on schwarz type smoothers for saddle point problems in this paper we consider additive schwarz type iteration methods for saddle point problems as smoothers in a multigrid method each iteration step of the additive schwarz method requires the solutions of several small local saddle point problems this method can be viewed as an additive version of a multiplicative vanka type iteration well known as a smoother for multigrid methods in computational fluid dynamics it is shown that under suitable conditions the iteration can be interpreted as a symmetric inexact uzawa method in the case of symmetric saddle point problems the smoothing property an important part in a multigrid convergence proof is analyzed for symmetric inexact uzawa methods including the special case of the additive schwarz type iterations as an example the theory is applied to the crouzeix raviart mixed finite element for the stokes equations and some numerical experiments are presented

use of group analysis in solving overdetermined systems of ordinary differential equations abstract group analysis is applied to overdetermined systems of odes if each ode of the system admits the same r parameter solvable lie group then the use of the corresponding differential invariants greatly simplifies the analysis of the system moreover this can even lead to its explicit solution examples are given

multigrid optimization in applications abstract iterative techniques are a key methodology for the numerical solution of optimization problems in differential equations in two practical application problems with different characteristics this paper shows how multigrid methods can be applied efficiently to this problem class problem formulations solution approaches as well as numerical results are presented

a general chebyshev complex function approximation procedure and an application to beamforming a new computational technique is described for the chebyshev or minimax approximation of a given complex valued function by means of linear combinations of given complex valued basis functions the domain of definition of all functions can be any finite set whatever neither the basis functions nor the function approximated need satisfy any special hypotheses beyond the requirement that they be defined on a common domain theoretical upper and lower bounds on the accuracy of the computed chebyshev error are derived these bounds permit both a priori and a posteriori error assessments efforts to extend the method to functions whose domain of definition is a continuum are discussed an application is presented involving re shading a 50 element antenna array to minimize the effects of a 10 element failure rate while maintaining full steering capability and mainlobe beamwidth

the evolution operator technique in solving the schrodinger equation and its application to disentangling exponential operators and solving the problem of a mass varying harmonic oscillator the authors present a method for finding the evolution operator for the schrodinger equation for the hamiltonian expressible as h t a1 t j4 a2 t j0 a3 t j where j j0 and j are the su 2 group generators such a method is applied to the disentangling technique for exponential operators which are not necessarily unitary as a demonstration of our general approach they solved the problem of a harmonic oscillator with a varying mass

optimal design of fir filters with the complex chebyshev error criteria we present an improved algorithm for an established filter design problem the design of an fir filter that best approximates in the complex chebyshev sense a desired complex valued frequency response the algorithm is a variant of the simplex algorithm of linear programming which has an interpretation as an implicit multiple exchange it is iterative robust and exhibits good convergence speed global optimum convergence is guaranteed both complex and real valued impulse responses can be designed with it the design of complex coefficient filters is new an example is given for each case the design of noncausal filters is new in addition to these new applications we conjecture that this new algorithm may have important advantages over existing techniques with respect to the maximum filter length possible speed and stability of convergence accuracy and memory requirements the ability to design long filters is among the more significant improvements over previous work filters of length 1000 have been designed with the new method

two families of superintegrable and isospectral potentials in two dimensions as an extension of the intertwining operator idea an algebraic method which provides a link between supersymmetric quantum mechanics and quantum super integrability is introduced by realization of the method in two dimensions two infinite families of superintegrable and isospectral stationary potentials are generated the method makes it possible to perform darboux transformations in such a way that in addition to the isospectral property they acquire the superintegrability preserving property symmetry generators are second and fourth order in derivatives and all potentials are isospectral with one of the smorodinsky winternitz potentials explicit expressions of the potentials their dynamical symmetry generators and the algebra they obey as well as their degenerate spectra and corresponding normalizable states are presented

higher symmetries and exact solutions of linear and nonlinear schrodinger equation a new approach for the analysis of partial differential equations is developed which is characterized by a simultaneous use of higher and conditional symmetries higher symmetries of the schrodinger equation with an arbitrary potential are investigated nonlinear determining equations for potentials are solved using reductions to weierstrass painleve and riccati forms algebraic properties of higher order symmetry operators are analyzed combinations of higher and conditional symmetries are used to generate families of exact solutions of linear and nonlinear schrodinger equations

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

parallel lagrange newton krylov schur methods for pde constrained optimization part i the krylov schur solver large scale optimization of systems governed by partial differential equations pdes is a frontier problem in scientific computation reduced quasi newton sequential quadratic programming sqp methods are state of the art approaches for such problems these methods take full advantage of existing pde solver technology and parallelize well however their algorithmic scalability is questionable for certain problem classes they can be very slow to converge in this two part article we propose a new method for steady state pde constrained optimization based on the idea of using a full space newton solver combined with an approximate reduced space quasi newton sqp preconditioner the basic components of the method are newton solution of the first order optimality conditions that characterize stationarity of the lagrangian function krylov solution of the karush kuhn tucker kkt linear systems arising at each newton iteration using a symmetric quasi minimum residual method preconditioning of the kkt system using an approximate state decision variable decomposition that replaces the forward pde jacobians by their own preconditioners and the decision space schur complement the reduced hessian by a bfgs approximation initialized by a two step stationary method accordingly we term the new method it lagrange newton krylov schur lnks it is fully parallelizable exploits the structure of available parallel algorithms for the pde forward problem and is locally quadratically convergent in part i of this two part article we investigate the effectiveness of the kkt linear system solver we test our method on two optimal control problems in which the state constraints are described by the steady state stokes equations the objective is to minimize dissipation or the deviation from a given velocity field the control variables are the boundary velocities numerical experiments on up to 256 cray t3e processors and on an sgi origin 2000 include scalability and performance assessment of the lnks algorithm and comparisons with reduced sqp for up to 1 000 000 state and 50 000 decision variables in part ii of the article we address globalization and inexactness issues and apply lnks to the optimal control of the steady incompressible navier stokes equations

a fast algorithm for linear complex chebyshev approximations we propose a new algorithm for finding best minimax polynomial approximations in the complex plane the algorithm is the first satisfactory generalization of the well known remez algorithm for real approximations among all available algorithms ours is the only quadratically convergent one numerical examples are presented to illustrate rapid convergence

equivalence and symmetries for variable coefficient linear heat type equations i a systematic and unified approach to transformations and symmetries of a class of variable coefficient heat type linear partial differential equations pdes is presented the complete symmetry group classification is re performed in the current context a useful criterion which is necessary and sufficient for being reducible to the standard heat equation by point transformations is established a similar criterion is also valid for the equations to have a four or six dimensional symmetry group nontrivial symmetry groups in this situation the basis elements are listed in terms of coefficients a number of illustrative examples are given in particular some applications from the recent literature are re examined in our new approach multidimensional parabolic pdes of heat and schrodinger type are also considered

complex chebyshev approximation for fir digital filters an adaptive approach to fir finite impulse response filter design is applied to a fully general complex specification showing it to be capable of giving an optimum chebyshev solution it is thought to be one of the first practical approaches to achieve this in the case of a complex function design examples include a bandpass filter with nearly linear phase and all pass filters with general shaping phase comparisons are drawn with equivalent results from other algorithms

factorization method and new potentials with the oscillator spectrum a one parameter family of potentials in one dimension is constructed with the energy spectrum coinciding with that of the harmonic oscillator this is a new derivation of a class of potentials previously obtained by abraham and moses with the help of the gelfand levitan formalism

nonlocal transformations of kolmogorov equations into the backward heat equation we extend and solve the classical kolmogorov problem of finding general classes of kolmogorov equations that can be transformed to the backward heat equation these new classes include kolmogorov equations with time independent and time dependent coefficients our main idea is to include nonlocal transformations we describe a step by step algorithm for determining such transformations we also show how all previously known results arise as particular cases in this wider framework

unitary transformation approach to the exact solutions of time dependent quantum systems with su 1 1 dynamical group a unitary transformation is introduced to determine the exact solution of the time dependent quantum system as examples two quantum systems with hamiltonian including the quadratic term of generators of su 1 1 are solved analytically

on the convergence of multi grid methods with transforming smoothers in the present paper we give a convergence theory for multi grid methods with transforming smoothers as introduced in 31 applied to a general system of partial differential equations the theory follows hackbusch s approach for scalar pde and allows a convergence proof for some well known multi grid methods for stokes and navier stokes equations as dgs by brandt dinar 5 tilu from 31 and the simple methods by patankar spalding 23

efficient preconditioners for optimality systems arising in connection with inverse problems this paper is devoted to the numerical treatment of linear optimality systems os that arise in connection with inverse problems for partial differential equations if such inverse problems are regularized by tikhonov regularization then it follows from standard theory that the associated os is well posed provided that the regularization parameter alpha is positive and that the involved state equation satisfies suitable assumptions we explain and analyze how certain mapping properties of the operators appearing in the os can be employed to define efficient preconditioners for finite element fe approximations of such systems the key feature of the scheme is that the number of iterations needed to solve the preconditioned problem by the minimal residual method is bounded independently of the mesh parameter h used in the fe discretization and increases only moderately as alpha rightarrow0 more specifically if the stopping criterion for the iteration process is defined in terms of the associated energy norm then the number of iterations required in the severely ill posed case cannot grow faster than o ln alpha 2 our analysis is based on a careful study of the operators involved which yields the distribution of the eigenvalues of the preconditioned os finally the theoretical results are illuminated by a number of numerical experiments addressing both a model problem studied by borzi kunisch and kwak siam j control optim 41 2003 pp 1477 1497 and an inverse problem arising in connection with electrocardiography nielsen cai and lysaker math biosci 210 2007 pp 523 553

block implicit multigrid solution of navier stokes equations in primitive variables abstract a calculation procedure for rapid computation of steady multidimensional viscous flows is presented the method solves the navier stokes equations in primitive variables using a coupled block implicit multigrid procedure the procedure is applicable to finite difference formulations using staggered locations of the flow variables a smoothing technique called symmetrical coupled gauss seidel scgs is proposed and is empirically observed to provide good smoothing rates the viscous flow in a square cavity with a moving top wall is calculated for a range of reynolds numbers calculations with finite difference grids as large as 321 321 nodes have been made to test the accuracy and efficiency of the calculation scheme the cpu times for these calculations are observed to be significantly smaller than other solution algorithms with primitive variable formulation the calculated flow fields in the cavity are in good agreement with earlier studies of the same flow situation

new classes of schrodinger equations equivalent to the free particle equation through non local transformations we introduce new classes of schrodinger equations with time dependent potentials which are transformable to the free particle equation through non local transformations these non local transformations arise when considering the potential systems of the schrodinger equation explicit formulae are given for the potentials and the corresponding solutions related to the solutions of the free particle equation

on schwarz type smoothers for saddle point problems with applications to pde constrained optimization problems in this paper we consider a one shot multigrid strategy for solving the discretized optimality system kkt system of a pde constrained optimization problem in particular we discuss the construction of an additive schwarz type smoother for a certain class of optimal control problems a rigorous multigrid convergence analysis is presented numerical experiments are shown which confirm the theoretical results

on one dimensional schrodinger problems allowing polynomial solutions we discuss the explicit construction of the schrodinger equations admitting a representation through some family of general polynomials almost all solvable quantum potentials are shown to be generated by this approach some generalization has also been performed in higher dimensional problems

sequences of related linear pdes abstract we introduce a transformation theory to generate chains of linear pdes and related solution sequences our procedure depends on representing a given linear pde in terms of a special equivalent system of two coupled linear pdes where the auxiliary dependent variable satisfies the next pde of a sequence the solution of a pde with variable coefficient depending on n 1 constants 1 2 n 1 is obtained from any solution of a pde of the same type with variable coefficient depending on n constants 1 2 n by a simple backlund transformation each sequence contains two inclusive chains since the pde with n constants is a special case of the pde with n 2 constants we generate solutions of wave equations with wave speeds c x 1 2 n fokker planck equations with drifts f x 1 2 n and diffusion equations with diffusivities k x 1 2 n where 1 2 n are arbitrary constants n 1 2 new explicit general solutions are obtained for a class of wave equations with wave speeds depending on three parameters

darboux transformations for schrodinger equations in two variables darboux transformations in one variable form the basis for the factorization methods and have numerous applications to geometry nonlinear equations and susy quantum mechanics in spite of this wide range of applications the theory of darboux transformations in two variables and its elegant relationship to analytic complex functions has not been recognized in the literature to close this gap we develop in this paper the theory of darboux transformation in the context of schrodinger equations in two variables this yields a constructive algorithm to determine the relationship between potential functions which are related by darboux transformations

analysis of the minimal residual method applied to ill posed optimality systems we analyze the performance of the minimal residual minres method applied to linear karush kuhn tucker systems arising in connection with inverse problems such optimality systems typically have a saddle point structure and have unique solutions for all alpha 0 where alpha is the parameter employed in the tikhonov regularization unfortunately the associated spectral condition number is very large for small values of alpha which strongly indicates that their numerical treatment is difficult our main result shows that a broad range of linear ill posed optimality systems can be solved efficiently with the minres method this result is obtained by carefully analyzing the spectrum of the associated saddle point operator except for a few isolated eigenvalues the spectrum consists of three bounded intervals krylov subspace methods handle such problems very well for severely ill posed cases techniques based on chebyshev polynomials are applied to prove that the number of iterations needed by

design of fir filters in the complex domain the design of fir digital filters with a complex valued desired frequency response using the chebyshev error is investigated the complex approximation problem is converted into a real approximation problem which is nearly equivalent to the complex problem a standard linear programming algorithm for the chebyshev solution of overdetermined equations is used to solve the real approximation problem additional constraints are introduced which allow weighting of the phase and or group delay of the approximation digital filters are designed which have nearly constant group delay in the passbands the desired constant group delay which gives the minimum chebyshev error is found to be smaller than that of a linear phase filter of the same length these filters in addition to having a smaller approximately constant group delay have better magnitude characteristics than exactly linear phase filters with the same length the filters have nearly equiripple magnitude and group delay

a robust multigrid method for elliptic optimal control problems we consider the discretized optimality system of a special class of elliptic optimal control problems and propose an all at once multigrid method for solving this discretized system under standard assumptions the convergence of the multigrid method and the robustness of the convergence rates with respect to the involved parameter are shown numerical experiments are presented for illustrating the theoretical results

normalizability of one dimensional quasi exactly solvable schrodinger operators we completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi exactly solvable schrodinger operator on the line methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi exactly solvable hamiltonians and explicit normalizability conditions in general coordinate systems

design and characterization of optimal fir filters with arbitrary phase an algorithm for designing a chebyshev optimal fir filter that approximates an arbitrary complex valued frequency response is presented this algorithm computes the optimal filter by solving the dual to the filter design problem it is guaranteed to converge theoretically and requires o n sup 2 computations per iteration for a filter of length n for the first time properties of the optimal filter are derived and the case where the desired filter has arbitrary constant group delay is studied in detail

gaussian decay of harmonic oscillators and related models abstract we prove that the decay of the eigenfunctions of harmonic oscillators uniform electric or magnetic fields is not stable under 0 order complex perturbations even if bounded of these hamiltonians in the sense that we can produce solutions to the evolutionary schrodinger flows associated to the hamiltonians with a stronger gaussian decay at two distinct times we then characterize in a quantitative way the sharpest possible gaussian decay of solutions as a function of the oscillation frequency or the strength of the field depending on the hamiltonian which is considered this is connected to the hardy s uncertainty principle for free schrodinger evolutions

optimal design of complex fir filters with arbitrary magnitude and group delay responses this paper presents a method for the frequency domain design of digital finite impulse response filters with arbitrary magnitude and group delay responses the method can deal with both the equiripple design problem and the peak constrained least squares pcls design problem consequently the method can also be applied to the equiripple passbands and pcls stopbands design problem as a special case of the pcls design both the equiripple and the pcls design problems are converted into weighted least squares optimization problems they are then solved iteratively with appropriately updated error weighting functions a novel scheme for updating the error weighting function is developed to incorporate the design requirements design examples are included in order to compare the performance of the filters designed using the proposed scheme and several other existing methods

on form preserving transformations for the time dependent schrodinger equation in this paper we point out a close connection between the darboux transformation and the group of point transformations which preserve the form of the time dependent schrodinger equation tdse in our main result we prove that any pair of time dependent real potentials related by a darboux transformation for the tdse may be transformed by a suitable point transformation into a pair of time independent potentials related by a usual darboux transformation for the stationary schrodinger equation thus any real potential solvable via a time dependent darboux transformation can alternatively be solved by applying an appropriate form preserving point transformation of the tdse to a time independent potential the pre eminent role of the latter type of transformations in the solution of the tdse is illustrated with a family of quasi exactly solvable time dependent anharmonic potentials

time dependent potentials associated with exceptional orthogonal polynomials we introduce a method for generating rational extensions of time dependent potentials such that the associated schrodinger equation admits solutions in terms of exceptional orthogonal polynomials our method is applicable to position dependent schrodinger equations as well as to their conventional counterparts for constant mass

multi grid methods for stokes and navier stokes equations in the present paper we introduce transforming iterations an approach to construct smoothers for indefinite systems this turns out to be a convenient tool to classify several well known smoothing iterations for stokes and navier stokes equations and to predict their convergence behaviour epecially in the case of high reynolds numbers using this approach we are able to construct a new smoother for the navier stokes equations based on incomplete lu decompositions yielding a highly effective and robust multi grid method besides some qualitative theoretical convergence results we give large numerical comparisons and tests for the stokes as well as for the navier stokes equations for a general convergence theory we refer to 29

a practical method for designing fir digital filters in the complex domain a new method is presented for the design of fir fillers in the complex domain the method converts the complex approximation problem into an equivalent real nonlinear optimization problem which is solved by successive linearization using an effective geometrical linearization technique the method converges rapidly and the solution has a guaranteed accuracy it also has the flexibility of allowing addition of extra constraints such as constant group delay constraints and transition band magnitude overshoot constraints numerical examples are presented to illustrate the performance of the method

how symmetries yield non invertible mappings of linear partial differential equations abstract nonlocally related partial differential equation pde systems are important in the analysis of a given pde system useful nonlocally related systems can be constructed through conservation law and symmetry based methods in this paper we focus on an application of the symmetry based method to linear pde systems in particular we show how to obtain systematically non invertible mappings of linear pdes to linear pdes as examples we obtain non invertible mappings of the kolmogorov equation with variable coefficients to the backward heat equation a pde with constant coefficients as well as non invertible mappings of linear hyperbolic pdes with variable coefficients to linear hyperbolic pdes with constant coefficients

exactly solvable three body systems with internal degrees of freedom the approach of multi dimensional susy quantum mechanics is used in an explicit construction of exactly solvable three body and quasi exactly solvable n body matrix problems on a line from intertwining relations with time dependent operators we build exactly solvable non stationary scalar and 2 2 matrix three body models which are time dependent extensions of the calogero model finally we investigate the invariant operators associated with these systems

symmetry of time dependent schrodinger equations i a classification of time dependent potentials by their maximal kinematical algebras potentials for the time dependent schrodinger equation 1 2 xx v x t x t i t x t are classified according to their space time or kinematical algebras in a search for exactly solvable time dependent models in addition it is shown that their dynamical algebras are isomorphic to their kinematical algebras on the solution space of the schrodinger equation

supersymmetric transformations of real potentials on the line a systematic study of supersymmetric or darboux factorizations on the line is performed all possible pairs of supersymmetric transformations with the same factorization energy are reviewed for such pairs there is no condition on this energy iterations of single transformations and of pairs both allow arbitrary modifications of the bound spectrum different iterative methods lead to compact analytical equations depending on the initial potential and its solutions iterations of single transformations are able to transform an even potential into an even potential particular cases of a method based on iteration of pairs conserve either the reflection coefficient at all energies or the norming constants of bound states these results are compared with previous methods established in other contexts the most general supersymmetric transformation of a given potential is finally described both with and without modifications of the bound spectrum

on invariance properties of the wave equation a complete group classification is given of both the wave equation c2 x uxx utt 0 i and its equivalent system vt ux c2 x vx ut ii when the wave speed c x const equations i and ii admit either a two or four parameter group for the exceptional case c x ax b 2 equation i admits an infinite group equations i and ii do not always admit the same group for a given c x the group for i can have more parameters or fewer parameters than that for ii moreover the groups can be different with the same number of parameters separately for i and ii all possible c x that admit a four parameter group are found explicitly the corresponding invariant similarity solutions are considered some of these wave speeds have realistic physical properties c x varies monotonically from one positive constant to another positive constant as x goes from to

on the design of fir filters by complex chebyshev approximation the long standing problem of approximating a complex valued desired function with a finite impulse response fir filter is considered it is formulated as an equalization to be solved using complex valued filters the proposed algorithm deals directly with the complex error function which depends linearly on the coefficients of the filter to be designed the magnitude of this error function is minimized in the chebshev sense using a generalization of the remez exchange algorithm the method can be used to design complex or real valued selective systems as well the well known design of optimal fir filters with linear phase is included here as a special case

on mapping linear partial differential equations to constant coefficient equations a constructive algorithm is developed to determine whether or not a given linear p d e can be mapped into a linear p d e with constant coefficients the algorithm is based on analyzing the infinitesimals of the lie group of transformations leaving invariant the given p d e as consequences in two dimensions necessary and sufficient conditions are given for mapping 1 a parabolic p d e into the heat equation 2 a hyperbolic p d e into the wave equation 3 an elliptic p d e into laplace s equation or the helmholtz equation the corresponding mappings are given explicitly

analysis of reflected diffusions via an exponential time based transformation let x t be a time homogeneous diffusion process with state space 0 infty where 0 is a reflecting or entrance endpoint and let z denote a random variable that describes the process x t evaluated at an exponentially distributed random time we propose a method to obtain closed form expressions for the conditional density and the mean of a new diffusion process y t with the same state space and with the same infinitesimal variance whose drift depends on the infinitesimal moments of x t and on the hazard rate function of z this method also allows us to obtain the laplace transform of the first passage time density of y t through a lower constant boundary we then discuss the relation between y t and the process x t subject to catastrophes as well as the interpretation of y t as a diffusion in a decreasing potential we study in detail some special cases concerning diffusion processes obtained when x t is the wiener ornstein uhlenbeck bessel and rayleigh process

three dimensional solutions of supersymmetrical intertwining relations and pairs of isospectral hamiltonians the general solution of supersymmetrical intertwining relations for three dimensional schrodinger operators is built using the class of second order supercharges with nondegenerate constant metric this solution includes several models with arbitrary parameters we are interested only in quantum systems which are not amenable to separation of variables i e cannot be reduced to lower dimensional problems all constructed hamiltonians are partially integrable each of them commutes with a symmetry operator of fourth order in momenta the same models can be considered also for complex values of parameters leading to a class of non hermitian isospectral hamiltonians

intertwined isospectral potentials in an arbitrary dimension the method of intertwining with n dimensional nd linear intertwining operator l is used to construct nd isospectral stationary potentials it has been proven that the differential part of l is a series in euclidean algebra generators integrability conditions of the consistency equations are investigated and the general form of a class of potentials respecting all these conditions have been specified for each n 2 3 4 5 the most general forms of 2d and 3d isospectral potentials are considered in detail and construction of their hierarchies is exhibited the followed approach provides coordinate systems which make it possible to perform separation of variables and to apply the known methods of supersymmetric quantum mechanics for 1d systems it has been shown that in choice of coordinates and l there are a number of alternatives increasing with n that enlarge the set of available potentials some salient features of higher dimensional extension as well as some applications of the results are presented

intertwining relations of non stationary schrodinger operators general first and higher order intertwining relations between non stationary one dimensional schrodinger operators are introduced for the first order case it is shown that the intertwining relations imply some hidden symmetry which in turn results in an r separation of variables the fokker planck and diffusion equations are briefly considered second order intertwining operators are also discussed within a general approach however due to its complicated structure only particular solutions are given in some detail

a comparison of the extrapolated successive overrelaxation and the preconditioned simultaneous displacement methods for augmented linear systems in this paper we study the impact of two types of preconditioning on the numerical solution of large sparse augmented linear systems the first preconditioning matrix is the lower triangular part whereas the second is the product of the lower triangular part with the upper triangular part of the augmented system s coefficient matrix for the first preconditioning matrix we form the generalized modified extrapolated successive overrelaxation gmesor method whereas the second preconditioning matrix yields the generalized modified preconditioned simultaneous displacement gmpsd method which is an extrapolated form of the symmetric successive overrelaxation method we find sufficient conditions for each aforementioned iterative method to converge in addition we develop a geometric approach for determining the optimum values of their parameters and corresponding spectral radii it is shown that both iterative methods studied gmesor and gmpsd attain the same rate of convergence numerical results confirm our theoretical expectations

accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system the finite difference multigrid solution of an optimal control problem associated with an elliptic equation is considered stability of the finite difference optimality system and optimal order error estimates in the discrete l2 norm and in the discrete h1 norm under minimum smoothness requirements on the exact solution are proved sharp convergence factor estimates of the two grid method for the optimality system are obtained by means of local fourier analysis a multigrid convergence theory is provided which guarantees convergence of the multigrid process towards weak solutions of the optimality system