a generalized variant of simplified hss preconditioner for generalized saddle point problems abstract based on the simplified hermitian and skew hermitian splitting shss preconditioner developed by cao ren and shi bit numer math 2016 a generalized variant of the shss gshss preconditioner is presented for solving generalized saddle point problems the convergence of the gshss iteration method and the spectral properties of the preconditioned matrix are studied in details eigenvalue bounds and the degree of minimal polynomials of the preconditioned matrix are obtained in particular a practical and feasible choice strategy of iteration parameter is given which also provides a feasible way to select the accelerated parameters for some other preconditioners numerical experiments are presented which show the effectiveness of the proposed preconditioner and the feasibility of the practical choice strategy of accelerated parameter

stability estimates and structural spectral properties of saddle point problems for a general class of saddle point problems sharp estimates for babuska s inf sup stability constants are derived in terms of the constants in brezzi s theory in the finite dimensional hermitian case more detailed spectral properties of preconditioned saddle point matrices are presented which are helpful for the convergence analysis of common krylov subspace methods the theoretical results are applied to two model problems from optimal control with time periodic state equations numerical experiments with the preconditioned minimal residual method are reported

a relaxed deteriorated pss preconditioner for nonsymmetric saddle point problems from the steady navier stokes equation for nonsymmetric saddle point problems arising from the steady navier stokes equations pan ng and bai presented a deteriorated positive definite and skew hermitian splitting dpss preconditioner pan et al 2006 to accelerate the convergence rates of the krylov subspace iteration methods such as gmres in this paper the unconditional convergence property of the dpss iteration method is proved and a relaxed dpss rdpss preconditioner is proposed the rdpss preconditioner is much closer to the coefficient matrix than the dpss preconditioner in certain norm which straightforwardly results in an rdpss iteration method the convergence conditions of the rdpss iteration are analyzed and the optimal parameter which minimizes the spectral radius of the rdpss iteration matrix is derived using the rdpss preconditioner to accelerate some krylov subspace methods like gmres is also studied the eigenproperty of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained finally numerical experiments of a model navier stokes equation are presented to illustrate the efficiency of the rdpss preconditioner

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

on one sided ideals of rings of continuous linear operators abstract let x be a real or complex locally convex vector space and l c x denote the ring in fact the algebra of continuous linear operators on x in this note we characterize certain one sided ideals of the ring l c x in terms of their rank one idempotents we use our main result to show that a one sided ideal of the ring of continuous linear operators on a real or complex locally convex space is triangularizable if and only if the one sided ideal is generated by a rank one idempotent if and only if rank ab ba 1 for all a b in the one sided ideal also a description of irreducible one sided ideals of the ring l c x in terms of their images or coimages will be given the counterparts of some of these results hold true for one sided ideals of the ring of all right resp left linear transformations on a right resp left vector space over a general division ring

generalized space time fractional diffusion equation with composite fractional time derivative abstract we investigate the solution of space time fractional diffusion equations with a generalized riemann liouville time fractional derivative and riesz feller space fractional derivative the laplace and fourier transform methods are applied to solve the proposed fractional diffusion equation the results are represented by using the mittag leffler functions and the fox h function special cases of the initial and boundary conditions are considered numerical scheme and grunwald letnikov approximation are also used to solve the space time fractional diffusion equation the fractional moments of the fundamental solution of the considered space time fractional diffusion equation are obtained many known results are special cases of those obtained in this paper we investigate also the solution of a space time fractional diffusion equations with a singular term of the form x t 1 0

levy walks in random scattering and growth of waves random spatial wave scattering and stochastic wave growth are studied where one or both of the random processes can be described by a levy walk this analysis extends previous work on randomly growing and scattering waves where both the random processes are modeled by gaussian diffusive statistics both random spatial scattering and stochastic wave growth modeled by levy walks are studied separately together and in combination with gaussian processes transmission coefficients lasing thresholds and energy densities in the medium are obtained for the different permutations

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

derivations and right ideals of algebras abstract let r be a k algebra acting densely on v d where k is a commutative ring with unity and v is a right vector space over a division k algebra d let be a nonzero right ideal of r and let f x 1 x t be a nonzero polynomial over k with constant term 0 such that r 0 for some coefficient of f x 1 x t suppose that d r r is a nonzero derivation it is proved that if rank d f x 1 x t m for all x 1 x t and for some positive integer m then either is generated by an idempotent of finite rank or d ad b for some b end v d of finite rank in addition if f x 1 x t is multilinear then b can be chosen such that rank b 2 6 t 13 m 2

optimization of random searches on defective lattice networks we study the general problem of how to search efficiently for targets randomly located on defective lattice networks i e regular lattices which have some fraction of its nodes randomly removed we consider large but finite triangular lattices and assume for the search dynamics that the walker chooses steps lengths lj from the power law distribution p lj approximately lj mu with the exponent mu regulating the strategy of the search process at each step lj the searcher moves in straight lines and constantly looks within a detection radius of vision rv for the targets along the way if there is contact with a defect the movement stops and a new step length is chosen hence the presence of defects decreases the efficiency of the overall process we study numerically how three different aspects of the lattice influence the optimization of the search efficiency i the type of boundary conditions ii the concentration of targets and defects and iii the category or class of search destructive nondestructive or regenerative motivated by the results we develop a type of mean field model for the problem and obtain an analytical approximation for the search efficiency function finally we discuss in the context of searches how defective lattices compare with perfect lattices and with continuous environments

right ideals generated by an idempotent of finite rank abstract let r be a k algebra acting densely on v d where k is a commutative ring with unity and v is a right vector space over a division k algebra d let f x 1 x t be an arbitrary and fixed polynomial over k in noncommuting indeterminates x 1 x t with constant term 0 such that r 0 for some k occurring in the coefficients of f x 1 x t it is proved that a right ideal of r is generated by an idempotent of finite rank if and only if the rank of f x 1 x t is bounded above by a same natural number for all x 1 x t in this case the rank of the idempotent that generates is also explicitly given the results are then applied to considering the triangularization of and the irreducibility of f where f denotes the additive subgroup of r generated by the elements f x 1 x t for x 1 x t

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

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

preconditioned hss methods for the solution of non hermitian positive definite linear systems and applications to the discrete convection diffusion equation we study the role of preconditioning strategies recently developed for coercive problems in connection with a two step iterative method based on the hermitian skew hermitian splitting hss of the coefficient matrix proposed by bai golub and ng for the solution of nonsymmetric linear systems whose real part is coercive as a model problem we consider finite differences fd matrix sequences an a p n discretizing the elliptic convection diffusion problem with being a plurirectangle of rd with a x being a uniformly positive function and p x denoting the reynolds function here for plurirectangle we mean a connected union of rectangles in d dimensions with edges parallel to the axes more precisely in connection with preconditioned hss gmres like methods we consider the preconditioning sequence pn a n pn a dn1 2 a an 1 0 dn1 2 a where dn a is the suitably scaled main diagonal of an a 0 if a x is positive and regular enough then the preconditioned sequence shows a strong clustering at unity so that the sequence pn a n turns out to be a superlinear preconditioning sequence for an a 0 n where an a 0 represents a good approximation of re an a p namely the real part of an a p the computational interest is due to the fact that the preconditioned hss method has a convergence behavior depending on the spectral properties of pn 1 a re an a p n pn 1 a an a 0 n therefore the solution of a linear system with coefficient matrix an a p is reduced to computations involving diagonals and to the use of fast poisson solvers for an 1 0 n some numerical experimentations confirm the optimality of the discussed proposal and its superiority with respect to existing techniques

spectral analysis of inexact constraint preconditioning for symmetric saddle point matrices abstract large symmetric linear systems in saddle point form arise in many scientific and engineering applications their efficient solution by means of iterative methods mainly relies on exploiting the matrix structure constraint preconditioners are among the most successful structure oriented preconditioning strategies especially when dealing with optimization problems in this paper we provide a full spectral characterization of the constraint based preconditioned matrix by means of the weyr canonical form we also derive estimates for the spectrum when so called inexact variants are used numerical experiments confirm our findings and illustrate that these theoretical results can be helpful in analyzing matrices stemming from real applications

a practical formula for computing optimal parameters in the hss iteration methods in the hss iteration methods proposed by bai golub and ng z z bai g h golub m k ng hermitian and skew hermitian splitting methods for non hermitian positive definite linear systems siam j matrix anal appl 24 2003 603 626 the determination of the optimal parameter is a tough task when solving a non hermitian positive definite linear system in this paper a new and simple strategy for obtaining the optimal parameter is proposed which computes the optimal parameter by solving a cubic polynomial equation the coefficients of this polynomial are determined by several traces of some matrices related to the symmetric and skew symmetric parts of the coefficient matrix of the real linear system numerical experiments show that our new strategy is very effective for approximating the optimal parameter in the hss iteration methods as it leads to fast convergence of the method

generalized shift splitting preconditioners for nonsingular and singular generalized saddle point problems based on the shift splitting technique a class of generalized shift splitting preconditioners are proposed for both nonsingular and singular generalized saddle point problems the generalized shift splitting preconditioner is induced by a generalized shift splitting of the generalized saddle point matrix resulting in a generalized shift splitting fixed point iteration theoretical analyses show that the generalized shift splitting iteration method is convergent and semi convergent unconditionally for solving the nonsingular and the singular generalized saddle point problems respectively numerical experiments of a model navier stokes problem are implemented to demonstrate the feasibility and effectiveness of the proposed preconditioners

the influence of the environment on levy random search efficiency fractality and memory effects an open problem in the field of random searches relates to optimizing the search efficiency in fractal environments here we address this issue through a systematic study of levy searches in landscapes encompassing several degrees of target aggregation and fractality for scarce resources non destructive searches with unrestricted revisits to targets are shown to present universal optimal behavior irrespective of the general scaling properties of the spatial distribution of targets in contrast no such universal behavior occurs in the destructive case with forbidden revisits in which the optimal strategy strongly depends on the degree of target aggregation we also investigate how the presence of memory and learning skills of the searcher affect the search efficiency by considering a limiting model in which the searcher learns through recent experience to recognize food rich areas we find that a statistical memory of previous encounters does not necessarily increase the rate of target findings in random searches instead there is an optimal extent of memory dependent on specific details of the search space and stochastic dynamics which maximizes the search efficiency this finding suggests a more general result namely that in some instances there are actual advantages to ignoring certain pieces of partial information while searching for objects

first passage time moments of asymmetric levy flights we investigate the first passage dynamics of symmetric and asymmetric l evy flights in a semi infinite and bounded intervals by solving the space fractional diffusion equation we analyse the fractional order moments of the first passage time probability density function for different values of the index of stability and the skewness parameter a comparison with results using the langevin approach to l evy flights is presented for the semi infinite domain in certain special cases analytic results are derived explicitly and in bounded intervals a general analytical expression for the mean first passage time of l evy flights with arbitrary skewness is presented these results are complemented with extensive numerical analyses

levy flights in an infinite potential well as a hypersingular fredholm problem we study levy flights with arbitrary index 0 2 inside a potential well of infinite depth such a problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in disordered quantum systems the major technical tool is a transformation of the eigenvalue problem for initial fractional schrodinger equation into that for fredholm integral equation with hypersingular kernel the latter equation is then solved by means of expansion over the complete set of orthogonal functions in the domain d reducing the problem to the spectrum of a matrix of infinite dimensions the eigenvalues and eigenfunctions are then obtained numerically with some analytical results regarding the structure of the spectrum

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

optimization of random searches on regular lattices we investigate random searches on isotropic and topologically regular square and triangular lattices with periodic boundary conditions and study the efficiency of search strategies based on a power law distribution p ensuremath ell ensuremath sim ensuremath ell ensuremath ensuremath mu of step lengths ensuremath ell we consider both destructive searches in which a target can be visited only once and nondestructive searches when a target site is always available for future visits we discuss i the dependence of the search efficiency on the choice of the lattice topology ii the relevance of the periodic boundary conditions iii the behavior of the optimal power law exponent ensuremath mu mathrm opt as a function of target site density iv the differences between destructive and nondestructive environments and finally v how the results for the discrete searches differ from the continuous cases previously studied

levy flight random searches in biological phenomena abstract there has been growing interest in the study of levy flights observed in the movements of biological organisms performing random walks while searching for other organisms here we approach the problem of what is the best statistical strategy for optimizing the encounter rate between searcher and target organisms either of the same or of different species in terms of a limiting generalized searcher target model e g predator prey mating partner pollinator flower in this context we discuss known results showing that for fixed targets an inverse square density distribution of step lengths can optimize the encounter rate for moving targets we review how the encounter rate depends on whether organisms move in levy or brownian random walks we discuss recent findings indicating that levy walks confer a significant advantage for increasing encounter rates only when the searcher is larger or moves rapidly relative to the target and when the target density is low

shift splitting preconditioners for saddle point problems in this paper we first present a shift splitting preconditioner for saddle point problems the preconditioner is based on a shift splitting of the saddle point matrix resulting in an unconditional convergent fixed point iteration based on the idea of the splitting we further propose a local shift splitting preconditioner some properties of the local shift splitting preconditioned matrix are studied these preconditioners extend those studied by bai yin and su for solving non hermitian positive definite linear systems bai et al 2006 finally numerical experiments of a model stokes problem are presented to show the effectiveness of the proposed preconditioners

optimal parameter in hermitian and skew hermitian splitting method for certain two by two block matrices the optimal parameter of the hermitian skew hermitian splitting hss iteration method for a real two by two linear system is obtained the result is used to determine the optimal parameters for linear systems associated with certain two by two block matrices and to estimate the optimal parameters of the hss iteration method for linear systems with n by n real coefficient matrices numerical examples are given to illustrate the results

levy flights and random searches in this work we discuss some recent contributions to the random search problem our analysis includes superdiffusive levy processes and correlated random walks in several regimes of target site density mobility and revisitability we present results in the context of mean field like and closed form average calculations as well as numerical simulations we then consider random searches performed in regular lattices and lattices with defects and we discuss a necessary criterion for distinguishing true superdiffusion from correlated random walk processes we invoke energy considerations in relation to critical survival states on the edge of extinction and we analyze the emergence of levy behavior in deterministic search walks finally we comment on the random search problem in the context of biological foraging

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

average time spent by levy flights and walks on an interval with absorbing boundaries we consider a levy flyer of order alpha that starts from a point x 0 on an interval o l with absorbing boundaries we find a closed form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed these two quantities are equivalent to the mean first passage times for levy flights and levy walks respectively using fractional differential equations with a riesz kernel we find exact analytical expressions for both quantities in the continuous limit we show that numerical solutions for the discrete levy processes converge to the continuous approximations in all cases except the case of alpha 2 and the cases of x 0 0 and x 0 l for alpha 2 when the second moment of the flight length distribution exists our result is replaced by known results of classical diffusion we show that if x 0 is placed in the vicinity of absorbing boundaries the average total length has a minimum at alpha 1 corresponding to the cauchy distribution we discuss the relevance of this result to the problem of foraging which has received recent attention in the statistical physics literature

the spectrum of large powers of the laplacian in bounded domains we present exact results for the spectrum of the nth power of the laplacian in a bounded domain we begin with the one dimensional case and show that the whole spectrum can be obtained in the limit of large n we also show that it is a useful numerical approach valid for any n finally we discuss implications of this work and present its possible extensions for non integer n and for 3d laplacian problems

continuous time multidimensional markovian description of levy walks the paper presents a multidimensional model for nonlinear markovian random walks that generalizes the one we developed previously i lubashevsky r friedrich and a heuer phys rev e 79 011110 2009 in order to describe the levy type stochastic processes in terms of continuous trajectories of walker motion this approach may open a way to treat levy flights or levy random walks in inhomogeneous media or systems with boundaries in the future the proposed model assumes the velocity of a wandering particle to be affected by a linear friction and a nonlinear langevin force whose intensity is proportional to the magnitude of the velocity for its large values based on the singular perturbation technique the corresponding fokker planck equation is analyzed and the relationship between the system parameters and the levy exponent is found following actually the previous paper we demonstrate also that anomalously long displacements of the wandering particle are caused by extremely large fluctuations in the particle velocity whose duration is determined by the system parameters rather than the duration of the observation interval in this way we overcome the problem of ascribing to levy random walk non markov properties