the velocity tracking problem for navier stokes flows with bounded distributed controls we present some systematic approaches to the mathematical analysis and numerical approximation of the time dependent optimal control problem of tracking the velocity for navier stokes flows in bounded two dimensional domains with bounded distributed controls we study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined we also define and analyze semidiscrete in time and fully space time discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system the results of some computational experiments are provided

fully holderian stable minimum with respect to both tilt and parameter perturbations when the objective function undergoes both a tilt perturbation and a general parameter perturbation this paper considers the notions of a fully stable holder minimizer a uniform holder growth con

second order characterizations of tilt stability with applications to nonlinear programming the paper is devoted to the study of tilt stable local minimizers of general optimization problems in finite dimensional spaces and its applications to classical nonlinear programs with twice continuously differentiable data the importance of tilt stability has been well recognized from both theoretical and numerical aspects of optimization and this notion has been extensively studied in the literature based on advanced tools of second order variational analysis and generalized differentiation we develop a new approach to tilt stability which allows us to derive not only qualitative but also quantitative characterizations of tilt stable minimizers with calculating the corresponding moduli the implementation of this approach and general results in the classical framework of nonlinear programming provides complete characterizations of tilt stable minimizers under new second order qualification and optimality conditions

full lipschitzian and holderian stability in optimization with applications to mathematical programming and optimal control the paper concerns a systematic study of full stability in general optimization models including its conventional lipschitzian version as well as the new holderian one we derive various characteri

reduced sqp methods for parameter identification problems parameter estimation problems are formulated as constrained regularized optimization problems reduced sqp methods with bfgs update are analyzed to solve these infinite dimensional optimization problems rate of convergence results are given and numerical feasibility of the resulting algorithms is demonstrated

pseudodifferential operators on supermanifolds and the atiyah singer index theorem fermionic quantization or clifford algebra is combined with pseudodifferential operators to simplify the proof of the atiyah singer index theorem for the dirac operator on a spin manifold

tilt stability uniform quadratic growth and strong metric regularity of the subdifferential we prove that uniform second order growth tilt stability and strong metric regularity of the subdifferential three notions that have appeared in entirely different settings are all essentially equivalent for any lower semicontinuous extended real valued function

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

partial smoothness tilt stability and generalized hessians we compare two recent variational analytic approaches to second order conditions and sensitivity analysis for nonsmooth optimization we describe a broad setting where computing the generalized hessian of mordukhovich is easy in this setting the idea of tilt stability introduced by poliquin and rockafellar is equivalent to a classical smooth second order condition

weak sharp minima in mathematical programming the notion of a sharp or strongly unique minimum is extended to include the possibility of a nonunique solution set these minima will be called weak sharp minima conditions necessary for the solution set of a minimization problem to be a set of weak sharp minima are developed in both the unconstrained and constrained cases these conditions are also shown to be sufficient under the appropriate convexity hypotheses the existence of weak sharp minima is characterized in the cases of linear and quadratic convex programming and for the linear complementarity problem in particular a result of mangasarian and meyer is reproduced that shows that the solution set of a linear program is always a set of weak sharp minima whenever it is nonempty consequences for the convergence theory of algorithms are also examined especially conditions yielding finite termination

weak sharp minima for semi infinite optimization problems with applications we study local weak sharp minima and sharp minima for smooth semi infinite optimization problems sip we provide several dual and primal characterizations for a point to be a sharp minimum or a weak sharp minimum of sip as applications we present several sufficient and necessary conditions of calmness for infinitely many smooth inequalities in particular we improve some calmness results in r henrion and j outrata math program 104 2005 pp 437 464

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

second order sufficient optimality conditions for a class of nonlinear parabolic boundary control problems in this paper sufficient second order optimality conditions are established for parabolic boundary control problems with nonlinear boundary condition and constraints on the control and the state the main idea is to extend the known theory for systems governed by ordinary differential equations to the case of partial differential equations this is performed by means of a semigroup approach and a two norm technique the verification of the second order conditions is discussed

holder weak sharp minimizers and holder tilt stability abstract in this paper we introduce and study the notions of holder weak sharp minimizers stable holder weak sharp minimizers and holder tilt stable weak minimizers for a proper lower semicontinuous function f on a banach space in terms of the holder metric subregularity regularity of f we consider optimality conditions for holder weak sharp minimizers and stable holder weak sharp minimizers we prove that x is a stable holder weak sharp minimizer resp a holder tilt stable weak minimizer of f if and only if it is a stable holder sharp minimizer resp a holder tilt stable minimizer of f

an adaptive finite element strategy for the three dimensional time dependent navier stokes equations abstract an adaptive strategy for three dimensional time dependent problems in the context of the fem is presented the basic tools are a mechanism for local refinement and coarsening of simplicial meshes and an unexpensive error estimator the algorithm for local grid modification is based on bisecting tetrahedra the method is applied to the navier stokes equations

second order variational analysis and characterizations of tilt stable optimal solutions in infinite dimensional spaces abstract the paper is devoted to developing second order tools of variational analysis and their applications to characterizing tilt stable local minimizers of constrained optimization problems infinite dimensional spaces with many results new also in finite dimensional settings the importance of tilt stability has been well recognized from both theoretical and numerical aspects of optimization based on second order generalized differentiation we obtain qualitative and quantitative characterizations of tilt stability in general frameworks of constrained optimization and establish its relationships with strong metric regularity of subgradient mappings and uniform second order growth the results obtained are applied to deriving new necessary and sufficient conditions for tilt stable minimizers in problems of nonlinear programming with twice continuously differentiable data in hilbert spaces

the higher index theorem for manifolds with boundary abstract the global calculus of pseudodifferential operators on manifolds with cylindrical ends is developed to study the higher indices of elliptic boundary value problems the higher index theorem is then proved by employing the asymptotic symbolic calculus of the global pseudodifferential operators based on a scaling similar to that of getzler

optimal control of two and three dimensional incompressible navier stokes flows the focus of this work is on the development of large scale numerical optimization methods for optimal control of steady incompressible navier stokes flows the control is affected by the suction or injection of fluid on portions of the boundary and the objective function represents the rate at which energy is dissipated in the fluid we develop reduced hessian sequential quadratic programming methods that avoid converging the flow equations at each iteration both quasi newton and newton variants are developed and compared to the approach of eliminating the flow equations and variables which is effectively the generalized reduced gradient method optimal control problems are solved for two dimensional flow around a cylinder and three dimensional flow around a sphere the examples demonstrate at least an order of magnitude reduction in time taken allowing the optimal solution of flow control problems in as little as half an hour on a desktop workstation

the local convergence of broyden like methods on lipschitzian problems in hilbert spaces the classical local convergence analysis for broyden s method requires a finite number of real variables holder continuity of the jacobian and a rather accurate initial approximation b 0 here these three conditions are relaxed to allow for lipschitz continuous hilbert space operators that can be reasonably well approximated by a linear operator b that is close to b 0 up to an arbitrary compact perturbation to achieve local and r linear convergence broyden s method is modified by a very mild line search condition in case of a continuously differentiable system in euclidean space we obtain r superlinear convergence without restricting b 0 or requiring the jacobian to be holder continuous

optimal control of thermally convected fluid flows we examine the optimal control of stationary thermally convected fluid flows from the theoretical and numerical point of view we use thermal convection as control mechanism that is control is effected through the temperature on part of the boundary control problems are formulated as constrained minimization problems existence of optimal control is given and a first order necessary condition of optimality from which optimal solutions can be obtained is established we develop numerical methods to solve the necessary condition of optimality and present computational results for control of cavity and channel type flows showing the feasibility of the proposed approach

an infinite dimensional convergence theory for reduced sqp methods in hilbert space we develop a general convergence theory for a class of reduced successive quadratic programming sqp methods for infinite dimensional equality constrained optimization problems in hilbert space the methods under consideration approximate second order information of the lagrangian restricted to the null space of h the derivative of the constraints in particular a sufficient condition for two step superlinear convergence will be given and a general result for secant methods will be established which can serve as a tool to prove superlinear convergence in an infinite dimensional framework this result is applied to prove local two step superlinear convergence of reduced sqp methods under the assumption that exact reduced second order information is known up to a compact perturbation to test our convergence theory we consider optimal control problems and we formulate a reduced quasi newton algorithm which presents a new approach to an efficient solution of these problems our main motivation for st

mesh independence of the gradient projection method for optimal control problems the gradient projection method and its application to optimal control have been analyzed with regard to the rate of convergence quite extensively here another aspect of this method is considered in finite dimension the set of active indices is identified after finitely many iterations under mild nondegeneracy assumptions however it is clear that this property is restricted to the finite dimensional case in this paper for example a sequence of discretized optimal control problems is considered and it is observed that the number of steps to identify all active indices increases with the refinement of the discretization a result analogous to the finite dimensional result is valid in this situation if identification of active indices is understood in the correct light this paper shows that if a different termination criterion is imposed then the number of necessary steps for termination is indeed mesh independent numerical observations illustrating this result are reported for various examples fr

numerical solution of a flow control problem vorticity reduction by dynamic boundary action in order to laminarize an unsteady internal flow the vorticity field is minimized in a least squares sense using an optimal control approach the flow model is the navier stokes equation for a viscous incompressible fluid and the flow is controlled by suction and blowing on a part of the boundary a quasi newton method is used for the minimization of a quadratic objective function involving a measure of the vorticity and a regularization term the navier stokes equations are approximated using a finite difference scheme in time and finite element approximations in space accurate expressions for the gradient of the discrete objective function are needed to obtain a satisfactory convergence rate of the minimization algorithm therefore first order necessary conditions for a minimizer of the objective function are derived in the fully discrete case a memory saving device is discussed without which problems of any realistic size especially in three space dimensions would remain computationally intractable the feasibility of the optimal control approach for flow control problems is demonstrated by numerical experiments for a two dimensional flow in a rectangular cavity at a reynolds number high enough for nonlinear effects to be important

newton s method for a class of optimal shape design problems a class of optimal shape design problems is considered where a part of the boundary of the domain represents the free parameter the variable domain is parametrized by a class of functions in such a way that the optimal design problem results in an optimal control problem on a fixed domain the functions for the parametrization of the domain are used as controls and the corresponding states are then given by the solution of an elliptic boundary value problem on a fixed domain discretizing this control problem normally leads to a large scale optimization problem where the corresponding solution methods are characterized by the requirement of solving many boundary value problems in spite of this interesting numerical challenge until now little work has been done to derive more efficient algorithms by taking advantage of the specific structure of this kind of problem in this report newton s method in function space is derived resulting in an efficient algorithm for the discretized optimization problems by using the specific structure of these optimal shape design problems an efficient implementation of the numerical algorithm is introduced the properties of this algorithm are compared with those of the gradient method using illustrative numerical examples