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

homotopy of unitaries in simple c algebras with tracial rank one let 0 be a positive number is there a number 0 satisfying the following given any pair of unitaries u and v in a unital simple c algebra a with v 0 in k1 a for which uv vu there is a continuous path of unitaries v t t 0 1 a such that v 0 v v 1 1and uv t v t u for allt 0 1 an answer is given to this question when a is assumed to be a unital simple c algebra with tracial rank no more than one let c be a unital separable amenable simple c algebra with tracial rank no more than one which also satisfies the uct suppose that c a is a unital monomorphism and suppose that v a is a unitary with v 0 in k1 a such that v almost commutes with it is shown that there is a continuous path of unitaries v t t 0 1 in a with v 0 v and v 1 1 such that the entire path v t almost commutes with provided that an induced bott map vanishes other versions of the so called basic homotopy lemma are also presented

minimal dynamical systems on the product of the cantor set and the circle we prove that a crossed product algebra arising from a minimal dynamical system on the product of the cantor set and the circle has real rank zero if and only if the set of invariant measures of the system come from the associated cantor minimal system in the case that cocycles take values in the rotation group it is also shown that this condition implies tracial rank zero and in particular the crossed product algebra is isomorphic to a unital simple at algebra of real rank zero under the same assumption we show that two systems are approximately k conjugate if and only if there exists a sequence of isomorphisms between two associated crossed products which approximately maps open image in new window

a framework for the adaptive finite element solution of large scale inverse problems since problems involving the estimation of distributed coefficients in partial differential equations are numerically very challenging efficient methods are indispensable in this paper we will introduce a framework for the efficient solution of such problems this comprises the use of adaptive finite element schemes solvers for the large linear systems arising from discretization and methods to treat additional information in the form of inequality constraints on the parameter to be recovered the methods to be developed will be based on an all at once approach in which the inverse problem is solved through a lagrangian formulation the main feature of the paper is the use of a continuous function space setting to formulate algorithms in order to allow for discretizations that are adaptively refined as nonlinear iterations proceed this entails that steps such as the description of a newton step or a line search are first formulated on continuous functions and only then evaluated for discrete functions on the other hand this approach avoids the dependence of finite dimensional norms on the mesh size making individual steps of the algorithm comparable even if they used differently refined meshes numerical examples will demonstrate the applicability and efficiency of the method for problems with several million unknowns and more than 10 000 parameters

a posteriori error control for distributed elliptic optimal control problems with control constraints discretized by hp finite elements abstract a distributed elliptic control problem with control constraints is considered which is formulated as a three field problem and consists of two variational equations for the state and the co state variables as well as of a variational inequality for the control variable two discretization approaches with h p finite elements are discussed in the first discretization approach all variables state co state and control are discretized a semi smooth newton method is introduced for solving the resulting algebraic system in the second discretization approach only the state and the co state variables are discretized whereas the control is determined by projection a simple fixed point scheme is presented for the iterative solution of this approach the main focus of the paper is on the derivation of reliable and efficient a posteriori error estimates which enables h p adaptive mesh refinements in particular the estimates can be applied to the iteration solutions so that they can be used as a stopping criterion of the iterative solution schemes in several numerical experiments the order of convergence of the adaptive discretization approaches and the efficiency as well as the reliability of the a posteriori error estimates are studied

moving mesh methods in multiple dimensions based on harmonic maps in practice there are three types of adaptive methods using the finite element approach namely the h method p method and r method in the h method the overall method contains two parts a solution algorithm and a mesh selection algorithm these two parts are independent of each other in the sense that the change of the pdes will affect the first part only however in some of the existing versions of the r method also known as the moving mesh method these two parts are strongly associated with each other and as a result any change of the pdes will result in the rewriting of the whole code in this work we will propose a moving mesh method which also contains two parts a solution algorithm and a mesh redistribution algorithm our efforts are to keep the advantages of the r method e g keep the number of nodes unchanged and of the h method e g the two parts in the code are independent a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in this work we will extend dvinsky s method to provide an efficient solver for the mesh redistribution algorithm the key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure each iteration step is to move the mesh closer to the harmonic map this procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration the numerical schemes are applied to a number of test problems in two dimensions it is observed that the mesh redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi dimensional computations

computing the inertias in symmetric matrix pencils abstract we define and study generalized pencil eigenvalues for a pencil p s t of real symmetric matrices s and t this allows us to find all occurring inertias among pencil members as bt theoretically and numerically computed examples based on a matlab code are included

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

multiscale asymptotic analysis and computation of optimal control for elliptic systems with constraints this paper reports the multiscale analysis and numerical algorithms for a class of distributed elliptic optimal control problems with constraints multiscale asymptotic expansions are presented and an explicit rate of convergence is derived a multiscale numerical method is developed numerical experiments are carried out to validate the predicted convergence results

inertia and eigenvalue relations between symmetrized and symmetrizing matrices for the real and the general field case abstract starting from a theorem of frobenius that every n n matrix is the product of two symmetric ones we study relations between the similarity invariants of a square matrix and the congruence invariants of its symmetric factors section 1 treats the real case sec 2 the arbitrary field case and sec 3 the indefinite inner product case for krein spaces the proofs are obtained from the real canonical pair form in secs 1 and 3 and from the recently found rational canonical pair form in sec 2 each time via combinatorial type arguments on weighted partitions of n the resulting theorems typically give bounds for the elementary divisor structure of a in terms of the index or signature of one or both of its symmetric factors or vice versa our results greatly extend and generalize the classic results of klein loewy taussky et al and in sec 2 put new light on waterhouse s recent characterization of hereditarily euclidean fields a short survey on the history of the subject from the early 1800s on completes the paper

extensions of c x by simple c algebras of real rank zero let ext c x a be the set of unitarily equivalence classes of essential c algebra exten sions of the following form 0 a e c x 0 where a is a nonunital separable simple c algebra of real rank zero stable rank one with unique normalized trace and x is a finite cw complex we show that there is a bijection ext c x a kk c x m a a where m a is the multiplier algebra of a in particular we determine when an extension is actually splitting we also in a more general setting give a condition when an essential extension is quasidiagonal 0 introduction we study essential extensions of the following form 0 a f c x 0 where a is a j unital nonunital simple c algebra of real rank zero and stable rank one and x is finite cw complex the brown douglas fillmore bdf theory bdf1 and bdf2 gives a classification of these extensions up to unitary equivalence in the case that a k the c algebra of compact operators on an infinite dimensional separable hubert space since then the theory of c algebra extensions developed rapidly it is very difficult to give a complete list we refer the reader to bl for references while using of theory ext c x a have been computed for the most interesting cases however ext c x a does not provide enough information in general to classify extensions up to unitary equivalence in fact as shown in ln6 and lnl2 there are usually many different unitary equivalence classes of trivial extensions and an extension r with r 0 in ext c x a may not be trivial we will only consider simple c algebras that share some properties with c we consider only those with real rank zero stable rank one and unique quasi trace up to scalar multiples there is an additional property that c has namely its corona algebra m k k the calkin algebra is simple we also as sume that m a a is simple it has been studied when a nonunital but cr unital simple c algebra a has the property that m a a is simple see lnl rrl and zh2 etc it is shown in lnl that a simple c algebra a has the property that m a a is simple if and only if a has continuous scale

morphisms of extensions of c algebras pushing forward the busby invariant abstract we study completions of diagrams of extensions of c algebras in which all three c algebras in one of the rows and either the ideal or the quotient in the other are given along with the three morphisms between them we find universal solutions to all four of these problems under restrictions of varying severity on the given vertical maps and describe the solutions in terms of push outs and pull backs of certain diagrams our characterization of the universal solution to one of the diagrams yields a concrete description of various amalgamated free products this leads to new results about the k theory of amalgamated free products verifying the cuntz conjecture in certain cases we also obtain new results about extensions of matricial field c algebras verifying partially a conjecture of blackadar and kirchberg finally we show that almost commuting unitary matrices can be uniformly approximated by commuting unitaries when an index obstruction vanishes

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

a priori and a posteriori error estimates of h 1 galerkin mixed finite element methods for elliptic optimal control problems in this paper we investigate numerical approximations of h 1 galerkin mixed finite element methods for elliptic optimal control problems the presented scheme is independent symmetric and positive definite for the state variables and the adjoint state variables moreover the matching relation i e lbb condition between the mixed element spaces v h and w h is not necessary thus we can choose the approximation spaces more flexibly the state and co state are approximated by the lowest order raviart thomas mixed finite element spaces and the standard finite element spaces the control variable is approximated by piecewise constant functions we derive a priori and a posteriori error estimates for the control variable the state variables and the adjoint state variables finally some numerical examples are given to demonstrate the theoretical results about a priori error estimates

invariants of almost commuting unitaries abstract two invariants of pairs of almost commuting unitary matrices have been defined and used to identify those pairs which are bounded away from commuting pairs the first denoted k u v for unitaries u v mn c involved k theory while the second was defined via winding numbers we show that these invariants coincide we also establish some upper bounds on k u v

the ext class of an approximately inner automorphism let a be a simple unital at algebra of real rank zero it is shown below that the range of the natural map from the approximately inner automorphism group to kk a a coincides with the kernel of the map kk a a l 1 0 hom ki a ki a 1 introduction and preliminaries 1 1 an automorphism of a unital c algebra a is said to be an approximately inner automorphism if there is a sequence of unitaries un 2 a such that a limadun a for all a 2 a it follows that the induced map on k a is the identity map there is however an invariant of k theoretical interest which can occur nontrivial extensions may arise in the six term periodic sequence for the k theory of the mapping torus we show below that every extension does arise if a is a simple unital at algebra of real rank zero as an immediate corollary we obtain a stronger form of elliott s classification theorem for such algebras an invertible kk element that preserves positivity and the class of the unit lifts to an isomorphism 1 2 recall that a unital c algebra a is said to be a unital at algebra if it is expressible as the inductive limit of finite direct sums of algebras of the form mn c t with unital embeddings let a be a unital at algebra and let be an approximately inner automorphism of a the mapping torus of is the c algebra

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

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

almost commuting orthogonal matrices abstract we show that two almost commuting real orthogonal matrices are uniformly close to exactly commuting real orthogonal matrices we prove the same for symplectic unitary matrices this is in contrast to the general complex case where not all pairs of almost commuting unitaries are close to commuting pairs our techniques also yield results about almost normal matrices over the reals and the quaternions we conclude with an example where the k theoretical obstructions to approximation cannot be avoided our example is inspired by the physical systems known as topological superconductors

a moving mesh finite element algorithm for singular problems in two and three space dimensions a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in a recent work 2001 j comput phys 170 562 588 we extended dvinsky s method to provide an efficient moving mesh algorithm which compared favorably with the previously proposed schemes in terms of simplicity and reliability in this work we will further extend the moving mesh methods based on harmonic maps to deal with mesh adaptation in three space dimensions in obtaining the variational mesh we will solve an optimization problem with some appropriate constraints which is in contrast to the traditional method of solving the euler lagrange equation directly the key idea of this approach is to update the interior and boundary grids simultaneously rather than considering them separately application of the proposed moving mesh scheme is illustrated with some two and three dimensional problems with large solution gradients the numerical experiments show that our methods can accurately resolve detail features of singular problems in 3d

a posteriori error estimates for discontinuous galerkin time stepping method for optimal control problems governed by parabolic equations in this paper we examine the discontinuous galerkin dg finite element approximation to convex distributed optimal control problems governed by linear parabolic equations where the discontinuous finite element method is used for the time discretization and the conforming finite element method is used for the space discretization we derive a posteriori error estimates for both the state and the control approximation assuming only that the underlying mesh in space is nondegenerate for problems with control constraints of obstacle type which are the kind most frequently met in applications further improved error estimates are obtained

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

fast intersections on nested tetrahedrons fint an algorithm for adaptive finite element based distributed parameter estimation a variety of biomedical imaging techniques such as optical and fluorescence tomography electrical impedance tomography and ultrasound imaging can be cast as inverse problems wherein image reconstruction involves the estimation of spatially distributed parameter s of the pde system describing the physics of the imaging process finite element discretization of imaged domain with tetrahedral elements is a popular way of solving the forward and inverse imaging problems on complicated geometries a dual adaptive mesh based approach wherein one mesh is used for solving the forward imaging problem and the other mesh used for iteratively estimating the unknown distributed parameter can result in high resolution image reconstruction at minimum computation effort if both the meshes are allowed to adapt independently till date no efficient method has been reported to identify and resolve intersection between tetrahedrons in independently refined or coarsened dual meshes herein we report a fast and robust algorithm to identify and resolve intersection of tetrahedrons within nested dual meshes generated by 8 similar subtetrahedron subdivision scheme the algorithm exploits finite element weight functions and gives rise to a set of weight functions on each vertex of disjoint tetrahedron pieces that completely cover up the intersection region of two tetrahedrons the procedure enables fully adaptive tetrahedral finite elements by supporting independent refinement and coarsening of each individual mesh while preserving fast identification and resolution of intersection the computational efficiency of the algorithm is demonstrated by diffuse photon density wave solutions obtained from a single and a dual mesh and by reconstructing a fluorescent inclusion in simulated phantom from boundary frequency domain fluorescence measurements

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

classifying quadratic maps from plane to plane let v be a 2 dimensional vector space over an algebraically closed field f of characteristic 2 3 the elements in s 2 v v under the natural action of gl 2 f are classified generi

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

af embeddings of c t2 with a prescribed k theory let a be a unital af algebra without finite dimensional quotients our main result is the classification of the group homomorphisms k0 c t2 k0 a which are induced by unital star homomorphisms c t2 a specifically for some exactly when preserves the order unit and maps the reduced k theory of the torus into the kernel of for all bounded traces a c

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

almost commuting unitaries and classification of purely infinite simplec algebras we show that for any irrational rotational algebraa a o2 o2 this is proved by combining recently established results forc algebras of real rank zero with the following result for anye 0 there is 0 such that for any pair of unitariesu vin any purely infinite simplec algebraawithk1 a 0 satisfies uv vu and u v 0 there is a pair of commuting unitariesu v awith finite spectrum such that u u eand v v e more general results concerning almost commuting unitaries are also obtained

homotopy of a pair of approximately commuting unitaries in a simple c algebra abstract it is shown that for a class of unital c algebras including purely infinite simple c algebras real rank zero simple a t algebras and af algebras ifuandvare almost commuting unitaries whereuhas trivialk1 class vhas full spectrum and a certaink0 valued obstruction associated to the pairu vis trivial thenucan be deformed to 1 through a path of unitaries in the algebra almost commuting withv and the length of the path can be estimated by a universal constant this result is used to identify the obstruction with loring s bott element associated to the pairu vand also to prove the more universal statement that if ui vi i 1 2 are two pairs of almost commuting unitaries with ui 1 vi 1 and bott ui vi each independent ofi then one pair can be deformed into the other along a path of pairs of almost commuting unitaries in the algebra the length of the path being bounded by a universal constant

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

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

solving a class of linear projection equations many interesting and important constrained optimization problems in mathematical programming can be translated into an equivalent linear projection equation u p omega u mu q here p omega cdot is the orthogonal projection on some convex set omega e g omega bbb r n and m is a positive semidefinite matrix this paper presents some new methods for solving a class of linear projection equations the search directions of these methods are straighforward extensions of the directions in traditional methods for unconstrained optimization based on the idea of a projection and contraction method 7 we get a simple step length rule and are able to obtain global linear convergence

conjugate gradient boundary element solution for distributed elliptic optimal control problems an optimality system of equations for the optimal control problem governed by helmholtz type equations is derived by the associated first order necessary optimality condition we obtain the conjugate gradient method cgm in the continuous case introducing the sequence of higher order fundamental solutions we propose an iterative algorithm based on the conjugate gradient boundary element method using the multiple reciprocity method cgm mrbem for solving the discrete control input this algorithm has an advantage over that of the existing literatures because the main attribute the reduced dimensionality of the boundary element method is fully utilized finally the local error estimates for this scheme are obtained and a test problem is given to illustrate the efficiency of the proposed method

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

unbounded derivations in at algebras letabe a simple unital at algebra of real rank zero such that it has a unique tracial state andk1 a is neither 0 norz for each hom k1 a r with dense range inrwe construct a closed derivation inawhich generates a one parameter automorphism group ofasuch that u u 2 i u for any unitaryu d furthermore we construct such an with the rohlin property which is defined in kishimoto comm math phys 179 1996 599 622 in this case the crossed producta ris a simple at algebra of real rank zero as an application we obtain that for such ac algebraathe kernel of the natural homomorphism of the groupinn a of approximately inner automorphisms intoext k1 a k0 a ext k0 a k1 a is the group hinn a of automorphisms homotopic to inner automorphisms combining with the result of kishimoto and kumjian trans amer math soc to appear inn a hinn a is isomorphic to the above direct sum as another application of the construction of derivations we show that ifais ac algebra of the above type and hinn a has the rohlin property and comes from hom k1 a r with dense range as in kishimoto and kumjian preprint then the crossed producta zis again of the same type in particulara zis an at algebra the other properties are known from kishimoto j operator theory40 1998

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

weak series bv convergence of a moving finite element method for singular axisymmetric harmonic maps we prove the convergence of a moving finite element method for the minimization of a relaxed dirichlet energy among axisymmetric maps from the disc into the sphere the optimal mesh and the discrete minimizer exist for a discrete relaxed energy similarly to the true solution the discrete minimizer has a boundary layer because of the consistency error introduced by the discretization of the energy this discrete minimizer is nonconforming we show that it converges to the solution of the continuous problem in an appropriate bv space the proof is done for three different piecewise linear finite element discretizations

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

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

expansion formulas for the inertias of hermitian matrix polynomials and matrix pencils of orthogonal projectors abstract this paper gives a group of expansion formulas for the inertias of hermitian matrix polynomials a a 2 i a 2 and a a 3 through some congruence transformations for block matrices where a is a hermitian matrix then the paper derives various expansion formulas for the ranks and inertias of some matrix pencils generated from two or three orthogonal projectors and hermitian unitary matrices as applications the paper establishes necessary and sufficient conditions for many matrix equalities to hold as well as many inequalities in the lowner partial ordering to hold

extending cellular cohomology to c algebras a filtration on the k theory of c algebras is introduced the relative quotients define groups h a n 0 for any c algebra a which we call the spherical homology of a this extends cellular cohomology in the sense that hn c x d q hn x q for x a finite cw complex while no extension of cellular cohomology which is derived from a filtration on k theory can be additive morita invariant and continuous hn is shown to be infinitely additive morita invariant for unital c algebras and continuous in limited cases

almost commuting unitary elements in purely infinite simple c algebras we consider the problem whether a pair of almost commuting unitaries selfadjoints is close to a pair of commuting unitaries selfadjoints it is an old problem see hi psh bo v1 v2 d bd ch lorl exli exl2 de lin4 etc dan voiculescu showed in v2 that there is a positive number d and a sequence of commuting unitary matrices un and v with i u v v u ji o

superconvergence of quadratic optimal control problems by triangular mixed finite element methods the aim of this work is to investigate the discretization of a quadratic convex optimal control problem using the mixed finite element method the state and co state are approximated by the order k 1 raviart thomas mixed finite element spaces and the control is approximated by piecewise constant functions we construct an interpolation of the exact control and a projection of the discrete scalar co state to be the approximated solution of the control variable for the continuous optimal control problem as a result it can be proved that the difference between the interpolation and the piecewise constant approximation has superconvergence property for the control of order h 3 2 for k 0 and of order h 2 for k 1 moreover only for the order k 1 raviart thomas mixed finite element approximation does the postprocessing technique possess the superconvergence property of order h 2 finally some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence

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

l error estimates of rectangular mixed finite element methods for bilinear optimal control problem in this paper we investigate l error estimates of the bilinear elliptic optimal control problem by rectangular raviartthomas mixed finite element methods the control variable enters the state equation as a coefficient the state and the co state variables are approximated by the raviartthomas mixed finite elements of order k 1 and the control variable is approximated by piecewise linear functions the l error estimates are obtained for the control variable and coupled state variable and the convergence rates of orders o h2 and o h32 lnh 12 are also gained for the control and state variables and the flux of the state and co state variables respectively in addition the performance of the error estimates is assessed by two numerical examples

quasi norm local error estimators for p laplacian in this paper we extend the quasi norm techniques used in a priori error estimation of finite element approximation of degenerate nonlinear systems in order to carry out an improved a posteriori error analysis for the p laplacian we derive quasi norm a posteriori error estimators of residual type which are shown to provide improved upper and lower bounds on the discretization error for sufficiently regular solutions these estimators are further shown to be equivalent on the discretization error in a quasi norm numerical results demonstrating these a posteriori estimators are also presented

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

when almost multiplicative morphisms are close to homomorphisms absrract it is shown that approximately multiplicative contractive positive morphisms from c x with dim x 2 into a simple c algebra a of real rank zero and of stable rank one are close to homomorphisms provided that certain k theoretical obstacles vanish as a corollary we show that a homomorphism h c x a is approximated by homomorphisms with finite dimensional range if h gives no k theoretical obstacle

a posteriori error estimates for some model boundary control problems in this paper we derive a posteriori error estimates for the finite element approximation of quadratic optimal boundary control problems we derive a posteriori error estimates for both the state and the control approximation on polygonal domains such estimates which are apparently not available in the literature can be used to construct reliable adaptive finite element approximation schemes for the control problems

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