mixed finite element method for dirichlet boundary control problem governed by elliptic pdes in this paper we study the finite element approximation of dirichlet boundary control problems governed by elliptic pdes based on a mixed variational scheme we establish a mixed finite element approximation to the underlying optimal control problem we consider the optimal control problems posed on both polygonal and general smooth domains and we derive a priori error estimates for optimal control state and adjoint state the optimal and quasi optimal error estimates are obtained for problems on polygonal and smooth domains respectively numerical experiments are provided to confirm our theoretical results

the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

mesh refinement and numerical sensitivity analysis for parameter calibration of partial differential equations we consider the calibration of parameters in physical models described by partial differential equations this task is formulated as a constrained optimization problem with a cost functional of least squares type using information obtained from measurements an important issue in the numerical solution of this type of problem is the control of the errors introduced first by discretization of the equations describing the physical model and second by measurement errors or other perturbations our strategy is as follows we suppose that the user defines an interest functional i which might depend on both the state variable and the parameters and which represents the goal of the computation first we propose an a posteriori error estimator which measures the error with respect to this functional this error estimator is used in an adaptive algorithm to construct economic meshes by local mesh refinement the proposed estimator requires the solution of an auxiliary linear equation second we address the question of sensitivity applying similar techniques as before we derive quantities which describe the influence of small changes in the measurements on the value of the interest functional these numbers which we call relative condition numbers give additional information on the problem under consideration they can be computed by means of the solution of the auxiliary problem determined before finally we demonstrate our approach at hand of a parameter calibration problem for a model flow problem

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

guaranteed computable bounds on quantities of interest in finite element computations we develop and compare a number of alternative approaches to obtain guaranteed and fully computable bounds on the error in quantities of interest of arbitrary order finite element approximations in the context of a linear second order elliptic problem in each case the bounds are fully computable and do not involve any unknown multiplicative factors guaranteed computable bounds are also obtained for the case when the dirichlet boundary conditions are non homogeneous this is achieved by taking account of the error incurred by the approximation of the dirichlet data in the functional used to approximate the quantity of interest itself which is found to generally give better results numerical examples are presented to show that the resulting estimators provide tight bounds with the effectivity index tending to unity from above

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

an all at once approach for the optimal control of the unsteady burgers equation we apply an all at once method for the optimal control of the unsteady burgers equation the nonlinear burgers equation is discretized in time using the semi implicit discretization and the resulting first order optimality conditions are solved iteratively by newton s method the discretize then optimize approach is used because it leads to a symmetric indefinite saddle point problem numerical results for the distributed unconstrained and control constrained problems illustrate the performance of the all at once approach with semi implicit time discretization

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

a posteriori error estimation for control constrained linear quadratic optimal control problems we derive a posteriori error estimates for control constrained linear quadratic optimal control problems the error is measured in a norm which is motivated by the objective our abstract error estimator is separated into three contributions the error in the variational inequality i e in the optimality condition for the control and the errors in the state and adjoint equation hence one can use well established estimators for the differential equations we show that the abstract error estimator is reliable and efficient if the utilized estimators for the differential equations have these properties we apply the error estimator to two distributed optimal control problems with distributed and boundary observation respectively numerical examples exhibit a good error reduction if we use the local error contributions for an adaptive mesh refinement

new universal spectral correlators we study the universal properties of distributions of eigenvalues of random matrices in the large n limit the distributions fall in universality classes characterized entirely by the support of the spectral density

constrained dirichlet boundary control in l 2 for a class of evolution equations optimal dirichlet boundary control based on the very weak solution of a parabolic state equation is analyzed this approach allows us to consider the boundary controls in l 2 which has advantages over approaches which consider control in sobolev spaces involving fractional derivatives pointwise constraints on the boundary are incorporated by the primal dual active set strategy its global and local superlinear convergences are shown a discretization based on space time finite elements is proposed and numerical examples are included

non commutative discretize then optimize algorithms for elliptic pde constrained optimal control problems abstract in this paper we analyze the convergence of several optimize then discretize and discretize then optimize algorithms based on either a second order or a fourth order finite difference discretization for solving elliptic pde constrained optimal control problems to ensure the convergence of a discretize then optimize algorithm one well accepted criterion is to design the discretization scheme such that the resulting discretize then optimize algorithm commutes with the corresponding optimize then discretize algorithm in other words both algorithms should give rise to exactly the same discrete optimality system however such a restrictive criterion is not trivial to fulfill by investigating a distributed control problem governed by an elliptic equation we first show that enforcing such a stringent condition of commutative property is only sufficient but not necessary for achieving the desired convergence we then introduce some suitable h 1 semi norm penalty regularization terms to recover the lost convergence due to the inconsistency caused by the loss of commutativity numerical experiments are carried out to verify our theoretical analysis and also validate the effectiveness of our proposed regularization techniques

some convergence results for howard s algorithm this paper deals with convergence results of howard s algorithm for the resolution of min a in mathcal a b a x c a 0 where b a is a matrix c a is a vector and mathcal a is a compact set we show a global superlinear convergence result under a monotonicity assumption on the matrices b a extensions of howard s algorithm for a max min problem of the form max b in mathcal b min a in mathcal a b a b x c a b 0 are also proposed in the particular case of an obstacle problem of the form min ax b x g 0 where a is an n times n matrix satisfying a monotonicity assumption we show the convergence of howard s algorithm in no more than n iterations instead of the usual 2 n bound still in the case of an obstacle problem we establish the equivalence between howard s algorithm and a primal dual active set algorithm m hintermuller k ito and k kunisch siam j optim 13 2002 pp 865 888 the algorithms are illustrated on the discretization of nonlinear pdes arising in the context of mathematical finance american option and merton s portfolio problem of front propagation problems and for the double obstacle problem

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

a partition of unity method for a class of fourth order elliptic variational inequalities abstract we consider a partition of unity method pum for a class of fourth order elliptic variational inequalities on convex polygonal domains that include obstacle problems of simply supported kirchhoff plates and elliptic distributed optimal control problems with pointwise state constraints as special cases by including singular functions in the local approximation spaces we are able to show that the partition of unity method converges optimally numerical results that corroborate the theoretical estimates are also presented

on the asymptotic behaviour of correlators of multi cut matrix models we consider invariant ensembles of n n hermitian random matrices known also as the matrix models in the case where the support of the limiting density of states dos consists of several intervals by using recent results on the asymptotics of orthogonal polynomials we find first that the amplitudes of the leading terms of the correlator of the normalized traces of resolvent of random matrices and of their densities of states are quasi periodic functions of n whose frequencies are the integrals of the limiting dos over the intervals of the support and whose form is uniquely determined by the edges of the support this suggest a certain parametrization of the universality classes of the correlator second we show that the leading terms of these correlators can be expressed correspondingly via the matrix elements of the resolvent and of the spectral kernel of a certain quasi periodic jacobi matrix whose coefficients are determined by the same frequencies

singular behavior of the density of states and the lyapunov coefficient in binary random harmonic chains we study the integrated density of statesh 2 of a chain of harmonic oscillators with a binary random distribution of the masses we show in particular that there is a dense set of values of the squared frequency for which the differenceh 2 h 2 has a singularity of the type 2 multiplied by a periodic function of ln where the exponent and the period depend continuously on 2 in the region where 1 2 h is not differentiate on a dense set of points the same type of singularities is also present in the lyapunov coefficient

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

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

regularized state constrained boundary optimal control of the navier stokes equations abstract the numerical solution of the dirichlet boundary optimal control problem of the navier stokes equations in presence of pointwise state constraints is investigated two different regularization techniques are considered first a moreau yosida regularization of the problem is studied optimality conditions are derived and the convergence of the regularized solutions towards the original one is proved a source representation of the control combined with a lavrentiev type regularization strategy is also presented the analysis concerning optimality conditions and convergence of the regularized solutions is carried out in the last part of the paper numerical experiments are presented for the numerical solution of each regularized problem a semi smooth newton method is applied

optimal control problem of the two dimensional modified anomalous subdiffusion equation with discontinuous galerkin approximation abstract in this paper we consider an optimal control problem ocp governed by a modified anomalous subdiffusion equation with box constraints on the control we apply space time discontinuous galerkin method to discretize the problem specifically symmetric interior penalty galerkin method and piecewise discontinuous constant temporal approximations are used then we discuss the equivalence of optimize then discretize and discretize then optimize approaches in a general setting we derive stability and convergence analysis for the fully discrete problem at the end we present some numerical results to measure the numerical rate of convergence

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

preconditioning pde constrained optimization with l1 sparsity and control constraints abstract pde constrained optimization aims at finding optimal setups for partial differential equations so that relevant quantities are minimized including nonsmooth l 1 sparsity promoting terms in the formulation of such problems results in more practically relevant computed controls but adds more challenges to the numerical solution of these problems the needed l 1 terms as well as additional inclusion of box control constraints require the use of semismooth newton methods we propose robust preconditioners for different formulations of the newton equation with the inclusion of a line search strategy and an inexact approach for the solution of the linear systems the resulting semismooth newton s method is reliable for practical problems our results are underpinned by a theoretical analysis of the preconditioned matrix numerical experiments illustrate the robustness of the proposed scheme

mathematical programs with complementarity constraints in function space c and strong stationarity and a path following algorithm an optimal control problem governed by an elliptic variational inequality is studied the feasible set of the problem is relaxed and a path following type method is used to regularize the constraint on the state variable first order optimality conditions for the relaxed regularized subproblems are derived and convergence of stationary points with respect to the relaxation and regularization parameters is shown in particular c and strong stationarity as well as variants thereof are studied the subproblems are solved by using semismooth newton methods the overall algorithmic concept is provided and its performance is discussed by means of examples including problems with bilateral constraints and a nonsymmetric operator

energy corrected fem for optimal dirichlet boundary control problems in the presence of re entrant corners the solution of pde constrained optimization problems in general has singular components even when the given data are smooth consequently the use of standard finite element discretization gives rise to the so called pollution effect this means that only a reduced convergence rate compared to the best approximation error is obtained we discuss how optimal convergence rates in weighted norms can be achieved using the idea of energy corrected finite element methods applied to optimal dirichlet boundary control problem in the energy space we present optimal error estimates in weighted norms for the state variable and for the control several numerical examples illustrate the obtained theoretical results

a feasible active set method for strictly convex quadratic problems with simple bounds a primal dual active set method for quadratic problems with bound constraints is presented which extends the infeasible active set approach of kunisch and rendl siam j optim 14 2003 pp 35 52 based on a guess of the active set a primal dual pair x alpha is computed that satisfies stationarity and the complementary condition if x is not feasible the variables connected to the infeasibilities are added to the active set and a new primal dual pair x alpha is computed this process is iterated until a primal feasible solution is generated then a new active set is determined based on the feasibility information of the dual variable alpha strict convexity of the quadratic problem is sufficient for the algorithm to stop after a finite number of steps with an optimal solution computational experience indicates that this approach also performs well in practice

primal dual active set strategy for a general class of constrained optimal control problems an algorithm for efficient solution of optimal control problems with pointwise constraints is introduced it is based on an active set strategy using primal as well as dual variables under certain assumptions a decrease of an appropriately chosen merit function and convergence of the algorithm are proved numerical examples for a parabolic optimal control problem demonstrate the efficiency of the proposed method for control constrained problems

primal dual active set strategy for large scale optimization of cardiac defibrillation a feasibility study of optimal control techniques for cardiac defibrillation on anatomical three spatial dimensional rabbit ventricle geometry in the presence of bilateral control constraints is presented the work addresses the numerical treatment of multi scale and multi domain simulations of the bidomain equations and is based on the primal dual active set method to solve the optimality system for this large scale optimization problem numerical results are presented for a successful defibrillation study robustness of the optimization algorithm w r t to variations in the model parameters is demonstrated a feasibility study for multiple small boundary control support is included as well finally the numerical convergence of the optimization algorithm and the parallel efficiency is demonstrated

multigrid methods for parabolic distributed optimal control problems multigrid schemes that solve parabolic distributed optimality systems discretized by finite differences are investigated accuracy properties of finite difference approximation are discussed and validated two multigrid methods are considered which are based on a robust relaxation technique and use two different coarsening strategies semicoarsening and standard coarsening the resulting multigrid algorithms show robustness with respect to changes of the value of v the weight of the cost of the control is sufficiently small fourier mode analysis is used to investigate the dependence of the linear twogrid convergence factor on v and on the discretization parameters results of numerical experiments are reported that demonstrate sharpness of fourier analysis estimates a multigrid algorithm that solves optimal control problems with box constraints on the control is considered

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

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

optimal control for elliptic systems with pointwise euclidean norm constraints on the controls optimal control for an elliptic system with pointwise euclidean norm constraints on the control variables is investigated first order optimality conditions are derived in a manner that is amenable for numerical realisation an efficient semismooth newton algorithm is proposed based on this optimality system numerical examples are given to validate the superlinear convergence of the semismooth newton algorithm

a priori error estimates for space time finite element discretization of parabolic optimal control problems in this article we summarize recent results on a priori error estimates for space time finite element discretizations of linear quadratic parabolic optimal control problems we consider the following three cases problems without inequality constraints problems with pointwise control constraints and problems with state constraints pointwise in time for all cases error estimates with respect to the temporal and to the spatial discretization parameters are derived the results are illustrated by numerical examples

first order conditions for nonsmooth discretized constrained optimal control problems this paper studies first order conditions karush kuhn tucker conditions for discretized optimal control problems with nonsmooth constraints we present a simple condition which can be used to verify that a local optimal point satisfies the first order conditions and that a point satisfying the first order conditions is a global or local optimal solution of the optimal control problem

robust error estimates for the finite element approximation of elliptic optimal control problems in this paper we consider the finite element approximation of an elliptic optimal control problem based on an assumption on the adjoint state of the continuous problem with a small parameter which represents a regularization of the bang bang type control problem we derive robust a priori error estimates for optimal control and state and a posteriori error estimate is also presented numerical experiments confirm our theoretical results

on the statistical mechanics approach in the random matrix theory integrated density of states we consider the ensemble of random symmetricn n matrices specified by an orthogonal invariant probability distribution we treat this distribution as a gibbs measure of a mean field type model this allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn and that this limiting distribution is the solution of a certain self consistent equation

a semi smooth newton method for control constrained boundary optimal control of the navier stokes equations abstract in this paper we study optimal control of the navier stokes equations when the control acts as a pointwise constrained boundary condition of dirichlet type the problem is analyzed in the control space h 00 1 2 the optimality system and second order sufficient optimality conditions are derived for the numerical solution we apply a semi smooth newton method to a regularized version of the original problem and show convergence properties of the method and of the regularized solutions towards the original one

numerical sensitivity analysis for the quantity of interest in pde constrained optimization in this paper we consider the efficient computation of derivatives of a functional the quantity of interest which depends on the solution of a pde constrained optimization problem with inequality constraints and which may be different from the cost functional the optimization problem is subject to perturbations in the data we derive conditions under with the quantity of interest possesses first and second order derivatives with respect to the perturbation parameters an algorithm for the efficient evaluation of these derivatives is developed with considerable savings over a direct approach especially in the case of high dimensional parameter spaces the computational cost is shown to be small compared to that of the overall optimization algorithm numerical experiments involving a parameter identification problem for navier stokes flow and an optimal control problem for a reaction diffusion system are presented which demonstrate the efficiency of the method

adaptive finite element methods for an optimal control problem involving dirac measures the purpose of this work is the design and analysis of a reliable and efficient a posteriori error estimator for the so called pointwise tracking optimal control problem this linear quadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the state thus leading to an adjoint problem with dirac measures on the right hand side control constraints are also considered the proposed error estimator relies on a posteriori error estimates in the maximum norm for the state and in muckenhoupt weighted sobolev spaces for the adjoint state we present an analysis that is valid for two and three dimensional domains we conclude by presenting several numerical experiments which reveal the competitive performance of adaptive methods based on the devised error estimator

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

boundary control of linear uncertain 1 d parabolic pde using approximate dynamic programming this paper develops a near optimal boundary control method for distributed parameter systems governed by uncertain linear 1 d parabolic partial differential equations pde by using approximate dynamic programming a quadratic surface integral is proposed to express the optimal cost functional for the infinite dimensional state space accordingly the hamilton jacobi bellman hjb equation is formulated in the infinite dimensional domain without using any model reduction subsequently a neural network identifier is developed to estimate the unknown spatially varying coefficient in pde dynamics novel tuning law is proposed to guarantee the boundedness of identifier approximation error in the pde domain a radial basis network rbn is subsequently proposed to generate an approximate solution for the optimal surface kernel function online the tuning law for near optimal rbn weights is created such that the hjb equation error is minimized while the dynamics are identified and closed loop system remains stable ultimate boundedness ub of the closed loop system is verified by using the lyapunov theory the performance of the proposed controller is successfully confirmed by simulation on an unstable diffusion reaction process

l infty estimates for approximated optimal control problems an optimal control problem for a two dimensional elliptic equation is investigated with pointwise control constraints this paper is concerned with discretization of the control by piecewise linear functions the state and the adjoint state are discretized by linear finite elements approximation of order h in the l infty norm is proved in the main result

deformations of calabi yau hypersurfaces arising from deformations of toric varieties one can deform the complex structure of calabi yau hypersurfaces in toric varieties by changing the coefficients of the defining polynomial however there must also exist non polynomial deformations of the calabi yau hypersurfaces which cannot be realized inside the ambient toric variety in this paper we have constructed the missing non polynomial deformations which are induced by an automorphism on the open part of the ambient toric variety

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

a primal dual active set strategy for optimal boundary control of a nonlinear reaction diffusion system this paper is concerned with optimal boundary control of an instationary reaction diffusion system in three spatial dimensions this problem involves a coupled nonlinear system of parabolic differential equations with bilateral as well as integral control constraints we include the integral constraint in the cost by a penalty term whereas the bilateral control constraints are handled explicitly first and second order conditions for the optimization problem are analyzed a primal dual active set strategy is utilized to compute optimal solutions numerically the algorithm is compared to a semismooth newton method

universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles this paper is devoted to the rigorous proof of the universality conjecture of random matrix theory according to which the limiting eigenvalue statistics ofn n random matrices within spectral intervals ofo n 1 is determined by the type of matrix real symmetric hermitian or quaternion real and by the density of states we prove this conjecture for a certain class of the hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function the potential in the quantum field theory satisfying some regularity conditions

adaptive optimal control of the obstacle problem this article is concerned with the derivation of a posteriori error estimates for optimization problems subject to an obstacle problem to circumvent the nondifferentiability inherent to this type of problem we introduce a sequence of penalized but differentiable problems we show differentiability of the central path and derive separate a posteriori dual weighted residual estimates for the errors due to penalization discretization and iterative solution of the discrete problems the effectivity of the derived estimates and of the adaptive algorithm is demonstrated on two numerical examples

an infeasible active set method for quadratic problems with simple bounds a primal dual active set method for quadratic problems with bound constraints is presented based on a guess on the active set a primal dual pair x s is computed that satisfies the first order optimality condition and the complementarity condition if x s is not feasible a new active set is determined and the process is iterated sufficient conditions for the iterations to stop in a finite number of steps with an optimal solution are provided computational experience indicates that this approach often requires only a few less than 10 iterations to find the optimal solution

discretization error analysis and adaptive meshing algorithms for fluorescence diffuse optical tomography in the presence of measurement noise quantitatively accurate fluorescence diffuse optical tomographic fdot image reconstruction is a computationally demanding problem that requires repeated numerical solutions of two coupled partial differential equations and an associated inverse problem recently adaptive finite element methods have been explored to reduce the computation requirements of the fdot image reconstruction however existing approaches ignore the ubiquitous presence of noise in boundary measurements in this paper we analyze the effect of finite element discretization on the fdot forward and inverse problems in the presence of measurement noise and develop novel adaptive meshing algorithms for fdot that take into account noise statistics we formulate the fdot inverse problem as an optimization problem in the maximum a posteriori framework to estimate the fluorophore concentration in a bounded domain we use the mean square error mse between the exact solution and the discretized solution as a figure of merit to evaluate the image reconstruction accuracy and derive an upper bound on the mse which depends upon the forward and inverse problem discretization parameters noise statistics a priori information of fluorophore concentration source and detector geometry as well as background optical properties next we use this error bound to develop adaptive meshing algorithms for the fdot forward and inverse problems to reduce the mse due to discretization in the reconstructed images finally we present a set of numerical simulations to illustrate the practical advantages of our adaptive meshing algorithms for fdot image reconstruction

a family of second order methods for convex ell _1 l1 regularized optimization this paper is concerned with the minimization of an objective that is the sum of a convex function f and an ell 1 l1 regularization term our interest is in active set methods that incorporate second order information about the function f to accelerate convergence we describe a semismooth newton framework that can be used to generate a variety of second order methods including block active set methods orthant based methods and a second order iterative soft thresholding method the paper proposes a new active set method that performs multiple changes in the active manifold estimate at every iteration and employs a mechanism for correcting these estimates when needed this corrective mechanism is also evaluated in an orthant based method numerical tests comparing the performance of three active set methods are presented

on the interplay between interior point approximation and parametric sensitivities in optimal control this paper is concerned with the sensitivities of function space oriented interior point approximations in parameter dependent optimization problems for an abstract setting that covers control constrained optimal control problems the convergence of interior point sensitivities to the sensitivities of the optimal solution is shown error bounds for lq norms are derived and illustrated with numerical examples ams msc 2000 90c51 90c31 49m30

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

a multigrid method for constrained optimal control problems we consider the fast and efficient numerical solution of linear quadratic optimal control problems with additional constraints on the control discretization of the first order conditions leads to an indefinite linear system of saddle point type with additional complementarity conditions due to the control constraints the complementarity conditions are treated by a primal dual active set strategy that serves as outer iteration at each iteration step a kkt system has to be solved here we develop a multigrid method for its fast solution to this end we use a smoother which is based on an inexact constraint preconditioner we present numerical results which show that the proposed multigrid method possesses convergence rates of the same order as for the underlying elliptic pde problem furthermore when combined with a nested iteration the solver is of optimal complexity and achieves the solution of the optimization problem at only a small multiple of the cost for the pde solution

a mixed multiscale finite element method for convex optimal control problems with oscillating coefficients abstract we study numerical approximation of convex optimal control problems governed by elliptic partial differential equations with oscillating coefficients since the objective functional contains flux we approximate the problems using the mixed finite element methods we first analyze the standard mixed finite element approximation schemes then motivated by the numerical simulation of the primal variable and the flux in highly heterogeneous porous media we use a multiscale mixed finite element method to solve the state equations the multiscale finite element bases are constructed by locally solving neumann boundary value problems the analysis of the approximate control problems is carried out under the assumption that the oscillating coefficients are locally periodic which allows us to use homogenization theory to obtain the asymptotic structure of the solutions although the numerical schemes are designed for the general case

a priori error estimates for elliptic optimal control problems with a bilinear state equation in this paper a priori error analysis for the finite element discretization of an optimal control problem governed by an elliptic state equation is considered the control variable enters the state equation as a coefficient and is subject to pointwise inequality constraints we derive a priori error estimates for the discretization error in the control variable and confirm our theoretical results by numerical examples

constrained optimal control of navier stokes flow by semismooth newton methods abstract we propose and analyze a semismooth newton type method for the solution of a pointwise constrained optimal control problem governed by the time dependent incompressible navier stokes equations the method is based on a reformulation of the optimality system as an equivalent nonsmooth operator equation we analyze the flow control problem and prove q superlinear convergence of the method in the numerical implementation adjoint techniques are combined with a truncated conjugate gradient method numerical results are presented that support our theoretical results and confirm the viability of the approach

a level set approach for the solution of a state constrained optimal control problem state constrained optimal control problems for linear elliptic partial differential equations are considered the corresponding first order optimality conditions in primal dual form are analyzed and linked to a free boundary problem resulting in a novel algorithmic approach with the boundary interface between the active and inactive sets as optimization variable the new algorithm is based on the level set methodology the speed function involved in the level set equation for propagating the interface is computed by utilizing techniques from shape optimization encouraging numerical results attained by the new algorithm are reported on

indirect multiple shooting for nonlinear parabolic optimal control problems with control constraints we discuss the indirect multiple shooting approach for the solution of pde based parabolic optimal control problems with control constraints the method is formulated within an abstract function space setting and uses a space time galerkin finite element discretization the emphasis is on the embedding of indirect multiple shooting into the optimal control framework as well as the detailed description of the discretization within the pde context numerical results for linear and nonlinear model problems with and without control constraints illustrate the efficient use of indirect multiple shooting particularly in cases where other standard methods fail

a partition of unity method for the displacement obstacle problem of clamped kirchhoff plates a partition of unity method for the displacement obstacle problem of clamped kirchhoff plates is considered in this paper we derive optimal error estimates and present numerical results that illustrate the performance of the method

nonconvergence of the plain newton min algorithm for linear complementarity problems with a p matrix the plain newton min algorithm to solve the linear complementarity problem lcp for short 0 leq x perp mx q geq0 can be viewed as a semismooth newton algorithm without globalization technique to solve the system of piecewise linear equations min x mx q 0 which is equivalent to the lcp when m is an m matrix of order n the algorithm is known to converge in at most n iterations we show in this paper that this result no longer holds when m is a p matrix of order 3 since then the algorithm may cycle p matrices are interesting since they are those ensuring the existence and uniqueness of the solution to the lcp for an arbitrary q incidentally convergence occurs for a p matrix of order 1 or 2

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

asymptotic properties of large random matrices with independent entries we study the normalized trace gn z n 1 tr h zi 1 of the resolvent of n n real symmetric matrices h 1 jk wjk n j k 1n assuming that their entries are independent but not necessarily identically distributed random variables we develop a rigorous method of asymptotic analysis of moments of gn z for iz 0 where 0 is determined by the second moment of wjk by using this method we find the asymptotic form of the expectation e gn z and of the connected correlator e gn z1 gn z2 e gn z1 e gn z2 we also prove that the centralized trace ngn z e ngn z has the gaussian distribution in the limit n based on these results we present heuristic arguments supporting the universality property of the local eigenvalue statistics for this class of random matrix ensembles

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

superconvergence properties of optimal control problems an optimal control problem for a two dimensional 2 d elliptic equation is investigated with pointwise control constraints this paper is concerned with discretization of the control by piecewise constant functions the state and the adjoint state are discretized by linear finite elements approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state it is proved that these approximations have convergence order h2

the primal dual active set method for nonlinear optimal control problems with bilateral constraints the primal dual active set method has proved to be an efficient numerical tool in the context of diverse applications so far it has been investigated mainly for linear problems this paper is devoted to the study of global convergence of the primal dual active set method for nonlinear problems with bilateral constraints utilizing the close relationship between the primal dual active set method and semismooth newton methods local superlinear convergence of the method is investigated as well

semismooth newton and augmented lagrangian methods for a simplified friction problem in this paper a simplified friction problem and iterative second order algorithms for its solution are analyzed in infinite dimensional function spaces motivated from the dual formulation a primal dual active set strategy and a semismooth newton method for a regularized problem as well as an augmented lagrangian method for the original problem are presented and their close relation is analyzed local as well as global convergence results are given by means of numerical tests we discuss among others convergence properties the dependence on the mesh and the role of the regularization and illustrate the efficiency of the proposed methodologies