stable three point wavelet bases on general meshes recently we introduced a wavelet basis on general possibly locally refined linear finite element spaces each wavelet is a linear combination of three nodal basis functions independently of the number of space dimensions in the present paper we show h s stability of this basis for a range of s that in any case includes s 1 which means that the corresponding additive schwarz preconditioner is optimal for second order problems furthermore we generalize the construction of the wavelet basis to manifolds we show that the wavelets have at least one and in areas where the manifold is smooth and the mesh is uniform even two vanishing moments because of these vanishing moments apart from preconditioning the basis can be used for compression purposes for a class of integral equation problems the stiffness matrix with respect to the wavelet basis will be close to a sparse one in the sense that a priori it can be compressed to a sparse matrix without the order of convergence being reduced

efficient algebraic multigrid algorithms and their convergence in this paper efficient algorithms for the algebraic multigrid amg method are advanced and amg methods are proposed for improving the convergence rate a new convergence theorem of amg is given and a complete theoretical study of new algorithms is subsequently presented various numerical results demonstrate the efficiency of the new approaches

h guaranteed cost control for uncertain continuous time linear systems abstract this paper deals with the h guaranteed cost control problems for continuous time uncertain systems it consist of the determination of a stabilizing state feedback gain which imposes on all possible closed loop models an h norm upper bound 0 assuming that the uncertain domain is convex bounded and the uncertain system is quadratic stabilizable with disturbance attenuation it is shown how to determine by means of a convex programming problem the global minimum of as a particular and important case for precisely known linear systems the last problem reduces to the classical h optimal control problem the results follow from the definition of a special parameter space on which the above mentioned problems are convex

riesz bases of wavelets and applications to numerical solutions of elliptic equations we investigate riesz bases of wavelets in sobolev spaces and their applications to numerical solutions of the biharmonic equation and general elliptic equations of fourth order first we study bi cubic splines on the unit square with homogeneous boundary conditions the approximation properties of these cubic splines are established and applied to convergence analysis of the finite element method for the biharmonic equation second we develop a fairly general theory for riesz bases of hilbert spaces equipped with induced norms under the guidance of the general theory we are able to construct wavelet bases for sobolev spaces on the unit square the condition numbers of the stiffness matrices associated with the wavelet bases are relatively small and uniformly bounded third we provide several numerical examples to show that the numerical schemes based on our wavelet bases are very efficient finally we extend our study to general elliptic equations of fourth order and demonstrate that our numerical schemes also have superb performance in the general case

on a convex parameter space method for linear control design of uncertain systems this paper presents a new procedure for continuous and discrete time linear control systems design it consists of the definition of a convex programming problem in the parameter space that when solved provides the feedback gain one of the most important features of the procedure is that additional design constraints are easily incorporated in the original formulation yielding solutions to problems that have raised a great deal of interest within the last few years this is precisely the case of the decentralized control problem and the quadratic stabilizability problem of uncertain systems with both dynamic and input uncertain matrices in this last case necessary and sufficient conditions for the existence of a linear stabilizing gain are provided and to the authors knowledge this is one of the first numerical procedures able to handle and solve this interesting design problem for high order continuous time or discrete time linear models the theory is illustrated by examples

robust control with structure perturbations the problem of making a given stabilizing controller robust so that the closed loop system remains stable for prescribed ranges of variations of a set of physical parameters in the plant the problem is treated in the state space and transfer function domains in the state space domain a stability hypersphere is determined in the parameter space using lyapunov theory the radius of this hypersphere is iteratively increased by adjusting the controller parameters until the prescribed perturbation ranges are contained in the stability hypersphere in the transfer function domain a corresponding stability margin is defined and optimized on the basis of the recently introduced concept of the largest stability hypersphere in the space of coefficients of the closed loop characteristic polynomial the design algorithms are illustrated by examples

c 1 spline wavelets on triangulations in this paper we investigate spline wavelets on general triangulations in particular we are interested in c 1 wavelets generated from piecewise quadratic polynomials by using the powell sabin elements we set up a nested family of spaces of c 1 quadratic splines which are suitable for multiresolution analysis of besov spaces consequently we construct c 1 wavelet bases on general triangulations and give explicit expressions for the wavelets on the three direction mesh a general theory is developed so as to verify the global stability of these wavelets in besov spaces the wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations

galerkin method with new quadratic spline wavelets for integral and integro differential equations abstract the paper is concerned with the wavelet galerkin method for the numerical solution of fredholm linear integral equations and second order integro differential equations we propose a construction of a quadratic spline wavelet basis on the unit interval such that the wavelets have three vanishing moments and the shortest support among such wavelets we prove that this basis is a riesz basis in the space l 2 0 1 we adapt the basis to homogeneous dirichlet boundary conditions and using a tensor product we construct a wavelet basis on the hyperrectangle we use the wavelet galerkin method with the constructed bases for solving integral and integro differential equations and we show that the matrices arising from discretization have uniformly bounded condition numbers and that they can be approximated by sparse matrices we present numerical examples and compare the results with the galerkin method using other quadratic spline wavelet bases and other methods

on compactly supported spline wavelets and a duality principle let v 1 v 0 v 1 be a multiresolution analysis of l 2 generated by the mth order b spline n m x we exhibit a compactly supported basic wavelet m x that generates the corresponding orthogonal complementary wavelet subspaces w 1 w 0 w 1 consequently the two finite sequences that describe the two scale relations of n m x and m x in terms of n m 2x j j z yield an efficient reconstruction algorithm to give an efficient wavelet decomposition algorithm based on these two finite sequences we derive a duality principle which also happens to yield the dual bases n m x j and 59 m x j relative to n m x j and m x j respectively

an alternate numerical solution to the linear quadratic problem this note proposes a new method based on convex programming for solving the linear quadratic problem lqp directly on the parameter space generated by the feedback control gain all stabilizing controllers are mapped into a convex set the problem is then formulated as a minimization of a linear function over this convex set its optimal solution furnishes under certain conditions the same feedback control gain obtained from the classical riccati equation generalizations to decentralized control and output feedback control design are included the theory is illustrated by some numerical examples

postprocessing galerkin method using quadratic spline wavelets and its efficiency abstract the wavelet galerkin method is a useful tool for solving differential equations mainly because the condition number of the stiffness matrix is independent of the matrix size and thus the number of iterations for solving the discrete problem by the conjugate gradient method is small we have recently proposed a quadratic spline wavelet basis that has a small condition number and a short support in this paper we use this basis in the galerkin method for solving the second order elliptic problems with dirichlet boundary conditions in one and two dimensions and by an appropriate post processing we achieve the l 2 error of order o h 4 and the h 1 error of order o h 3 where h is the step size the rate of convergence is the same as the rate of convergence for the galerkin method with cubic spline wavelets we show theoretically as well as numerically that the presented method outperforms the galerkin method with other quadratic or cubic spline wavelets furthermore we present local post processing for example of the equation with dirac measure on the right hand side

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

convex analysis of output feedback control problems robust stability and performance this paper addresses the problem of optimal h sub 2 control by output feedback necessary and sufficient conditions on the existence of a linear stabilizing output feedback gain are provided in terms of the intersection of a convex set and a set defined by a nonlinear real valued function the results can be easily extended to deal with linear uncertain systems where uncertainties are supposed to belong to convex bounded domains providing an h sub 2 guaranteed cost output feedback control thanks to the properties of the above mentioned function we show that under certain conditions convex programming tools can be used for numerical purposes examples illustrate the theoretical results

on the algebraic multigrid method new formulations for the algebraic multigrid amg method are presented a new interpolation operator is developed in which the weighting could be negative numerical experiments demonstrate that the use of negative interpolation weights is necessary in some applications new approaches to construct the restriction operator and the coarse grid equations are discussed two new amg methods are proposed theoretical study and convergence analysis of the amg methods are presented the main contributions of this paper are to improve the convergence rate and to extend the range of applications of an amg method numerical experiments are reported for matrix computations that resulted from partial differential equations signal processing and queueing network problems the success of the proposed new amg algorithms is clearly demonstrated by applications to non diagonally dominant matrix problems for which the standard amg method fails to converge

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

cubic spline wavelets with complementary boundary conditions abstract we propose a new construction of a stable cubic spline wavelet basis on the interval satisfying complementary boundary conditions of the second order it means that the primal wavelet basis is adapted to homogeneous dirichlet boundary conditions of the second order while the dual wavelet basis preserves the full degree of polynomial exactness we present quantitative properties of the constructed bases and we show superiority of our construction in comparison to some other known spline wavelet bases in an adaptive wavelet method for the partial differential equation with the biharmonic operator

three dimensional unconditionally stable lod fdtd methods with low numerical dispersion in the desired directions in order to get low numerical dispersion in the desired directions four step locally one dimensional finite difference time domain lod4 fdtd five step lod fdtd lod5 fdtd and six step lod fdtd lod6 fdtd methods are developed here these methods are derived from the three step lod fdtd method and each of their substeps has variation in either the x y or z directions the lod4 fdtd lod5 fdtd and lod6 fdtd methods have two substeps in one two and three directions respectively these methods give low numerical dispersion in the directions in which they have two substeps in addition the lod5 fdtd method gives low anisotropy error ae for the directions in which it has two substeps and the lod6 fdtd method gives low 3 d ae furthermore to validate the analytical results the simulation results are given here

computational methods for parametric lq problems a survey iterative methods for finding the optimal constant feedback gains for parametric lq problems notably for optimal constant output feedback problems are surveyed the connections of several methods to loss function expansions are discussed with important implications to the understanding of their convergence properties especially the descent anderson moore method levine athans like methods and the newton method are considered convergence results are also included the initialization problem and the output feedback stabilization problem are also discussed furthermore it is shown that the concepts and methods surveyed in this paper are useful in solving many realistic generalized parametric lq problems as well notably so called robust parametric lq problems

cubic spline wavelets with short support for fourth order problems in the paper we propose a construction of new cubic spline wavelet bases on the unit cube satisfying homogeneous dirichlet boundary conditions of the second order the basis functions have small supports and wavelets have vanishing moments we show that stiffness matrices arising from discretization of the biharmonic problem using a constructed wavelet basis have uniformly bounded condition numbers and these condition numbers are very small we present quantitative properties of the constructed bases and we show a superiority of our construction in comparison to some other cubic spline wavelet bases satisfying boundary conditions of the same type