on the global krylov subspace methods for solving general coupled matrix equations in the present paper we propose the global full orthogonalization method gl fom and global generalized minimum residual gl gmres method for solving large and sparse general coupled matrix equations j 1pa i jx jb i j c i i 1 p where a i j r m x m b i j r n x n c i r m x n i j 1 2 p are given matrices and x i r m x n i 1 2 p are the unknown matrices to do so first a new inner product and its corresponding matrix norm are defined then using a linear operator equation and new matrix product we demonstrate how to employ gl fom and gl gmres algorithms for solving general coupled matrix equations finally some numerical experiments are given to illustrate the validity and applicability of the results obtained in this work

eigenvalue generalized eigenvector assignment by output feedback this note generalizes the previous results of the closed loop eigenstructure assignment via output feedback in linear multivariable systems necessary and sufficient conditions for the closed loop eigenstructure assignment by output feedback are presented some known results on entire eigenstructure assignment are deduced from this result

consistency for bi skew symmetric solutions to systems of generalized sylvester equations over a finite central algebra let be a finite dimensional central algebra with an involutorial antiautomorphism and char 2 n n be the set of all n n matrices over a aij n n is called bisymmetric if aij an i 1 n j 1 aji and biskewsymmetric if aij an i 1 n j 1 aji the following systems of generalized sylvester equations over a1x yb1 c1 i asx ybs cs a1xb1 c1xd1 e1 ii asxbs csxds es are considered necessary and sufficient conditions are given for the existence of constant solutions with bi skew symmetric constrains to i and ii as a particular case auxiliary results dealing with the system of sylvester equations are also presented

computation of discontinuous optical flow by domain decomposition and shape optimization a well known method for the reconstruction of motion fields from noisy image data is to determine flow fields by the minimization of a quadratic functional the first approach of this class has been proposed by horn and schunck 1981 a drawback of such approaches is that an explicit representation of the discontinuities of the motion field is lacking and that in general the resulting flow fields approximate the motion fields only badly at the corresponding locations in the image plane in this article we discuss the possibility to improve the results by hypothesizing the qualitative structure of the motion field in terms of certain parameters we decompose the image plane into disjoint sets restrict the domain of definition of the functionals to these sets and use the hypotheses to deform and to move the boundaries of the sets within the image plane we discuss the range of applicability of this new technique and illustrate the algorithm by numerical examples this article is a revised and extended version of schnorr 1990

gradient based iterative solutions for general linear matrix equations in this paper we present a gradient based iterative algorithm for solving general linear matrix equations by extending the jacobi iteration and by applying the hierarchical identification principle convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices two numerical examples are provided to show that the proposed algorithm is effective

iterative algorithms for solving the matrix equation axb cx td e in this paper we propose two iterative algorithms to solve the matrix equation axb cx td e the first algorithm is applied when the matrix equation is consistent in this case for any special initial matrix x 1 a solution the minimal frobenius norm solution can be obtained within finite iteration steps in the absence of roundoff errors the second algorithm is applied when the matrix equation is inconsistent in this case for any special initial matrix x 1 a least squares solution the minimal frobenius norm least squares solution can be obtained within finite iteration steps in the absence of roundoff errors some examples verify the efficiency of these algorithms

iterative solutions of the generalized sylvester matrix equations by using the hierarchical identification principle abstract in this paper by extending the well known jacobi and gauss seidel iterations for ax b we study iterative solutions of matrix equations axb f and generalized sylvester matrix equations axb cxd f including the sylvester equation ax xb f as a special case and present a gradient based and a least squares based iterative algorithms for the solution it is proved that the iterative solution always converges to the exact solution for any initial values the basic idea is to regard the unknown matrix x to be solved as the parameters of a system to be identified and to obtain the iterative solutions by applying the hierarchical identification principle finally we test the algorithms and show their effectiveness using a numerical example

the general coupled matrix equations over generalized bisymmetric matrices abstract in the present paper by extending the idea of conjugate gradient cg method we construct an iterative method to solve the general coupled matrix equations j 1 p a ij x j b ij m i i 1 2 p including the generalized coupled lyapunov and sylvester matrix equations as special cases over generalized bisymmetric matrix group x 1 x 2 x p by using the iterative method the solvability of the general coupled matrix equations over generalized bisymmetric matrix group can be determined in the absence of roundoff errors when the general coupled matrix equations are consistent over generalized bisymmetric matrices a generalized bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors the least frobenius norm generalized bisymmetric solution group of the general coupled matrix equations can be derived when an appropriate initial iterative matrix group is chosen in addition the optimal approximation generalized bisymmetric solution group to a given matrix group x 1 x 2 x p in frobenius norm can be obtained by finding the least frobenius norm generalized bisymmetric solution group of new general coupled matrix equations the numerical results indicate that the iterative method works quite well in practice

an iterative algorithm for the least frobenius norm least squares solution of a class of generalized coupled sylvester transpose linear matrix equations abstract the iterative algorithm of a class of generalized coupled sylvester transpose matrix equations is presented we prove that if the system is consistent a solution can be obtained within finite iterative steps in the absence of round off errors for any initial matrices if the system is inconsistent the least squares solution can be obtained within finite iterative steps in the absence of round off errors furthermore we provide a method for choosing the initial matrices to obtain the least frobenius norm least squares solution of the problem finally numerical examples are presented to demonstrate that the algorithm is efficient

on iterative solutions of general coupled matrix equations in this paper we study coupled matrix equations which are encountered in many systems and control applications first we extend the well known jacobi and gauss seidel iterations and present a large family of iterative methods which are then applied to develop iterative solutions to coupled sylvester matrix equations the basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified and to obtain the iterative solutions by applying a hierarchical identification principle next we generalize the sylvester equations to general coupled matrix equations and propose a gradient based iterative algorithm for the solutions using a block matrix inner product the star star product we prove that the iterative algorithm always converges to the unique solutions for any initial values one advantage of the algorithms proposed is that they require less storage space in implementation than existing numerical methods finally we test the algorithms and show their effectiveness using numerical examples

on global hessenberg based methods for solving sylvester matrix equations abstract in the first part of this paper we investigate the use of hessenberg based methods for solving the sylvester matrix equation a x x b c to achieve this goal the sylvester form of the global generalized hessenberg process is presented using this process different methods based on a petrov galerkin or on a minimal norm condition are derived in the second part we focus on the sgl cmrh method which is based on the sylvester form of the hessenberg process with pivoting strategy combined with a minimal norm condition in order to accelerate the sgl cmrh method a preconditioned framework of this method is also considered it includes both fixed and flexible variants of the sgl cmrh method moreover the connection between the flexible preconditioned sgl cmrh method and the fixed one is studied and some upper bounds for the residual norm are obtained in particular application of the obtained theoretical results is investigated for the special case of solving linear systems of equations with several right hand sides finally some numerical experiments are given in order to evaluate the effectiveness of the proposed methods

on the generalized sylvester mapping and matrix equations general parametric solution to a family of generalized sylvester matrix equations arising in linear system theory is presented by using the so called generalized sylvester mapping which has some elegant properties the solution consists of some polynomial matrices satisfying certain conditions and a parametric matrix representing the degree of freedom in the solution the results provide great convenience to the computation and analysis of the solutions to this family of equations and can perform important functions in many analysis and design problems in linear system theory it is also expected that this so called generalized sylvester mapping tool may have some other applications in control system theory

technical communique complete parametric approach for eigenstructure assignment in a class of second order linear systems this note deals with eigenstructure assignment in the second order linear system email protected email protected cq bu using the proportional plus derivative feedback controller u k 0q k email protected under the controllability condition of the matrix pair ab very simple general and complete parametric expressions in direct closed forms for both the closed loop eigenvector matrix and the feedback gains are established in terms of the closed loop eigenvalues and a group of parameter vectors the main computations are two sets of elementary matrix transformations which can be replaced by a series of singular value decompositions when the closed loop eigenvalues are chosen a priori the approach utilizes directly the original system data a b and c and involves manipulations on only n dimensional matrices an example illustrates the effect of the proposed approach

hydrodynamics beyond navier stokes the slip flow model recently analytical solutions for the nonlinear couette flow demonstrated the relevance of the lattice boltzmann lb models to hydrodynamics beyond the continuum limit s ansumali et al phys rev lett 98 124502 2007 in this paper we present a systematic study of the simplest lb kinetic equation the nine bit model in two dimensions in order to quantify it as a slip flow approximation details of the aforementioned analytical solution are presented and results are extended to include a general shear and force driven unidirectional flow in confined geometry exact solutions for the velocity as well as for pertinent higher order moments of the distribution functions are obtained in both couette and poiseuille steady state flows for all values of rarefaction parameter knudsen number results are compared with the slip flow solution by cercignani and a good quantitative agreement is found for both flow situations thus the standard nine bit lb model is characterized as a valid and self consistent slip flow model for simulations beyond the navier stokes approximation

lattice boltzmann models based on half range gauss hermite quadratures we discuss general features of thermal lattice boltzmann models based on half range gauss quadratures specialising to the half range gauss hermite and gauss laguerre cases the main focus of the paper is on the construction of high order half range hermite lattice boltzmann hhlb models the performance of the hhlb models is compared with that of laguerre lattice boltzmann llb and full range hermite lattice boltzmann hlb models by conducting convergence tests with respect to the quadrature order on stationary profiles of the particle number density macroscopic velocity temperature and heat fluxes in the two dimensional couette flow the bhatnagar gross krook bgk collision term is used throughout the paper to reduce the computational costs of the numerical simulations we use mixed lattice boltzmann models constructed using different quadrature methods on each cartesian axis for kn 0 01 the hlb models require the least number of velocities to satisfy our convergence test when kn 0 05 the hlb models are outperformed in terms of number of velocities employed by both the llb and the hhlb models moreover we find that the hhlb models require less quadrature points than the llb models at all tested values of kn which we attribute to the maxwellian form of the weight function for the half range hermite polynomials

eigenstructure assignment in descriptor systems coordinate free conditions are given for pole assignment by feedback in linear descriptor singular systems which guarantee closed loop regularity these conditions are shown to be both necessary and sufficient for assignment of the maximum possible number of finite poles transformation to special coordinates are not used and the results provide a robust algorithm for the computation of the required feedback

dense estimation and object based segmentation of the optical flow with robust techniques we address the issue of recovering and segmenting the apparent velocity field in sequences of images as for motion estimation we minimize an objective function involving two robust terms the first one cautiously captures the optical flow constraint while the second a priori term incorporates a discontinuity preserving smoothness constraint to cope with the nonconvex minimization problem thus defined we design an efficient deterministic multigrid procedure it converges fast toward estimates of good quality while revealing the large discontinuity structures of flow fields we then propose an extension of the model by attaching to it a flexible object based segmentation device based on deformable closed curves different families of curve equipped with different kinds of prior can be easily supported experimental results on synthetic and natural sequences are presented including an analysis of sensitivity to parameter tuning

consistent lattice boltzmann modeling of low speed isothermal flows at finite knudsen numbers in slip flow regime ii application to curved boundaries gaseous flows inside microfluidic devices often fall in the slip flow regime according to this theoretical description the navier stokes model remains applicable in bulk while at solid walls a slip velocity boundary model shall be considered physically it is well established that to properly account for the wall curvature the wall slip velocity must be determined by the shear stress rather than the normal component of the velocity derivative alone as commonly applied to planar surfaces it follows that the numerical transcription of this type of boundary condition is generally a challenging task for standard computational fluid dynamics cfd techniques this paper aims to show that the realization of the slip velocity condition on arbitrarily shaped boundaries can be accomplished in a natural way with the lattice boltzmann method lbm to substantiate this conclusion this work undertakes the following three studies first we examine the conditions under which the generic reflection type boundary rules used by lbm become consistent models for the slip velocity boundary condition this effort makes use of the second order chapman enskog expansion method where we address both planar and curved boundaries the analysis also clarifies the capabilities and limitations behind the considered reflection type slip schemes second we revisit the family of parabolic accurate lbm slip boundary schemes originally formulated in phys rev e 96 013311 2017 2470 004510 1103 physreve 96 013311 on the basis of the multireflection framework and discuss their characteristics when operating on curved boundaries as well as the limitations of other less accurate lbm slip boundary formulations such as the linearly accurate slip schemes and the widely popular kinetic based boundary schemes in addition we also discuss the numerical stability of the parabolic slip schemes previously developed providing an heuristic strategy to improve their stable range of operation third we evaluate the performance of the several slip boundary schemes debated in this paper the numerical tests correspond to two classical 2d benchmark flow problems of slip over non planar solid surfaces namely i the velocity profile of the cylindrical couette flow and ii the permeability of a slow rarefied gas over a periodic array of circular cylindrical obstacles the obtained numerical results confirm the competitiveness of the lbm when equipped with slip boundary schemes of parabolic accuracy as cfd tool to simulate slippage phenomena over arbitrarily non planar surfaces indeed although operating on a simple uniform mesh discretization the lbm yields a similar or even superior level of accuracy compared to state of the art fem simulations conducted on hardworking body fitted meshes this conclusion establishes the lbm as a very appealing cfd technique for simulating microfluidic flows in the slip flow regime a result that deserves further exploration in future studies

an iterative algorithm for the reflexive solutions of the generalized coupled sylvester matrix equations and its optimal approximation abstract the generalized coupled sylvester matrix equations ay zb cy zd e f with unknown matrices y z are encountered in many systems and control applications also these matrix equations have several applications relating to the problem of computing stable eigendecompositions of matrix pencils in this work we construct an iterative algorithm to solve the generalized coupled sylvester matrix equations over reflexive matrices y z and when the matrix equations are consistent for any initial matrix pair y 0 z 0 a reflexive solution pair can be obtained within finite iteration steps in the absence of roundoff errors and the least frobenius norm reflexive solution pair can be obtained by choosing a special kind of initial matrix pair also we obtain the optimal approximation reflexive solution pair to a given matrix pair y z in the reflexive solution pair set of the generalized coupled sylvester matrix equations ay zb cy zd e f moreover several numerical examples are given to show the efficiency of the presented iterative algorithm

solutions to a family of matrix equations by using the kronecker matrix polynomials closed form solutions to a family of generalized sylvester matrix equation in form of i 0 fa ixf email protected i 0 jb kyf email protected j 0 fe jrf j are given by using the so called kronecker matrix polynomials it is found that the structure of the solutions is independent of the orders f j and f this type of uniform closed form solutions includes our early results as special cases the results provide great convenience to the computation and analysis of the solutions to this class of equations and can perform important functions in many analysis and design problems in linear systems

new approach to robust observer design this paper shows that based on the recent development of observer design solution an observer pole selection method can be formulated to minimize the observer gain to the system input it is proved that this method is a deterministic approach to the recovery of the loop transfer function and robustness of direct state feedback systems

gradient based and least squares based iterative algorithms for matrix equations axb cxtd f abstract this paper develops a gradient based and a least squares based iterative algorithms for solving matrix equation axb cx t d f the basic idea is to decompose the matrix equation system under consideration into two subsystems by applying the hierarchical identification principle and to derive the iterative algorithms by extending the iterative methods for solving ax b and axb f the analysis shows that when the matrix equation has a unique solution under the sense of least squares the iterative solution converges to the exact solution for any initial values a numerical example verifies the proposed theorems

iterative method to solve the generalized coupled sylvester transpose linear matrix equations over reflexive or anti reflexive matrix abstract the iterative method of generalized coupled sylvester transpose linear matrix equations a x b c y t d s 1 e x t f g y h s 2 over reflexive or anti reflexive matrix pair x y is presented on the condition that the coupled matrix equations are consistent we show that the solution pair x y proposed by the iterative method can be obtained within finite iterative steps in the absence of roundoff error for any initial value given a reflexive or anti reflexive matrix moreover the optimal approximation reflexive or anti reflexive matrix solution pair to an arbitrary given reflexive or anti reflexive matrix pair can be derived by searching the least frobenius norm solution pair of the new generalized coupled sylvester transpose linear matrix equations finally some numerical examples are given which illustrate that the introduced iterative algorithm is quite efficient

the modified conjugate gradient methods for solving a class of generalized coupled sylvester transpose matrix equations in this paper we consider the iteration solutions of generalized coupled sylvester transpose matrix equations a1xb1 c1ytd1 f1 a2yb2 c2xtd2 f2 when the coupled matrix equations are consistent we propose a modified conjugate gradient method to solve the equations and prove that a solution x y can be obtained within finite iterative steps in the absence of roundoff error for any initial value furthermore we show that the minimum norm solution can be got by choosing a special kind of initial matrices when the coupled matrix equations are inconsistent we present another modified conjugate gradient method to find the least squares solution with the minimum norm finally some numerical examples are given to show the behavior of the considered algorithms

parallel computation of the solutions of coupled algebraic lyapunov equations a parallel algorithm for solving systems of coupled lyapunov equations associated with linear jump parameter systems is introduced the recursive scheme is based on solving independent reduced order lyapunov equations monotonicity of convergence is established

least squares solution with the minimum norm to general matrix equations via iteration two iterative algorithms are presented in this paper to solve the minimal norm least squares solution to a general linear matrix equations including the well known sylvester matrix equation and lyapunov matrix equation as special cases the first algorithm is based on the gradient based searching principle and the other one can be viewed as its dual form necessary and sufficient conditions for the step sizes in these two algorithms are proposed to guarantee the convergence of the algorithms for arbitrary initial conditions sufficient condition that is easy to compute is also given moreover two methods are proposed to choose the optimal step sizes such that the convergence speeds of the algorithms are maximized between these two methods the first one is to minimize the spectral radius of the iteration matrix and explicit expression for the optimal step size is obtained the second method is to minimize the square sum of the f norm of the error matrices produced by the algorithm and it is shown that the optimal step size exits uniquely and lies in an interval several numerical examples are given to illustrate the efficiency of the proposed approach

semiclassical lattice hydrodynamics of rarefied channel flows the two dimensional channel flows of gas of arbitrary statistics in the slip and transition regimes as characterized by the knudsen number are studied using a newly developed semiclassical lattice boltzmann method the method is directly derived by projecting the uehling uhlenbeck boltzmann bgk equations onto the tensor hermite polynomials using moment expansion method the intrinsic discrete nodes of the gauss hermite quadrature provide the natural lattice velocities for the semiclassical lattice boltzmann method the mass flow rates and the velocity profiles are calculated for the three particle statistics over wide range of knudsen numbers and the knudsen minimum can be captured the results indicate distinct characteristics of the effects of quantum statistics

accuracy analysis of high order lattice boltzmann models for rarefied gas flows in this work we have theoretically analyzed and numerically evaluated the accuracy of high order lattice boltzmann lb models for capturing non equilibrium effects in rarefied gas flows in the incompressible limit the lb equation is shown to be able to reduce to the linearized bhatnagar gross krook bgk equation therefore when the same gauss hermite quadrature is used lb method closely resembles the discrete velocity method dvm in addition the order of hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the knudsen number to the bgk equation for incompressible flows meanwhile we have numerically evaluated the lb models for a standing shear wave problem which is designed specifically for assessing model accuracy by excluding the influence of gas molecule surface interactions at wall boundaries the numerical simulation results confirm that the high order terms in the discrete equilibrium distribution function play a negligible role in capturing non equilibrium effect for low speed flows by contrast appropriate gauss hermite quadrature has the most significant effect on whether lb models can describe the essential flow physics of rarefied gas accurately our simulation results where the effect of wall gas interactions is excluded can lead to conclusion on the lb modeling capability that the models with higher order quadratures provide more accurate results for the same order gauss hermite quadrature the exact abscissae will also modestly influence numerical accuracy using the same gauss hermite quadrature the numerical results of both lb and dvm methods are in excellent agreement for flows across a broad range of the knudsen numbers which confirms that the lb simulation is similar to the dvm process therefore lb method can offer flexible models suitable for simulating continuum flows at the navier stokes level and rarefied gas flows at the linearized boltzmann model equation level