infinite time exact observability of volterra systems in hilbert spaces abstract we study infinite time exact observability of volterra systems in hilbert spaces sufficient conditions are established under which infinite time exact observability of a volterra system follows from that of the corresponding cauchy system without convolution term this is done by combining the composition operator method with a result about bounded below composition operators on the hardy space

factorizing the factorization a spectral element solver for elliptic equations with linear operation count abstract the paper proposes a novel factorization technique for static condensation of a spectral element discretization matrix that yields a linear operation count of just 13n multiplications for the residual evaluation where n is the total number of unknowns in comparison to previous work it saves a factor larger than 3 and outpaces unfactored variants for all polynomial degrees using the new technique as a building block for a preconditioned conjugate gradient method yields linear scaling of the runtime with n which is demonstrated for polynomial degrees from 2 to 32 this makes the spectral element method cost effective even for low polynomial degrees moreover the dependence of the iterative solution on the element aspect ratio is addressed showing only a slight increase in the number of iterations for aspect ratios up to 128 hence the solver is very robust for practical applications

controllability and observability for non autonomous evolution equations the averaged hautus test abstract we consider the observability problem for non autonomous evolution systems i e the operators governing the system depend on time we introduce an averaged hautus condition and prove that for skew adjoint operators it characterizes exact observability next we extend this to more general class of operators under a growth condition on the associated evolution family we give an application to the schrodinger equation with time dependent potential and the damped wave equation with a time dependent damping coefficient

exact observability conditions for hilbert space dynamical systems connected with riesz basis of divided differences abstract in the paper we consider the problem of exact observability of a general class of distributed parameter systems in hilbert spaces we show that under some conditions on asymptotic behavior of the spectrum of the differential operator the system is not exact observable in default topologies and find a stronger topology for state observation for which the system becomes exactly observable we illustrate this result with a vibrating clamped free timoshenko beam model

an excmg accelerated multiscale multigrid computation for 3d poisson equation abstract multiscale multigrid msmg method is an effective computational framework for efficiently computing high accuracy solutions for elliptic partial differential equations in the current msmg method compared to the cpu cost on computing sixth order solutions by applying extrapolation and other techniques on two fourth order solutions from different scales grids much more cpu time is spent on computing fourth order solutions themselves on coarse and fine grids particularly for high dimensional problems here we propose to embed extrapolation cascadic multigrid excmg method into the msmg framework to accelerate the whole process numerical results on 3d poisson equations show that the new excmg msmg method is more efficient than the existing msmg method and the excmg method for sixth order solution computation

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

a cascadic multigrid algorithm for the stokes equations a variant of multigrid schemes for the stokes problem is discussed in particular we propose and analyse a cascadic version for the stokes problem the analysis of the transfer between the grids requires special care in order to establish that the complexity is the same as that for classical multigrid algorithms

functional approach to the error control in adaptive iga schemes for elliptic boundary value problems abstract this work presents a numerical study of functional type a posteriori error estimates for iga approximation schemes in the context of elliptic boundary value problems along with the detailed discussion of the most crucial properties of such estimates we present the algorithm of a reliable solution approximation together with the scheme of an efficient a posteriori error bound generation in this approach we take advantage of b thb spline s high smoothness for the auxiliary vector function reconstruction which at the same time allows to use much coarser meshes and decrease the number of unknowns substantially the most representative numerical results obtained during a systematic testing of error estimates are presented in the second part of the paper the efficiency of the obtained error bounds is analysed from both the error estimation indication and the computational expenses points of view several examples illustrate that functional error estimates alternatively referred to as the majorants and minorants of deviation from an exact solution perform a much sharper error control than for instance residual based error estimates simultaneously assembling and solving routines for an auxiliary variables reconstruction which generate the majorant or minorant of an error can be executed several times faster than the routines for a primal unknown

a cascadic multigrid algorithm for semilinear elliptic problems summary we propose a cascadic multigrid algorithm for a semilinear elliptic problem the nonlinear equations arising from linear finite element discretizations are solved by newton s method given an approximate solution on the coarsest grid on each finer grid we perform exactly one newton step taking the approximate solution from the previous grid as initial guess the newton systems are solved iteratively by an appropriate smoothing method we prove that the algorithm yields an approximate solution within the discretization error on the finest grid provided that the start approximation is sufficiently accurate and that the initial grid size is sufficiently small moreover we show that the method has multigrid complexity

an iterative multi level algorithm for solving nonlinear ill posed problems the convergence analysis of landweber s iteration for the solution of nonlinear ill posed problem has been developed recently by hanke neubauer and scherzer in concrete applications sufficient conditions for convergence of the landweber iterates developed there although quite natural turned out to be complicated to verify analytically however in numerical realizations when discretizations are considered sufficient conditions for local convergence can usually easily be stated this paper is motivated by these observations initially a discretization is fixed and a discrete landweber iteration is implemented until an appropriate stopping criterion becomes active the output is used as an initial guess for a finer discretization an advantage of this method is that the convergence analysis can be considered in a family of finite dimensional spaces the numerical performance of this multi level algorithm is compared with landweber s iteration

resolvent conditions for the control of parabolic equations abstract since the seminal work of russell and weiss in 1994 resolvent conditions for various notions of admissibility observability and controllability and for various notions of linear evolution equations have been investigated intensively sometimes under the name of infinite dimensional hautus test this paper sets out resolvent conditions for null controllability in arbitrary time necessary for general semigroups sufficient for analytic normal semigroups for a positive self adjoint operator a it gives a sufficient condition for the null controllability of the semigroup generated by a which is only logarithmically stronger than the usual condition for the unitary group generated by ia this condition is sharp when the observation operator is bounded the proof combines the so called control transmutation method and a new version of the direct lebeau robbiano strategy the improvement of this strategy also yields interior null controllability of new logarithmic anomalous diffusions

domain decomposition methods for large linearly elliptic three dimensional problems abstract the idea of solving large problems using domain decomposition technique appears particularly attractive on present day large scale parallel computers but the performance of such techniques used on a parallel computer depends on both the numerical efficiency of the proposed algorithm and the efficiency of its parallel implementation the approach proposed herein splits the computational domain in unstructured subdomains of arbitrary shape and solves for unknown on the interface using the associated trace operator the steklov poincare operator on the continuous level or the schur complement matrix after a finite element discretization and a preconditioned conjugate gradient method this algorithm involves the solution of dirichlet and of neumann problems defined on each subdomain and which can be solved in parallel this method has been implemented on a cray 2 computer using multitasking and on an intel hypercube it was tested on a large scale industrial ill conditioned three dimensional linear elasticity problem which gives a fair indication of its performance in a real life environment in such situations the proposed method appears operational and competitive on both machines compared to standard techniques it yields faster results with far less memory requirements

the convergence of the cascadic conjugate gradient method applied to elliptic problems in domains with re entrant corners we study the convergence properties of the cascadic conjugate gradient method ccg method which can be considered as a multilevel method without coarse grid correction nevertheless the ccg method converges with a rate that is independent of the number of unknowns and the number of grid levels we prove this property for two dimensional elliptic second order dirichlet problems in a polygonal domain with an interior angle greater than for plecewise linear finite elements we construct special nested triangulations that satisfy the conditions of a triangulation of type h l in the sense of i babuska r b kellogg and j pitkaranta in this way we can guarantee both the same order of accuracy in the energy norm of the discrete solution and the same convergence rate of the ccg method as in the case of quasiuniform triangulations of a convex polygonal domain

a new parallel cascadic multigrid method in this paper a new parallel cascadic multigrid method is studied with domain decomposition technique and extrapolation technique convergence analysis is given based on the superconvergence of the finite element the numerical results show that the method is efficient

substructuring preconditioners for the q_1 mortar element method the q 1 mortar element method is a non conforming finite element method with q 1 elements based on domain decomposition for the laplace equation it yields an ill conditioned linear system for solving the linear system the so called preconditioned conjugate gradient method in a subspace is used preconditioners are proposed and estimates on condition numbers and arithmetical complexity are given finally numerical experiments are presented

hierarchical a posteriori error estimators for mortar finite element methods with lagrange multipliers hierarchical a posteriori error estimators are introduced and analyzed for mortar finite element methods a weak continuity condition at the interfaces is enforced by means of lagrange multipliers the two proposed error estimators are based on a defect correction in higher order finite element spaces and an adequate hierarchical two level splitting the first provides upper and lower bounds for the discrete energy norm of the mortar finite element solution whereas the second also estimates the error for the lagrange multiplier it is shown that an appropriate measure for the nonconformity of the mortar finite element solution is the weighted l2 norm of the jumps across the interfaces

multiplier spaces for the mortar finite element method in three dimensions we consider the construction of multiplier spaces for use with the mortar finite element method in three spatial dimensions on globally or locally quasi uniform meshes a set of abstract conditions is given for the multiplier spaces which are sufficient to guarantee a stable and convergent mortar approximation three examples of multipliers satisfying these conditions are presented the first one is a dual basis example while the remaining two are based on finite volumes finally the results of computational examples illustrating the theory are reported

on the convergence of an extrapolation cascadic multigrid method for elliptic problems abstract classical cascadic multigrid method is optimal under the energy norm for h 2 regular elliptic problems this paper analyzes an extrapolation cascadic multigrid excmg method originally proposed by chen et al 2008 for solving second order elliptic equations following the idea of bornemann and deuflhard 1996 we present a superconvergence result for the excmg method which enables us to show that the excmg method with the conjugate gradient method as a basic iterative scheme is optimal for h 3 regular elliptic problems in three dimension with respect to the l 2 norm moreover we also prove the super optimality of the excmg method under the energy norm for h 2 regular 0 1 problems in both two and three dimensions with a reasonable assumption on asymptotic error expansions finally numerical results are presented to verify our theoretical analysis

on the multi level splitting of finite element spaces in this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases we show that the condition number of such a stiffness matrix behaves like o log 2 where is the condition number of the stiffness matrix with respect to a nodal basis in the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like o left left log frac 1 h right 2 right instead of o left left frac 1 h right 2 right for a nodal basis the proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization especially we do not need the quasiuniformity of the employed triangulations as the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis our results mean that the method of conjugate gradients needs onlyo log n steps ando n log n computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures heren denotes the dimension of the finite element space and of the discrete linear problem to be solved

a stabilized domain decomposition method with nonmatching grids for the stokes problem in three dimensions we present and study a nonconforming domain decomposition method for the discretization of the three dimensional stokes problem in the velocity pressure formulation the approximation is based on some local mixed finite elements for nonmatching tetrahedral grids the aim pursued is a systematic construction of the mortared discrete velocity space the pressure being not subjected to any matching constraints across the interfaces using the bubble stabilization techniques applied in brezzi and marini s paper to the three fields method math comp 70 2001 pp 911 934 allows us to define an algorithm which is easy to implement the numerical analysis relies on the pressure splitting argument of boland and nicolaides and allows us to establish an inf sup condition with a constant that does not depend on the mesh size or on the total number of the subdomains then by the berger scott strang lemma written down for our saddle point system we derive optimal accuracy results

korn s inequalities for piecewise h 1 vector fields korn s inequalities for piecewise h 1 vector fields are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

analysis of parallel finite element algorithm based on three linearization methods for the steady incompressible mhd flow abstract based on full domain partition parallel finite element algorithm based on three linearizations is designed to solve the stationary incompressible magnetohydrodynamics mhd in every algorithm each processor is in charge of a subproblem which is linearized by three options stokes type newton s and picard s iterations each subproblem is defined in the whole domain with vast majority of the degrees of freedom associated with the particular subdomain the error estimates of these proposed algorithms with respect to iterative step and small mesh sizes are analyzed finally some numerical experiments are provided to show the validity and high efficiency of our algorithms

error estimates of the lumped mass finite element method for semilinear elliptic problems we are concerned with the semilinear elliptic problems we first investigate the l 2 error estimate for the lumped mass finite element method we then use the cascadic multigrid method to solve the corresponding discrete problem on the basis of the finite element error estimates we prove the optimality of the proposed multigrid method we also report some numerical results to support the theory

convergence estimates for multigrid algorithms without regularity assumptions a new technique for proving rate of convergence estimates of multi grid algorithms for symmetric positive definite problems will be given in this paper the standard multigrid theory requires a regularity and approxima tion assumption in contrast the new theory requires only an easily verified approximation assumption this leads to convergence results for multigrid re finement applications problems with irregular coefficients and problems whose coefficients have large jumps in addition the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

error estimates for the three field formulation with bubble stabilization in this paper we prove convergence and error estimates for the so called 3 field formulation using piecewise linear finite elements stabilized with boundary bubbles optimal error bounds are proved in l 2 and in the broken h 1 norm for the internal variable u and in suitable weighted l 2 norms for the other variables a and

a posteriori error estimates for elliptic variational inequalities we derive hierarchical a posteriori error estimates for elliptic variational inequalities the evaluation amounts to the solution of corresponding scalar local subproblems we derive some upper bounds for the effectivity rates and the numerical properties are illustrated by typical examples

cascadic multilevel methods for ill posed problems multilevel methods are popular for the solution of well posed problems such as certain boundary value problems for partial differential equations and fredholm integral equations of the second kind however little is known about the behavior of multilevel methods when applied to the solution of linear ill posed problems such as fredholm integral equations of the first kind with a right hand side that is contaminated by error this paper shows that cascadic multilevel methods with a conjugate gradient type method as basic iterative scheme are regularization methods the iterations are terminated by a stopping rule based on the discrepancy principle

some new discretization and adaptation and multigrid methods for 2 d 3 t diffusion equations in the simulation of laser driven implosion of a fuel capsule in inertial confinement fusion experiments a system of two dimensional diffusion equations coupled with electron iron and photon temperature are widely used to approximately describe the process of energy across multiple materials and the exchange of energy among electrons irons and photons the numerical solution of such equations is always challenging because of its strong nonlinear phenomena and strong discontinuous interfaces in this article we design a symmetric finite volume method develop the corresponding preconditioning technique and propose a mesh adaptation algorithm based on hessian matrix and a two grid method using these new methods we demonstrate that the energy conservation error and computation efficiency of the integrated algorithm are much better than classical method

formulation and numerical solution of finite level quantum optimal control problems optimal control of finite level quantum systems is investigated and iterative solution schemes for the optimization of a control representing laser pulses are developed the purpose of this external field is to channel the system s wavefunction between given states in its most efficient way physically motivated constraints such as limited laser resources or population suppression of certain states are accounted for through an appropriately chosen cost functional first order necessary optimality conditions and second order sufficient optimality conditions are investigated for solving the optimal control problems a cascadic non linear conjugate gradient scheme and a monotonic scheme are discussed results of numerical experiments with a representative finite level quantum system demonstrate the effectiveness of the optimal control formulation and efficiency and robustness of the proposed approaches

a two grid discretization method for decoupling systems of partial differential equations in this paper we propose a two grid finite element method for solving coupled partial differential equations e g the schrodinger type equation with this method the solution of the coupled equations on a fine grid is reduced to the solution of coupled equations on a much coarser grid together with the solution of decoupled equations on the fine grid it is shown both theoretically and numerically that the resulting solution still achieves asymptotically optimal accuracy

cyclic schemes for pde based image analysis we investigate a class of efficient numerical algorithms for many partial differential equations pdes in image analysis they are applicable to parabolic or elliptic pdes that have bounded coefficients and lead to space discretisations with symmetric matrices our schemes are easy to implement and well suited for parallel implementations on gpus since they are based on the explicit diffusion scheme in the parabolic case and the jacobi method in the elliptic case by supplementing these methods with cyclically varying time step sizes or relaxation parameters we achieve efficiency gains of several orders of magnitude we call the resulting algorithms fast explicit diffusion fed and fast jacobi fj methods to achieve a good compromise between efficiency and accuracy we show that one should use parameter cycles that result from factorisations of box filters for these cycles we establish stability results in the euclidean norm our schemes perform favourably in a number of applications including isotropic nonlinear diffusion filters with widely varying diffusivities as well as anisotropic diffusion methods for image filtering inpainting and regularisation in computer vision moreover they are equally suited for higher dimensional problems as well as higher order pdes and they can also be interpreted as efficient first order methods for smooth optimisation problems

a basic norm equivalence for the theory of multilevel methods subspace decompositions of finite element spaces based onl 2 like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods recently several authors have proved the equivalence of the associated discrete norms with theh 1 norm the present paper gives an elementary self contained derivation of this result which is based on the use ofk functionals known from the theory of interpolation spaces

simplified conditions of initial observability for infinite dimensional second order damped dynamical systems abstract in the paper the initial observability of abstract second order infinite dimensional dynamical systems is considered two cases of observation operator are explored first the observation operator is assumed to be bounded then results are extended to more general class of observations related to the observation operators that are assumed to be bounded with respect to square root of elasticity operator which covers the case of some unboundedness in observation using the frequency domain method it is proved that initial observability of second order system under discussion can be verified by the initial observability conditions for the corresponding simplified first order system general results are then applied for initial observability investigation of infinite dimensional systems described by partial differential equations modelling flexible mechanical structures some special cases are also discussed and practical comments and remarks are given the paper extends earlier results on initial observability for a class of infinite dimensional second order abstract dynamical systems

multigrid smoothers for ultraparallel computing this paper investigates the properties of smoothers in the context of algebraic multigrid amg running on parallel computers with potentially millions of processors the development of multigrid smoothers in this case is challenging because some of the best relaxation schemes such as the gauss seidel gs algorithm are inherently sequential based on the sharp two grid multigrid theory from r d falgout and p s vassilevski siam j numer anal 42 2004 pp 1669 1693 and r d falgout p s vassilevski and l t zikatanov numer linear algebra appl 12 2005 pp 471 494 we characterize the smoothing properties of a number of practical candidates for parallel smoothers including several c f polynomial and hybrid schemes we show in particular that the popular hybrid gs algorithm has multigrid smoothing properties which are independent of the number of processors in many practical applications provided that the problem size per processor is large enough this is encouraging news for the scalability of amg on ultraparallel computers we also introduce the more robust ell 1 smoothers which are always convergent and have already proven essential for the parallel solution of some electromagnetic problems t kolev and p vassilevski j comput math 27 2009 pp 604 623

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

multiscale linear solvers for very large systems derived from pdes we present a novel linear solver that works well for large systems obtained from discretizing pdes it is robust and for the examples we studied the computational effort scales linearly with the number of equations the algorithm is based on a wavelength decomposition that combines conjugate gradient multiscaling and iterative splitting methods into a single approach on the surface the algorithm is a simple preconditioned conjugate gradient with all the sophistication of the algorithm in the choice of the preconditioning matrix the preconditioner is a very good approximate inverse of the linear operator it is constructed from the inverse of the coarse grained linear operator and from smoothing operators that are based on an operator splitting on the fine grid the coarse graining captures the long wavelength behavior of the inverse operator while the smoothing operator captures the short wavelength behavior the conjugate gradient iteration accounts for the coupling between long and short wavelengths the coarse grained operator corresponds to a lower resolution approximation to the pdes while the coarse grained inverse is not known explicitly the algorithm requires only that the preconditioner can be a applied to a vector the coarse inverse applied to a vector can be obtained as the solution of another preconditioned conjugate gradient solver that applies the same algorithm to the smaller problem thus the method is naturally recursive the recursion ends when the matrix is sufficiently small for a solution to be obtained efficiently with a standard solver the local feedback provided by the conjugate gradient step at every level makes the algorithm very robust in spite of the effort required for the coarse inverse the algorithm is efficient because the increased quality of the approximate inverse greatly reduces the number of times the preconditioner needs to be evaluated a feature of the algorithm is that the transition between coarse grids is determined dynamically by the accuracy requirement of the conjugate gradient solver at each level typically later iterations on the finer scales need fewer iterations on the coarser scales and the computational effort is proportional to n rather than n log n where n is the number of equations we have tested our solver on the porous flow equation on a workstation we have solved problems on grids ranging in dimension over three orders of magnitude from 103 to 106 and found that the linear scaling holds the algorithm works well even when the permeability tensor has spatial variations exceeding a factor of 109

adaptive multivlevel methods in three space dimensions we consider the approximate solution of self adjoint elliptic problems in three space dimensions by piecewise linear finite elements with respect to a highly non uniform tetrahedral mesh which is generated adaptively the arising linear systems are solved iteratively by the conjugate gradient method provided with a multilevel preconditioner here the accuracy of the iterative solution is coupled with the discretization error as the performance of hierarchical bases preconditioners deteriorates in three space dimensions the bpx preconditioner is used taking special care of an efficient implementation reliable a posteriori estimates for the discretization error are derived from a local comparison with the approximation resulting from piecewise quadratic elements to illustrate the theoretical results we consider a familiar model problem involving reentrant corners and a real life problem arising from hyperthermia a recent clinical method for cancer therapy

from functional analysis to iterative methods we examine condition numbers preconditioners and iterative methods for finite element discretizations of coercive pdes in the context of the fundamental solvability result the lax milgram lemma working in this hilbert space context is justified because finite element operators are restrictions of infinite dimensional hilbert space operators to finite dimensional subspaces moreover useful insight is gained as to the relationship between hilbert space and matrix condition numbers and translating hilbert space fixed point iterations into matrix computations provides new ways of motivating and explaining some classic iteration schemes in this framework the simplest preconditioner for an operator from a hilbert space into its dual is the riesz isomorphism simple analysis gives spectral bounds and iteration counts bounded independent of the finite element subspaces chosen moreover the abstraction allows us to consider not only riesz map preconditioning for convection diffusion equations in h 1 but also operators on other hilbert spaces such as planar elasticity in left h 1 right 2

an economical cascadic multigrid method for the weak galerkin finite element approximation of second order elliptic problems abstract in this paper we present both error and computational complexity estimates of economical cascadic multigrid solver for the linear system of equations arising from weak galerkin finite element approximation of second order elliptic problems on triangular meshes our analysis shows that the proposed economical cascadic multigrid method with the conjugate gradient smoother is optimal in both accuracy and computational complexity for two dimensional problems in addition an elimination technique is utilized to further improve the computation efficiency the interior degrees of freedom on each element are removed from the resulting linear system so that the size of the problem becomes much smaller we demonstrate the accuracy and efficiency of our proposed economical cascadic multigrid methods through ample numerical experiments the numerical results show that compared with the usual cascadic multigrid method the elimination economical cascadic multigrid method saves computational cost significantly

convergence of nonconforming v cycle and f cycle multigrid algorithms for second order elliptic boundary value problems the convergence of v cycle and f cycle multigrid algorithms with a sufficiently large number of smoothing steps is established for nonconforming finite element methods for second order elliptic boundary value problems

on the hautus test for exponentially stable c_0 groups for finite dimensional systems the hautus test is a well known and easy checkable condition for observability russell and weiss siam j control optim 32 1994 pp 1 23 suggested an infinite dimensional version of the hautus test which is necessary for exact observability and sufficient for approximate observability of exponentially stable systems in this paper it is shown that this hautus test is sufficient for exact observability of certain exponentially stable systems generated by a c 0 group and it is proved that the hautus test is in general not sufficient for approximate observability of strongly stable systems even if the system is modeled by a contraction semigroup and the observation operator is bounded

two scale dirichlet neumann preconditioners for elastic problems with boundary refinements abstract the present work deals with the efficient resolution of elastostatics problems on domains with boundary refinements the proposed approach separates the boundary refinements from the interior of the domain by the mortar method and uses dirichlet neumann preconditioners to solve the corresponding algebraic system we prove that the simplest dirichlet neumann algorithm achieves independence of the condition number of the preconditioned system with respect to the number and the size of the small details nevertheless the situation no longer prevails when the refined boundary is clamped an enhanced preconditioner is then designed by the introduction of a coarse space to mitigate the aforementioned sensitivity some numerical tests are performed to confirm the analysis and the tools are extended by the proposition of a quasi newton method in the case of nonlinear elasticity this paper is an extended version of a work presented at the dd16 conference with proofs and complete numerical results

an optimal cascadic multigrid method for the radiative transfer equation this paper presents a fast and optimal multigrid solver for the radiative transfer equation a discrete ordinate discontinuous streamline diffusion method is employed to discretize the radiative transfer equation instead of utilizing conventional multigrid methods for spatial variables only a spatial cascadic multigrid method and a full cascadic multigrid method are developed to achieve rapid convergence in iterative calculation preliminary analysis is also conducted suggesting the optimal convergence rate numerical tests show a significant reduction of the computational time compared to conventional iterative methods for the radiative transfer equation

asymptotic error expansion and richardson extrapolation for linear fine elements the elliptic ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes this justifies the application of richardson extrapolation for increasing the accuracy

exact observability square functions and spectral theory in the first part of this article we introduce the notion of a backward forward conditioning bfc system that generalises the notion of zero class admissibility introduced in xu et al 2008 30 we can show that unless the spectrum contains a half plane the bfc property occurs only in situations where the underlying semigroup extends to a group in a second part we present a sufficient condition for exact and final state observability in banach spaces that is designed for infinite dimensional output spaces and general strongly continuous semigroups to obtain this we make use of certain weighted square function estimates specialising to the hilbert space situation we obtain a result for contraction semigroups without an analyticity condition on the semigroup

a posteriori error estimates for elliptic problems in two and three space dimensions let u in h be the exact solution of a given selfadjoint elliptic boundary value problem which is approximated by some tilde u in mathcal s mathcal s being a suitable finite element space efficient and reliable a posteriors estimates of the error u tilde u measuring the local quality of tilde u play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh a well known class of error estimates can be derived systematically by localizing the discretized defect problem by using domain decomposition techniques in this paper we provide a guideline for the theoretical analysis of such error estimates we further clarify the relation to other concepts our analysis leads to new error estimates which are specially suited to three space dimensions the theoretical results are illustrated by numerical computations

variational optical flow computation in real time this paper investigates the usefulness of bidirectional multigrid methods for variational optical flow computations although these numerical schemes are among the fastest methods for solving equation systems they are rarely applied in the field of computer vision we demonstrate how to employ those numerical methods for the treatment of variational optical flow formulations and show that the efficiency of this approach even allows for real time performance on standard pcs as a representative for variational optic flow methods we consider the recently introduced combined local global method it can be considered as a noise robust generalization of the horn and schunck technique we present a decoupled as well as a coupled version of the classical gau spl szlig seidel solver and we develop several multigrid implementations based on a discretization coarse grid approximation in contrast with standard bidirectional multigrid algorithms we take advantage of intergrid transfer operators that allow for nondyadic grid hierarchies as a consequence no restrictions concerning the image size or the number of traversed levels have to be imposed in the experimental section we juxtapose the developed multigrid schemes and demonstrate their superior performance when compared to unidirectional multigrid methods and nonhierachical solvers for the well known 316 spl times 252 yosemite sequence we succeeded in computing the complete set of dense flow fields in three quarters of a second on a 3 06 ghz pentium4 pc this corresponds to a frame rate of 18 flow fields per second which outperforms the widely used gau spl szlig seidel method by almost three orders of magnitude

cascadic multilevel algorithms for symmetric saddle point systems abstract in this paper we introduce a multilevel algorithm for approximating variational formulations of symmetric saddle point systems the algorithm is based on availability of families of stable finite element pairs and on the availability of fast and accurate solvers for symmetric positive definite systems on each fixed level an efficient solver such as the gradient or the conjugate gradient algorithm for inverting a schur complement is implemented the level change criterion follows the cascade principle and requires that the iteration error be close to the expected discretization error we prove new estimates that relate the iteration error and the residual for the constraint equation the new estimates are the key ingredients in imposing an efficient level change criterion the first iteration on each new level uses information about the best approximation of the discrete solution from the previous level the theoretical results and experiments show that the algorithms achieve optimal or close to optimal approximation rates by performing a non increasing number of iterations on each level even though numerical results supporting the efficiency of the algorithms are presented for the stokes system the algorithms can be applied to a large class of boundary value problems including first order systems that can be reformulated at the continuous level as symmetric saddle point problems such as the maxwell equations

stopping criteria for iterations in finite element methods this work extends the results of arioli 1 2 on stopping criteria for iterative solution methods for linear finite element problems to the case of nonsymmetric positive definite problems we show that the residual measured in the norm induced by the symmetric part of the inverse of the system matrix is relevant to convergence in a finite element context we then use krylov solvers to provide alternative ways of calculating or estimating this quantity and present numerical experiments which validate our criteria

nonconforming finite elements and the cascadic multi grid method we derive sufficient conditions under which the cascadic multi grid method applied to nonconforming finite element discretizations yields an optimal solver key ingredients are optimal error estimates of such discretizations which we therefore study in detail we derive a new efficient modified morley finite element method optimal cascadic multi grid methods are obtained for problems of second and using a new smoother of fourth order as well as for the stokes problem

a multigrid method enhanced by krylov subspace iteration for discrete helmholtz equations standard multigrid algorithms have proven ineffective for the solution of discretizations of helmholtz equations in this work we modify the standard algorithm by adding gmres iterations at coarse levels and as an outer iteration we demonstrate the algorithm s effectiveness through theoretical analysis of a model problem and experimental results in particular we show that the combined use of gmres as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200 for fixed wave numbers it displays grid independent convergence rates and has costs proportional to the number of unknowns

iterative substructuring preconditioners for mortar element methods in two dimensions the mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains the resulting methods are nonconforming but still yield optimal approximations in this paper we will discuss iterative substructuring algorithms for the algebraic systems arising from the discretization of symmetric second order elliptic equations in two dimensions both spectral and finite element methods for geometrically conforming as well as nonconforming domain decompositions are studied in each case we obtain a polylogarithmic bound on the condition number of the preconditioned matrix

a mortar finite element method using dual spaces for the lagrange multiplier the mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries in the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers in this paper this lagrange multiplier space is replaced by a dual space without losing the optimality of the method the advantage of this new approach is that the matching condition is much easier to realize in particular all the basis functions of the new method are supported in a few elements the mortar map can be represented by a diagonal matrix in the standard mortar method a linear system of equations must be solved the problem is considered in a positive definite nonconforming variational as well as an equivalent saddle point formulation

analysis of two grid methods for reaction diffusion equations by expanded mixed finite element methods we present two efficient methods of two grid scheme for the approximation of two dimensional semi linear reaction diffusion equations using an expanded mixed finite element method to linearize the discretized equations we use two newton iterations on the fine grid in our methods firstly we solve an original non linear problem on the coarse grid then we use twice newton iterations on the fine grid in our first method and while in second method we make a correction on the coarse grid between two newton iterations on the fine grid these two grid ideas are from xu s work siam j sci comput 1994 15 231 237 siam j numer anal 1996 33 1759 1777 on standard finite element method we extend the ideas to the mixed finite element method moreover we obtain the error estimates for two algorithms of two grid method it is showed that coarse space can be extremely coarse and we achieve asymptotically optimal approximation as long as the mesh sizes satisfy h h in the first algorithm and h h in second algorithm copyright 2006 john wiley sons ltd

a multilevel successive iteration method for nonlinear elliptic problems in this paper a multilevel successive iteration method for solving nonlinear elliptic problems is proposed by combining a multilevel linearization technique and the cascadic multigrid approach the error analysis and the complexity analysis for the proposed method are carried out based on the two grid theory and its multilevel extension a superconvergence result for the multilevel linearization algorithm is established which besides being interesting for its own sake enables us to obtain the error estimates for the multilevel successive iteration method the optimal complexity is established for nonlinear elliptic problems in 2 d provided that the number of grid levels is fixed

analysis of a multilevel iterative method for nonlinear finite element equations the multilevel iterative technique is a powerful technique for solving the systems of equations associated with discretized partial differential equations we describe how this technique can be combined with a globally convergent approximate newton method to solve nonlinear partial differential equations we show that asymptotically only one newton iteration per level is required thus the complexity for linear and nonlinear problems is essentially equal

a novel two grid method for semilinear elliptic equations a new finite element discretization technique based on two coarse and fine subspaces is presented for a semilinear elliptic boundary value problem the solution of a nonlinear system on the fine space is reduced to the solution of two small one linear and one nonlinear systems on the coarse space and a linear system on the fine space it is shown both theoretically and numerically that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation as a result the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

on the convergence of a cascadic multigrid method for semilinear elliptic problem a cascadic multigrid method for semilinear elliptic problem is analysed it has been proved that the algorithm has optimal order of convergence in energy norm and quasi optimal computational complexity under more general conditions

some estimates of the rate of convergence for the cascadic conjugate gradient method abstract the paper deals with a cascadic conjugate gradient method shortly called the ccg algorithm which was proposed by p deuflhard and can be considered as a simpler version of a multigrid multilevel method we define it recurrently for discrete self adjoint positive definite problems on a sequence of grids on the coarsest grid the linear discrete algebraic system is solved directly on the finer grids the system is iteratively solved by the conjugate gradient method where the starting guess is an interpolation of the approximated solution on the previous grid any preconditioning or restriction to coarser grids is not implemented nevertheless the ccg algorithm has the same optimal property compared to multigrid methods namely the algorithm converges with a rate which is independent of the number of unknowns and the number of grids as an example this property is proved for elliptic second order dirichlet problems in two dimensional convex polygonal bounded domains for ensuring convergence the number of iterations on each grid level has to increase from finer to coarser grids the optimal dependence of these numbers is established with respect to the mesh size and the number of unknowns the theory has been presented in an abstract setting which allows the application to both finite element and finite difference methods

the cascadic multigrid method for elliptic problems the paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of 10 from the multigrid point of view no correction cycles are needed from the cascade principle view a basic iteration method without any preconditioner is used at successive refinement levels for a prescribed error tolerance on the final level more iterations must be spent on coarser grids in order to allow for less iterations on finer grids a first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of 9 in short ccg method whichused the cg method as basic iteration method on each level in 18 it has been proven that the ccg method is accurate with optimal complexity for elliptic problems in 2d and quasi uniform triangulations the present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid smoothers moreover an adaptive control strategy for the number of iterations on successive refinement levels for possibly highly non uniform grids is worked out on the basis of a posteriori estimates numerical tests confirm the efficiency and robustness of the cascadic multigrid method

a continuum mechanical approach to geodesics in shape space in this paper concepts from continuum mechanics are used to define geodesic paths in the space of shapes where shapes are implicitly described as boundary contours of objects the proposed shape metric is derived from a continuum mechanical notion of viscous dissipation a geodesic path is defined as the family of shapes such that the total amount of viscous dissipation caused by an optimal material transport along the path is minimized the approach can easily be generalized to shapes given as segment contours of multi labeled images and to geodesic paths between partially occluded objects the proposed computational framework for finding such a minimizer is based on the time discretization of a geodesic path as a sequence of pairwise matching problems which is strictly invariant with respect to rigid body motions and ensures a 1 1 correspondence along the induced flow in shape space when decreasing the time step size the proposed model leads to the minimization of the actual geodesic length where the hessian of the pairwise matching energy reflects the chosen riemannian metric on the underlying shape space if the constraint of pairwise shape correspondence is replaced by the volume of the shape mismatch as a penalty functional one obtains for decreasing time step size an optical flow term controlling the transport of the shape by the underlying motion field the method is implemented via a level set representation of shapes and a finite element approximation is employed as spatial discretization both for the pairwise matching deformations and for the level set representations the numerical relaxation of the energy is performed via an efficient multi scale procedure in space and time various examples for 2d and 3d shapes underline the effectiveness and robustness of the proposed approach