linear maps on algebraic groups abstract let g be an irreducible simple classical algebraic subgroup of sl n c the n n matrices of determinant one we discuss the problem of classifying all invertible linear maps t on m n c such that t g g this has already been solved with two general exceptions we show how to eliminate one of these two cases in general t must have the form t x pxq or t x px t q for suitable invertible matrices p and q

noncore square matrices miscellany a noncore square matrix is one whose unique core nilpotent decomposition does not exclusively consist of the core part if the core part is idempotent the matrix itself is called quasiidempotent this paper makes a brief survey of these and other noncore matrices and their properties

orthogonal sets of normal or conjugate normal matrices abstract we point out several implications of a result of djokovic 1971 8 concerning orthogonality of normal matrices that satisfy a certain condition on the eigenvalues of their sum we obtain an analogous result in the setting of conjugate normal matrices

classification of voronoi and delone tiles of quasicrystals iii decagonal acceptance window of any size this paper is the last of a series of three articles presenting a classification of vornoi and delone tilings determined by point sets quasicrystals built by the standard projection of the root lattice of type a4 to a two dimensional plane spanned by the roots of the coxeter group h2 dihedral group of order 10 the acceptance window for in the present paper is a regular decagon of any radius 0 r there are 14 distinct vt sets of voronoi tiles and 6 sets dt of delone tiles up to a uniform scaling by the factor and the number of voronoi tiles in different quasicrystal tilings varies between 3 and 12 similarly the number of delone tiles is varying between 4 and 6 there are 7 vt sets of the generic type and 7 of the singular type the latter occur for seven precise values of the radius of the acceptance window quasicrystals with acceptance windows with radii in between these values have constant vt sets only the relative densities and arrangement of the tiles in the tilings change similarly we distinguish singular and generic sets dt of delone tiles

bpx type preconditioners for second and fourth order elliptic problems on the sphere we develop two bramble pasciak xu type preconditioners for second resp fourth order elliptic problems on the surface of the two sphere to discretize the second order problem we construct c 0 linear elements on the sphere and for the fourth order problem we construct c 1 finite elements of powell sabin type on the sphere the main reason these bpx preconditioners work depends on this particular choice of basis we prove optimality and provide numerical examples furthermore we numerically compare the bpx preconditioners with the suboptimal hierarchical basis preconditioners

a further algebraic version of cochran s theorem and matrix partial orderings abstract a new version of cochran s theorem for rectangular matrices is established being oriented toward partial isometries the new version parallels corresponding results concerned with arbitrary tripotent matrices and covers results concerned with hermitian tripotent matrices a discussion of a related new matrix partial ordering is also given

multigrid methods for isogeometric analysis with thb splines we propose a geometric multigrid algorithm for the solution of the linear systems which arise when using hierarchical spline spaces for isogeometric discretizations of elliptic partial differential equations in particular we focus on the use of hierarchical b spline hb and truncated hierarchical b spline thb bases to implement adaptively refined isogeometric discretizations we describe an approach for constructing the nested sequences of hierarchical spline spaces which are required for our geometric multigrid solver furthermore we give an algorithm for the computation of the so called prolongation matrices which map coarser space functions to their finer space representations for both hb and thb spline bases in several numerical experiments we study two dimensional poisson problems with singularities and employ a posteriori error estimators for the adaptive refinement of the hierarchical spline spaces using standard multigrid smoothers we observe that using thb spline bases consistently confers a dramatic advantage in terms of iteration numbers over the use of hb spline bases in particular for higher spline degrees when used as a preconditioner for conjugate gradient iteration the proposed multigrid method exhibits practical iteration numbers for two dimensional problems up to the tested case of fourth degree splines

classification of voronoi and delone tiles in quasicrystals i general method a new general method is presented which allows one to find all distinct voronoi and delone tiles in any quasicrystal from a large family this includes the tiles which may be present with arbitrarily low density 0 at all stages the method requires only consideration of a possibly large finite number of cases our method is applicable in principle to quasicrystals in any dimension and with any irrationality this is the first of three papers where the voronoi and delone tilings are studied two dimensional point sets quasicrystals arising from the a4 root lattice by means of the standard projection to a two dimensional plane with the irrationality 1 2 1 5 are considered in general we require that the acceptance window be bounded with non empty interior specific results are provided here for rhombic acceptance windows of any size oriented along the direction of simple roots of the coxeter group h2 within one quasicrystal the tiles are distinguished by their shape size and orientation the rhombic window case is indispensable for subsequent classification of voronoi and delone tiles in quasicrystals with general shape of the acceptance window voronoi and delone tiles of quasicrystals with circular and decagonal windows of any size are given in subsequent papers let vt denote the set of distinct voronoi tiles making up a quasicrystal with a given acceptance window there are three vt sets of the generic type and three of the singular type the latter occur for one precise value of the size of the acceptance window any other vt set is a uniform scaling of the tiles listed here similar results differing in detail are provided for the sets of distinct delone tiles dt altogether there are four different sets dt of delone tiles

partial orderings preorderings and the polar decomposition of matrices abstract when a unique decomposition of a complex rectangular matrix into two components is given then a partial ordering on the domain of one of the components induces a preordering on the whole set of matrices and partial orderings on the domains of both components induce a partial ordering on the whole set in this not we consider the three main partial orderings lo wner star and rank subtractivity on the respective domains of the two components of matrices subjected to polar decomposition a ben israel t n e greville generalized inverses theory and applications wiley new york 1974 p 255 and investigate the resulting pre and partial orderings

isogeometric overlapping schwarz preconditioners for the bidomain reaction diffusion system abstract we present and study an overlapping additive schwarz preconditioner for the isogeometric discretization of a reaction diffusion system modeling the heart bioelectrical activity known as the bidomain model consisting of a degenerate system of parabolic and elliptic pde this model includes intramural fiber rotation anisotropic conductivity coefficients and is coupled through the reaction term with a system of odes which models the ionic currents of the cellular membrane the overlapping schwarz preconditioner is applied with a pcg accelerator to solve the linear system arising at each time step from the isogeometric discretization in space and a semi implicit adaptive method in time a theoretical convergence rate analysis shows that the resulting solver is scalable optimal in the ratio of subdomain element size and the convergence rate improves with increasing overlap size numerical tests in three dimensional ellipsoidal domains confirm the theoretical estimates and additionally show the robustness with respect to jump discontinuities of the orthotropic conductivity coefficients

linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

on sets of elements in rickart rings induced by partial orders abstract let a be a rickart ring and let a 1 be the set of all regular elements in a the set of all a a such that a b are characterized where b a 1 is given and is the minus partial order in case when a is a rickart ring such sets are characterized for the diamond the left star the right star the left sharp and the right sharp partial orders some recent results of mosic et al on partial orders in b h the algebra of all bounded linear operators on a hilbert space h are thus generalized

thb splines an effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis local refinement with hierarchical b spline structures is an active topic of research in the context of geometric modeling and isogeometric analysis by exploiting a multilevel control structure we show that truncated hierarchical b spline thb spline representations support interactive modeling tools while simultaneously providing effective approximation schemes for the manipulation of complex data sets and the solution of partial differential equations via isogeometric analysis a selection of illustrative 2d and 3d numerical examples demonstrates the potential of the hierarchical framework

on some characterizations of pairwise star orthogonality using rank and dagger additivity and subtractivity abstract the main results are that pairwise star orthogonality a i a j 0 and a i a j 0 for all i j where a 1 a k are complex m n matrices is equivalent to i a i a j and to ii a i rs a j and a j a j where i 1 k here the superscript denotes the moore penrose inverse while and rs denote respectively the star and rank subtractivity or minus partial orderings five more characterizations of the pairwise star orthogonality of k complex m n matrices are also presented our characterizations extend earlier results by hartwig and styan 1986 for k 2

spaces of matrices of equal rank abstract let mm n f denote the space of all mxn matrices over the algebraically closed field f a subspace of mm n f all of whose nonzero elements have rank k is said to be essentially decomposable if there exist nonsingular mxn matrices u and v respectively such that for any element a uav has the form uav a 1 a 2 a 3 0 where a1 is ix k i for some i k theorem if k is a space of rank k matrices then either k is essentially decomposable or dim k k 1 an example shows that the above bound on non essentially decomposable spaces of rank k matrices is sharp whenever n 2k 1

isogeometric block feti dp preconditioners for the stokes and mixed linear elasticity systems abstract the aim of this work is to construct and analyze a feti dp type domain decomposition preconditioner for isogeometric discretizations of the stokes and mixed linear elasticity systems this method extends to the isogeometric analysis context the preconditioner previously proposed by tu and li 2015 for finite element discretizations of the stokes system the resulting isogeometric feti dp algorithm is proven to be scalable in the number of subdomains and has a quasi optimal convergence rate bound which is polylogarithmic in the ratio of subdomain and element sizes extensive two dimensional numerical experiments validate the theory investigate the behavior of the preconditioner with respect to both the spline polynomial degree and regularity and show its robustness with respect to domain deformation material incompressibility and presence of elastic coefficient discontinuities across subdomain interfaces

maps preserving the diamond partial order the present paper is devoted to the study of the diamond partial order in c algebras we characterize linear maps preserving this partial order

a note on the rank subtractivity ordering abstract some well known matrix partial orderings can be created by supplementing the so called rank subtractivity partial ordering by an appropriate preordering in this note a new partial ordering with the same property is introduced this new ordering is seen to be weaker than the other orderings but still preserving some inheritance properties

robust and optimal multi iterative techniques for iga galerkin linear systems we consider fast solvers for the large linear systems coming from the isogeometric analysis iga collocation approximation based on b splines of full elliptic d dimensional partial differential equations pdes we are interested in designing iterative algorithms which are optimal and robust the former property implies that the computational cost is linear with respect to the number of degrees of freedom i e the matrix size the latter property means that the convergence rate is completely independent of or only mildly dependent on all the relevant parameters in our setting we can mention the coefficients of the pde the dimensionality d the geometric map g describing the physical domain the matrix size related to the fineness parameters and the spline degrees associated with the iga approximation order our approach is based on the spectral symbol which describes the global eigenvalue behavior of the iga collocation matrices it is precisely the spectral information contained in the symbol that is exploited for the design of an optimal and robust multi iterative solver of multigrid type several numerical experiments are presented and discussed in view of recent theoretical findings concerning the symbol and its properties

quasicrystals and icosians a family of quasicrystals of dimensions 1 2 3 4 governed by the root lattice e8 is constructed the use of the icosian ring found in the quaternions with coefficients in q square root 5 allows simultaneous interpretation of the construction both in physical space and as a result of the standard cut and projection method in double dimension icosians are seen to provide a natural co ordination scheme for these quasicrystals nested sequences of quasicrystals form systems whose symmetries are all derivable from inflational and reflective symmetries directly related to the arithmetic of the icosians the use of coxeter diagrams clarifies the amazing relationship of e8 and quasicrystal symmetries and leads to the fundamental chain e8 contains implies d6 contains implies a4 contains implies a1 a1 that underlies five fold symmetry in quasicrystals decomposition of quasicrystals into concentric shells and a counting formula for the cardinalities of these shells is discussed

optimal additive schwarz preconditioning for adaptive 2d iga boundary element methods abstract we define and analyze local multilevel diagonal preconditioners for isogeometric boundary elements on locally refined meshes in two dimensions hypersingular and weakly singular integral equations are considered we prove that the condition number of the preconditioned systems of linear equations is independent of the mesh size and the refinement level therefore the computational complexity when using appropriate iterative solvers is optimal our analysis is carried out for closed and open boundaries and numerical examples confirm our theoretical results

on rank additivity and the polynomial equation p a 0 for an arbitrary polynomial p two characterizations of a square matrix a satisfying the polynomial equation p a 0 are obtained some results on rank additivity closely related to the matrix theoretic version of cochran s statistical theorem are given

a robust multigrid method for isogeometric analysis in two dimensions using boundary correction we consider geometric multigrid methods for the solution of linear systems arising from isogeometric discretizations of elliptic partial differential equations for classical finite elements such methods are well known to be fast solvers showing optimal convergence behavior however the naive application of multigrid to the isogeometric case results in significant deterioration of the convergence rates if the spline degree is increased recently a robust approximation error estimate and a corresponding inverse inequality for b splines of maximum smoothness have been shown both with constants independent of the spline degree we use these results to construct multigrid solvers for discretizations of two dimensional problems based on tensor product b splines with maximum smoothness which exhibit robust convergence rates

overlapping schwarz methods for isogeometric analysis we construct and analyze an overlapping schwarz preconditioner for elliptic problems discretized with isogeometric analysis the preconditioner is based on partitioning the domain of the problem into overlapping subdomains solving local isogeometric problems on these subdomains and solving an additional coarse isogeometric problem associated with the subdomain mesh we develop an h analysis of the preconditioner showing in particular that the resulting algorithm is scalable and its convergence rate depends linearly on the ratio between subdomain and overlap sizes for fixed polynomial degree p and regularity k of the basis functions numerical results in two and three dimensional tests show the good convergence properties of the preconditioner with respect to the isogeometric discretization parameters h p k number of subdomains n overlap size and also jumps in the coefficients of the elliptic operator

symbol based multigrid methods for galerkin b spline isogeometric analysis we consider the stiffness matrices arising from the galerkin b spline isogeometric analysis discretization of classical elliptic problems by exploiting their specific spectral properties compactly described by a symbol we design an efficient multigrid method for the fast solution of the related linear systems the convergence rate of general purpose multigrid methods based on classical stationary smoothers is optimal i e bounded independently of the matrix size but it also worsens exponentially with respect to the spline degree the symbol allows us to give a detailed theoretical explanation of this exponential worsening in the case of the two grid scheme in addition thanks to a specific factorization of the symbol we are able to design an ad hoc multigrid method with an effective preconditioned cg or gmres smoother at the finest level in the spirit of the multi iterative idea the convergence rate of this multi iterative multigrid method is not only optimal but also robust i e bounded su

linear preservers for matrix inequalities and partial orderings we characterize the linear preservers of minus partial order on matrix algebras the developed approach allows us to classify linear transformations that live fixed several matrix relations arising as extremal cases in some classical matrix inequalities including rank additivity relation and related properties applications to the determinant preservers are considered

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

some orders for operators on hilbert spaces in this paper we present some interesting properties of the diamond left right star and sharp orders for operators on hilbert spaces

robust multigrid solvers for the biharmonic problem in isogeometric analysis abstract in this paper we develop multigrid solvers for the biharmonic problem in the framework of isogeometric analysis iga in this framework one typically sets up b splines on the unit square or cube and transforms them to the domain of interest by a global smooth geometry function with this approach it is feasible to set up h 2 conforming discretizations we propose two multigrid methods for such a discretization one based on gauss seidel smoothing and one based on mass smoothing we prove that both are robust in the grid size the latter is also robust in the spline degree numerical experiments illustrate the convergence theory and indicate the efficiency of the proposed multigrid approaches particularly of a hybrid approach combining both smoothers

the minus partial order and the shorted matrix the minus partial order was defined by hartwig 6 weakening the conditions of the star partial order of drazin 5 several new properties of the minus partial order are established the minus partial order is used to redefine the shorted matrix 15 17 and to define the infimum a b and the supremum a b of a pair a b of matrices of the same order the definition of the shorted matrix given here is similar to the krein anderson trapp definition of the shorted positive operator

a note on rank additivity and range additivity abstract some rank additivity results for matrices are extended to range additivity results for three bounded linear operators acting on an infinite dimensional hilbert space we also give counterexamples showing which of these statements fail for a larger number of operators

multigrid methods for isogeometric discretization abstract we present geometric multigrid methods for isogeometric discretization of scalar second order elliptic problems the smoothing property of the relaxation method and the approximation property of the intergrid transfer operators are analyzed these properties when used in the framework of classical multigrid theory imply uniform convergence of two grid and multigrid methods supporting numerical results are provided for the smoothing property the approximation property convergence factor and iterations count for v w and f cycles and the linear dependence of v cycle convergence on the smoothing steps for two dimensions numerical results include the problems with variable coefficients simple multi patch geometry a quarter annulus and the dependence of convergence behavior on refinement levels l whereas for three dimensions only the constant coefficient problem in a unit cube is considered the numerical results are complete up to polynomial order p 4 and for c 0 and c p 1 smoothness

overlapping schwarz methods for isogeometric analysis based on generalized b splines abstract generalized b splines gb splines are a special class of tchebycheff b splines that are smooth piecewise function with sections in more general spaces gb splines allow for an exact representation of conic sections as well as transcendental curves and thus they become very attractive for geometrical modeling and numerical simulation in this paper we present overlapping schwarz preconditioners for elliptic problems discretized with isogeometric analysis based on gb splines an h analysis of the proposed preconditioners shows an optimal convergence rate bound that is scalable in the number of subdomains and that is linear in the ratio between subdomain and overlap sizes numerical results in two and three dimensional tests confirm this analysis and also illustrate the good convergence properties of the preconditioner with respect to the discretization parameters the domain deformation and the jumping coefficients

overlapping schwarz preconditioners for isogeometric collocation methods isogeometric collocation methods are very recent and promising numerical schemes that preserve the advantages of isogeometric analysis but often exhibit better performances than their galerkin counterparts in the present paper an additive overlapping schwarz method for isogeometric collocation discretizations is introduced and studied the resulting preconditioner accelerated by gmres is shown to be scalable with respect to the number of subdomains and very robust with respect to the isogeometric discretization parameters such as the mesh size and polynomial degree as well as with respect to the presence of discontinuous elliptic coefficients and domain deformations

partial orderings of matrices referring to singular values or eigenvalues abstract a new partial ordering in the set of complex matrices is defined which is weaker than the star ordering introduced by drazin in 1978 and stronger than the minus ordering introduced by hartwig in 1980 this ordering refers to singular values of matrices and the interest in it was generated by canonical interpretations of the minus and star orderings given by hartwig and styan in 1986 for hermitian matrices a similar ordering referring to eigenvalues is also considered and its connection with a problem concerning distributions of quadratic forms in normal variables is pointed out

spaces of symmetric matrices of bounded rank abstract let i n denote the space of all n n symmetric matrices over a field f let t be a positive integer such that t a subspace w of i n f is said to be a t subspace provided that the rank of every matrix in w is bounded by t meshulam showed under the assumption f n 1 that the maximal dimension of a t subspace of i n f is given by max t 1 2 k 1 2 k n k if t 2k max t 1 2 k 1 2 k n k 1 if t 2k 1 provided that we also assume char f 2 we show here that any t subspace of i n f of maximal dimension is congruent to w 1 n t a i n f a ij 0 if i t or j t

on the wedderburn guttman theorem abstract let a be a u by v matrix of rank a and let m and n be u by g and v by g matrices respectively such that m an is nonsingular then rank a n m an 1m a a g where g rank an m an 1m a rank m an this is called wedderburn guttman theorem what happens if m an is rectangular and or singular in this paper we investigate conditions under which the regular inverse m an 1 can be replaced by a g inverse m an of some kind thereby extending the wedderburn guttman theorem the resultant conditions look similar to those arising in seemingly unrelated contexts namely cochran s and related theorems on distributions of quadratic forms involving a normal random vector

linear transformations which preserve fixed rank abstract let m m n f denote the set of all m n matrices over the algebraically closed field f let t denote a linear transformation t m m n f m m n f theorem if max m n 2 k 2 k 1 and t preserves rank k matrices i e a k implies t a k then there exist nonsingular m m and n n matrices u and v respectively such that either i t a uav for all a m m n f or ii m n and t a ua t v for all a m n f where a t denotes the transpose of a

upper and lower bounds for ranks of matrix expressions using generalized inverses abstract the maximal and minimal ranks of the matrix expression a 1 b 1 x 1 c 1 b 2 x 2 c 2 with respect to x 1 and x 2 are presented as applications the maximal and minimal ranks of a 1 b 1 xc 1 subject to a consistent matrix equation b 2 xc 2 a 2 are also determined in addition the maximal and minimal ranks of the schur complement d ca b with respect to the generalized inverse a of a and their various consequences are also considered

b spline based monotone multigrid methods for the efficient numerical solution of elliptic variational inequalities on closed convex sets multigrid methods based on piecewise linear finite elements have been investigated over the past decades essential to their success is the appropriate approximation of the constraint set on coarser grids which is based on function values for piecewise linear finite elements on the other hand there are a number of problems which profit from higher order approximations among these are the problem of pricing american options formulated as a parabolic boundary value problem involving black scholes equation with a free boundary in addition to computing the free boundary the optimal exercise price of the option of particular importance are accurate pointwise derivatives of the value of the stock option up to order two the so called greek letters in this paper we propose a monotone multigrid method for discretizations in terms of b splines of arbitrary order to solve elliptic variational inequalities on a closed convex set in order to maintain monotonicity upper bound and quasi optimality lower bound of the coarse grid corrections we propose an optimized coarse grid correction ocgc algorithm which is based on b spline expansion coefficients we prove that the ocgc algorithm is of optimal complexity of the degrees of freedom of the coarse grid and therefore the resulting monotone multigrid method is of asymptotically optimal multigrid complexity finally the method is applied to a standard model for the valuation of american options in particular it is shown that a discretization based on b splines of order four enables us to compute the second derivative of the value of the stock option to high precision

classification of voronoi and delone tiles of quasicrystals ii circular acceptance window of arbitrary size in this paper we describe all the voronoi and delone tiles arising in tilings of point sets quasicrystals built by the standard projection of the root lattice of type a4 onto a two dimensional plane spanned by the roots of the coxeter group h2 dihedral group of order 10 the acceptance window for is a disc of any radius 0 r there are 22 distinct sets vtj j 1 22 of voronoi tiles and eight sets dtk k 1 8 of delone tiles up to a uniform scaling by a factor n where 1 2 1 5 is the golden ratio and n

a relationship between the star and minus orderings abstract the result of hartwig and styan 1986 stating that matrices a and b are star ordered if and only if they are minus ordered and b a b a 1 3 or b a b a 1 4 is extended by considering generalized inverses of a linear combination b a with any 0

algebraic multilevel preconditioning methods i a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

on the spectrum of stiffness matrices arising from isogeometric analysis we study the spectral properties of stiffness matrices that arise in the context of isogeometric analysis for the numerical solution of classical second order elliptic problems motivated by the applicative interest in the fast solution of the related linear systems we are looking for a spectral characterization of the involved matrices in particular we investigate non singularity conditioning extremal behavior spectral distribution in the weyl sense as well as clustering of the eigenvalues to a certain compact subset of mathbb c c all the analysis is related to the notion of symbol in the toeplitz setting and is carried out both for the cases of 1d and 2d problems

on a matrix version of cochran s statistical theorem abstract cochran s theorem on the distribution of quadratic forms in normal random variables can be equivalently formulated as a rank additivity result for symmetric idempotent matrices a generalization of this theorem to matrices satisfying a general matrix polynomial equation p a 0 is given

some properties of matrix partial orderings abstract the matrix partial orderings considered are 1 the star ordering and 2 the minus ordering or rank subtractivity both in the set of m n complex matrices and 3 the lowner ordering in the set of m m matrices the problems discussed are 1 inheriting certain properties under a given ordering 2 preserving an ordering under some matrix multiplications 3 relationships between an ordering among direct or kronecker and hadamard products and the corresponding orderings between the factors involved 4 orderings between generalized inverses of a given matrix and 5 preserving or reversing a given ordering under generalized inversions several generalizations of results known in the literature and a number of new results are derived

anisotropic nurbs approximation in isogeometric analysis abstract we investigate the approximation error of mapped nurbs spaces such as those used in isogeometric analysis in accordance to the isoparametric paradigm we extend the results of 6 to the case of anisotropic meshes such as those needed in the presence of boundary layers the theoretical results cover anisotropic h refinement and are supported by numerical tests involving different geometries

bpx preconditioning for isogeometric analysis we consider elliptic pdes partial differential equations in the framework of isogeometric analysis i e we treat the physical domain by means of a b spline or nurbs mapping which we assume to be regular the numerical solution of the pde is computed by means of tensor product b splines mapped onto the physical domain we construct additive multilevel preconditioners and show that they are asymptotically optimal i e the spectral condition number of the resulting stiffness matrix is independent of h together with a nested iteration scheme this enables an iterative solution scheme of optimal linear complexity the theoretical results are substantiated by numerical examples in two and three space dimensions

bpx preconditioners for the bidomain model of electrocardiology the aim of this work is to develop a bpx preconditioner for the bidomain model of electrocardiology this model describes the bioelectrical activity of the cardiac tissue and consists of a system of a non linear parabolic reaction diffusion partial differential equation pde and an elliptic linear pde modeling at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue the evolution equation is coupled through the non linear reaction term with a stiff system of ordinary differential equations the so called membrane model describing the ionic currents through the cellular membrane the discretization of the coupled system by finite elements in space and semi implicit finite differences in time yields at each time step the solution of an ill conditioned linear system the goal of the present study is to construct analyze and numerically test a bpx preconditioner for the linear system arising from the discretization of the bidomain model optimal convergence rate estimates are established and verified by two and three dimensional numerical tests on both structured and unstructured meshes moreover in a full heartbeat simulation on a three dimensional wedge of ventricular tissue the bpx preconditioner is about 35 faster in terms of cpu times than ilu 0 and an algebraic multigrid preconditioner

a note on an algebraic version of cochran s theorem we characterize complex square matrices with the property that each of their orthogonal decompositions is rank additive

some general techniques on linear preserver problems abstract several general techniques on linear preserver problems are described the first one is based on a transfer principle in model theoretic algebra that allows one to extend linear preserver results on complex matrices to matrices over other algebraically closed fields of characteristic 0 the second one concerns the use of some simple geometric technique to reduce linear preserver problems to standard types so that known results can be applied the third one is about solving linear preserver problems on more general operator algebras by reducing the problems to idempotent preservers numerous examples will be given to demonstrate the proposed techniques

algebraic multilevel preconditioning methods ii a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance