an optimization based approach for the design of pde solution algorithms we develop and analyze an optimization based approach for the robust and efficient solution of pde problems consisting of multiple physics operators with fundamentally different mathematical properties our approach relies on three essential steps decomposition of the original problem into subproblems for which robust solution algorithms are available integration of the subproblems into an equivalent pde constrained optimization problem and solution of the resulting optimization problem either directly as a fully coupled algebraic system or in the null space of the pde constraints this strategy gives rise to a general approach for synthesizing robust solvers for complex coupled problems from solvers for their simpler physics components

a reduced order model based preconditioner for the efficient solution of transient diffusion equations this paper presents a novel class of preconditioners for the iterative solution of the sequence of symmetric positive definite linear systems arising from the numerical discretization of transient parabolic and selfadjoint partial differential equations the preconditioners are obtained by nesting appropriate projections of reduced order models into the classical iteration of the preconditioned conjugate gradient pcg the main idea is to employ the reduced order solver to project the residual associated with the conjugate gradient iterations onto the space spanned by the reduced bases this approach is particularly appealing for transient systems where the full model solution has to be computed at each time step in these cases the natural reduced space is the one generated by full model solutions at previous time steps when increasing the size of the projection space the proposed methodology highly reduces the system conditioning number and the number of pcg iterations at every time step the cost of the application of the preconditioner linearly increases with the size of the projection basis and a trade off must be found to effectively reduce the pcg computational cost the quality and efficiency of the proposed approach is finally tested in the solution of groundwater flow models c 2016 the authors international journal for numerical methods in engineering published by john wiley sons ltd

a rational function preconditioner for indefinite sparse linear systems this paper introduces a rational function preconditioner for linear systems with indefinite sparse matrices a by resorting to rational functions of a the algorithm decomposes the spectrum of a into two disjoint regions and approximates the restriction of a 1 on these regions separately we show a systematic way to construct these rational functions so that they can be applied stably and inexpensively an attractive feature of the proposed approach is that the construction and application of the preconditioner can exploit two levels of parallelism moreover the proposed preconditioner can be modified at a negligible cost into a preconditioner for a nearby matrix of the form a ci which can be useful in some applications the efficiency and robustness of the proposed preconditioner are demonstrated on a few tests with challenging model problems including problems arising from the helmholtz equation in three dimensions

optimal l sub spl infin to l sub spl infin estimation for periodic systems in this paper we consider the problem of finding a filter that minimizes the worst case magnitude l sub spl infin of the estimation error in the case of linear periodically time varying systems subjected to unknown but magnitude bounded l sub spl infin inputs these inputs consist of process and observation noises and the optimization problem is considered over an infinite time horizon lifting techniques are utilized to transform the problem to a time invariant l sub 1 model matching problem subject to additional constraints taking advantage of the particular structure of the estimation problem it is shown how standard methods of l sub 1 optimization in particular the delay augmentation technique can be suitably modified to solve this nonstandard problem

polynomial matrix solution of the h infinity filtering problem and the relationship to riccati equation state space results a solution to the standard suboptimal h filtering problem is presented using a new polynomial systems ap proach the polynomial matrix solution is closely related to the recent h state space based riccati equation results and the links between the polynomial and state equation methodologies are demonstrated the h filter is derived by solving a special type of minimum variance filtering problem using a technique based on game theory the solution of this game problem in a stochastic setting by polynomial methods is novel the cal culation of the h filter involves a j spectral factorization and the solution of two coupled diophantine equations there are several computational advantages over previous polynomial methods for calculating h filters

flexible variants of block restarted gmres methods with application to geophysics in a wide number of applications in computational science and engineering the solution of large linear systems of equations with several right hand sides given at once is required direct methods based on gaussian elimination are known to be especially appealing in that setting nevertheless when the dimension of the problem is very large preconditioned block krylov space solvers are often considered as the method of choice the purpose of this paper is thus to present iterative methods based on block restarted gmres that allow variable preconditioning for the solution of linear systems with multiple right hand sides the use of flexible methods is especially of interest when approximate possibly iterative solvers are considered in the preconditioning phase first we introduce a new variant of block flexible restarted gmres that includes a strategy for detecting when a linear combination of the systems has approximately converged this explicit block size reduction is often called deflation we analyze the main properties of this flexible method based on deflation and notably prove that the frobenius norm of the block residual is always nonincreasing we also present a flexible variant based on both deflation and truncation to especially be used in case of limited memory finally we illustrate the numerical behavior of these flexible block methods for large industrial simulations arising in geophysics where indefinite linear systems of size up to 1 billion unknowns with multiple right hand sides have been successfully solved in a parallel distributed memory environment

h sup spl infin control and estimation with preview part i matrix are solutions in continuous time preview control and fixed lag smoothing allow a noncausal component in the controller estimator time domain variational analysis is used in a reduction to an open loop differential game leading to a complete necessary and sufficient characterization of suboptimal values and an explicit state space design in terms of a parameterized nonstandard algebraic matrix riccati equation in a general continuous time linear system setting the solution offers insight into the appropriate structure of the associated hamiltonian where the state and co state are not the usual state of the original dynamic system and that of its adjoint rather the state and co state are selected to capture the respective lumped effects of initial data and future input selection in the allied game

reduced order modeling for time dependent optimization problems with initial value controls this paper presents a new reduced order model rom hessian approximation for linear quadratic optimal control problems where the optimal control is the initial value such problems arise in parameter identification and data assimilation where the parameters to be identified appear in the initial data and also as subproblems in multiple shooting formulations of more general optimal control problems the new rom hessians can provide a substantially better approximation than the underlying basic rom approximation and thus can substantially reduce the computing time needed to solve these optimal control problems the computation of a hessian vector product requires the solution of the linearized state equation with initial value given by the vector to which the hessian is applied followed by the solution of the second order adjoint equation projection based roms of these two linear differential equations are used to generate the hessian approximation while the objective function and gradient are compute

an efficient multigrid method for graph laplacian systems ii robust aggregation we consider the iterative solution of linear systems whose matrices are laplacians of undirected graphs designing robust solvers for this class of problems is challenging due to the diversity of connectivity patterns encountered in practical applications our starting point is a recently proposed aggregation based algebraic multigrid method that combines the recursive static elimination of the vertices of degree 1 with the degree aware rooted aggregation dra algorithm the latter always produces aggregates big enough to ensure that the preconditioner cost per iteration is low here we further improve the robustness of the method by controlling the quality of the aggregates more precisely bad vertices are removed from the aggregates formed by the dra algorithm until a quality test is passed this ensures that the two grid condition number is nicely bounded whereas the cost per iteration is kept low by reforming too small aggregates when it happens that the mean aggregate size is not large enough t

intervalwise receding horizon h sub spl infin tracking control for discrete linear periodic systems in this paper a fixed horizon h sub spl infin tracking control htc for discrete time varying systems is obtained in a state feedback form from the htc an intervalwise receding horizon htc ihtc is suggested for discrete periodic systems conditions are proposed under which ihtc can stabilize a closed loop system it is shown that under these conditions ihtc guarantees an h sub spl infin norm bound and that ihtc with integral action provides zero offset for a constant command signal the performance of ihtc is compared with those of existing controls via an example the results in this paper are also applicable to time invariant systems which belong to the class of periodic systems

conserved quantities of the discrete finite toda equation and lower bounds of the minimal singular value of upper bidiagonal matrices some numerical algorithms are known to be related to discrete time integrable systems where it is essential that quantities to be computed for example eigenvalues and singular values of a matrix poles of a continued fraction are conserved quantities in this paper a new application of conserved quantities of integrable systems to numerical algorithms is presented for an n n n 2 real upper bidiagonal matrix b where all the diagonals and the upper subdiagonals are positive conserved quantities tr btb m 1 m 1 2 of the discrete finite toda equation give a sequence of lower bounds of the minimal singular value of b recurrence relations for computing higher order conserved quantities tr btb m 1 are also derived

algebraic multilevel iteration method for lowest order raviart thomas space and applications an optimal order algebraic multilevel iterative method for solving system of linear algebraic equations arising from the finite element discretization of certain boundary value problems that have their weak formulation in the space h div is presented the algorithm is developed for the discrete problem obtained by using the lowest order raviart thomas space the method is theoretically analyzed and supporting numerical examples are presented furthermore as a particular application the algorithm is used for the solution of the discrete minimization problem which arises in the functional type a posteriori error estimates for the discontinuous galerkin approximation of elliptic problems copyright 2011 john wiley sons ltd

accurate singular values and differential qd algorithms in september 1991 j w demmel and w m kahan were awarded the second siam prize in numerical linear algebra for their paper accurate singular values of bidiagonal matrices 1 referred to as dk hereafter among several valuable results was the observation that the standard bidiagonal qr algorithm used in linpack 2 and in many other svd programs can be simplified when the shift is zero and of greater importance no subtractions occur the last feature permits very small singular values to be found with almost all the accuracy permitted by the data and at no extra cost

robust multilevel solvers for high contrast anisotropic multiscale problems a robust multilevel method for computing the solution of a scalar elliptic equation with anisotropic highly varying tensor coefficients is presented the method which belongs to the class of nonlinear algebraic multilevel iterations amlis uses an abstract framework for general symmetric positive definite bilinear forms previously presented in another publication by the author the crucial ingredient for obtaining robustness with respect to the variations in the coefficients and the anisotropies is the design of a nested sequence of spaces based on local generalized eigenvalue problems a discussion on how to achieve large coarsening factors in this sequence of spaces which is desirable in terms of computational complexity is included particular emphasis is put on how to handle the situation when the computed generating sets of the nested spaces are not minimal i e do not constitute bases several numerical examples are provided verifying the theoretically established robustness results

weighted and flexible versions of block cmrh method for solving nonsymmetric linear systems with multiple right hand sides abstract block krylov subspace methods are the most popular algorithms for solving large nonsymmetric linear systems with multiple right hand sides one of them is the block cmrh method this method generates a non orthogonal basis of the krylov subspace through the block hessenberg process to accelerate the convergence of the block cmrh method we will introduce two new methods first we present the block cmrh method with weighting strategy in this method the block cmrh method uses a different product at each restart second we introduce a flexible version of the block cmrh algorithm that allows varying preconditioning at every step of the algorithm numerical experiments illustrate the benefits of the presented methods

algebraic multilevel preconditioning in isogeometric analysis construction and numerical studies abstract we present algebraic multilevel iteration amli methods for isogeometric discretization of scalar second order elliptic problems the construction of coarse grid operators and hierarchical complementary operators are given moreover for a uniform mesh on a unit interval the explicit representation of b spline basis functions for a fixed mesh size h is given for p 2 3 4 and for c 0 and c p 1 continuity the presented methods show h and almost p independent convergence rates supporting numerical results for convergence factor and iterations count for amli cycles v linear w nonlinear w are provided numerical tests are performed in two dimensions on square domain and quarter annulus and in three dimensions on quarter thick ring

robust multilevel methods for general symmetric positive definite operators a general robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed the term robust refers to the convergence rate of the method being independent of discretization parameters i e the problem size and problem parameters important instances of such problem parameters are in particular highly varying coefficients the method belongs to the class of nonlinear algebraic multilevel iterations the crucial ingredient for obtaining robustness is the construction of a nested sequence of spaces based on local generalized eigenvalue problems the method is analyzed in a rather general setting applying to the scalar elliptic equation the equations of linear elasticity and equations arising in the solution of maxwell s equations numerical results for the scalar elliptic equation are presented showing its robust convergence behavior

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

flexible bicg and flexible bi cgstab for nonsymmetric linear systems flexible versions of the bicg and bi cgstab algorithms are presented which allow for the use of a different preconditioner at each step of the algorithm in particular inexact solutions of the preconditioned equations are allowed as well as the use of some inner iterative method as a preconditioner flexible bicg is investigated theoretically and a theorem is presented comparing the residual norms of flexible bicg and flexible qmr that generalizes a well known relation between the residual norms of bicg and qmr numerical experiments are conducted for flexible bi cgstab for several matrices including some indefinite and highly nonsymmetric matrices these experiments illustrate the applicability of the flexible method

a game theoretic approach to a finite time disturbance attenuation problem a disturbance attenuation problem over a finite time interval is considered by a game theoretic approach where the control restricted to a function of the measurement history plays against adversaries composed of the process and measurement disturbances and the initial state a zero sum game formulated as a quadratic cost criterion subject to linear time varying dynamics and measurements is solved by a calculus of variation technique by first maximizing the quadratic cost criterion with respect to the process disturbance and initial state a full information game between the control and the measurement residual subject to the estimator dynamics results the resulting solution produces an n dimensional compensator which expresses the controller as a linear combination of the measurement history a disturbance attenuation problem is solved based on the results of the game problem for time invariant systems it is shown that under certain conditions the time varying controller becomes time invariant on the infinite time interval the resulting controller satisfies an h sub infinity norm bound

additive schur complement approximation and application to multilevel preconditioning this paper introduces an algorithm for additive schur complement approximation asca which can be applied in various iterative methods for solving systems of linear algebraic equations arising from finite element fe discretization of partial differential equations pde it is shown how the asca can be used to set up a nonlinear algebraic multilevel iteration amli method this requires the construction of a linear multiplicative two level preconditioner at each level the latter is computed in the course of a simultaneous exact two by two block factorization of local stiffness matrices associated with a covering of the entire domain by overlapping subdomains unlike in schwarz type domain decomposition methods this method does not require a global coarse problem but instead uses local coarse problems to provide global communication a robust condition number bound is proved for a particular partitioning of the nodes of a uniform grid the presented numerical tests demonstrate that the asca whe

algebraic multigrid for moderate order finite elements we investigate the use of algebraic multigrid amg methods for the solution of large sparse linear systems arising from the discretization of scalar elliptic partial differential equations with lagrangian finite elements of order at most 4 the resulting system matrices do not have the m matrix property that is required by standard analyses of classical amg and aggregation based amg methods a unified approach is presented that allows us to extend these analyses it uses an intermediate m matrix and highlights the role of the spectral equivalence constant that relates this matrix to the original system matrix this constant is shown to be bounded independently of the problem size and jumps in the coefficients of the partial differential equations provided that jumps are located at elements boundaries for two dimensional problems it is further shown to be uniformly bounded if the angles in the triangulation also satisfy a uniform bound this analysis validates the application of the amg methods to the

flexible inner outer krylov subspace methods flexible krylov methods refers to a class of methods which accept preconditioning that can change from one step to the next given a krylov subspace method such as cg gmres qmr etc for the solution of a linear system ax b instead of having a fixed preconditioner m and the right preconditioned equation am 1 y b mx y one may have a different matrix say mk at each step in this paper the case where the preconditioner itself is a krylov subspace method is studied there are several papers in the literature where such a situation is presented and numerical examples given a general theory is provided encompassing many of these cases including truncated methods the overall space where the solution is approximated is no longer a krylov subspace but a subspace of a larger krylov space we show how this subspace keeps growing as the outer iteration progresses thus providing a convergence theory for these inner outer methods numerical tests illustrate some important implementation aspects that make the discussed inner outer methods very appealing in practical circumstances

the deflated conjugate gradient method convergence perturbation and accuracy abstract deflation techniques for krylov subspace methods have seen a lot of attention in recent years they provide means to improve the convergence speed of these methods by enriching the krylov subspace with a deflation subspace the most common approach for the construction of deflation subspaces is to use approximate eigenvectors but also more general subspaces are applicable in this paper we discuss two results concerning the accuracy requirements within the deflated cg method first we show that the effective condition number which bounds the convergence rate of the deflated conjugate gradient method depends asymptotically linearly on the size of the perturbations in the deflation subspace second we discuss the accuracy required in calculating the deflating projection this is crucial concerning the overall convergence of the method and also allows to save some computational work to show these results we use the fact that as a projection approach deflation has many similarities to multigrid methods in particular recent results relate the spectra of the deflated matrix to the spectra of the error propagator of twogrid methods in the spirit of these results we show that the effective condition number can be bounded by the constant of a weak approximation property

critical value computation for h riccati difference equations abstract in designing finite horizon discrete time h controllers the associated h riccati difference equations must be solved but the riccati equation has a non negative solution only when 2 is small enough so it is important to get the upper bound of the parameter i e the critical value that ensures the existence of the solution to the riccati equation the solution sequence of the riccati difference equation can be constructed by the conjoined basis of an associated linear hamiltonian difference system based on this expression and the hamiltonian difference system eigenvalue theorems the equivalence between the critical value and the first order eigenvalue of the linear hamiltonian difference system is presented since the critical value is also shown to be the fundamental eigenvalue of a generalized rayleigh quotient an extended form of wittrick williams algorithm is presented to search this value

minimum h sub infinity norm regulation of linear discrete time systems and its relation to linear quadratic discrete games a solution is derived to the h sub infinity optimization problem that arises in multivariable discrete time regulation when the controller has full access to the state vector the solution method is based on the close relations that exist between linear quadratic differential game theory and h sub infinity optimization the existing theory of discrete time quadratic games is readily applied in order to derive the solution to a finite time horizon version of the h sub infinity optimization problem the solution of the infinite time horizon h sub infinity optimization problem is obtained by formally taking the limit of the number of stages to infinity

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

algebraic multigrid for stokes equations a method is investigated for solving stationary or time dependent discrete stokes equations it uses one of the standard flavors of algebraic multigrid for coupled partial differential equations which however is not applied directly to the linear system stemming from discretization but to an equivalent system obtained with a simple algebraic transformation which may be seen as a form of preconditioning in the literal sense a two grid analysis is provided showing that the eigenvalues of the preconditioned matrix are within a region of the complex plane that is both bounded and away from the origin independently of the mesh or grid size as well as of other main problem parameters on the other hand whereas the approach can in principle be combined with any type of algebraic multigrid scheme an investigation of the properties of the coarse grid matrices reveals that plain aggregation has to be preferred to maintain nice two grid convergence at coarser levels eventually numerical experiments are

ℋ control in discrete time state feedback control and norm bounds h norm bounds are useful in control design for disturbance rejection for discrete systems h norm optimizing controls can be characterized via the solution of the discrete algebraic riccati equation dare in addition the dare can be used to characterize controls that guarantee a uniform h norm bound in this paper one identifies an entire class of state feedback controls that guarantee an h norm bound and provides new lower bounds for the h norm of stable systems as well as for the best achievable h norm using state feedback controls

the orthogonal qd algorithm the orthogonal qd algorithm is presented to compute the singular value decomposition of a bidiagonal matrix this algorithm represents a modification of rutishauser s qd algorithm and it is capable of determining all the singular values and their corresponding singular vectors to high relative accuracy a generalization of the givens transformation which has applications besides the orthogonal qd algorithm is also introduced the shift strategy of the orthogonal qd algorithm is based on laguerre s method which is used to compute a lower bound on the smallest singular value of the bidiagonal matrix special attention is devoted to the numerically stable evaluation of this shift

the effect of non optimal bases on the convergence of krylov subspace methods there are many examples where non orthogonality of a basis for krylov subspace methods arises naturally these methods usually require less storage or computational effort per iteration than methods using an orthonormal basis optimal methods but the convergence may be delayed truncated krylov subspace methods and other examples of non optimal methods have been shown to converge in many situations often with small delay but not in others we explore the question of what is the effect of having a non optimal basis we prove certain identities for the relative residual gap i e the relative difference between the residuals of the optimal and non optimal methods these identities and related bounds provide insight into when the delay is small and convergence is achieved further understanding is gained by using a general theory of superlinear convergence recently developed our analysis confirms the observed fact that in exact arithmetic the orthogonality of the basis is not important only the need to maintain linear independence is numerical examples illustrate our theoretical results

h control with output feedback for time varying discrete systems h problems with state feedback are considered for time varying discrete systems first quadratic games related to h problems are studied a necessary and sufficient condition for the existence of a stabilizing controller such that the norm of the operator from the disturbance to the output is less than a prescribed number is given in terms of a riccati equation

towards a less conservative model predictive control based on mixed ℋ2 ℋ control approach a novel approach to the design of a less conservative model predictive control is proposed using mixed 2 design method for time invariant discrete time linear systems the controller has the form of a state feedback satisfies input and state constraints and is constructed from the solution of a set of feasibility linear matrix inequalities the control law takes into account the disturbances naturally and the input constraint is less conservative in the sense that the full control is used numerical examples comparing the proposed algorithm with some existing techniques are provided to demonstrate the applicability and the advantage of the proposed algorithm

quadratic games and h type problems for time varying systems abstract h type control problems with state feedback are considered for time varying systems first general quadratic differential games are studied and then the relationship between h problems and differential games is clarified a necessary and sufficient condition for the existence of a stabilizing controller such that the norm of the operator from the disturbance to the output is less than a prescribed number is given in terms of a riccati equation earlier results on time invariant systems are found to be special cases of our main results the finite horizon problem is also studied

a transfer function approach to the problems of discrete time systems h sub infinity optimal linear control and filtering a solution is derived to the problems of h sub infinity optimal linear state regulation and filtering the solution method for both problems is based on a transfer function approach which applies standard spectral factorization return difference relations are given which are extensions of the relations associated with the linear quadratic problem

conjugate gradient for nonsingular saddle point systems with a maximally rank deficient leading block abstract we consider iterative solvers for large sparse symmetric linear systems with a saddle point structure since such systems are indefinite the conjugate gradient cg method is not naturally designed for solving them however in the case of a maximally rank deficient leading block we prove that there are two sufficient conditions that allow for cg to be used we show that the conditions are satisfied for a model time harmonic maxwell problem to support our analysis we present several numerical experiments for three dimensional problems on complicated computational domains with constant and variable coefficients

flexible conjugate gradients we analyze the conjugate gradient cg method with preconditioning slightly variable from one iteration to the next to maintain the optimal convergence properties we consider a variant proposed by axelsson that performs an explicit orthogonalization of the search directions vectors for this method which we refer to as flexible cg we develop a theoretical analysis that shows that the convergence rate is essentially independent of the variations in the preconditioner as long as the latter are kept sufficiently small we further discuss the real convergence rate on the basis of some heuristic arguments supported by numerical experiments depending on the eigenvalue distribution corresponding to the fixed reference preconditioner several situations have to be distinguished in some cases the convergence is as fast with truncated versions of the algorithm or even with the standard cg method whereas quite large variations are allowed without too much penalty in other cases the flexible variant effectively outperforms the standard method while the need for truncation limits the size of the variations that can be reasonably allowed

on the discrete time bounded real lemma with application in the characterization of static state feedback h controllers abstract this paper deals with the discrete time bounded real lemma and its application in the characterization of all static state feedback h controllers for discrete time systems we first present two versions of the discrete time bounded real lemma the results apply to non minimal stable state space realizations and are given in terms of either the positive semi definite strong solution or the positive semi definite stabilizing solution of a discrete algebraic riccati equation based on the discrete time bounded real lemma we obtain a characterization of all static state feedback h controllers for discrete time systems as an application of the parameterization of all static state feedback h controllers we develop a solution to the discrete time h control problem via static output feedback also a numerical procedure is proposed to determine an output feedback gain

closed form solutions to discrete time lq optimal control and disturbance attenuation abstract this paper presents straightforward derivations of closed form solutions to two discrete time problems linear quadratic optimal control and disturbance attenuation then as the main results it shows how to reduce directly the closed form solutions to their recursive forms these recursive solutions are in the form of riccati equation and hence establish a close relation between the closed form solutions and the riccati equations used in both problems

on optimal e to e filtering abstract in this paper we consider the problem of finding a filter that minimizes the worst case magnitude l of the estimation error in the case of linear time invariant systems subjected to unknown but magnitude bounded l inputs these inputs consist of process and observation noise as well as initial conditions also the optimization problem is considered over an infinite time horizon taking a model matching approach suboptimal solutions are presented that stem from the resulting l induced norm minimization problem

h sub infinity optimal one step ahead output feedback control of discrete time systems the discrete time one step ahead standard four blocks h sub infinity optimal control problem is considered the output feedback problem is transformed into a one step ahead h sub infinity optimal state estimation problem and the existing solution is used to derive the controller it is assumed that only previous measurements are available for control two results are obtained the first requires the solutions of two coupled riccati equations where the necessary and sufficient condition for the existence of the h sub infinity optimal result requires only the positive semidefiniteness of these solutions the second result applies two decoupled riccati equations with a coupling constraint as in the continuous time case and two additional conditions the derivation procedure is most simple and may thus provide considerable insight into the solution of the discrete time standard problem a similar procedure can also be used to obtain a solution to problems where the current measurement is available

flexibly preconditioned extended least squares migration in shot record domain abstractwe have developed a method for accelerating the convergence of iterative least squares migration the algorithm uses a pseudodifferential scaling dip and spatially varying filter preconditioner together with a variant of conjugate gradient cg iteration with iterate dependent flexible preconditioning the migration is formulated without the image stack thus producing a shot dependent image volume that retains offset information useful for velocity updating and amplitude variation with offset analysis numerical experiments indicate that flexible preconditioning with pseudodifferential scaling not only attains considerably smaller data misfit and gradient error for a given computational effort but also produces higher resolution image volumes with more balanced amplitude and fewer artifacts than is achieved with a nonpreconditioned cg method

analysis of aggregation based multigrid aggregation based multigrid with standard piecewise constant like prolongation is investigated unknowns are aggregated either by pairs or by quadruplets in the latter case the grouping may be either linewise or boxwise a fourier analysis is developed for a model two dimensional anisotropic problem most of the results are stated for an arbitrary smoother which fits with the fourier analysis framework it turns out that the convergence factor of two grid schemes can be bounded independently of the grid size with a sensible choice of the linewise or boxwise coarsening the bound is also uniform with respect to the anisotropy ratio without requiring a specialized smoother the bound is too large to guarantee optimal convergence properties with the v cycle or the standard w cycle but a w cycle scheme accelerated by the recursive use of the conjugate gradient method exhibits near grid independent convergence