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

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

drinfel d algebra deformations homotopy comodules and the associahedra the aim of this work is to construct a cohomology theory controlling the deformations of a general drinfel d algebra a and thus finish the program which began in 13 14 the task is accomplished in three steps the first step which was taken in the aforementioned articles is the construction of a modified cobar complex adapted to a non coassociative comultiplication the following two steps each involve a new highly non trivial construction the first construction essentially combinatorial defines a differential graded lie algebra structure on the simplicial chain complex of the associahedra the second construction of a more algebraic nature is the definition of map of differential graded lie algebras from the complex defined above to the algebra of derivations on the bar resolution using the existence of this map and the acyclicity of the associahedra we can define a so called homotopy comodule structure definition 3 3 below on the bar resolution of a general drinfeld algebra this in turn allows us to define the desired cohomology theory in terms of a complex which consists roughly speaking of the bimodule and bicomodule maps from the bar resolution to the modified cobar resolution the complex is bigraded but not a bicomplex as in the gerstenhaber schack theory for bialgebra deformations the new components of the coboundary operator are defined via the constructions mentioned above the results of the paper were announced in 12

preconditioning of active set newton methods for pde constrained optimal control problems we address the problem of preconditioning a sequence of saddle point linear systems arising in the solution of pde constrained optimal control problems via active set newton methods with control and regularized state constraints we present two new preconditioners based on a full block matrix factorization of the schur complement of the jacobian matrices where the active set blocks are merged into the constraint blocks we discuss the robustness of the new preconditioners with respect to the parameters of the continuous and discrete problems numerical experiments on three dimensional problems are presented including comparisons with existing approaches based on preconditioned conjugate gradients in a nonstandard inner product

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

optimal control of a semilinear pde with nonlocal radiation interface conditions we consider a control constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions these conditions occur during the modeling of diffuse gray conductive radiative heat transfer the problem arises when attempting to optimize the temperature gradient within crystal growth by the physical vapor transport pvt method based on a minimum principle for the semilinear equation as well as l infty estimates for the weak solution we establish the existence of an optimal solution as well as necessary optimality conditions the theoretical results are illustrated by numerical computations

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

basic hopf algebras and quantum groups this paper investigates the structure of basic finite dimensional hopf algebras h over an algebraically closed field k the algebra h is basic provided h modulo its jacobson radical is a product of the field k in this case h is isomorphic to a path algebra given by a finite quiver with relations necessary conditions on the quiver and on the coalgebra structure are found in particular it is shown that only the quivers gamma g w given in terms of a finite group g and sequence w w 1 w 2 ldots w r of elements of g in the following way can occur the quiver gamma g w has vertices v g g in g and arrows a i g colon v g 1 to v w ig 1 mid g in g w i in w where the set w 1 w 2 ldots w r is closed under conjugation with elements in g and for each g in g the sequences w and gw 1g 1 gw 2g 1 ldots gw rg 1 are the same up to a permutation we show how k gamma g w is a kg bimodule and study properties of the left and right actions of g on the path algebra furthermore it is shown that the conditions we find can be used to give the path algebras k gamma g w themselves a hopf algebra structure for an arbitrary field k the results are also translated into the language of coverings finally a new class of finite dimensional basic hopf algebras are constructed over a not necessarily algebraically closed field most of which are quantum groups the construction is not characteristic free all the quivers gamma g w where the elements of w generates an abelian subgroup of g are shown to occur for finite dimensional hopf algebras the existence of such algebras is shown by explicit construction for closely related results of cibils and rosso see ci r

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 weak continuity of elliptic operators and applications to potential theory in this paper we establish weak continuity results for quasilinear elliptic and subellip tic operators of divergence form acting on corresponding classes of subharmonic functions these results are analogous to our earlier results for fully nonlinear fc hessian operators from the weak continuity we derive various potential theoretic results including capacity estimates potential esti mates and the wiener criterion for regular boundary points our methods make substantial use of harnack inequalities for solutions 1 introduction there are numerous papers on the local behavior of solu tions subsolutions and supersolutions of second order quasilinear elliptic diver gence structure partial differential equations as well as extensions to degenerate elliptic and subelliptic equations in this paper we show that many properties including the quasi continuity of subharmonic functions asymptotic behavior of fundamental solutions removability of singularities wiener criteria for regular boundary points can be derived via harnack inequalities for solutions the two main ingredients in our approach which follow from our treatment of the fully nonlinear hessian operators in 45 46 47 together with that of labutin 30 are a weak convergence result and a pointwise potential estimate for subharmonic functions the weak convergence result is analogous to that for hessian operators 45 46 while the potential estimate extends that found for quasilinear elliptic operators by kilpelainen and maly 27 weak convergence results in the theory of nonlinear partial differential equa tions have been addressed in the survey 13 in this paper we treat nonlinear elliptic divergence structure operators for which harnack type inequalities are available to state our result we need an appropriate notion of subharmonic func tion in the fully nonlinear case 46 we employed the viscosity notion based on differential inequalities for smooth test functions for quasilinear divergence structure equations the traditional definition using the homogeneous equation is more convenient accordingly we shall say that a continuous function w in a domain q in euclidean space ir is harmonic with respect to an operator or c harmonic in short if it is a solution of the homogeneous equation c u 0

existence of minimal solutions to quasilinear elliptic equations with several sub natural growth terms abstract we study the existence of positive solutions to quasilinear elliptic equations of the type p u u q in r n in the sub natural growth case 0 q p 1 where p u u p 2 u is the p laplacian with 1 p n and and are nonnegative radon measures on r n we construct minimal generalized solutions under certain generalized energy conditions on and to prove this we give new estimates for interaction between measures we also construct solutions to equations with several sub natural growth terms using the same methods

elliptic problems with nonlinear terms depending on the gradient and singular on the boundary abstract in this paper we consider the problem p u u q u q f x in u 0 on where r n is a bounded domain 1 q 2 r and f 0 we prove that 1 if q 1 then problem p has a distributional solution for all f l 1 and all 0 2 if 1 q 0 then problem p has a solution for all f l s where s n q if n 2 and without any restriction on 3 if q 2 and 1 q 0 then problem p has infinitely many solutions under suitable hypotheses on f 4 if 0 q and f l 1 satisfies 1 f inf w 0 1 2 2 d x f 2 d x 0 then problem p has a positive solution if 0 1 f and no positive solution for large

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

state constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions we consider a control and state constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions these conditions occur during the modeling of diffuse gray conductive radiative heat transfer the nonlocal radiation interface condition and the pointwise state constraints represent the particular features of this problem to deal with the state constraints continuity of the state is shown which allows us to derive first order necessary conditions afterwards we establish second order sufficient conditions that account for strongly active sets and ensure local optimality in an l 2 neighborhood

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

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

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

a priori error estimates for space time finite element discretization of parabolic optimal control problems part i problems without control constraints in this paper we develop a priori error analysis for galerkin finite element discretizations of optimal control problems governed by linear parabolic equations the space discretization of the state variable is done using usual conforming finite elements whereas the time discretization is based on discontinuous galerkin methods for different types of control discretizations we provide error estimates of optimal order with respect to both space and time discretization parameters the paper is divided into two parts in the first part we develop some stability and error estimates for space time discretization of the state equation and provide error estimates for optimal control problems without control constraints in the second part of the paper the techniques and results of the first part are used to develop a priori error analysis for optimal control problems with pointwise inequality constraints on the control variable

optimal control of the convection diffusion equation using stabilized finite element methods in this paper we analyze the discretization of optimal control problems governed by convection diffusion equations which are subject to pointwise control constraints we present a stabilization scheme which leads to improved approximate solutions even on corse meshes in the convection dominated case moreover the in general different approaches optimize then discretize and discretize then optimize coincide for the proposed discretization scheme this allows for a symmetric optimality system at the discrete level and optimal order of convergence

wolff s inequality for intrinsic nonlinear potentials and quasilinear elliptic equations abstract we prove an analogue of wolff s inequality for the so called intrinsic nonlinear potentials associated with the quasilinear elliptic equation p u u q in r n in the case 0 q p 1 here p u div u p 2 u is the p laplacian and is a nonnegative measurable function or measure as an application we give a necessary and sufficient condition for the existence of a positive solution u l r r n 0 r to this problem which was open even in the case p 2 our version of wolff s inequality for intrinsic nonlinear potentials relies on a new characterization of discrete littlewood paley spaces f p q defined in terms of characteristic functions of dyadic cubes in r n

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

quiver approaches to quasi hopf algebras we provide a quiver setting for quasi hopf algebras generalizing the hopf quiver theory as applications we obtain some general structure theorems in particular the quasi hopf analog of the cartier theorem and the cartier gabriel decomposition theorem

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

preconditioning for allen cahn variational inequalities with non local constraints the solution of allen cahn variational inequalities with mass constraints is of interest in many applications this problem can be solved both in its scalar and vector valued form as a pde constrained optimization problem by means of a primal dual active set method at the heart of this method lies the solution of linear systems in saddle point form in this paper we propose the use of krylov subspace solvers and suitable preconditioners for the saddle point systems numerical results illustrate the competitiveness of this approach

optimal distributed control of a nonlocal convective cahn hilliard equation by the velocity in three dimensions in this paper we study a distributed optimal control problem for a nonlocal convective cahn hilliard equation with degenerate mobility and singular potential in three dimensions of space while the cost functional is of standard tracking type the control problem under investigation cannot easily be treated via standard techniques for two reasons the state system is a highly nonlinear system of pdes containing singular and degenerating terms and the control variable which is given by the velocity of the motion occurring in the convective term is nonlinearly coupled to the state variable the latter fact makes it necessary to state rather special regularity assumptions for the admissible controls which while looking a bit nonstandard are however quite natural in the corresponding analytical framework in fact they are indispensable prerequisites to guarantee the well posedness of the associated state system in this paper we employ recently proved existence uniqueness and regularity results f

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

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

preconditioned solution of state gradient constrained elliptic optimal control problems elliptic optimal control problems with pointwise state gradient constraints are considered a quadratic penalty approach is employed together with a semismooth newton iteration three different preconditioners are proposed and the ensuing spectral properties of the preconditioned linear newton saddle point systems are analyzed dependent on the penalty parameter a new bound for the smallest positive eigenvalue is proved since the analysis is carried out in function space it will ensure mesh independent convergence behavior of suitable krylov subspace methods such as minres also in discretized settings a path following strategy with a preconditioned inexact newton solver is implemented and numerical results are provided

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

bosonization for dual quasi bialgebras and preantipode abstract to every dual quasi bialgebra h and every bialgebra r in the category of yetter drinfeld modules over h one can associate a dual quasi bialgebra called bosonization in this paper using the fundamental theorem we characterize as bosonizations the dual quasi bialgebras with a projection onto a dual quasi bialgebra with a preantipode as an application we investigate the structure of the graded coalgebra gr a associated to a dual quasi bialgebra a with the dual chevalley property e g a is pointed

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

bialgebra cohomology deformations and quantum groups abstract we introduce cohomology and deformation theories for a bialgebra a over a commutative unital ring k such that the second cohomology group is the space of infinitesimal deformations our theory gives a natural identification between the underlying k modules of the original and the deformed bialgebra certain explicit deformation formulas are given for the construction of quantum groups i e hopf algebras that are neither commutative nor cocommutative whether or not they arise from quantum yang baxter operators these formulas yield in particular all glq n and slq n as deformations of gl n and sl n using a hodge decomposition of the underlying cochain complex we compute our cohomology for gl n with this we show that every deformation of gl n is equivalent to one in which the comultiplication is unchanged not merely on elements of degree one but on all elements settling in the strongest way a decade old conjecture and in which the quantum determinant as an element of the underlying k module is identical with the usual one

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

nonlinear elliptic equations and intrinsic potentials of wolff type abstract we give necessary and sufficient conditions for the existence of weak solutions to the model equation p u u q on r n in the case 0 q p 1 where 0 is an arbitrary locally integrable function or measure and p u div u u p 2 is the p laplacian sharp global pointwise estimates and regularity properties of solutions are obtained as well as a consequence we characterize the solvability of the equation p v b v p v on r n where b 0 these results are new even in the classical case p 2 our approach is based on the use of special nonlinear potentials of wolff type adapted for sublinear problems and related integral inequalities it allows us to treat simultaneously several problems of this type such as equations with general quasilinear operators div a x u fractional laplacians or fully nonlinear k hessian operators

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

optimal boundary control of a steady state heat transfer model accounting for radiative effects abstract a boundary control problem for a nonlinear steady state heat transfer model accounting for heat radiation effects is considered the aim of control consists in obtaining a prescribed temperature distribution in a part of the model domain by controlling the boundary temperature the solvability of this control problem is proven and optimality conditions are derived numerical simulations are presented

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

quivers quasi quantum groups and finite tensor categories we study finite quasi quantum groups in their quiver setting developed recently by the first author we obtain a classification of finite dimensional pointed majid algebras of finite corepresentation type or equivalently a classification of elementary quasi hopf algebras of finite representation type over the field of complex numbers by the tannaka krein duality principle this provides a classification of the finite tensor categories in which every simple object has frobenius perron dimension 1 and there are finitely many indecomposable objects up to isomorphism some interesting information of these finite tensor categories is given by making use of the quiver representation theory

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

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 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

hessian measures ii in our previous paper on this topic we introduced the notion of k hessian measure associated with a continuous k convex function in a domain q in euclidean n space k 1 n and proved a weak continuity result with respect to local uniform convergence in this paper we consider k convex functions not necessarily continuous and prove the weak continuity of the associated k hessian measure with respect to convergence in measure the proof depends upon local integral estimates for the gradients of k convex functions

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

optimal control of the stokes equations a priori error analysis for finite element discretization with postprocessing an optimal control problem for 2d and 3d stokes equations is investigated with pointwise control constraints this paper is concerned with the discretization of the control by piecewise constant functions the state and the adjoint state are discretized by finite element schemes in the paper a postprocessing strategy is suggested which allows for significant improvement of the accuracy

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

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

analysis on inexact block diagonal preconditioners for elliptic pde constrained optimization problems abstract by using the galerkin finite element method the distributed control problems can be discretized into a saddle point problem with a coefficient matrix of block three by three which can be reduced to a linear system with lower order we first introduce a class of inexact block diagonal preconditioners and estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices respectively based on the cholesky decomposition of the known matrices we also analyze a lower triangular preconditioner to accelerate the minimal residual method for the reduced linear system and discuss its real and complex eigenvalues respectively moreover these bounds do not rely on the regularization parameter and the eigenvalues of the matrices in the discrete system numerical experiments are also presented to demonstrate the effectiveness and robustness of the two new preconditioners

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

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

some elliptic problems with singular natural growth lower order terms abstract in this paper we deal with a nonlinear elliptic problem whose model is u b d u 2 u f in u 0 on where b 0 and f 0 belongs to a lebesgue space we prove the existence of positive solutions in suitable sobolev spaces depending on f and b

quiver hopf algebras abstract in this paper we study subhopfalgebras of graded hopf algebra structures on a path coalgebra k q c we prove that a hopf structure on some subhopfquivers can be lifted to a hopf structure on the whole hopf quiver if q is a schurian hopf quiver then we classified all simple pointed subhopfalgebras of a graded hopf structure on k q c we also prove a dual gabriel theorem for pointed hopf algebras

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