence rate the parallel speedup and the memory requirements of the proposed method are reported and evaluated

the monadic second order logic of graphs i recognizable sets of finite graphs abstract the notion of a recognizable set of finite graphs is introduced every set of finite graphs that is definable in monadic second order logic is recognizable but not vice versa the monadic second order theory of a context free set of graphs is decidable

stabilization of unstable procedures the recursive projection method fixed point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes the recursive projection method rpm which stabilizes such procedures by computing a projection onto the unstable subspace is presented on this subspace a newton or special newton iteration is performed and the fixed point iteration is used on the complement as continuation in the parameter proceeds the projection is efficiently updated possibly increasing or decreasing the dimension of the unstable subspace the method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system convergence proofs are given and pseudo arclength continuation on the unstable subspace is introduced to allow continuation past folds examples are presented for an important application of the rpm in which a black box time integration scheme is stabilized enabling it to compute unstable steady states the rpm can also be used to accelerate iterative procedures when slow convergence is due to a few slowly decaying modes

energy estimates for the von karman model of thin film blistering we consider the behavior of buckling driven thin film blisterings using von karman s plate theory our focus is on the setting where the blistered region is the unit square with clamped boundary conditions at the vertical sides and periodic ones along the horizontal sides in this setting we prove rigorous upper and lower bounds for the elastic energy which are of the same order as the film thickness we also present a convincing argument for the necessity of branching of folds near the boundary as has been observed in experiments

bicgstab vpastab and an adaptation to mildly nonlinear systems the key equations of bicgstab are summarised to show its connections with pade and vector pade approximation these considerations lead naturally to stabilised vector pade approximation of a vector valued function vpastab and an algorithm for the acceleration of convergence of a linearly generated sequence of vectors a generalisation of this algorithm for the acceleration of convergence of a nonlinearly generated system is proposed here and comparative numerical results are given

trace class multipliers and spectral variation of normal matrices abstract in this article we show how to estimate the trace multiplier norm of a rank 2 matrix as an application an alternative proof of a theorem of holbrook et al maximal spectral distance linear algebra appl 249 1996 197 205 on the maximal spectral distance between two normal matrices with prescribed eigenvalues is given

variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion a model for the adhesive quasistatic and frictionless contact between a viscoelastic body and a deformable foundation is described the adhesion process is modelled by a bonding field on the contact surface and contact is described by a modified normal compliance condition the problem is formulated as a coupled system of a variational equality for the displacements and a differential equation for the bonding field the existence of a unique weak solution for the problem and its continuous dependence on the adhesion parameters are established then the numerical analysis of the problem is conducted for the fully discrete approximation the convergence of the scheme is established and error estimates derived finally representative numerical simulations are presented depicting the evolution of the state of the system and in particular the evolution of the bonding field

the superlinear convergence behaviour of gmres abstract gmres is a rather popular iterative method for the solution of nonsingular nonsymmetric linear systems it is well known that gmres often has a so called superlinear convergence behaviour i e the rate of convergence seems to improve as the iteration proceeds for the conjugate gradients method this phenomenon has been related to a modest degree of convergence of the ritz values it has been observed in experiments that for gmres too changes in the convergence behaviour seem to be related to the convergence of ritz values in this paper we prove that as soon as eigenvalues of the original operator are sufficiently well approximated by ritz values gmres from then on converges at least as fast as for a related system in which these eigenvalues and their eigenvector components are missing

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

an axiomatic definition of context free rewriting and its application to nlc graph grammars abstract an abstract notion of context free grammar is introduced it deals with abstract objects that can be words trees graphs or other combinatorial objects it is applied to nlc graph grammars introduced by rozenberg and janssens the monadic second order theory of a context free nlc set of graphs is decidable

efficient computation of representative families with applications in parameterized and exact algorithms let m e i be a matroid and let s ls1 str be a family of subsets of e of size p a subfamily s s is q representative for s if for every set y e of size at most q if there is a set x s disjoint from y with x y i then there is a set x s disjoint from y with x y i by the classic result of bollobas in a uniform matroid every family of sets of size p has a q representative family with at most p qp sets in his famous two families theorem from 1977 lovasz proved that the same bound also holds for any matroid representable over a field f we give an efficient construction of a q representative family of size at most p qp in time bounded by a polynomial in p qp t and the time required for field operations we demonstrate how the efficient construction of representative families can be a powerful tool for designing single exponential parameterized and exact exponential time algorithms the applications of our approach include the following in the l ong d irected c ycle problem the input is a directed n vertex graph g and the positive integer k the task is to find a directed cycle of length at least k in g if such a cycle exists as a consequence of our 6 75k o k no 1 time algorithm we have that a directed cycle of length at least log n if such a cycle exists can be found in polynomial time in the m inimum e quivalent g raph meg problem we are seeking a spanning subdigraph d of a given n vertex digraph d with as few arcs as possible in which the reachability relation is the same as in the original digraph d we provide an alternative proof of the recent results for algorithms on graphs of bounded treewidth showing that many connectivity problems such as h amiltonian c ycle or s teiner t ree can be solved in time 2o t n on n vertex graphs of treewidth at most t for the special case of uniform matroids on n elements we give a faster algorithm to compute a representative family we use this algorithm to provide the fastest known deterministic parameterized algorithms for k p ath k t ree and more generally k s ubgraph i somorphism where the k vertex pattern graph is of constant treewidth

a note on the superlinear convergence of gmres in the paper it is shown how the rate of convergence of the gmres method for solving a linear operator equation lambda i k u f in a hilbert space is related to the degree of compactness of k measured by the products of its singular values

a cubic hermite finite element continuation method for numerical solutions of the von karman equations abstract we study a cubic hermite finite element method for numerical solutions of the von karman equations defined in a rectangular domain with totally clamped boundary conditions a novel iterative method combined with a predictor corrector continuation algorithm is exploited to trace solution curves of the von karman equations the fourth order finite element approximations compute accurate numerical solutions for the deformation and the airy stress function as well as their first order partial derivatives and the mixed second order partial derivatives in this regard the classical predictor corrector continuation method is interpreted in a different way our numerical results show that the bifurcation scenario of the von karman equations with totally clamped boundary conditions is different from those with simply supported and partially clamped boundary conditions

mesh independence of matrix free methods for path following in this paper we consider a matrix free path following algorithm for nonlinear parameter dependent compact fixed point problems we show that if these problems are discretized so that certain collective compactness and strong convergence properties hold then this algorithm can follow smooth folds and capture simple bifurcations in a mesh independent way

on the parameterized complexity of recognizing well covered r l graph an r ell partition of a graph g is a partition of its vertex set into r independent sets and ell cliques a graph is r ell if it admits an r ell partition a graph is well covered if every maximal independent set is also maximum a graph is r ell well covered if it is both r ell and well covered in this paper we consider two different decision problems in the r ell well covered graph problem r ell wcg for short we are given a graph g and the question is whether g is an r ell well covered graph in the well covered r ell graph problem wc r ell g for short we are given an r ell graph g together with an r ell partition of v g into r independent sets and ell cliques and the question is whether g is well covered we classify most of these problems into p conp complete np complete np hard or conp hard only the cases wc r 0 g for r ge 3 remain open in addition we consider the parameterized complexity of these problems for several choices of parameters such as the size alpha of a maximum independent set of the input graph its neighborhood diversity or the number ell of cliques in an r ell partition in particular we show that the parameterized problem of deciding whether a general graph is well covered parameterized by alpha can be reduced to the wc 0 ell g problem parameterized by ell and we prove that this latter problem is in xp but does not admit polynomial kernels unless mathsf conp subseteq mathsf np mathsf poly

a biparametric perturbation method for the foppl von karman equations of bimodular thin plates abstract in this study a biparametric perturbation method is proposed to solve the foppl von karman equations of bimodular thin plates subjected to a single load first by using two small parameters one describes the bimodular effect and another stands for the central deflection we expanded the unknown deflection and stress in double power series with respect to the two parameters and obtained the approximate analytical solutions under various edge conditions due to the diversity of selection of parameters and its combination by using the bimodular parameter and the load as two perturbation parameters we elucidated further the application of this method the use of two sets of parameter schemes both can obtain satisfactory perturbation solutions the numerical simulations also verify this idea the results indicate that in a biparametric perturbation method the selection and its combination of parameters may reflect the combined effects introduced by nonlinear factors the method proposed in this study may be used for solving other mathematical equations established in some application problems

bifurcation detection with the un preconditioned gmres m abstract when iterative solvers are used for the solution of the linearized equations sets arising from newton s method detection of bifurcation points in the solution space can be done by solving several eigenvalue problems with the jacobian matrix during parameter continuation monitoring the condition number of appropriate matrices involved in the gmres method employed here replaces or significantly reduces the costly eigenvalue computations the effectiveness of the detection method is examined in connection with the usage of unpre and preconditioned iterative solvers