recovery type superconvergence and a posteriori error estimates for control problems governed by stokes equations in this paper we derive recovery type superconvergence analysis and a posteriori error estimates for the finite element approximation of the distributed optimal control governed by stokes equations we obtain superconvergence results and asymptotically exact a posteriori error estimates by applying two recovery methods which are the patch recovery technique and the least squares surface fitting method our results are based on some regularity assumption for the stokes control problems and are applicable to the first order conforming finite element method with regular but nonuniform partitions

a framework for the adaptive finite element solution of large scale inverse problems since problems involving the estimation of distributed coefficients in partial differential equations are numerically very challenging efficient methods are indispensable in this paper we will introduce a framework for the efficient solution of such problems this comprises the use of adaptive finite element schemes solvers for the large linear systems arising from discretization and methods to treat additional information in the form of inequality constraints on the parameter to be recovered the methods to be developed will be based on an all at once approach in which the inverse problem is solved through a lagrangian formulation the main feature of the paper is the use of a continuous function space setting to formulate algorithms in order to allow for discretizations that are adaptively refined as nonlinear iterations proceed this entails that steps such as the description of a newton step or a line search are first formulated on continuous functions and only then evaluated for discrete functions on the other hand this approach avoids the dependence of finite dimensional norms on the mesh size making individual steps of the algorithm comparable even if they used differently refined meshes numerical examples will demonstrate the applicability and efficiency of the method for problems with several million unknowns and more than 10 000 parameters

a posteriori error control for distributed elliptic optimal control problems with control constraints discretized by hp finite elements abstract a distributed elliptic control problem with control constraints is considered which is formulated as a three field problem and consists of two variational equations for the state and the co state variables as well as of a variational inequality for the control variable two discretization approaches with h p finite elements are discussed in the first discretization approach all variables state co state and control are discretized a semi smooth newton method is introduced for solving the resulting algebraic system in the second discretization approach only the state and the co state variables are discretized whereas the control is determined by projection a simple fixed point scheme is presented for the iterative solution of this approach the main focus of the paper is on the derivation of reliable and efficient a posteriori error estimates which enables h p adaptive mesh refinements in particular the estimates can be applied to the iteration solutions so that they can be used as a stopping criterion of the iterative solution schemes in several numerical experiments the order of convergence of the adaptive discretization approaches and the efficiency as well as the reliability of the a posteriori error estimates are studied

edge connectivity in regular multigraphs from eigenvalues abstract let g be a d regular multigraph and let 2 g be the second largest eigenvalue of g in this paper we prove that if 2 g d 1 9 d 2 10 d 17 4 then g is 2 edge connected furthermore for t 2 we show that g is t 1 edge connected when 2 g d t and in fact when 2 g d t 1 if t is odd

error analysis for optimal control problem governed by convection diffusion equations dg method in the current paper we study the convergence properties of the dgfe approximation of optimal control problem governed by convection diffusion equations we derive a posteriori error estimates and a priori error estimates for both the states ad joint and the control variable approximation for the optimal control problem these estimates are apparently not available in the literature

edge disjoint spanning trees and eigenvalues of regular graphs abstract partially answering a question of paul seymour we obtain a sufficient eigenvalue condition for the existence of k edge disjoint spanning trees in a regular graph when k 2 3 more precisely we show that if the second largest eigenvalue of a d regular graph g is less than d 2 k 1 d 1 then g contains at least k edge disjoint spanning trees when k 2 3 we construct examples of graphs that show our bounds are essentially best possible we conjecture that the above statement is true for any k d 2

a posteriori error estimation for finite element discretization of parameter identification problems in this paper we develop an a posteriori error estimator for parameter identification problems the state equation is given by a partial differential equation involving a finite number of unknown parameters the presented error estimator aims to control the error in the parameters due to discretization by finite elements for this we consider the general setting of a partial differential equation written in weak form with abstract parameter dependence exploiting the special structure of the parameter identification problem allows us to derive an error estimator which is cheap in comparison to the overall optimization algorithm several examples illustrating the behavior of an adaptive mesh refinement algorithm based on our error estimator are discussed in the numerical section for the problems considered here both the efficiency of the estimator and the quality of the generated meshes are satisfactory

edge disjoint spanning trees and eigenvalues of graphs it is proven that a graph of order n with minimum degree 2k has k edge disjoint spanning trees if its second largest eigenvalue is less than 2k 1 1 o 1 n this extends earlier results of cioab and wong and improves results of gu lai li and yao and also confirms approximately their conjectures for sufficiently large graphs with respect to the minor term we also conjecture that the minor term can be removed

an interior point algorithm with inexact step computation in function space for state constrained optimal control we consider an interior point method in function space for pde constrained optimal control problems with state constraints our emphasis is on the construction and analysis of an algorithm that integrates a newton path following method with adaptive grid refinement this is done in the framework of inexact newton methods in function space where the discretization error of each newton step is controlled by adaptive grid refinement in the innermost loop this allows to perform most of the required newton steps on coarse grids such that the overall computational time is dominated by the last few steps for this purpose we propose an a posteriori error estimator for a problem suited norm

blockwise adaptivity for time dependent problems based on coarse scale adjoint solutions we describe and test an adaptive algorithm for evolution problems that employs a sequence of blocks consisting of fixed though nonuniform space meshes this approach offers the advantages of adaptive mesh refinement but with reduced overhead costs associated with load balancing remeshing matrix reassembly and the solution of adjoint problems used to estimate discretization error and the effects of mesh changes a major issue with a block adaptive approach is determining block discretizations from coarse scale solution information that achieve the desired accuracy we describe several strategies for achieving this goal using adjoint based a posteriori error estimates and we demonstrate the behavior of the proposed algorithms as well as several technical issues in a set of examples

adaptive finite element approximation for a class of parameter estimation problems in this paper we investigate the finite element approximation of a class of parameter estimation problems which are the form of performance as the optimal control problems governed by bilinear parabolic equations we propose to use adaptive multi meshes in developing efficient algorithms for the estimation problems we derive a posteriori error estimators for both the control and state approximations which particularly suit an adaptive multi meshes finite element scheme the error estimators are then implemented and tested with promising numerical results

note on edge disjoint spanning trees and eigenvalues abstract let g 2 g n 1 g and 2 g be the maximum number of edge disjoint spanning trees the second largest adjacency eigenvalue the algebraic connectivity and the second largest signless laplace eigenvalue of g respectively in this note we prove that for any graph g with minimum degree 2 k if 2 g 2 k 1 1 or n 1 g 2 k 1 1 or 2 g 2 2 k 1 1 then g k which confirms a conjecture of liu hong and lai and also implies a conjecture of cioab and wong

analysis and approximation of the ginzburg landau model of superconductivity the authors consider the ginzburg landau model for superconductivity first some well known features of superconducting materials are reviewed and then various results concerning the model the resultant differential equations and their solution on bounded domains are derived then finite element approximations of the solutions of the ginzburg landau equations are considered and error estimates of optimal order are derived

adaptive finite element methods for optimal control of partial differential equations basic concept a new approach to error control and mesh adaptivity is described for the discretization of optimal control problems governed by elliptic partial differential equations the lagrangian formalism yields the first order necessary optimality condition in form of an indefinite boundary value problem which is approximated by an adaptive galerkin finite element method the mesh design in the resulting reduced models is controlled by residual based a posteriori error estimates these are derived by duality arguments employing the cost functional of the optimization problem for controlling the discretization error in this case the computed state and costate variables can be used as sensitivity factors multiplying the local cell residuals in the error estimators this results in a generic and simple algorithm for mesh adaptation within the optimization process this method is developed and tested for simple boundary control problems in semiconductor models

adaptive finite element approximation for distributed elliptic optimal control problems in this paper sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems these error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach our numerical results indicate that the sharp error estimators work satisfactorily in guiding the mesh adjustments and can save substantial computational work

goal oriented adaptivity in pointwise state constrained optimal control of partial differential equations we derive primal dual weighted goal oriented a posteriori error estimates for pointwise state constrained optimal control problems for second order elliptic partial differential equations the constraints give rise to a primal dual weighted error term representing the mismatch in the complementarity system due to discretization in the case of sufficiently regular active or coincidence sets and problem data a further decomposition of the multiplier into a regular l 2 part on the active set and a singular part concentrated on the boundary between the active and inactive set allows us to further characterize the mismatch error the paper ends with a report on the behavior of the error estimates for test cases including the case of singular active sets consisting of only smooth curves or points

a posteriori error estimates for discontinuous galerkin time stepping method for optimal control problems governed by parabolic equations in this paper we examine the discontinuous galerkin dg finite element approximation to convex distributed optimal control problems governed by linear parabolic equations where the discontinuous finite element method is used for the time discretization and the conforming finite element method is used for the space discretization we derive a posteriori error estimates for both the state and the control approximation assuming only that the underlying mesh in space is nondegenerate for problems with control constraints of obstacle type which are the kind most frequently met in applications further improved error estimates are obtained

a posteriori error estimates for control problems governed by stokes equations in this paper we derive a posteriori error estimates for the finite element approximation of distributed optimal control problems governed by the stokes equations we obtain a posteriori error estimators for both the state and the control approximation in the l2 norm and the h1 norm these estimates can be used to construct reliable adaptive finite element approximation for the control problems

a strategy for input estimation with sensitivity analysis the estimation of unknown excitation or input signals from measured response is essentially the solution of an optimization problem a function representing the input is sought such that the discrepancy between measured and simulated response is minimized from the view of optimization it is convenient to consider the general situation where the governing differential equations may be time variant and non linear moreover the choice of parameterization for the sought input sampling instances of measurement data and discretization of the state problem are all decoupled in contrast to standard methods such as dynamic programming in this paper the optimization problem is solved by seeking a stationary point of the pertinent lagrangian functional resulting in a newton type iterative algorithm moreover the sensitivity of the solution with respect to measurement noise may also be estimated from a perturbation analysis of the optimality conditions two numerical examples show that the approach produces results in agreement with those of standard methods however with the additional advantage the numerical properties of the considered problem may be investigated in particular such knowledge may be used for choosing a proper amount of regularization copyright 2006 john wiley sons ltd

consistent approximations and approximate functions and gradients in optimal control because of the unavoidable use of numerical integration methods such as runge kutta or finite elements the numerical solution of optimal control problems with either ode or pde dynamics is governed by a discretization parameter such as the integration mesh size usually when explicit integration techniques are used function and derivative values can be computed exactly for the discretized problems recently we have come across some examples where function and derivative values of the explicitly discretized problems had to be approximated by the outcome of n iterations of a solver consequently the discretization of these problems is controlled by two parameters the mesh size and the number of iterations of the solver referring to e polak optimization algorithms and consistent approximations springer verlag 1997 we find a theory for solving optimization problems that require discretization it deals with two situations in the first which is referred to as that of consistent approximations it is assumed that an infinite dimensional optimization problem can be suitably approximated by a family of progressively higher dimensional optimization problems in this case strategies in the form of algorithm models are presented for diagonalizing the solution process in the second situation it is assumed that numerical solution of the dynamic equations does not result in a family of finite dimensional consistent approximations e g when implicit integration methods are used for this case the theory provides models for the implementation of conceptual algorithms unfortunately neither of these situations envisions the possibility of two discretization parameters in this paper we present new algorithm models that can be used with two discretization parameters the first one controls the mesh size of an explicit integration scheme and the second one controls the precision with which functions and gradients associated with a fixed mesh size are computed the result can be seen as a framework of quasi consistent approximations we implemented these new algorithm models using an approximate steepest descent method for the solution of two problems a two point boundary value problem in which the discretized linear ode dynamics are solved approximately using the gauss seidel method and a distributed control problem in which the discretized dynamics are solved using a domain decomposition algorithm which can be implemented on parallelized computers our numerical results show that these new algorithms perform quite well and are fairly insensitive to the selection of user set parameters also they appear to be superior to some alternative ad hoc schemes

augmented lagrangian sqp methods for nonlinear optimalcontrol problems of tracking type an augmented lagrangian method with second order update is developed and its relationship to the sequential quadratic programming method is described the rate of convergence proof depends on a second order sufficient optimality condition which is shown to be satisfied for a class of nonlinear optimal control problems of tracking type numerical examples are included which demonstrate the globalizing effect of the augmented lagrangian method

interlacing eigenvalues and graphs abstract we give several old and some new applications of eigenvalue interlacing to matrices associated to graphs bounds are obtained for characteristic numbers of graphs such as the size of a maximal co clique the chromatic number the diameter and the bandwidth in terms of the eigenvalues of the standard adjacency matrix or the laplacian matrix we also deal with inequalities and regularity results concerning the structure of graphs and block designs

edge disjoint spanning trees and eigenvalues abstract let g and 2 g be the maximum number of edge disjoint spanning trees and the second largest eigenvalue of a graph g respectively motivated by a question of seymour on the relationship between eigenvalues of a graph g and g cioab and wong conjectured that for any integers k 2 d 2 k and a d regular graph g if 2 g d 2 k 1 d 1 then g k they proved this conjecture for k 2 3 gu lai li and yao generalized this conjecture to simple graph and conjectured that for any integer k 2 and a graph g with minimum degree and maximum degree if 2 g 2 2 k 1 1 then g k in this paper we prove that 2 g 2 k 2 k 1 implies g k and show the two conjectures hold for sufficiently large n

a high accuracy nfem for constrained optimal control problems governed by elliptic equations in this paper a new nonconforming finite element method nfem is proposed for the constrained optimal control problems ocps governed by elliptic equations the state and co state are approximated by the nonconforming eq 1 rot element and the control is approximated by the piecewise constant element respectively the superclose properties for the above three variables are obtained moreover for the state and co state the superconvergence results are achieved in a broken energy norm by using the post processing technique lastly some numerical examples are provided to verify the theoretical analysis

a posteriori error estimates for optimal control problems governed by parabolic equations in this paper we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation we obtain a posteriori error estimates for both the state and the control approximation such estimates which are apparently not available in the literature are an important step towards developing reliable adaptive finite element approximation schemes for the control problem

eigenvalues and edge connectivity of regular graphs abstract in this paper we show that if the second largest eigenvalue of a d regular graph is less than d 2 k 1 d 1 then the graph is k edge connected when k is 2 or 3 we prove stronger results let d denote the largest root of x 3 d 3 x 2 3 d 2 x 2 0 we show that if the second largest eigenvalue of a d regular graph g is less than d then g is 2 edge connected and we prove that if the second largest eigenvalue of g is less than d 3 d 3 2 16 2 then g is 3 edge connected

adaptive finite element method for elliptic optimal control problems convergence and optimality in this paper we consider the convergence analysis of adaptive finite element method for elliptic optimal control problems with pointwise control constraints we use variational discretization concept to discretize the control variable and piecewise linear and continuous finite elements to approximate the state variable based on the well established convergence theory of afem for elliptic boundary value problems we rigorously prove the convergence and quasi optimality of afem for optimal control problems with respect to the state and adjoint state variables by using the so called perturbation argument numerical experiments confirm our theoretical analysis

conjugate gradient boundary element solution for distributed elliptic optimal control problems an optimality system of equations for the optimal control problem governed by helmholtz type equations is derived by the associated first order necessary optimality condition we obtain the conjugate gradient method cgm in the continuous case introducing the sequence of higher order fundamental solutions we propose an iterative algorithm based on the conjugate gradient boundary element method using the multiple reciprocity method cgm mrbem for solving the discrete control input this algorithm has an advantage over that of the existing literatures because the main attribute the reduced dimensionality of the boundary element method is fully utilized finally the local error estimates for this scheme are obtained and a test problem is given to illustrate the efficiency of the proposed method

adaptive space time finite element methods for parabolic optimization problems in this paper we derive a posteriori error estimates for space time finite element discretizations of parabolic optimization problems the provided error estimates assess the discretization error with respect to a given quantity of interest and separate the influences of different parts of the discretization time space and control discretization this allows us to set up an efficient adaptive algorithm which successively improves the accuracy of the computed solution by construction of locally refined meshes for time and space discretizations

on stabilized mixed methods for generalized stokes problem based on the velocity pseudostress formulation a priori error estimates abstract in this paper we present an augmented mixed formulation applied to generalized stokes problem and uses it as state equation in an optimal control problem the augmented scheme is obtained adding suitable least squares terms to the corresponding velocity pseudostress formulation of the generalized stokes problem to ensure the existence and uniqueness of solution at continuous and discrete levels we prove coerciveness of the corresponding augmented bilinear form and using approximation properties of the respective discrete subspaces we deduce the optimal rate of convergence as by product and considering the associated optimal control problem we derive error estimates for the approximated control unknown finally we present several numerical examples confirming the theoretical properties of this approach

a new approach to numerical solution of defective boundary value problems in incompressible fluid dynamics we consider the incompressible navier stokes equations where on a part of the boundary flow rate and mean pressure boundary conditions are prescribed there are basically two strategies for solving these defective boundary problems the variational approach see j heywood r rannacher and s turek internat j numer methods fluids 22 1996 pp 325 352 and the augmented formulation see l formaggia j f gerbeau f nobile and a quarteroni siam j numer anal 40 2002 pp 376 401 and a veneziani and c vergara internat j numer methods fluids 47 2005 pp 803 816 however both of these approaches present some drawbacks for the flow rate problem the former resorts to nonstandard functional spaces which are quite difficult to discretize on the other hand for the mean pressure problem it yields exact solutions only in very special cases the latter is applicable only to the flow rate problem since for the mean pressure problem it provides unfeasible boundary conditions in this paper we propose a new strategy based on a reformulation of the problems at hand in terms of the minimization of an appropriate functional this approach allows us to treat the two kinds of problems flow rate and mean pressure successfully within the same framework which can be useful in view of a mixed problem where the two conditions are simultaneously prescribed on different artificial boundaries moreover it is more versatile being prone to be extended to other kinds of defective conditions we analyze the problems obtained with this approach and propose some algorithms for their numerical solution several numerical results are presented supporting the effectiveness of our approach

spectral conditions for edge connectivity and packing spanning trees in multigraphs abstract a multigraph is a graph with possible multiple edges but no loops the multiplicity of a multigraph is the maximum number of edges between any pair of vertices we prove that for a multigraph g with multiplicity m and minimum degree 2 k if the algebraic connectivity is greater than min 2 k 1 1 m 2 k 1 2 then g has at least k edge disjoint spanning trees for a multigraph g with multiplicity m and minimum degree k if the algebraic connectivity is greater than min 2 k 1 1 m k 1 then the edge connectivity is at least k these extend some earlier results a balloon of a graph g is a maximal 2 edge connected subgraph that is joined to the rest of g by exactly one cut edge we provide spectral conditions for the number of balloons in a multigraph which also generalizes an earlier result

galerkin spectral approximations for optimal control problems governed by the fourth order equation with an integral constraint on state abstract an integral constraint on state optimal control problem governed by the first bi harmonic equation is analyzed and its legendre galerkin spectral approximations are constructed the optimality conditions of the continuous and the discretized optimal control problems are presented by the kuhn tucker condition respectively the a priori error estimates of high accuracy are obtained furthermore an efficient projection algorithm is proposed and the convergent rate is discussed with precise details ample numerical examples are carried out to validate the theoretical results

space time adaptive wavelet methods for optimal control problems constrained by parabolic evolution equations an adaptive algorithm based on wavelets is proposed for the efficient numerical solution of a control problem governed by a linear parabolic evolution equation first the constraints are represented by means of a full weak space time formulation as a linear system in ell 2 in wavelet coordinates following a recent approach by schwab and stevenson second a quadratic cost functional involving a tracking type term for the state and a regularization term for the distributed control is also formulated in terms of ell 2 sequence norms of wavelet coordinates this functional serves as a representer for a functional involving different sobolev norms with possibly nonintegral smoothness parameter standard techniques from optimization are then used to derive the first order necessary conditions as a coupled system in ell 2 coordinates for this purpose an adaptive method is proposed which can be interpreted as an inexact gradient method for the control in each iteration step the primal and adjoint systems are solved up to a prescribed accuracy by the adaptive algorithm it is shown that the adaptive algorithm converges moreover the algorithm is proved to be asymptotically optimal the convergence rate achieved for computing each of the components of the solution state adjoint state and control up to a desired target tolerance is asymptotically the same as the wavelet best n term approximation of each solution component and the total computational work is proportional to the number of computational unknowns

a posteriori error estimates for hp finite element solutions of convex optimal control problems in this paper we present a posteriori error analysis for hp finite element approximation of convex optimal control problems we derive a new quasi interpolation operator of clement type and a new quasi interpolation operator of scott zhang type that preserves homogeneous boundary condition the scott zhang type quasi interpolation is suitable for an application in bounding the errors in l 2 norm then hp a posteriori error estimators are obtained for the coupled state and control approximations such estimators can be used to construct reliable adaptive finite elements for the control problems

direct solvers performance on h adapted grids we analyse the performance of direct solvers when applied to a system of linear equations arising from an h adapted c 0 finite element space theoretical estimates are derived for typical h refinement patterns arising as a result of a point edge or face singularity as well as boundary layers they are based on the elimination trees constructed specifically for the considered grids theoretical estimates are compared with experiments performed with mumps using the nested dissection algorithm for construction of the elimination tree from metis library the numerical experiments provide the same performance for the cases where our trees are identical with those constructed by the nested dissection algorithm and worse performance for some cases where our trees are different we also present numerical experiments for the cases with mixed singularities where how to construct optimal elimination trees is unknown in all analysed cases the use of h adaptive grids significantly reduces the cost of the direct solver algorithm per unknown as compared to uniform grids the theoretical estimates predict and the experimental data confirm that the computational complexity is linear for various refinement patterns in most cases the cost of the direct solver per unknown is lower when employing anisotropic refinements as opposed to isotropic ones

feedback optimal control of distributed parameter systems by using finite dimensional approximation schemes optimal control for systems described by partial differential equations is investigated by proposing a methodology to design feedback controllers in approximate form the approximation stems from constraining the control law to take on a fixed structure where a finite number of free parameters can be suitably chosen the original infinite dimensional optimization problem is then reduced to a mathematical programming one of finite dimension that consists in optimizing the parameters the solution of such a problem is performed by using sequential quadratic programming linear combinations of fixed and parameterized basis functions are used as the structure for the control law thus giving rise to two different finite dimensional approximation schemes the proposed paradigm is general since it allows one to treat problems with distributed and boundary controls within the same approximation framework it can be applied to systems described by either linear or nonlinear elliptic parabolic and hyperbolic equations in arbitrary multidimensional domains simulation results obtained in two case studies show the potentials of the proposed approach as compared with dynamic programming

parameter identification in constitutive models via optimization with a posteriori error control in this paper we outline a computational technique for the calibration of macroscopic constitutive laws with automatic error control in the most general situation the state variables of the constitutive law as well as the material parameters are spatially non homogeneous the experimental observations are given in space time based on an appropriate dual problem we compute a posteriori the discretization error contributions from approximations of the parameter state and costate elds in space time for an arbitrarily chosen goal oriented error measure of engineering signi cance such a measure can be used in an adaptive strategy not discussed in this paper to meet a prede ned error tolerance an important observation is that the jacobian matrix associated with the resulting newton method is used in principle in solving the dual problem rather than treating the jacobian in a monolithic fashion we utilize a sequential solution strategy whereby the fe topology of the discretized state problem is used repeatedly moreover the proposed solution strategy lends itself naturally to the computation of rst and second order sensitivities which are obtained with little extra computational effort numerical results are given for the prototype model of con ned aquifer ow with spatially non homogeneous permeability the ef ciency of the optimization strategy and the effectivity of the error computation are assessed

multigoal oriented optimal control problems with nonlinear pde constraints abstract in this work we consider an optimal control problem subject to a nonlinear pde constraint and apply it to a semi linear monotone pde and the regularized p laplace equation to this end a reduced unconstrained optimization problem in terms of the control variable is formulated based on the reduced approach we then derive an a posteriori error representation and mesh adaptivity for multiple quantities of interest all quantities are combined to one and then the dual weighted residual dwr method is applied to this combined functional furthermore the estimator allows for balancing the discretization error and the nonlinear iteration error these developments allow us to formulate an adaptive solution strategy which is finally substantiated with the help of several numerical examples

fast iterative solution of elliptic control problems in wavelet discretization we investigate wavelet methods for the efficient numerical solution of a class of control problems constrained by a linear elliptic boundary value problem where the cost functional may contain fractional sobolev norms of the control and the state starting point is the formulation of the infinite dimensional control problem in terms of boundary adapted biorthogonal spline wavelets involving only l2 norms of wavelet expansion coefficients where different norms are realized by a diagonal scaling together with a riesz map and constraints in form of an l2 isomorphism the coupled system of equations resulting from optimization is solved by an inexact conjugate gradient cg method for the control which involves the approximate inversion of the primal and the adjoint operator using again cg iterations starting from a coarse discretization level we use nested iteration to solve the coupled system on successively finer uniform discretizations up to discretization error accuracy on each level the resulting inexact cg scheme is a fast solver it is of asymptotic optimal complexity in the sense that the overall computational effort to compute the solution up to discretization error on the finest grid is proportional to the number of unknowns on that grid a consequence of grid independent condition numbers of the linear operators in wavelet coordinates in the numerical examples we study the choice of different norms and the regularization parameter in the cost functional and their effect on the solution moreover for different situations the performance of the fully iterative inexact cg scheme is investigated confirming the theoretical results

some aspects of adaptive grid technology related to boundary and interior layers we consider the use of adaptive mesh strategies for solution of problems exhibiting boundary and interior layer solutions as the presence of these layer structures suggests reliable and accurate solution of this class of problems using finite difference finite volume or finite element schemes requires grading the mesh into the layers and due attention to the associated algorithms when the nature and structure of the layer is known mesh grading can be achieved during the grid generation by specifying an appropriate grading function however in many applications the location and nature of the layer behavior is not known in advance consequently adaptive mesh techniques that employ feedback from intermediate grid solutions are an appealing approach in this paper we provide a brief overview of the main adaptive grid strategies in the context of problems with layers associated error indicators that guide the refinement feedback control grid optimization process are also covered and there is a brief commentary on the supporting data structure requirements some current issues concerning the use of stabilization in conjunction with adaptive mesh refinement amr the question of pollution effects in computation of local error indicators the influence of nonlinearities and the design of meshes for targeted optimization of specific quantities are considered the application of amr for layer problems is illustrated by means of case studies from semiconductor device transport drift diffusion nonlinear reaction diffusion layers due to surface capillary effects and shockwaves in compressible gas dynamics

local a posteriori error estimates for convex boundary control problems in this paper we derive some improved a posteriori error estimates for finite element approximation of neumann boundary control problems we first establish local upper a posteriori error estimates for both the state and the control approximation of the general convex boundary control problems we then derive local upper and lower a posteriori error estimates for a class of control problems that frequently appear in applications

numerical approximation of an sqp type method for parameter identification this paper deals with the numerical approximation of the levenberg marquardt sqp lmsqp method for parameter identification problems which has been presented and analyzed in m burger and w muhlhuber inverse problems 18 2002 pp 943 969 it is shown that a galerkin type discretization leads to a convergent approximation and that the indefinite system arising from the karush kuhn tucker kkt system is well posed in addition we present a multilevel version of the levenberg marquardt method and discuss the simultaneous solution of the discretized kkt system by preconditioned iteration methods for indefinite problems from a discussion of the numerical effort we conclude that these approaches may lead to a considerable speed up with respect to standard iterative regularization methods that eliminate the underlying state equation the numerical efficiency of the lmsqp method is confirmed by numerical examples

finite dimensional approximation of a class of constrained nonlinear optimal control problems an abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed nonlinearities occur in both the objective functional and the constraints the framework includes an abstract nonlinear optimization problem posed on infinite dimensional spaces an approximate problem posed on finite dimensional spaces together with a number of hypotheses concerning the two problems the framework is used to show that optimal solutions exist to show that lagrange multipliers may be used to enforce the constraints to derive an optimality system from which optimal states and controls may be deduced and to derive existence results and error estimates for solutions of the approximate problem the abstract framework and the results derived from that framework are then applied to three concrete control or optimization problems and their approximation by finite element methods the first involves the von k arm an plate equations of nonlinear elasticity the second the ginzburg landau equations of superconductivity and the third the navier stokes equations for incompressible viscous flows

finite element methods for optimal control problems governed by integral equations and integro differential equations in this paper we analyze finite element galerkin discretizations for a class of constrained optimal control problems that are governed by fredholm integral or integro differential equations the analysis focuses on the derivation of a priori error estimates and a posteriori error estimators for the approximation schemes

adaptive finite elements for elliptic optimization problems with control constraints in this paper we develop a posteriori error estimates for finite element discretization of elliptic optimization problems with pointwise inequality constraints on the control variable we derive error estimators for assessing the discretization error with respect to the cost functional as well as with respect to a given quantity of interest these error estimators provide quantitative information about the discretization error and guide an adaptive mesh refinement algorithm allowing for substantial saving in degrees of freedom the behavior of the method is demonstrated on numerical examples

equivalent a posteriori error estimates for elliptic optimal control problems with l1 control cost abstract an elliptic optimal control problem involving the l 1 norm of the control in the cost functional is considered in this paper we use the full discretization and the variational discretization to approximate the control problem and the efficient and reliable a posteriori error estimates are obtained for the two cases for the variational discretization we also analyze the convergence of adaptive finite element methods in the end some examples are provided to validate our analysis

adaptive multilevel inexact sqp methods for pde constrained optimization we present a class of inexact adaptive multilevel trust region sqp methods for the efficient solution of optimization problems governed by nonlinear pdes the algorithm starts with a coarse discretization of the underlying optimization problem and provides during the optimization process 1 implementable criteria for an adaptive refinement strategy of the current discretization based on local error estimators and 2 implementable accuracy requirements for iterative solvers of the linearized pde and adjoint pde on the current grid we prove global convergence to a stationary point of the infinite dimensional problem moreover we illustrate how the adaptive refinement strategy of the algorithm can be implemented by using existing reliable a posteriori error estimators for the state and the adjoint equations numerical results are presented

a stable nonconforming mixed finite element scheme for elliptic optimal control problems a new stable nonconforming mixed finite element mfe scheme is proposed for the constrained optimal control problem ocp governed by elliptic equations the nonconforming fes are employed to approximate the co state the piecewise constant element for the state and control respectively the corresponding optimal order error estimates are derived by full use of the distinguish properties of the nonconforming element i e a the divergence of the function in the fe space is a constant b the consistency error is one order higher than its interpolation error in the broken energy norm finally some numerical results are carried out to verify the theoretical analysis