a new monte carlo power method for the eigenvalue problem of transfer matrices we propose a new monte carlo method for calculating eigenvalues of transfer matrices leading to free energies and to correlation lengths of classical and quantum many body systems generally this method can be applied to the calculation of the maximum eigenvalue of a nonnegative matrix such that all the matrix elements of k are strictly positive for an integerk this method is based on a new representation of the maximum eigenvalue of the matrix as the thermal average of a certain observable of a many body system therefore one can easily calculate the maximum eigenvalue of a transfer matrix leading to the free energy in the standard monte carlo simulations such as the metropolis algorithm as test cases we calculate the free energies of the square lattice ising model and of the spin 1 2xy heisenberg chain we also prove two useful theorems on the ergodicity in quantum monte carlo algorithms or more generally on the ergodicity of monte carlo algorithms using our new representation of the maximum eigenvalue of the matrix

error estimates of mixed finite element methods for quadratic optimal control problems in this paper we investigate the error estimates for the solutions of optimal control problems by mixed finite element methods the state and costate are approximated by raviart thomas mixed finite element spaces of order k and the control is approximated by piecewise polynomials of order k under the special constraint set we will show that the control variable can be smooth in the whole domain we derive error estimates of optimal order both for the state variables and the control variable

goal oriented adaptivity in control constrained optimal control of partial differential equations dual weighted goal oriented error estimates for a class of pointwise control constrained optimal control problems for second order elliptic partial differential equations are derived it is demonstrated that the constraints give rise to a primal dual weighted error term representing the mismatch in the complementarity system due to discretization the paper also contains a posteriori error estimators for the l 2 norm of the error in the state and in the adjoint state

on the asymptotic exactness of error estimators for linear triangular finite elements this paper deals with a posteriori error estimates for piecewise linear finite element approximations of elliptic problems we analyze two estimators based on recovery operators for the gradient of the approximate solution by using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution one of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation this estimator is equivalent to the error in the energy norm under rather general conditions however we show that for the asymptotic exactness the regularity assumption on the mesh is not merely technical while doing this we analyze the relation between superconvergence and asymptotic exactness for some particular examples

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

finite size corrections and numerical calculations for long spin 1 2 heisenberg chains in the critical region leading and next to leading order finite size corrections to the ground and first excited states are calculated for the spin 1 2 anisotropic heisenberg model in the critical region the analytic results are compared to numerical data obtained for chains up to a length of n 1024 it is found that near the isotropic point the asymptotic region where the results obtained for n to infinity are applicable sets in at very large n values and for obtaining good accuracy in fitting the numerical data one has to take into account several correction terms even at large n 100 chain lengths

augemented lagrangian techniques for elliptic state constrained optimal control problems we propose augmented lagrangian methods to solve state and control constrained optimal control problems the approach is based on the lagrangian formulation of nonsmooth convex optimization in hilbert spaces developed in k ito and k kunisch augmented lagrangian methods for nonsmooth convex optimization in hilbert spaces preprint 1994 we investigate a linear optimal control problem with a boundary control function as in m bergounioux numer funct anal optim 14 1993 pp 515 543 both the equation and the constraints are augmented the proposed methods are general and can be adapted to a much wider class of problems

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

conformal spectrum of the six vertex model by way of a generalization of analytical methods recently introduced the conformal spectrum of the six vertex model with twisted boundary conditions is analysed nonlinear integral equations are derived from which all scaling dimensions and the spins can be extracted analytically in terms of rogers dilogarithms the scaling dimensions of low lying excitations are given by the so called gaussian dimensions

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

conformal anomaly and critical exponents of the spin 1 takhtajan babujian model the authors consider the critical behaviour of the spin 1 hamiltonian of the takhtajan babujian model the bethe ansatz equations for this antiferromagnet in a finite chain are investigated analytically and numerically the conformal anomaly and critical exponents are obtained by exploiting their relations with the eigenspectrum of the finite system the results strongly support the conjecture that the wess zumino witten non linear sigma model with topological charge k 2 is the underlying field theory for this statistical mechanics model

gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes in this paper the gradient recovery type a posteriori error estimators for finite element approximations are proposed for irregular meshes both the global and the local a posteriori error estimates are derived moreover it is shown that the a posteriori error estimates is asymptotically exact on where the mesh is regular enough and the exact solution is smooth

convergence of finite element approximations to state constrained convex parabolic boundary control problems this paper is concerned with the numerical solution of state constrained parabolic boundary control problems by finite element approximations error estimates for the optimal values and in the coercive case for the optimal solutions are derived these estimates are used to prove convergence results under rather weak assumptions in the noncoercive case a bang bang principle is shown to obtain convergence results for the discrete controls also a discussion of several numerical examples concludes the paper

constrained lqr problems in elliptic distributed control systems with point observations in this paper we study bound constrained lqr problems in distributed control systems governed by the elliptic equation with point observations which are motivated by problems in corrosion engineering and contemporary smart materials several regularity and characterization theorems have been established in particular three decomposition formulas are obtained to characterize the optimal control and optimal layer density and are used to direct the numerical computations these results cannot be obtained by the traditional galerkin variational method in the process several useful lemmas are established which are of independent interest we point out that the classical lagrangian multiplier method lmm may fail to provide a reliable numerical algorithm based on our characterization results and the boundary element method two algorithms are proposed to carry out numerical computations it has been shown by our numerical experiments that both algorithms are efficient and insensitive to the partition number of the boundary an adaptive local refinement scheme has also been designed to handle the rough behavior of the optimal solution around sensor locations

infinite conservation laws in the one dimensional hubbard model we show that the hamiltonian of the 1d hubbard model commutes with a one parameter family of transfer matrices of a new 2d classical model corresponding to two coupled six vertex models central to this result is a new local algebraic relation a generalization of the infinitesimal startriangle relation

pointwise accuracy of a finite element method for nonlinear variational inequalities consider the following quasilinear elliptic pde which is equivalent to a nonlinear variational inequality divf u u f here is a singular maximal monotone graph and the nonlinear differential operator is only assumed to be monotone surfaces of prescribed mean curvature over obstacles may thus be viewed as relevant examples the numerical approximation proposed in this paper consists of combining continuous piecewise linear finite elements with a preliminary regularization of the resulting scheme is shown to be quasi optimally accurate inl the underlying analysis makes use of both a topological technique and a sharpl p duality argument

grothendieck s problems concerning profinite completions and representations of groups in 1970 alexander grothendieck 6 posed the following problem let 1 and 2 be finitely presented residually finite groups and let u 1 2 be a homomorphism such that the induced map of profinite completions u 1 2 is an isomorphism does it follow that u is an isomorphism in this paper we settle this problem by exhibiting pairs of groups u p such that is a direct product of two residually finite hyperbolic groups p is a finitely presented subgroup of infinite index p is not

finite size scaling for correlations of quantum spin chains at criticality we study the finite size scaling behavior of two point correlation functions of translationally invariant many body systems at criticality we propose an efficient method for calculating the two point correlation functions in the thermodynamic limit from numerical data of finite systems our method is most effective when applied to a two dimensional classical system which possesses a conformal invariance by using this method with numerical data obtained from exact diagonalizations and monte carlo simulations we study the spin spin correlations of the quantum spin 1 2 and 3 2 antifierromagnetic chains in particular the logarithmic corrections to power law decay of the correlation of the spin 1 2 isotropic heisenberg antiferromagnetic chain are studied thoroughly we clarify the cause of the discrepancy in previous calculations for the logarithmic corrections our result strongly supports the field theoretic prediction based on the mappings to the wess zumino witten nonlinear model or the sine gordon model we also treat logarithmic corrections and crossover phenomena in the spin spin correlation of the spin 3 2 isotropic heisenberg antiferromagnetic chain our results are consistent with the affleck haldane prediction that the correlation of the spin 3 2 chain exhibits a crossover to the same asymptotic behavior as in the spin 1 2 chain

nonhomogeneous boundary value problems the diffusion equation is solved under stochastic nonhomogeneity using eigen function expansion and the georges method the statistical moments of the solution process are computed through the two previously mentioned techniques and proved to be the same a general solution is obtained under general initial and boundary conditions a random source composed of deterministic and stochastic parts is taken into consideration the stochastic part is then restricted to a generalized gaussian field mainly modulated white noise a special case is considered under constant noise level and constant average noise a numerical case study concerning pollution in a stream is solved and a parametric study is achieved through various figures

new elliptic projections and a priori error estimates of h1 galerkin mixed finite element methods for optimal control problems governed by parabolic integro differential equations in this paper we discuss a priori error estimates of h1 galerkin mixed finite element methods for optimal control problems governed by parabolic integro differential equations the state variables and co state variables are approximated by the lowest order raviart thomas mixed finite element and linear finite element and the control variable is approximated by piecewise constant functions both semidiscrete and fully discrete schemes are considered based on some new elliptic projections we derive a priori error estimates for the control variable the state variables and the adjoint state variables the related a priori error estimates for the new projections error are also established a numerical example is given to demonstrate the theoretical results

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

finite element approximation of parabolic time optimal control problems the numerical approximation of a parabolic time optimal control problem via piecewise linear splines is considered at each stage of the approximation a bang bang approximate control is selected by solving for its switching times as the solution of a constrained nonlinear least squares optimization problem the well posedness of the approximation scheme is shown and the rate of convergence to the exact solution investigated numerical results for some one and two dimensional parabolic control problems are given

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

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

monte carlo determination of multiple extremal eigenpairs we present a monte carlo algorithm that allows the simultaneous determination of a few extremal eigenpairs of a very large matrix without the need to compute the inner product of two vectors or store all the components of any one vector the algorithm a monte carlo implementation of a deterministic one we recently bench marked is an extension of the power method in the implementation presented we used a basic monte carlo splitting and termination method called the comb incorporated the weight cancellation method of arnow et al and exploited a sampling method the sewing method that does a large state space sampling as a succession of small state space samplings we illustrate the effectiveness of the algorithm by its determination of the two largest eigenvalues of the transfer matrices for variously sized two dimensional zero field ising models while very likely useful for other transfer matrix problems the algorithm is however quite general and should find application to a larger variety of problems requiring a few dominant eigenvalues of a matrix

subgroup growth and congruence subgroups letk be a global field o its ring of integers g an almost simple simply connected connected algebraic subgroups ofgl m defined overk and g o which is assumed to be infinite let n resp n be the number of all resp congruence subgroups of index at mostn in we show a if char k 0 then i c 1 frac log 2 n log log n leqq log gamma n gamma leqq c 2 frac log 2 n log log n for suitable constantsc1 andc2 under some mild assumptions we also have ii has the congruence subgroup property if and only if log n o log2n iii if is boundedly generated the has the congruence subgroup property this confirms a conjecture of rapinchuk r1 which was also proved by platonov and rapinchuk pr3 b if char k 0 and under somewhat stronger conditions ong then for suitable constantsc3 andc4 c3 log2n log n c4log3n

groups of oscillating intermediate growth we construct an uncountable family of nitely generated groups of intermediate growth with growth functions of new type these functions can have large oscillations between lower and upper bounds both of which come from a wide class of functions in particular we can have growth oscillating between e n and any prescribed function growing as rapidly as desired our construction is built on top of any of the grigorchuk groups of intermediate growth and is a variation on the limit of permutational wreath product

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

a posteriori error estimates for elliptic variational inequalities we derive hierarchical a posteriori error estimates for elliptic variational inequalities the evaluation amounts to the solution of corresponding scalar local subproblems we derive some upper bounds for the effectivity rates and the numerical properties are illustrated by typical examples

error estimates and superconvergence of mixed finite element methods for fourth order hyperbolic control problems abstract in this paper we investigate the error estimates and superconvergence of the semidiscrete mixed finite element methods for quadratic optimal control problems governed by linear fourth order hyperbolic equations the state and the co state are discretized by the order k raviart thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k k 0 we derive error estimates for both the state and the control approximation moreover we present the superconvergence analysis for mixed finite element approximation of the optimal control problems

analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls an optimal control problem for impressed cathodic systems in electrochemistry is studied the control in this problem is the current density on the anode a matching objective functional is considered we first demonstrate the existence and uniqueness of solutions for the governing partial differential equation with a nonlinear boundary condition we then prove the existence of an optimal solution next we derive a necessary condition of optimality and establish an optimality system of equations finally we define a finite element algorithm and derive optimal error estimates

ritz galerkin approximation of the time optimal boundary control problem for parabolic systems with dirichlet boundary conditions this paper deals with ritz galerkin approximations of the following two problems i boundary value problems with l 2 boundary data given in the form of dirichlet boundary conditions ii time optimal control problems for parabolic systems with control acting on the boundary in both cases optimal rates of convergence of approximation are obtained also the specialization is considered to approximating with nonconformal elements which are not required to satisfy boundary conditions corresponding rates of convergence are also proved

on the structure of lagrange multipliers for state constrained optimal control problems abstract state constrained optimal control problems represent severe analytical and numerical challenges it is shown that under certain conditions on the active set the measure representing the lagrange multiplier associated to the state constraint can be decomposed in a distributed part in l2 with support on the active set and a boundary measure concentrated on the interface between active and inactive sets

a posteriori error estimators for the stokes equations we present two a posteriori error estimators for the mini element discretization of the stokes equations one is based on a suitable evaluation of the residual of the finite element solution the other one is based on the solution of suitable local stokes problems involving the residual of the finite element solution both estimators are globally upper and locally lower bounds for the error of the finite element discretization numerical examples show their efficiency both in estimating the error and in controlling an automatic self adaptive mesh refinement process the methods presented here can easily be generalized to the navier stokes equations and to other discretization schemes

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

arbitrarily large residual finiteness growth the residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients in this paper we construct groups with arbitrarily large residual finiteness growth we also demonstrate a new relationship between residual finiteness growth and some decision problems in groups which we apply to our new groups let be a residually finite group generated by a finite set x an element is detected by a finite quotient q of if there exists a homomorphism q such that ker set f x n to be the minimal natural number n such that any element of word length n with respect to x can be detected in a quotient q of cardinality n the study of f x the residual finiteness growth is an area of active research see for instance b10 b11 bm1 bk10 and km11 our main result is the following theorem 1 for any function f n n there exists a residually finite group bf and a two element generating set x for bf such that fbf x n f n for all n 8 our proof is constructive elementary and demonstrates the versatility of b h neumann s construction of a continuous family of non isomorphic two generated groups n37 our work is of the same flavor as lps96 and p04 where neumann s construction is modified to find groups with interesting subgroup growth properties our remaining results exhibit how residual finiteness growth may be applied to illicit decision problem data on groups the first of these shows that a variant of theorem 1 does not exist for groups that are finitely presented in particular the main result on residual finiteness growth stated in kms13 is the strongest of its kind for finitely presented groups proposition 1 if is a residually finite finitely presented group then the residual finiteness growth function for is computable o kharlampovich showed that there exists a finitely presented solvable group with undecidable word problem k81 of which many variations have been found see b83 in ls77 iv 4 3 theorem 19 it is shown that f2 f2 does not have decidable membership problem with respect to finitely generated subgroups we address a weaker membership problem for the groups bf for a finitely generated group g we say the finite index normal subgroup membership problem in g is solvable if there exists an algorithm which for every element g g and every finiteindex normal subgroup h g will determine in in a finite amount of time whether g h 1991 mathematics subject classification 20e26

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

symmetric groups and expander graphs we construct explicit generating sets sn and tilde s n of the alternating and the symmetric groups which turn the cayley graphs mathcal c text alt n s n and mathcal c text sym n tilde s n into a family of bounded degree expanders for all n

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

superconvergence of mixed finite element methods for optimal control problems t in this paper we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods the state and co state variables are approximated by the lowest order raviart thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique we derive l 2 superconvergence properties for the flux functions along the gauss lines and for the scalar functions at the gauss points via mixed projections moreover global l 2 superconvergence results are obtained by virtue of an interpolation postprocessing technique thus based on these superconvergence estimates some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods finally some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence

commensurability growths of algebraic groups fixing a subgroup vargamma in a group g the full commensurability growth function assigns to each n the cardinality of the set of subgroups vardelta of g with vargamma vargamma cap vardelta vardelta vargamma cap vardelta le n for pairs vargamma le g where g is a higher rank chevalley group scheme defined over mathbb z and vargamma is an arithmetic lattice in g we give precise estimates for the full commensurability growth relating it to subgroup growth and a computable invariant that depends only on g

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

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

nonarithmetic superrigid groups counterexamples to platonov s conjecture margulis showed that most arithmetic groups are superrigid platonov conjectured conversely that finitely generated linear groups which are superrigid must be of arithmetic type we construct counterexamples to platonov s conjecture

adaptive symmetric interior penalty galerkin method for boundary control problems we investigate an a posteriori error analysis of adaptive finite element approximations of linear quadratic boundary optimal control problems under bilateral box constraints which act on a neumann boundary control we use a symmetric interior galerkin method as discretization technique an efficient and reliable residual type error estimator is introduced by invoking data oscillations we then derive local upper and lower a posteriori error estimates for the boundary control problem adaptive mesh refinement indicated by a posteriori error estimates is applied numerical results are presented to illustrate the performance of the adaptive finite element approximation

decision problems and profinite completions of groups abstract we consider pairs of finitely presented residually finite groups p for which the induced map of profinite completions p is an isomorphism we prove that there is no algorithm that given an arbitrary such pair can determine whether or not p is isomorphic to we construct pairs for which the conjugacy problem in can be solved in quadratic time but the conjugacy problem in p is unsolvable let j be the class of super perfect groups that have a compact classifying space and no proper subgroups of finite index we prove that there does not exist an algorithm that given a finite presentation of a group and a guarantee that j can determine whether or not 1 we construct a finitely presented acyclic group h and an integer k such that there is no algorithm that can determine which k generator subgroups of h are perfect

a priori and a posteriori error estimates of h 1 galerkin mixed finite element methods for optimal control problems governed by pseudo hyperbolic integro differential equations abstract in this paper we investigate a priori and a posteriori error estimates of h1 galerkin mixed finite element methods for optimal control problems governed by pseudo hyperbolic integro differential equations the state variables and co state variables are approximated by the lowest order raviart thomas mixed finite element and linear finite element and the control variable is approximated by piecewise constant functions based on two new elliptic projections we derive a priori error estimates both for the control variable the state variable and the co state variable the related a priori error estimates for the new projections error are also established moreover a posteriori error estimates for all variables are derived via energy method such a posteriori error estimates which are apparently not available in the literature are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem

a posteriori error estimates for control problems governed by nonlinear elliptic equations in hp fem abstract in this paper we derive a posteriori error estimates for the hp finite element approximation of the distributed convex optimal control problems governed by nonlinear elliptic equations we obtain hp a posteriori error estimators for both the state and the control approximation these estimates can be used to construct reliable adaptive finite element approximation for the control problems

optimal control for variational inequalities consider the problem of maximizing a functional which depends on a control function k and on the solution of an elliptic variational inequality with k appearing in the data the variational problem for k is nondiflerentiable and nonconvex we obtain necessary conditions on a maximizes k 0 and then use them to determine the structure of k 0 in some cases

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

a priori and a posteriori error analysis of edge stabilization galerkin method for the optimal control problem governed by convection dominated diffusion equation in this paper we study an edge stabilization galerkin approximation scheme for the constrained optimal control problem governed by convection dominated diffusion equation the method uses least square stabilization of the gradient jumps across element edges a priori and a posteriori error estimates are derived for both the state co state and the control the theoretical results are illustrated by two numerical experiments

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

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

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

approximations of elliptic optimal control problems with controls acting on a lower dimensional manifold in this paper we study finite element approximations to some elliptic optimal control problems with controls acting on a lower dimensional manifold which can be a point a curve or a surface we use piecewise linear finite elements to approximate state variables while utilizing the variational discretization to approximate control variables we derive several a priori error estimates for optimal controls from different cases depending on the dimensions of the computational domain and the manifold where controls act we ends up with extensive numerical experiments which confirm our theoretical findings

convergence and quasi optimality of an adaptive finite element method for elliptic robin boundary control problem abstract in this paper we focus on the convergence and quasi optimality of an adaptive finite element method for elliptic robin boundary control problems we use piecewise linear finite elements to approximate the state and the adjoint state variables and the variational discretization to approximate the control variable under mild assumption on the initial mesh we prove the contraction property for the sum of the energy errors of the state and adjoint state and the scaled error estimator on two consecutive adaptive loops the resulting linear convergence yields the quasi optimal convergence rate for the afem algorithm applied to our problem additionally some numerical results are provided to support our theoretical analysis