an algebraic multigrid method for a class of elliptic differential systems the development numerical investigation and analysis of an algebraic multigrid amg method for the numerical solution of a representative class of elliptic differential systems are presented our amg scheme is based on the amg strategy of brandt ruge and stuben however the present approach may be implemented within other amg schemes by means of numerical experiments with model problems arising from some application areas it is demonstrated that the computational performance of the proposed amg scheme is comparable to that of amg applied to a single scalar equation an amg convergence theory is presented that extends convergence results for amg methods for scalar problems to the case of amg applied to elliptic systems some details of our amg implementation are given

the numerical solution of the steady state solid fuel ignition model and its optimal control we present a finite difference multigrid technique that efficiently solves optimal control problems associated with the steady state solid fuel ignition model this is a nonlinear indefinite problem which gives rise to singular optimal control systems regarding the direct problem necessary and sufficient conditions for existence of solutions are given and second order convergence of numerical solution errors with respect to the mesh size is proven a robust nonlinear multigrid method is implemented to validate this convergence estimate two optimal control strategies for solid fuel ignition phenomena are considered and compared the corresponding optimality systems are solved to second order accuracy by a multigrid method whose convergence properties are independent of the values of the weights in the cost functionals and of the number of grid points

stabilized finite element methods based on multiscale enrichment for the stokes problem this work concerns the development of stabilized finite element methods for the stokes problem considering nonstable different or equal order of velocity and pressure interpolations the approach is based on the enrichment of the standard polynomial space for the velocity component with multiscale functions which no longer vanish on the element boundary on the other hand since the test function space is enriched with bubble like functions a petrov galerkin approach is employed we use such a strategy to propose stable variational formulations for continuous piecewise linear in velocity and pressure and for piecewise linear piecewise constant interpolation pairs optimal order convergence results are derived and numerical tests validate the proposed methods

a numerical study of the limited memory bfgs method and the truncated newton method for large scale optimization this paper examines the numerical performances of two methods for large scale optimization a limited memory quasi newton method l bfgs and a discrete truncated newton method tn various ways of classifying test problems are discussed in order to better understand the types of problems that each algorithm solves well the l bfgs and tn methods are also compared with the polak ribiere conjugate gradient method

superlinear convergence and implicit filtering in this paper we show how the implicit filtering algorithm can be coupled with the bfgs quasi newton update to obtain a superlinearly convergent iteration if the noise in the objective function decays sufficiently rapidly as the optimal point is approached in this way we give insight into the observations of good performance in practice of quasi newton methods when they are coupled with implicit filtering we also report on numerical experiments that show how an implementation of implicit filtering that exploits these new results can improve the performance of the algorithm

primal dual strategy for constrained optimal control problems an algorithm for efficient solution of control constrained optimal control problems is proposed and analyzed it is based on an active set strategy involving primal as well as dual variables for discretized problems sufficient conditions for convergence in finitely many iterations are given numerical examples are given and the role of the strict complementarity condition is discussed

projected sequential quadratic programming methods in this paper we introduce and analyze a class of optimization methods called projected sequential quadratic programming sqp methods for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on parts of the variables such problems frequently arise in the numerical solution of optimal control problems projected sqp methods combine the ideas of projected newton methods and sqp methods they use the simple projection onto the set defined by the bound constraints and maintain feasibility with respect to these constraints the iterates are computed using an extension of sqp methods and require only the solution of the linearized equality constraint global convergence of these methods is enforced using a constrained merit function and an armijo like line search we discuss global and local convergence properties of these methods the identification of active indices and we present numerical examples for an optimal control problem governed by a nonlinear h

a posteriori error estimation for a pde constrained optimization problem involving the generalized oseen equations we derive globally reliable a posteriori error estimators for a linear quadratic optimal control problem involving the generalized oseen equations as state equations control constraints are also considered the corresponding local error indicators are locally efficient the assumptions under which we perform the analysis are such that they can be satisfied for a wide variety of stabilized finite element methods as well as for standard finite element methods when stabilized methods are considered no a priori relation between the stabilization terms for the state and adjoint equations is required if a lower bound for the inf sup constant is available a posteriori error estimators that are fully computable and provide guaranteed upper bounds on the norm of the error can be obtained we illustrate the theory with numerical examples

shaped circularly symmetric dual reflector antennas by combining local conventional dual reflector systems this paper presents a geometrical optics go shaping method for circularly symmetric dual reflector antennas the method is a one step procedure obtaining a local dual reflector system including both reflector surfaces and caustic simultaneously such that a given feed power is transformed through a corresponding local reflector system into a desired aperture distribution a shaped reflector system consists of an electrically small section of distinctive local reflector surfaces and its own caustic each caustic location varies and explains how the power is redistributed through the shaped reflector system each local reflector system is found iteratively with a previously known local dual reflector system several nonlinear algebraic equations are formulated based on go principals and geometrical properties of a dual reflector system the method reduces the solution to one simple non differential equation with one unknown in order to find a local dual reflector system physical optics po analysis is used to verify the solution

on the convergence of recursive trust region methods for multiscale nonlinear optimization and applications to nonlinear mechanics we prove new convergence results for a class of multiscale trust region algorithms originally introduced by gratton sartenaer and toint in siam j optim 19 2008 pp 414 444 to solve unconstrained minimization problems within the euclidean space mathbb r n we will state less restrictive assumptions on the objective function and on the iteratively computed trust region corrections which allow for proving first order convergence moreover we propose a novel projection approach for obtaining the initial coarse level iterates which are needed within the nonlinear multiscale iteration we show the efficiency and robustness of our approach by means of numerical examples from nonlinear continuum mechanics where stored energy functions for materials of ogden type and for materials with visco plastic behavior are minimized

a multigrid optimization algorithm for the numerical solution of quasilinear variational inequalities involving the p laplacian abstract in this paper we propose a multigrid optimization algorithm mg opt for the numerical solution of a class of quasilinear variational inequalities of the second kind this approach is enabled by the fact that the solution of the variational inequality is given by the minimizer of a nonsmooth energy functional involving the p laplace operator we propose a huber regularization of the functional and a finite element discretization for the problem further we analyze the regularity of the discretized energy functional and we are able to prove that its jacobian is slantly differentiable this regularity property is useful to analyze the convergence of the mg opt algorithm in fact we demonstrate that the algorithm is globally convergent by using a mean value theorem for semismooth functions finally we apply the mg opt algorithm to the numerical simulation of the viscoplastic flow of bingham casson and herschel bulkley fluids in a pipe several experiments are carried out to show the efficiency of the proposed algorithm when solving this kind of fluid mechanics problems

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

on the convergence theory of trust region based algorithms for equality constrained optimization in a recent paper dennis el alem and maciel proved global convergence to a stationary point for a general trust region based algorithm for equality constrained optimization this general algorithm is based on appropriate choices of trust region subproblems and seems particularly suitable for large problems this paper shows global convergence to a point satisfying the second order necessary optimality conditions for the same general trust region based algorithm the results given here can be seen as a generalization of the convergence results for trust regions methods for unconstrained optimization obtained by more and sorensen the behavior of the trust radius and the local rate of convergence are analyzed some interesting facts concerning the trust region subproblem for the linearized constraints the quasi normal component of the step and the hard case are presented it is shown how these results can be applied to a class of discretized optimal control problems

provably convergent multifidelity optimization algorithm not requiring high fidelity derivatives functions does not alter the con vergence proof of the optimization algorithm and is shown to be robust to poor low delity information the algorithm is compared with an unconstrained single delity quasi newton algorithm and two

penalty approximation of stokes flow abstract the particular class of problems considered is stokes flow in which the constraint is the incompressibility condition u 0 on the velocity field u in part i we present first of all a simple constructive method for determining the spurious modes this is followed by an analysis of the discrete stability condition for the penalty method showing that this condition holds with a parameter that is o h we develop improved estimates of the dependence of the stability parameter on mesh size h in a more general proof than previously available in part ii we prove error estimates for the velocity and computed pressure approximations and use an aubin nitsche technique to get l 2 estimates for the velocity in the concluding numerical studies we examine the stability of the computed pressure and convergence rates with uniform mesh refinement for problems having data with varying degree of regularity

on the lagrange newton sqp method for the optimal control of semilinear parabolic equations a class of lagrange newton sqp methods is investigated for optimal control problems governed by semilinear parabolic initial boundary value problems distributed and boundary controls are given restricted by pointwise upper and lower bounds the convergence of the method is discussed in appropriate banach spaces based on a weak second order sufficient optimality condition for the reference solution local quadratic convergence is proved the proof is based on the theory of newton methods for generalized equations in banach spaces

smoothing sqp methods for solving degenerate nonsmooth constrained optimization problems with applications to bilevel programs we consider a degenerate nonsmooth and nonconvex optimization problem for which the standard constraint qualification such as the generalized mangasarian fromovitz constraint qualification gmfcq may not hold we use smoothing functions with the gradient consistency property to approximate the nonsmooth functions and introduce a smoothing sequential quadratic programming sqp algorithm under the l infty penalty framework we show that any accumulation point of a selected subsequence of the iteration sequence generated by the smoothing sqp algorithm is a clarke stationary point provided that the sequence of multipliers and the sequence of penalty parameters are bounded furthermore we propose a new condition called the weakly generalized mangasarian fromovitz constraint qualification wgmfcq that is weaker than the gmfcq we show that the extended version of the wgmfcq guarantees the boundedness of the sequence of multipliers and the sequence of penalty parameters and thus guarantees the global c

a taxonomy of consistently stabilized finite element methods for the stokes problem stabilized mixed methods can circumvent the restrictive inf sup condition without introducing penalty errors for properly chosen stabilization parameters these methods are well posed for all conforming velocity pressure pairs however their variational forms have widely varying properties first stabilization offers a choice between weakly or strongly coercive bilinear forms that give rise to linear systems with identical solutions but very different matrix properties second coercivity may be conditional upon a proper choice of a stabilizing parameter here we focus on how these two aspects of stabilized methods affect their accuracy and efficient iterative solution we present results that indicate a preference of krylov subspace solvers for strongly coercive formulations stability criteria obtained by finite element and algebraic analyses are compared with numerical experiments while for two popular classes of stabilized methods sufficient stability bounds correlate well with numerical stability our experiments indicate the intriguing possibility that the pressure stabilized galerkin method is unconditionally stable

multigrid methods for pde optimization research on multigrid methods for optimization problems is reviewed optimization problems considered include shape design parameter optimization and optimal control problems governed by partial differential equations of elliptic parabolic and hyperbolic type

evolutionary optimization of computationally expensive problems via surrogate modeling we present a parallel evolutionary optimization algorithm that leverages surrogate models for solving computationally expensive design problems with general constraints on a limited computational budget the essential backbone of our framework is an evolutionary algorithm coupled with a feasible sequential quadratic programming solver in the spirit of lamarckian learning we employ a trust region approach for interleaving use of exact modelsfortheobjectiveandconstraintfunctionswithcomputationallycheapsurrogatemodelsduringlocalsearch in contrastto earlier work we construct local surrogatemodels using radial basis functionsmotivated by theprinciple of transductive inference further the present approach retains the intrinsic parallelism of evolutionary algorithms and can hence be readily implemented on grid computing infrastructures experimental results are presented for some benchmark test functions and an aerodynamic wing design problem to demonstrate that our algorithm converges to good designs on a limited computational budget

global convergence of general derivative free trust region algorithms to first and second order critical points in this paper we prove global convergence for first and second order stationary points of a class of derivative free trust region methods for unconstrained optimization these methods are based on the sequential minimization of quadratic or linear models built from evaluating the objective function at sample sets the derivative free models are required to satisfy taylor type bounds but apart from that the analysis is independent of the sampling techniques a number of new issues are addressed including global convergence when acceptance of iterates is based on simple decrease of the objective function trust region radius maintenance at the criticality step and global convergence for second order critical points

reliable a posteriori error control for nonconforming finite element approximation of stokes flow we derive computable a posteriori error estimates for the lowest order nonconforming crouzeix raviart element applied to the approximation of incompressible stokes flow the estimator provides an explicit upper bound that is free of any unknown constants provided that a reasonable lower bound for the inf sup constant of the underlying problem is available in addition it is shown that the estimator provides an equivalent lower bound on the error up to a generic constant

multilevel algorithms for ill posed problems in this paper new multilevel algorithms are proposed for the numerical solution of first kind operator equations convergence estimates are established for multilevel algorithms applied to tikhonov type regularization methods our theory relates the convergence rate of these algorithms to the minimal eigenvalue of the discrete version of the operator and the regularization parameter the algorithms and analysis are presented in an abstract setting that can be applied to first kind integral equations

second order methods for optimal control of time dependent fluid flow second order methods for open loop optimal control problems governed by the two dimensional instationary navier stokes equations are investigated optimality systems based on a lagrangian formulation and adjoint equations are derived the newton and quasi newton methods as well as various variants of sqp methods are developed for applications to optimal flow control and their complexity in terms of system solves is discussed local convergence and rate of convergence are proved a numerical example illustrates the feasibility of solving optimal control problems for two dimensional instationary navier stokes equations by second order numerical methods in a standard workstation environment

adaptive inexact iterative algorithms based on polynomial degree robust a posteriori estimates for the stokes problem in this paper we develop adaptive inexact versions of iterative algorithms applied to finite element discretizations of the linear stokes problem we base our developments on an equilibrated stress a posteriori error estimate distinguishing the different error components namely the discretization error component the inner algebraic solver error component and possibly the outer algebraic solver error component for algorithms of the uzawa type we prove that our estimate gives a guaranteed upper bound on the total error as well as a polynomial degree robust local efficiency and this on each step of the employed iterative algorithm our adaptive algorithms stop the iterations when the corresponding error components do not have a significant influence on the total error the developed framework covers all standard conforming and conforming stabilized finite element methods on simplicial and rectangular parallelepipeds meshes in two or three space dimensions and an arbitrary algebraic solver implementation into the freefem programming language is invoked and numerical examples showcase the performance of our a posteriori estimates and of the proposed adaptive strategies as example we choose here the unpreconditioned and preconditioned uzawa algorithm and the preconditioned minimum residual minres algorithm in combination with the taylor hood discretization

geometry of interpolation sets in derivative free optimization we consider derivative free methods based on sampling approaches for nonlinear optimization problems where derivatives of the objective function are not available and cannot be directly approximated we show how the bounds on the error between an interpolating polynomial and the true function can be used in the convergence theory of derivative free sampling methods these bounds involve a constant that reflects the quality of the interpolation set the main task of such a derivative free algorithm is to maintain an interpolation sampling set so that this constant remains small and at least uniformly bounded this constant is often described through the basis of lagrange polynomials associated with the interpolation set we provide an alternative more intuitive definition for this concept and show how this constant is related to the condition number of a certain matrix this relation enables us to provide a range of algorithms whilst maintaining the interpolation set so that this condition number or the geometry constant remain uniformly bounded we also derive bounds on the error between the model and the function and between their derivatives directly in terms of this condition number and of this geometry constant

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

model problems for the multigrid optimization of systems governed by differential equations we discuss a multigrid approach to the optimization of systems governed by differential equations such optimization problems appear in many applications and are of a different nature than systems of equations our approach uses an optimization based multigrid algorithm in which the multigrid algorithm relies explicitly on nonlinear optimization models as subproblems on coarser grids our goal is not to argue for a particular optimization based multigrid algorithm but instead to demonstrate how multigrid can be used to accelerate nonlinear programming algorithms furthermore using several model problems we give evidence both theoretical and numerical that the optimization setting is well suited to multigrid algorithms some of the model problems show that the optimization problem may be more amenable to multigrid than the governing differential equation in addition we relate the multigrid approach to more traditional optimization methods as further justification for the use of an optimization based multigrid algorithm

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

a trust region approach to nonlinear systems of equalities and inequalities in this paper two new trust region algorithms for the numerical solution of systems of nonlinear equalities and inequalities are introduced the formulation is free of arbitrary parameters and possesses sufficient smoothness to exploit the robustness of the trust region approach the proposed algorithms are one sided least squares trust region algorithms the first algorithm is a single model algorithm and the second one is a multimodel algorithm where the cauchy point computation is a model selection procedure global convergence analysis for the two algorithms is presented our analysis generalizes to nonlinear systems of equalities and inequalities the well developed theory for nonlinear least squares problems numerical experiments on the two algorithms are also presented the performance of the two algorithms is reported the numerical results validate the effectiveness of our approach

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

superlinear convergence of affine scaling interior point newton methods for infinite dimensional nonlinear problems with pointwise bounds we develop and analyze a superlinearly convergent affine scaling interior point newton method for infinite dimensional problems with pointwise bounds in lp space the problem formulation is motivated by optimal control problems with lp controls and pointwise control constraints the finite dimensional convergence theory by coleman and li siam j optim 6 1996 pp 418 445 makes essential use of the equivalence of norms and the exact identifiability of the active constraints close to an optimizer with strict complementarity since these features are not available in our infinite dimensional framework algorithmic changes are necessary to ensure fast local convergence the main building block is a newton like iteration for an affine scaling formulation of the kkt condition we demonstrate in an example that a stepsize rule to obtain an interior iterate may require very small stepsizes even arbitrarily close to a nondegenerate solution using a pointwise projection instead we prove superlinear convergence under a weak strict complementarity condition and convergence with mbox q rate 1 under a slightly stronger condition if a smoothing step is available we discuss how the algorithm can be embedded in the class of globally convergent trust region interior point methods recently developed by m heinkenschloss and the authors numerical results for the control of a heating process confirm our theoretical findings

a pde sensitivity equation method for optimal aerodynamic design the use of gradient based optimization algorithms in inverse design is well established as a practical approach to aerodynamic design a typical procedure uses a simulation scheme to evaluate the objective function from the approximate states and its gradient then passes this information to an optimization algorithm once the simulation scheme cfd flow solver has been selected and used to provide approximate function evaluations there are several possible approaches to the problem of computing gradients one popular method is to differentiate the simulation scheme and compute design sensitivities that are then used to obtain gradients although this black box approach has many advantages in shape optimization problems one must compute mesh sensitivities in order to compute the design sensitivity in this paper we present an alternative approach using the pde sensitivity equation to develop algorithms for computing gradients this approach has the advantage that mesh sensitivities need not be computed moreover when it is possible to use the cfd scheme for both the forward problem and the sensitivity equation then there are computational advantages an apparent disadvantage of this approach is that it does not always produce consistent derivatives however for a proper combination of discretization schemes one can showasymptotic consistencyunder mesh refinement which is often sufficient to guarantee convergence of the optimal design algorithm in particular we show that when asymptotically consistent schemes are combined with a trust region optimization algorithm the resulting optimal design method converges we denote this approach as thesensitivity equation method the sensitivity equation method is presented convergence results are given and the approach is illustrated on two optimal design problems involving shocks

neumann neumann domain decomposition preconditioners for linear quadratic elliptic optimal control problems we present a class of domain decomposition dd preconditioners for the solution of elliptic linear quadratic optimal control problems our dd preconditioners are extensions of neumann neumann dd preconditioners which have been successfully applied to the solution of single pdes the dd preconditioners are based on a decomposition of the optimality conditions for the elliptic linear quadratic optimal control problem into smaller subdomain optimality conditions with dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces these subdomain optimality conditions are coupled through neumann interface conditions for the states and the adjoints this decomposition leads to a schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces the schur complement operator is the sum of subdomain schur complement operators the application of which is shown to correspond to the solution of subdomain elliptic linear quadratic optimal control problems which are essentially smaller copies of the original optimal control problem we show that under suitable conditions the application of the inverse of the subdomain schur complement operators requires the solution of a subdomain elliptic linear quadratic optimal control problem with neumann interface conditions for the state the subdomain schur complement operators are analyzed in the variational setting of the problem as well as the algebraic setting obtained after a finite element discretization of the problem definiteness properties of the algebraic form of the subdomain schur complement operator s are studied numerical tests show that the dependence of these preconditioners on mesh size and subdomain size is comparable to its counterpart applied to elliptic equations only these tests also show that the preconditioners are insensitive to the size of the control regularization parameter

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

space mapping for optimal control of partial differential equations solving optimal control problems for nonlinear partial differential equations represents a significant numerical challenge due to the tremendous size and possible model difficulties e g nonlinearities of the discretized problems in this paper a novel space mapping technique for solving the aforementioned problem class is introduced analyzed and tested the advantage of the space mapping approach compared to classical multigrid techniques lies in the flexibility of not only using grid coarsening as a model reduction but also employing perhaps less nonlinear surrogates the space mapping is based on a regularization approach which in contrast to other space mapping techniques results in a smooth mapping and thus avoids certain irregular situations at kinks a new broyden update formula for the sensitivities of the space map is also introduced this quasi newton update is motivated by the usual secant condition combined with a secant condition resulting from differentiating the space mapping surrogate the overall algorithm employs a trust region framework for global convergence issues involved in the computations are highlighted and a report on a few illustrative numerical tests is given

newton s method with a model trust region modification a modified newton method for unconstrained minimization is presented and analyzed the modification is based upon the model trust region approach this report contains a thorough analysis of the locally constrained quadratic minimizations that arise as subproblems in the modified newton iteration several promising alternatives are presented for solving these subproblems in ways that overcome certain theoretical difficulties exposed by this analysis very strong convergence results are presented concerning the minimization algorithm in particular the explicit use of second order information is justified by demonstrating that the iterates converge to a point which satisfies the second order necessary conditions for minimization with the exception of very pathological cases this occurs whenever the algorithm is applied to problems with continuous second partial derivatives

on the adaptive selection of the parameter in stabilized finite element approximations a systematic approach is developed for the selection of the stabilization parameter for stabilized finite element approximation of the stokes problem whereby the parameter is chosen to minimize a computable upper bound for the error in the approximation the approach is applied in the context of both a single fixed mesh and an adaptive mesh refinement procedure the optimization is carried out by a derivative free optimization algorithm and is based on minimizing a new fully computable error estimator numerical results are presented illustrating the theory and the performance of the estimator together with the optimization algorithm

space time adaptive solution of inverse problems with the discrete adjoint method this paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent adaptive grid adaptive step models discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation however this approach brings several important challenges the space time adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations a reduction in accuracy of the discrete adjoint sensitivities may appear due to the inter grid transfer operators moreover the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations this work shows that several of these potential issues can be avoided through a multi level optimization strategy using discontinuous galerkin dg hp adaptive discretizations paired with runge kutta rk time integration we extend the concept of dual adjoint consistency to space time rk dg discretizations which are then shown to be well suited for the adaptive solution of time dependent inverse problems furthermore we prove that dg mesh transfer operators on general meshes are also dual consistent this allows the simultaneous derivation of the discrete adjoint for both the numerical solver and the mesh transfer logic with an automatic code generation mechanism such as algorithmic differentiation ad potentially speeding up development of large scale simulation codes the theoretical analysis is supported by numerical results reported for a two dimensional non stationary inverse problem

a trust region method for norm constrained problems a trust region method for the solution of nonlinear optimization problems with norm constraints is analyzed such optimization problems arise in parameter identification and nonlinear eigenvalue problems the characteristics of these applications motivate the formulation and analysis of the trust region method the algorithms studied here allow for inexact gradient information and the use of subspace methods for the approximate solution of subproblems characterizations and the descent properties of trust region steps are given criteria for the existence of successful iterations under inexact gradient information and under the use of subspace methods are established and global convergence of the method is proven

a class of trust region methods for parallel optimization we present a new class of optimization methods that incorporates a parallel direct search pds method within a trust region newton framework this approach combines the inherent parallelism of pds with the rapid and robust convergence properties of newton methods numerical tests have yielded favorable results for both standard test problems and engineering applications in addition the new method appears to be more robust in the presence of noisy functions which are inherent in many engineering simulations

a posteriori optimization of parameters in stabilized methods for convection diffusion problems part i abstract stabilized finite element methods for convection dominated problems require the choice of appropriate stabilization parameters from numerical analysis often only their asymptotic values are known this paper presents a general framework for optimizing stabilization parameters with respect to the minimization of a target functional exemplarily this framework is applied to the supg finite element method and the minimization of a residual based error estimator an error indicator and a functional including the crosswind derivative of the computed solution benefits of the basic approach are demonstrated by means of numerical results

a unified framework for a posteriori error estimation for the stokes problem in this paper a unified framework for a posteriori error estimation for the stokes problem is developed it is based on h 1 0 omega d conforming velocity reconstruction and underline boldsymbol h mathrm div omega conforming locally conservative flux stress reconstruction it gives guaranteed fully computable global upper bounds as well as local lower bounds on the energy error in order to apply this framework to a given numerical method two simple conditions need to be checked we show how to do this for various conforming and conforming stabilized finite element methods the discontinuous galerkin method the crouzeix raviart nonconforming finite element method the mixed finite element method and a general class of finite volume methods the tools developed and used include a new simple equilibration on dual meshes and the solution of local poisson type neumann problems by the mixed finite element method numerical experiments illustrate the theoretical developments

local analysis of a new multipliers method abstract in this paper we introduce a penalty function and a corresponding multipliers method for the solution of a class of nonlinear programming problems where the equality constraints have a particular structure the class models optimal control and engineering design problems with bounds on the state and control variables and has wide applicability the multipliers method updates multipliers corresponding to inequality constraints maintaining their nonnegativity instead of dealing with multipliers associated with equality constraints the basic local convergence properties of the method are proved and a dual framework is introduced we also analyze the properties of the penalized problem related with the penalty function

a priori error analysis of residual free bubbles for advection diffusion problems we develop an a priori error analysis of a finite element approximation to the elliptic advection diffusion equation eps delta u conv cdot nabla u f subject to a homogeneous dirichlet boundary condition based on the use of residual free bubble functions an optimal order error bound is derived in the so called stability norm biggl eps nabla v 2 l 2 omega sum t h t conv cdot nabla v 2 l 2 t biggr 1 2 where ht denotes the diameter of element t in the subdivision of the computational domain

a line search multigrid method for large scale nonlinear optimization we present a line search multigrid method for solving discretized versions of general unconstrained infinite dimensional optimization problems at each iteration on each level the algorithm computes either a direct search direction on the current level or a recursive search direction from coarser level models introducing a new condition that must be satisfied by a backtracking line search procedure the recursive search direction is guaranteed to be a descent direction global convergence is proved under fairly minimal requirements on the minimization method used at all grid levels using a limited memory bfgs quasi newton method to produce the direct search direction preliminary numerical experiments show that our line search multigrid approach is promising

a feasible trust region sequential quadratic programming algorithm an algorithm for smooth nonlinear constrained optimization problems is described in which a sequence of feasible iterates is generated by solving a trust region sequential quadratic programming sqp subproblem at each iteration and by perturbing the resulting step to retain feasibility of each iterate by retaining feasibility the algorithm avoids several complications of other trust region sqp approaches the objective function can be used as a merit function and the sqp subproblems are feasible for all choices of the trust region radius global convergence properties are analyzed under various assumptions on the approximate hessian under additional assumptions superlinear convergence to points satisfying second order sufficient conditions is proved

virtual bubbles and galerkin least squares type methods ga l s abstract the equivalence between stabilized finite element methods or galerkin least squares tyoe methods ga l s and the standard galerkin method with bubble functions is established in an abstract framework the results are applicable to various finite element spaces including high order elements and applications to the advective diffusive model and to the stokes problem are presented illustrating the potential of the abstract theory introduced here

solving elliptic control problems with interior point and sqp methods control and state constraints abstract we study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints both boundary control and distributed control problems are considered with boundary conditions of dirichlet or neumann type by introducing suitable discretization schemes the control problem is transcribed into a nonlinear programming problem necessary conditions of optimality are discussed both for the continuous and the discretized control problem it is shown that the recently developed interior point method loqo of 35 is capable of solving these problems even for high discretizations four numerical examples with dirichlet and neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang bang controls

optimal solvers for pde constrained optimization optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering particularly in problems of design the solution of such pde constrained optimization problems is usually a major computational task here we consider simple problems of this type distributed control problems in which the 2 and 3 dimensional poisson problem is the pde the large dimensional linear systems which result from discretization and which need to be solved are of saddle point type we introduce two optimal preconditioners for these systems which lead to convergence of symmetric krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem these preconditioners are block structured and involve standard multigrid cycles the optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases the theoretical proof indicates that these approaches may have much broader applicability for other pdes

optimal low order finite element methods for incompressible flow abstract in this paper we consider the characterization of optimal stabilisation parameters associated with mixed finite element approximations of stokes flow we show how the optimal parameter in terms of the approximating power of the method is related to the parameter value which minimises the condition number of the associated schur complement system for the pressure field we illustrate the tightness of our abstract theoretical results in the cases of the q 1 p 0 quadrilateral and p 1 p 1 triangular elements

convergence properties of minimization algorithms for convex constraints using a structured trust region in this paper we present a class of trust region algorithms for minimization problems within convex feasible regions in which the structure of the problem is explicitly used in the definition of the trust region this development is intended to reflect the possibility that some parts of the problem may be more accurately modelled than others a common occurrence in large scale nonlinear applications after describing the structured trust region mechanism we prove global convergence for all algorithms in our class

adjoint based optimization of pde systems with alternative gradients in this work we investigate a technique for accelerating convergence of adjoint based optimization of pde systems based on a nonlinear change of variables in the control space this change of variables is accomplished in the differentiate then discretize approach by constructing the descent directions in a control space not equipped with the hilbert structure we show how such descent directions can be computed in general lebesgue and besov spaces and argue that in the besov space case determination of descent directions can be interpreted as nonlinear wavelet filtering of the adjoint field the freedom involved in choosing parameters characterizing the spaces in which the steepest descent directions are constructed can be leveraged to accelerate convergence of iterations our computational examples involving state estimation problems for the 1d kuramoto sivashinsky and 3d navier stokes equations indeed show significantly improved performance of the proposed method as compared to the standard approaches

towards discrete velte decompositions and narrow bounds for inf sup constants abstract for the velte decomposition of the function space h 1 0 into three orthogonal subspaces the possibility of discrete equivalents is investigated and some first results are derived for conforming finite element methods for the difference approximation on a staggered grid a full description of the discrete velte decomposition is given in the continuous case we consider the connection of the velte decomposition to an orthogonal decomposition of the pressure space and to the inf sup stability condition of the stokes problem some bounds for inf sup constants in the continuous two dimensional case and numerous discrete inf sup constants have been computed

second order necessary and sufficient optimality conditions for optimization problems and applications to control theory this paper deals with a class of nonlinear optimization problems in a function space where the solution is restricted by pointwise upper and lower bounds and by finitely many equality and inequality constraints of functional type second order necessary and sufficient optimality conditions are established where the cone of critical directions is arbitrarily close to the form which is expected from the optimization in finite dimensional spaces the results are applied to some optimal control problems for ordinary and partial differential equations

on multilevel iterative methods for optimization problems this paper is concerned with multilevel iterative methods which combine a descent scheme with a hierarchy of auxiliary problems in lower dimensional subspaces the construction of auxiliary problems as well as applications to elasto plastic model and linear programming are described the auxiliary problem for the dual of a perturbed linear program is interpreted as a dual of perturbed aggregated linear program coercivity of the objective function over the feasible set is sufficient for the boundedness of the iterates equivalents of this condition are presented in special cases

uniform convergence of multigrid v cycle iterations for indefinite and nonsymmetric problems in this paper an analysis of a multigrid method for nonsymmetric and or indefinite elliptic problems is presented in this multigrid method various types of smoothers may be used one type of smoother considered is defined in terms of an associated symmetric problem and includes point and line jacobi and gauss seidel iterations smoothers based entirely on the original operator are also considered one smoother is based on the normal form that is the product of the operator and its transpose other smoothers studied include point and line jacobi and gauss seidel it is shown that the uniform estimates of j h bramble and j e pasciak math comp 60 1993 pp 447 471 for symmetric positive definite problems carry over to these algorithms more precisely the multigrid iteration for the nonsymmetric and or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine but not dependent on the number of multigrid levels

v cycle convergence of some multigrid methods for ill posed problems for ill posed linear operator equations we consider some v cycle multigrid approaches that in the framework of bramble pasciak wang and xu 1991 we prove to yield level independent contraction factor estimates consequently we can incorporate these multigrid operators in a full multigrid method that together with a discrepancy principle is shown to act as an iterative regularization method for the underlying infinite dimensional ill posed problem numerical experiments illustrate the theoretical results

optimal control formulation for determining optical flow an optimal control formulation for determining optical flow is presented the new framework differs from preceding approaches in that it does not require differentiation of the data and does combine optical flow with image reconstruction it can be considered as a control in the coefficients problem with a cost functional of tracking type a numerical algorithm that solves the optimality system consisting of hyperbolic and elliptic partial differential equations is presented numerical experiments demonstrate the effectiveness of the optimal control approach

a survey of projection based model reduction methods for parametric dynamical systems numerical simulation of large scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena however the inherent large scale nature of the models often leads to unmanageable demands on computational resources model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate yet accurately represent the original large scale system behavior model reduction of linear nonparametric dynamical systems has reached a considerable level of maturity as reflected by several survey papers and books however parametric model reduction has emerged only more recently as an important and vibrant research area with several recent advances making a survey paper timely thus this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods parametric model reduction targets the broad class of problems for wh

on the global convergence of trust region algorithms using inexact gradient information trust region algorithms are an important class of methods that can be used to solve unconstrained optimization problems strong global convergence results are demonstrated for a class of methods where the gradient values are approximated rather than computed exactly provided they obey a simple relative error condition no requirement is made that gradients be recomputed to successively greater accuracy after unsuccessful iterations

shaping axis symmetric dual reflector antennas by combining conic sections a simple procedure for the shaping of axis symmetric dual reflector antennas is described the shaping procedure is based on the consecutive concatenation of local conic sections suited to provide under geometrical optics go principles an aperture field with uniform phase together with a prescribed amplitude distribution the procedure has fast numerical convergence and is valid for any circularly symmetric dual reflector configuration to illustrate the procedure two representative configurations are investigated the go shaping results are validated using accurate method of moments analysis

adaptive computation for boundary control of radiative heat transfer in glass this paper is concerned with an optimal boundary control of the cooling down process of glass an important step in glass manufacturing since the computation of the complete radiative heat transfer equations is too complex for optimization purposes we use simplified approximations of spherical harmonics including a practically relevant frequency bands model the optimal control problem is considered as a constrained optimization problem a first order optimality system is derived and decoupled with the help of a gradient method based on the solution to the adjoint equations the arising partial differential algebraic equations of mixed parabolic elliptic type are numerically solved by a self adaptive method of lines approach of rothe type adaptive finite elements in space and one step methods of rosenbrock type with variable step sizes in time are applied we present numerical results for a two dimensional glass cooling problem

consistent local projection stabilized finite element methods this work establishes a formal derivation of local projection stabilized methods as a result of an enriched petrov galerkin strategy for the stokes problem the initial unstable mathbb p 1 times mathbb p l l 0 1 finite element space is enhanced with solutions of residual based local problems and then the static condensation procedure is applied to derive new methods the approach keeps degrees of freedom unchanged while giving rise to new stable and consistent methods for continuous and discontinuous approximation spaces for the pressure the resulting methods do not need the use of a macroelement grid structure and are parameter free the numerical analysis is carried out showing optimal convergence in natural norms and moreover two ways of rendering the velocity field locally mass conservative are proposed some numerics validate the theoretical results

multigrid one shot method for state constrained aerodynamic shape optimization this paper presents applications of an optimization based multigrid method to state constrained aerodynamic shape optimization problems the one shot simultaneous pseudo time stepping method has been applied successfully to such problems reducing the cost of computation up to 70 from that of a traditional gradient method however the number of optimization iterations is comparatively large in this method since we update the design parameters in each time step of state and costate runs in this paper we use an optimization based multigrid strategy to reduce the total number of optimization iterations while keeping the idea of the one shot pseudo time stepping method design examples of drag reduction with constant lift together with geometrical constraints for an rae2822 airfoil are included the total effort required for convergence of the optimization problem is between four and six times that of forward simulation runs depending on computational grid size

numerical experience with a class of algorithms for nonlinear optimization using inexact function and gradient information for optimization problems associated with engineering design parameter estimation image reconstruction and other optimization simulation applications low accuracy function and gradient values are frequently much less expensive to obtain than high accuracy values the computational performance of trust region methods for nonlinear optimization is investigated for cases when high accuracy evaluations are unavailable or prohibitively expensive and earlier theoretical predictions that such methods are convergent even with relative gradient errors of 0 5 or more is confirmed the proper choice of the amount of accuracy to use in function and gradient evaluations can result in orders of magnitude savings in computational cost

mesh independence for an augmented lagrangian sqp method in hilbert spaces an augmented lagrangian sqp algorithm for optimal control of differential equations in hilbert spaces is analyzed this algorithm has second order convergence rate provided that a second order sufficient optimality condition is satisfied the internal approximation of this method is investigated and convergence results are presented a mesh independence principle for the augmented lagrangian sqp method is proved which asserts that asymptotically the infinite dimensional algorithm and finite dimensional discretizations have the same convergence property more precisely for sufficiently small mesh size there is at most a difference of one iteration step between the number of steps required by the infinite dimensional method and its discretization to converge within a given tolerance varepsilon 0 the theoretical results are demonstrated by two optimal control problems for the burgers equation

inexact objective function evaluations in a trust region algorithm for pde constrained optimization under uncertainty this paper improves the trust region algorithm with adaptive sparse grids introduced in siam j sci comput 35 2013 pp a1847 a1879 for the solution of optimization problems governed by partial differential equations pdes with uncertain coefficients the previous algorithm used adaptive sparse grid discretizations to generate models that are applied in a trust region framework to generate a trial step the decision whether to accept this trial step as the new iterate however required relatively high fidelity adaptive discretizations of the objective function in this paper we extend the algorithm and convergence theory to allow the use of low fidelity adaptive sparse grid models in objective function evaluations this is accomplished by extending conditions on inexact function evaluations used in previous trust region frameworks our algorithm adaptively builds two separate sparse grids one to generate optimization models for the step computation and one to approximate the objective function

rigorous quantitative analysis of multigrid i constant coefficients two level cycle with l 2 norm exact numerical convergence factors for any multigrid cycle can be predicted by local mode fourier analysis for general linear elliptic partial differential equation pde systems with piecewise smooth coefficients in general domains discretized by uniform grids it is proved that in the limit of small meshsizes these predicted factors are indeed obtained provided the cycle is supplemented with a proper processing at and near the boundaries that processing it is proved costs negligible extra computer work apart from mode analysis a coarse grid approximation cga condition is introduced which is both necessary and sufficient for the multigrid algorithm to work properly the present part studies the l 2 convergence in one cycle for equations with constant coefficients in the sequel brandt rigorous quantitative analysis of multigrid ii extensions and practical implications manuscript extensions are discussed to many cycles asymptotic convergence to more levels with arbitrary cycle ty

recursive trust region methods for multiscale nonlinear optimization a class of trust region methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems like those arising in systems governed by partial differential equations the algorithms in this class make use of the discretization level as a means of speeding up the computation of the step this use is recursive leading to true multilevel multiscale optimization methods reminiscent of multigrid methods in linear algebra and the solution of partial differential equations a simple algorithm of the class is then described and its numerical performance is shown to be numerically promising this observation then motivates a proof of global convergence to first order stationary points on the fine grid that is valid for all algorithms in the class

a posteriori error estimators for stabilized finite element approximations of an optimal control problem abstract we derive a posteriori error estimators for an optimal control problem governed by a convection reaction diffusion equation control constraints are also considered we consider a family of low order stabilized finite element methods to approximate the solutions to the state and adjoint equations we obtain a fully computable a posteriori error estimator for the optimal control problem all the constants that appear in the upper bound for the error are fully specified therefore the proposed estimator can be used as a stopping criterion in adaptive algorithms we also obtain a robust a posteriori error estimator for when the error is measured in a norm that involves the dual norm of the convective derivative numerical examples in two and three dimensions are presented to illustrate the theory

stabilization of low order mixed finite elements for the stokes equations we present a new family of stabilized methods for the stokes problem the focus of the paper is on the lowest order velocity pressure pairs while not lbb compliant their simplicity and attractive computational properties make these pairs a popular choice in engineering practice our stabilization approach is motivated by terms that characterize the lbb deficiency of the unstable spaces the stabilized methods are defined by using these terms to modify the saddle point lagrangian associated with the stokes equations the new stabilized methods offer a number of attractive computational properties in contrast to other stabilization procedures they are parameter free do not require calculation of higher order derivatives or edge based data structures and always lead to symmetric linear systems furthermore the new methods are unconditionally stable achieve optimal accuracy with respect to solution regularity and have simple and straightforward implementations we present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods

main reflector shaping of omnidirectional dual reflectors using local conic sections this work presents an alternative method for go synthesis of dual reflector antennas suited for omnidirectional coverage both reflectors are bodies of revolution where a single conic section generates the sub reflector while a concatenation of local conic sections represent the main reflector generatrix which is shaped to provide an arbitrary radiation pattern in the antenna s elevation plane to illustrate the method omnidirectional axis displaced ellipse oade configurations are synthesized to provide a cosecant squared radiation pattern in the elevation plane two different ray structures are considered with real or virtual main reflector caustics the go shaping results are validated by method of moments simulations

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