block diagonal and constraint preconditioners for nonsymmetric indefinite linear systems part i theory we study block diagonal preconditioners and an efficient variant of constraint preconditioners for general two by two block linear systems with zero 2 2 block we derive block diagonal preconditioners from a splitting of the 1 1 block of the matrix from the resulting preconditioned system we derive a smaller so called related system that yields the solution of the original problem solving the related system corresponds to an efficient implementation of constraint preconditioning we analyze the properties of both classes of preconditioned matrices in particular their spectra using analytical results we show that the related system matrix has the more favorable spectrum which in many applications translates into faster convergence for krylov subspace methods we show that fast convergence depends mainly on the quality of the splitting a topic for which a substantial body of theory exists our analysis also provides a number of new relations between block diagonal preconditioners and constraint preconditioners for constrained problems solving the related system produces iterates that satisfy the constraints exactly just as for systems with a constraint preconditioner finally for the lagrange multiplier formulation of a constrained optimization problem we show how scaling nonlinear constraints can dramatically improve the convergence for linear systems in a newton iteration our theoretical results are confirmed by numerical experiments on a constrained optimization problem we consider the general nonsymmetric nonsingular case our only additional requirement is the nonsingularity of the schur complement type matrix derived from the splitting that defines the preconditioners in particular the 1 2 block need not equal the transposed 2 1 block and the 1 1 block might be indefinite or even singular this is the first paper in a two part sequence in the second paper we will study the use of our preconditioners in a variety of applications

a block product preconditioner for saddle point problems abstract in this paper a block product bp preconditioner is established for saddle point problems spectral properties of the bp preconditioned matrix are investigated a strategy for practical choice of quasi optimal parameter is given numerical results on saddle point linear systems arising from stokes problems and weighted least square problems show that the proposed bp preconditioner is more economic to implement within krylov subspace acceleration than some extensively studied preconditioners

the isometries of certain maximum norms abstract let a 1 a m be nonsingular n n matrices over f r or c and let 1 p j for j 1 m we characterize the linear isometries for the norm n on f n defined by n x max l p j a j x 1 j m where l p y denotes the l p norm of y f n as an application we study the conditions on matrices a 1 a m for which n is absolute or a symmetric gauge function

calculating functions of fuzzy numbers abstract this paper discusses the calculation of functions of fuzzy numbers our general approach follows that of the fuzzy weighted algorithm fwa of wong and dong however by developing a classification scheme for functions and their arguments we show that there are many cases for which the fwa and all other published methods will give wrong results we suggest an alternative approach which will work in all cases for certain restricted classes of functions we develop methods which require less computation than the fwa these algorithms have been implemented in a constraint based logic programming language examples are given to illustrate the method and its relation to other discrete methods and the exact nonlinear programming approach

a block alternating splitting iteration method for a class of block two by two complex linear systems optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering in this paper we focus on solving a class of block two by two complex linear systems arising from the distributed optimal control with time periodic parabolic equations a new block alternating splitting bas iteration method is presented for solving the class of complex linear systems the convergence theory and the spectral properties of the bas iteration method are discussed numerical experiments are presented to illustrate the efficiency of the bas iteration as a solver as well as a preconditioner for krylov subspace methods

fuzzy differential systems and malliavin calculus abstract we consider fuzzy differential systems or differential equations with fuzzy inputs and fuzzy initial conditions using a fuzzy sample path approach we derive partial differential equations for the resulting membership functions we also demonstrate sufficient continuity conditions under which the solutions exist and are fuzzy convex the membership functions are shown to be identical to those for a time varying deterministic differential equation with a fuzzy initial condition under sufficient stability conditions we also consider the membership functions in steady state

certain isometries on rn abstract let r n be the linear space of all real column vectors with n coordinates given x x 1 x n t r n its p k norm is defined by x p k max x i 1 p x i k p 1 p 1 i 1 k n where k is a positive integer satisfying 1 k n and p satisfies 1 p whereas its c norm is defined by x c max c t px p is a generalized permutation matrix for any nonzero c r n this paper characterizes the isometries for c norms and p k norms some related results are also considered

computation of the output of a function with fuzzy inputs based on a low rank tensor approximation we apply a derivative free optimization method based on novel low rank tensor methods to the problem of propagating fuzzy uncertainty through a continuous real valued function adhering to zadeh s extension principle such a problem can be reformulated as a sequence of optimization problems over nested search spaces the optimization method we use is based on a low rank tensor approximation of the function sampled on a grid and a search for the minimal and maximal entries in this low rank tensor in contrast to classical fuzzy uncertainty propagation methods such as the vertex method and the transformation method the method we propose does not exhibit an inherent exponential scaling for increasing dimension of the search space obviously no derivative free optimization algorithm can exist which shows sub exponential scaling with the dimension for all possible continuous functions the algorithm that we present here however can exploit a specific type of structure and regularity beyond continuity that is often present in real world optimization problems we illustrate this with some high dimensional numerical examples where the presented method clearly outperforms some established derivative free optimization codes

on inexact preconditioners for nonsymmetric matrices inexact versions of the block triangular preconditioners for nonsymmetric matrices of block two by two structures in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 and i c f ipsen siam j sci comput 23 2001 pp 1050 1051 are presented and the two preconditioners for symmetric block two by two matrices in c durazzi and v ruggiero numer linear algebra appl 10 2003 pp 673 688 are extended to general nonsymmetric matrices moreover we precisely describe the spectral properties of the preconditioned matrices and the finite step termination properties of the preconditioned krylov subspace iteration methods with an optimal or galerkin property with respect to these preconditioners several numerical examples are performed to illustrate the effectiveness of the proposed preconditioners

a theory of pseudoskeleton approximations abstract let an m n matrix a be approximated by a rank r matrix with an accuracy e we prove that it is possible to choose r columns and r rows of a forming a so called pseudoskeleton component which approximates a with o e r m n accuracy in the sense of the 2 norm on the way to this estimate we study the interconnection between the volume i e the determinant in the absolute value and the minimal singular value r of r r submatrices of an n r matrix with orthogonal columns we propose a lower bound better than one given by chandrasekaran and ipsen and by hong and pan for the maximum of r over all these submatrices and formulate a hypothesis on a tighter bound

operator norms multiplicativity factors and c numerical radii abstract let v be a normed vector space over c let b v denote the algebra of linear bounded operators on v and let n be an arbitrary seminorm or norm on b v in this paper we discuss multiplicativity factors for n i e constants 0 for which n n is submultiplicative we find that while in the finite dimensional case nontrivial indefinite seminorms have no multiplicativity factors and norms do have multiplicativity factors in the infinite dimensional case n may or may not have such factors our results are then applied in order to compute multiplicativity factors for certain generalizations of the classical numerical radius called c numerical radii this is done with the help of a combinatorial inequality which seems to be of independent interest

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

fuzzy differential equations an approach via fuzzification of the derivative operator abstract in this paper we study fuzzy differential equations fdes in terms of derivative for fuzzy functions in a different way from the traditional hukuhara derivative defined for set valued functions the derivative we use is obtained by means of fuzzification of the classical derivative operator for standard functions we discuss the relation of this approach to fuzzy differential inclusions fdis and hukuhara and strongly generalized derivatives a theorem of existence of a solution is studied with hypothesis similar to those assumed for fdis some examples are explored in order to illustrate the theory

fuzzy weighted averages and implementation of the extension principle abstract this paper addresses the computational aspect of the extension principle when the principle is applied to algebraic mappings and in particular to weighted average operations in risk and decision analysis a computational algorithm based on the cut representation of fuzzy sets and interval analysis is described the method provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner examples are given to illustrate the method and its relation to other discrete methods and the exact approach by non linear programming the algorithm has been implemented in a computer program which can handle very general extended algebraic operations on fuzzy numbers

g invariant norms and g c radii abstract let v be a finite dimensional inner product space over f r or c and let g be a closed subgroup of the group of unitary operators on v a norm or a seminorm on v is said to be g invariant if g x x for all g g x v the concept of g invariant norm specializes to many interesting particular cases such as the absolute norms on f n symmetric gauge functions on r n unitarily invariant norms on f m n etc which are of wide research interest in this paper we study the general properties of g invariant norms our main strategy is to study g invariant norms via the g c radius rg c on v defined by r g c x max x g c g g where c v it is shown that the g c radii are very important g invariant seminorms because every g invariant norm or seminorm admits a representation in terms of them as a result one may focus attention on g c radii in order to get results on g invariant norms we study the norm properties of g c radii and obtain various results relating g invariant norms and g c radii the linear operators on v that preserve g invariant norms g invariant seminorms or g c radii are also investigated several open questions are mentioned

faces of the unit ball of a unitarily invariant norm abstract let be a unitarily invariant norm on the space of all n m complex matrices and let g r n r be the symmetric gauge function associated with that is to say we have a g s a for any a where s a is the nonincreasing sequence of singular values of a in this article we consider the relationship between the facial structures of the closed unit balls of g and which we denote by bg and b our main result gives a complete characterization of the faces of b in terms of the faces of bg for with each face e of b we associate a standard face f e of bg i e f e is a face of bg whose barycenter is a nonnegative nonincreasing n vector conversely each standard face f of bg originates a set of n m matrices say the set m f which is proven to be a face of b our main result also asserts that any face e of b is of the form q m fr where q and r are unitary matrices of appropriate orders and f f e we then explore the main result and draw some of its consequences for example we show that the singular values of the barycenter of a face e of b have the same multiplicities as the barycenter of f e formulas are given which relate the dimensions of e and f e the relative interior and the relative boundary of e are characterized in terms of the relative interior and the relative boundary of f e we completely describe the face of b generated by a matrix a our results also cover the case of real matrices

schur products and matrix completions we prove that a necessary and sufficient condition for a given partially positive matrix to have a positive completion is that a certain schur product map defined on a certain subspace of matrices is a positive map by analyzing the positive elements of this subspace we obtain new proofs of the results of h dym and i gohberg and grone johnson sa and wolkowitz linear algebra appl 58 1984 109 124 linear algebra appl 36 1981 1 24 we also obtain a new proof of a result of u haagerup decomposition of completely bounded maps on operation algebras preprint characterizing the norm of schur product maps and a new hahn banach type extension theorem for these maps finally we obtain generalizations of many of these results to matrices of operators which we apply to the study of representations of certain subalgebras of the n n matrices

preconditioning of complex linear systems from the helmholtz equation abstract in this paper for solving a class of complex linear systems from the helmholtz equation efficiently a new splitting preconditioner is established and a real valued preconditioned iterative method is presented spectral properties of the preconditioned matrix are discussed and bound on the eigenvalues of the preconditioned matrix is given theorem which provides the dimension of the krylov subspace methods for the preconditioned iteration method is obtained the implementation of the preconditioned method is given and the optimal choice of the accelerated parameter is derived in particular a more practical way to choose the accelerated parameter is also proposed numerical experiments arising from the helmholtz equation are used to illustrate the performance of the preconditioner which show the effectiveness and robustness of the new preconditioned gmres method and demonstrate meshsize independent and wave number insensitive convergence behavior

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

towards fuzzy differential calculus part 1 integration of fuzzy mappings abstract this paper deals with fuzzy set valued mappings of the real line and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line these mappings can also be viewed as fuzzy relations using zadeh s extension principle the integral of such fuzzy mappings over a crisp interval is defined provided a special analytical representation of the fuzzy mapping the practical computation of such an integral is shown to be easy practically speaking it yields the fuzzy surface of a fuzzily bounded area

block triangular preconditioners for saddle point problems with a penalty term block triangular preconditioners for a class of saddle point problems with a penalty term are considered an important example is the mixed formulation of the pure displacement problem in linear elasticity it is shown that the spectrum of the preconditioned system is contained in a real positive interval and that the interval bounds can be made independent of the discretization and penalty parameters this fact is used to construct bounds of the convergence rate of the gmres method with respect to an energy norm numerical results are given for gmres and bi cgstab

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

on the higher numerical radius and spectral norm abstract let c n n be the vector space of n n complex matrices and denote by u n the group of n n unitary matrices in c n n for 1 m n set jm im 0m n whereim is the m m identity matrix and 0k denotes a k k zero matrix with this for ain c n n and 1 m n we define r m a max tr j m uau u u n and a m max tr j m uav u v u n as the m numerical radius and m spectral norm of a respectively when m 1 these quantities are reduced to the classical numerical radius and spectral norm in this note we obtain inequalities concerning m and rm for 1 m n examples are given to show that the inequalities obtained cannot be improved we also study multiplicativity factors for rm 1 m n

on augmentation block triangular preconditioners for regularized saddle point problems two augmentation block triangular preconditioners were introduced by shen et al 2012 for the regularized saddle point problem however the spectral analysis of the preconditioner based on the augmentation of the 2 2 block was not throughly derived there in this short paper we give a detailed spectral analysis on this preconditioner a numerical example is employed to validate the efficiency of the presented theoretical results

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

on parameterized inexact uzawa methods for generalized saddle point problems abstract for the large sparse saddle point problems bai et al recently studied a class of parameterized inexact uzawa piu methods z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 for a special case that the 1 1 block is solved exactly they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method in this paper we develop these methods to the large sparse generalized saddle point problems for the obtained parameterized inexact uzawa method we prove its convergence under suitable restrictions on the iteration parameters in particular we determine its quasi optimal iteration parameters and the corresponding quasi optimal convergence factor for the saddle point problems furthermore this piu method is generalized to obtain a framework of accelerated variants of the parameterized inexact uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation matrix relaxation and inexact iteration

iterative solvers for a spectral galerkin approach to elliptic partial differential equations with fuzzy coefficients mathematical models of physical systems often contain parameters with an imprecisely known and uncertain character it is quite common to represent these parameters by means of random variables numerous methods have been developed to compute accurate approximations to solutions of equations with such parameters this approach however may not be entirely justified when the uncertainty is due to vagueness or incomplete knowledge for such cases alternative uncertainty representations using tools from imprecise probability theory have been suggested among those the fuzzy representation is probably most popular in this paper we consider numerical methods for solving partial differential equations with fuzzy coefficients we demonstrate that spectral expansion methods quite common in the random variable approach can also be used effectively for solving fuzzy equations we motivate the use of chebyshev polynomials in the spectral representation and apply a galerkin projection to convert the fuzzy prob

some aspects of the theory of norms abstract let v be a finite dimensional vector space motivated by theory or applications one might want to consider different kinds of norms on v in this paper we discuss some results and problems involving different classes of norms on a vector space studied by this author in the past few years the paper consists of five sections section 1 concerns the conditions on two vectors x y v satisfying x y for all in a certain class of norms section 2 concerns the isometry groups of g invariant norms i e norms that satisfy g x x for all x v and for all g g where g is a group of unitary orthogonal operators on v section 3 concerns g invariant norms that satisfy some special properties section 4 concerns the best approximation s x 0 t of y where y t v with respect to different kinds of norms additional open problems topics and references are mentioned in section 5

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

duality between some linear preserver problems iii c spectral norms and skew symmetric matrices with fixed singular values abstract let f denote either the complex field c or the real field r let v be sn f or kn f the vector spaces of all n n symmetric and skew symmetric matrices respectively over f for c c1 cn 0 with c1 cn 0 the c spectral norm of a matrix a v is the quantity a c i l n c i i a where 1 a n a are the singular values of a let d d1 dn 0 with d1 dn 0 we study the linear isometries between the normed spaces v c and v d by using the fact that they are dual transformations of the linear operators which map d onto c where c x v x has singular values c 1 c n it is shown that such isometries and hence their dual transformations exist if and only if c and d are scalar multiples of each other in such case we completely determine the structure of such isometries and prove that they and their dual transformations belong to a same class of operators in the proof we obtain characterizations of the extreme points of the unit ball in v for different cases with respect to c which is of independent interest

the isometries and the g invariance of certain seminorms abstract let f be the field of real or complex numbers and let g be a subgroup of the general linear group gl n f if a is an m n matrix over f let a be the seminorm on f n defined by x a ax for all x f n in this paper we characterize the linear isometries for the seminorm a and study the conditions on a for which a is g invariant that is sx a x a for all x f n and all s g as a special case we describe all matrices a for which a is absolute or a symmetric gauge function

computing expensive multivariate functions of fuzzy numbers using sparse grids fuzzy arithmetic provides a powerful tool to introduce uncertainty into mathematical models with zadeh s extension principle one can obtain a fuzzy extension of any objective function computing expensive multivariate functions of fuzzy numbers however often poses a difficult problem due to non applicability of common fuzzy arithmetic algorithms severe overestimation or very high computational complexity this paper proposes a new approach based on sparse grids consisting of two parts first we compute a surrogate function using sparse grid interpolation second we perform the fuzzy valued evaluation of the surrogate function by a suitable implementation of the extension principle based on real or interval arithmetic the new approach gives accurate results and requires only few function evaluations

approximate iterations for structured matrices important matrix valued functions f a are e g the inverse a 1 the square root sqrt a and the sign function their evaluation for large matrices arising from pdes is not an easy task and needs techniques exploiting appropriate structures of the matrices a and f a often f a possesses this structure only approximately however intermediate matrices arising during the evaluation may lose the structure of the initial matrix this would make the computations inefficient and even infeasible however the main result of this paper is that an iterative fixed point like process for the evaluation of f a can be transformed under certain general assumptions into another process which preserves the convergence rate and benefits from the underlying structure it is shown how this result applies to matrices in a tensor format with a bounded tensor rank and to the structure of the hierarchical matrix technique we demonstrate our results by verifying all requirements in the case of the iterative computation of a 1 and sqrt a

zadeh s extension principle for continuous functions of non interactive variables a parallel optimization approach there is a growing interest in the use of fuzzy intervals in many engineering applications however a direct implementation of zadeh s extension principle which forms the basis for computing with fuzzy intervals is still computationally too demanding for practical use in the case of a continuous function and fuzzy intervals that describe non interactive variables as inputs the output is a fuzzy interval as well and can be determined for each cut separately the problem thus reduces to finding the endpoints of these cuts which amounts to a number of interwoven optimization problems in the case of a non monotone continuous function however these optimization problems are non trivial in this paper different optimization algorithms are applied for that purpose gradient descent based on sequential quadratic programming simplex simulated annealing particle swarm optimization and particle swarm optimization combined with gradient descent in addition two approaches are followed to determine a suitable number of cuts either a fixed predetermined number is used or an initially very small number is chosen that is subsequently increased according to a linearity criterion both a non parallel and a parallel implementation are designed the parallel version is restricted to work with particle swarm optimization and employs communication to optimize its internal performance by exploiting the dependence between the various optimization problems different configurations are evaluated on a set of benchmark functions in terms of the mean area under the output fuzzy interval and the number of function evaluations particle swarm optimization combined with gradient descent starting from a small number of cuts leads to the most accurate fuzzy intervals at the cost of a relatively large number of function evaluations

assessment of uncertainty on structural dynamic responses with the short transformation method abstract the predictive capability of finite element models is limited by their deterministic nature typically not all model parameters are exactly known while even small deviations may have significant effects on the predicted response parameter uncertainty should therefore be taken into account e g with fuzzy arithmetic the absence of fuzzy solvers led to interval arithmetic as a numerical alternative the transformation method tm presented by m hanss replaces interval arithmetic with a set of deterministic computations for each interval all parameter extrema are combined in every possible way in a design of experiments terminology the tm is a so called full factorial design the tm is applicable if the output is monotonic in the inputs unlike interval arithmetic it does not overestimate the response uncertainty as only parameter combinations are evaluated that actually occur in this paper the tm has been applied to visualise uncertain frequency response functions frfs obtained with modal superposition this yields accurate results when validated against monte carlo data but the computation time is rather high the short transformation method stm is proposed as an attractive alternative to the original tm a full set of deterministic computations combining all interval extrema is only performed at the lowest interval for higher levels a smaller set is evaluated this allows reconstructing the fuzzy frf from a much lower number of deterministic computations with only a small reduction in the accuracy of frfs both methods are demonstrated on a clamped plate and a car front cradle with uncertain design parameters

a shifted nested splitting iterative method with applications to ill posed problems and image restoration we present a shifted nested iterative method for solving systems of linear equations with a coefficient matrix that contains a dominant skew hermitian part this new scheme is practically the inner outer iterations which employs the cgnr method as inner iteration to approximate each outer iterate while each outer iteration is induced by a convergent splitting of the coefficient matrix convergence properties of the new scheme are studied in depth and possible choices of the shift parameter are discussed moreover an adapted version of the method is used for ill posed problems and image restoration at the last numerical examples are used to further examine the effectiveness and robustness of the new method

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

faces and traces of the unit ball of a symmetric gauge function abstract we investigate the faces of the closed unit ball bg x g x 1 of a symmetric gauge function g r n r firstly we describe the group of symmetries of a face f of bg in terms of its barycenter we introduce the concept of standard face of bg i e a face whose relative interior meets the set d x r n x 1 x n 0 called here the standard cone of r n and the concept of trace of a face f which is the intersection of f with d characterizations of standard faces are given and we show that the trace of a face if nonempty is the trace of a standard face it is proved that a face f and its trace have the same dimension iff f is a standard face a relation is found between the dimension of f and the dimension of its symmetry core i e the set of points of f that are fixed by all symmetries of f other results of this type are given the main result has a long and delicate proof it completely characterizes in several different ways the subsets of bg that are traces of faces of bg

on inexact hermitian and skew hermitian splitting methods for non hermitian positive definite linear systems we study theoretical properties of two inexact hermitian skew hermitian splitting ihss iteration methods for the large sparse non hermitian positive definite system of linear equations in the inner iteration processes we employ the conjugate gradient cg method to solve the linear systems associated with the hermitian part and the lanczos or conjugate gradient for normal equations cgne method to solve the linear systems associated with the skew hermitian part respectively resulting in ihss cg lanczos and ihss cg cgne iteration methods correspondingly theoretical analyses show that both ihss cg lanczos and ihss cg cgne converge unconditionally to the exact solution of the non hermitian positive definite linear system moreover their contraction factors and asymptotic convergence rates are dominantly dependent on the spectrum of the hermitian part but are less dependent on the spectrum of the skew hermitian part and are independent of the eigenvectors of the matrices involved optimal choices of the inner iteration steps in the ihss cg lanczos and ihss cg cgne iterations are discussed in detail by considering both global convergence speed and overall computation workload and computational efficiencies of both inexact iterations are analyzed and compared deliberately

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

an efficient algorithm for fuzzy weighted average in multisensor intelligent systems the information fusion plays an important role several algorithms have been proposed for the purpose of aggregating imprecise sensory information represented by fuzzy numbers this paper proposes an efficient algorithm to compute fuzzy weighted average which turned out to be superior to the previous works by reducing the number of comparisons and arithmetic operations to o n log n

weighted toeplitz regularized least squares computation for image restoration the main aim of this paper is to develop a fast algorithm for solving weighted toeplitz regularized least squares problems arising from image restoration based on augmented system formulation we develop new hermitian and skew hermitian splitting hss preconditioners for solving such linear systems the advantage of the proposed preconditioner is that the blurring matrix weighting matrix and regularization matrix can be decoupled such that the resulting preconditioner is not expensive to use we show that for a preconditioned system that is derived from a saddle point structure of size m n times m n the preconditioned matrix has an eigenvalue at 1 with multiplicity n and the other m eigenvalues of the form 1 lambda with lambda 1 we also study how to choose the hss parameter to minimize the magnitude of lambda and therefore the krylov subspace method applied to solving the preconditioned system converges very quickly experimental results for image restoration problems are re

a generalization of spectral radius numerical radius and spectral norm abstract let c n be the linear space of complex column vectors with n coordinates associated with the usual inner product denote by k the set of all k tuples of orthonormal vectors in c n given a real number p 1and an n n complex matrix a with eigenvalues 1 n we define p k max j 1 k i j p 1 p 1 i 1 k n r p k max i 1 k x i ax i p 1 p x 1 x k k and a p k max i 1 k x i ay i p 1 p x 1 x k y 1 y k k as the p k spectral radius p k numerical radius and p k spectral norm of a respectively for k 1 they are the classical spectral radius numerical radius and spectral norm of a in this note we study the matrices for which p k a rp k a p k a a p k or rp k a a p k moreover the norm properties of rp k and p k are investigated

motivations and realizations of krylov subspace methods for large sparse linear systems we briefly introduce typical and important direct and iterative methods for solving systems of linear equations concretely describe their fundamental characteristics in viewpoints of both theory and applications and clearly clarify the substantial differences among these methods in particular the motivations of searching the solution of a linear system in a krylov subspace are described and the algorithmic realizations of the generalized minimal residual gmres method are shown and several classes of state of the art algebraic preconditioners are briefly reviewed all this is useful for correctly deeply and completely understanding the application scopes theoretical properties and numerical behaviors of these methods and is also helpful in designing new methods for solving systems of linear equations

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

new variant of the hss iteration method for weighted toeplitz regularized least squares problems from image restoration abstract for fast solving weighted toeplitz least squares problems from image restoration ng and pan 2014 studied a new hermitian and skew hermitian splitting nhss preconditioner in this paper a generalization of the nhss preconditioner and the corresponding iterative method are presented convergence for the new iteration method is studied and optimal choice of the parameters is discussed bounds on the eigenvalues and the corresponding eigenvector distributions are proposed the degree of the minimal polynomial of the preconditioned matrix is obtained numerical experiments arising from image restoration are provided which show the proposed iteration method is effective and confirm our theoretical results are correct

black box approximation of tensors in hierarchical tucker format we derive and analyse a scheme for the approximation of order d tensors a rn1 nd in the hierarchical h tucker format a dimension multilevel variant of the tucker format and strongly related to the tt tensor train format for a fixed rank parameter k the storage complexity of a tensor in h tucker format is odk3 k i 1dni and we present a heuristic algorithm that finds an approximation to a tensor in the h tucker format in odk4 log d k2 i 1dni by inspection of only odk3 log d k2 i 1dni entries under mild assumptions tensors in the h tucker format are reconstructed for general tensors we derive error bounds that are based on the approximability of matrices matricizations of the tensor by few outer products of its rows and columns the construction parallelizes with respect to the order d and we also propose an adaptive approach that aims at finding the rank parameter for a given target accuracy e automatically

a generalization of the inexact parameterized uzawa methods for saddle point problems abstract for large sparse saddle point problems bai and wang recently studied a class of parameterized inexact uzawa methods see z z bai z q wang on paramaterized inexact uzawa methods for generalized saddle point problems linear algebra appl 428 2008 2900 2932 in this paper we generalize these methods and propose a class of generalized inexact parameterized iterative schemes for solving the saddle point problems we derive conditions for guaranteeing the convergence of these iterative methods with different choices of the parameter matrices the generalized iterative methods lead to a series of existing and new iterative methods including the classical uzawa method the inexact uzawa method the gsor method and the giaor method

the transformation method for the simulation and analysis of systems with uncertain parameters in this paper the transformation method is introduced as a powerful approach for both the simulation and the analysis of systems with uncertain model parameters based on the concept of cuts the method represents a special implementation of fuzzy arithmetic that avoids the well known effect of overestimation which usually arises when fuzzy arithmetic is reduced to interval computation systems with uncertain model parameters can thus be simulated without any artificial widening of the simulation results as a by product of the implementation scheme the transformation method also provides a measure of influence to quantitatively analyze the uncertain system with respect to the effect of each uncertain model parameter on the overall uncertainty of the model output by this a special kind of sensitivity analysis can be defined on the basis of fuzzy arithmetic finally to show the efficiency of the transformation method the method is applied to the simulation and analysis of a model for the friction interface between the sliding surfaces of a bolted joint connection