sufficient second order optimality conditions for semilinear control problems with pointwise state constraints second order sufficient optimality conditions are established for the optimal control of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state in contrast to former publications on this subject the cone of critical directions is the smallest possible in the sense that the second order sufficient conditions are the closest to the associated necessary ones the theory is developed for elliptic distributed controls in domains up to dimension three moreover problems of elliptic boundary control and parabolic distributed control are discussed in spatial domains of dimension two and one respectively

optimal control of an eddy current problem with a dipole source abstract this paper is concerned with the analysis of a class of optimal control problems governed by a time harmonic eddy current system with a dipole source which is taken as the control variable a mathematical model is set up for the state equation where the dipole source takes the form of a dirac mass located in the interior of the conducting domain a non standard approach featuring the fundamental solution of a curl curl id operator is proposed to address the well posedness of the state problem leading to a split structure of the state field as the sum of a singular part and a regular part the aim of the control is the best approximation of desired electric and magnetic fields via a suitable l 2 quadratic tracking cost functional here special attention is devoted to establishing an adjoint calculus which is consistent with the form of the state variable and in this way first order optimality conditions are eventually derived

second order analysis for optimal control problems improving results expected from abstract theory an abstract optimization problem of minimizing a functional on a convex subset of a banach space is considered we discuss natural assumptions on the functional that permit establishing sufficient second order optimality conditions with minimal gap with respect to the associated necessary ones though the two norm discrepancy is taken into account the obtained results exhibit the same formulation as the classical ones known from finite dimensional optimization we demonstrate that these assumptions are fulfilled in particular by important optimal control problems for partial differential equations we prove that in contrast to a widespread common belief the standard second order conditions formulated for these control problems imply strict local optimality of the controls not only in the sense of l infty but also of l 2

on a family of linear maps from mn c to mn2 c abstract bhat 1 characterizes the family of linear maps defined on b h which preserve unitary conjugation we generalize this idea and study the maps with a similar equivariance property on finite dimensional matrix algebras we show that the maps with equivariance property are significant to study k positivity of linear maps defined on finite dimensional matrix algebras in 5 choi showed that n positivity is different from n 1 positivity for the linear maps defined on n by n matrix algebras in this paper we present a parametric family of linear maps n m n c m n 2 c and study the properties of positivity completely positivity decomposability etc we determine values of parameters and for which the family of maps n is positive for any natural number n 3 we focus on the case of n 3 that is 3 and study the properties of 2 positivity completely positivity and decomposability in particular we give values of parameters and for which the family of maps 3 is 2 positive and not completely positive

clot norm minimization for continuous hands off control abstract in this paper we propose optimal control that is both sparse and continuous unlike previously proposed alternatives to maximum hands off control the maximum hands off control is the l 0 optimal or sparsest control among all feasible controls that are bounded by a specified value and transfer the state from a given initial state to the origin within a fixed time duration the proposed control is obtained via minimization of the clot combined l one and two norm of the control input along with the constraints on the state variable the constraints on the state variable ensures that the states are not blown up while achieving the optimal control by using the non smooth maximum principle we prove that the clot norm optimal control is unique and it is continuous in the time variable we show by numerical simulations that the clot control is continuous unlike l 0 optimal control or maximum hands off control and much sparser i e has longer time duration on which the control equals 0 than the conventional en elastic net control which is a convex combination of l 1 and squared l 2 norms

maximal parabolic regularity for divergence operators including mixed boundary conditions abstract we show that elliptic second order operators a of divergence type fulfill maximal parabolic regularity on distribution spaces even if the underlying domain is highly non smooth the coefficients of a are discontinuous and a is complemented with mixed boundary conditions applications to quasilinear parabolic equations with non smooth data are presented

preconditioning for pde constrained optimization with total variation regularization abstract the efficient solution of pde constrained optimization problems is of great significance in many scientific and engineering applications the key point of achieving this goal is to solve the linear systems arising from such problems more efficiently by preconditioning in this paper we consider preconditioning the linear systems arising from pde constrained optimization problems with total variation tv regularization based on the block structure of the coefficient matrix or its row and column transformed forms we propose three kinds of new preconditioners by approximating the schur complements in different ways we analyze the eigenvalue properties of the preconditioned coefficient matrices and provide the eigenvalue bounds independent of mesh size we also show the relations of eigenvalue bounds corresponding to different preconditioners numerical results illustrate the efficiency of the proposed preconditioners by comparison with other existing methods

structure of irreducibly covariant quantum channels for finite groups we obtain an explicit characterization of linear maps in particular quantum channels which are covariant with respect to an irreducible representation u of a finite group g whenever u uc is simply reducible with uc being the contragradient representation using the theory of group representations we obtain the spectral decomposition of any such linear map the eigenvalues and orthogonal projections arising in this decomposition are expressed entirely in terms of representation characteristics of the group g this in turn yields necessary and sufficient conditions on the eigenvalues of any such linear map for it to be a quantum channel we also obtain a wide class of quantum channels which are irreducibly covariant by construction for two dimensional irrreducible representations of the symmetric group s 3 and the quaternion group q we also characterize quantum channels which are both irreducibly covariant and entanglement breaking

second order and stability analysis for state constrained elliptic optimal control problems with sparse controls an optimal control problem for a semilinear elliptic partial differential equation is discussed subject to pointwise control constraints on the control and the state the main novelty of the paper is the presence of the l 1 norm of the control as part of the objective functional that eventually leads to sparsity of the optimal control functions second order sufficient optimality conditions are analyzed they are applied to show the convergence of optimal solutions for vanishing l 2 regularization parameter for the control the associated convergence rate is estimated

a priori error estimates for three dimensional parabolic optimal control problems with pointwise control in this paper we provide an a priori error analysis for parabolic optimal control problems with a pointwise dirac type control in space on three dimensional domains the two dimensional case was treated in d leykekhman and b vexler siam j numer anal 51 2013 pp 2797 2821 however the three dimensional case is technically much more involved to approximate the problem we use standard continuous piecewise linear elements in space and the piecewise constant discontinuous galerkin method in time despite low regularity of the state equation we establish mathcal o sqrt k h order of convergence rate for the control in the l 2 norm this result improves almost twice the previously known estimate in w gong m hinze and z zhou siam j control optim 52 2014 pp 97 119 and in addition does not require any relationship between the time step k and the mesh size h the main technical tools are discrete maximal parabolic regularity results and the best approximation type estim

on the geometry of positive maps in matrix algebras ii abstract geometric relations among the identity map the transpose map and the map x 1 n x tr x 1 n in the matrix algebra mn are illustrated providing parametric examples of maps which are not k 1 positive but which satisfy the matrix schwartz inequality of order k for 1 k n 1

a convex penalty for switching control of partial differential equations abstract a convex penalty for promoting switching controls for partial differential equations is introduced such controls consist of an arbitrary number of components of which at most one should be simultaneously active using a moreau yosida approximation a family of approximating problems is obtained that is amenable to solution by a semismooth newton method the efficiency of this approach and the structure of the obtained controls are demonstrated by numerical examples

parabolic control problems in measure spaces with sparse solutions optimal control problems in measure spaces lead to controls that have small support which is desirable e g in the context of optimal actuator placement for problems governed by parabolic partial differential equations well posedness is guaranteed in the space of square integrable measure valued functions which leads to controls with a spatial sparsity structure a conforming approximation framework allows one to derive numerically accessible optimality conditions as well as convergence rates in particular although the state is discretized the control problem can still be formulated and solved in the measure space numerical examples illustrate the structural features of the optimal controls

l2 estimates for the finite element method for the dirichlet problem with singular data in this paper we consider the approximation by the finite element method of second order elliptic problems on convex domains and homogeneous dirichlet condition on the boundary in these problems the data are borel measures using a quasiuniform mesh of finite elements and polynomials of degree 1 we prove that in two dimensions the convergence is of orderh inl 2 and in three dimensions of orderh 1 2

approximation of sparse controls in semilinear equations by piecewise linear functions semilinear elliptic optimal control problems involving the l 1 norm of the control in the objective are considered a priori finite element error estimates for piecewise linear discretizations for the control and the state are proved these are obtained by a new technique based on an appropriate discretization of the objective function numerical experiments confirm the convergence rates

forecasting enrollments with fuzzy time series part ii abstract there have been a good many methods to forecast university enrollments in the literature however none of them could be applied when the historical data are linguistic values fuzzy time series is an effective tool to deal with such problems in this paper as an application of fuzzy time series in educational research the forecast of the enrollments of the university of alabama is carried out in so doing a fuzzy time series model is developed using historical data a complete procedure is proposed which includes fuzzifying the historical data developing a fuzzy time series model and calculating and interpreting the outputs to evaluate the forecasting model the robustness of the fuzzy time series model is tested advantages and problems of the forecasting method are also discussed

optimal control of pdes in a complex space setting application to the schrodinger equation in this paper we discuss optimality conditions for abstract optimization problems over complex spaces we then apply these results to optimal control problems with a semigroup structure as an application we detail the case when the state equation is the schrodinger one with pointwise constraints on the bilinear control we derive first and second order optimality conditions and address in particular the case that the control enters the state equation and cost function linearly

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

directional sparsity in optimal control of partial differential equations we study optimal control problems in which controls with certain sparsity patterns are preferred for time dependent problems the approach can be used to find locations for control devices that allow controlling the system in an optimal way over the entire time interval the approach uses a nondifferentiable cost functional to implement the sparsity requirements additionally bound constraints for the optimal controls can be included we study the resulting problem in appropriate function spaces and present two solution methods of newton type based on different formulations of the optimality system using elliptic and parabolic test problems we research the sparsity properties of the optimal controls and analyze the behavior of the proposed solution algorithms

adjoint optimization of one dimensional hyperbolic equations with constrained periodic boundary conditions abstract this paper presents a continuous adjoint based optimization theory for a general closed loop transport hyperbolic model controlled via a periodic boundary control to minimize a cost functional the periodic boundary control is subject to a nonlinear differential equation constraint thus resulting in a coupling between the hyperbolic equation and the ordinary differential equation variational principles are used to derive the pontryagin s minimum principle for optimality that results in a dual adjoint system a numerical optimization method is implemented using the adjoint based second order gradient method to solve for the optimal trajectory of the control numerical methods for solving the hyperbolic equation using an explicit scheme wave splitting method and for solving the adjoint equation using an implicit scheme and a quasi steady state method are described

quantum channels irreducibly covariant with respect to the finite group generated by the weyl operators in matrix algebras we introduce a class of linear maps that are irreducibly covariant with respect to the finite group generated by the weyl operators in particular we analyze the irreducibly covariant quantum channels that is the completely positive and trace preserving linear maps interestingly imposing additional symmetries leads to the so called generalized pauli channels which were recently considered in the context of the non markovian quantum evolution finally we provide examples of irreducibly covariant positive but not necessarily completely positive maps

a new hybrid artificial neural networks and fuzzy regression model for time series forecasting quantitative methods have nowadays become very important tools for forecasting purposes in financial markets as for improved decisions and investments forecasting accuracy is one of the most important factors involved in selecting a forecasting method hence never has research directed at improving upon the effectiveness of time series models stopped artificial neural networks anns are flexible computing frameworks and universal approximators that can be applied to a wide range of forecasting problems with a high degree of accuracy however anns need a large amount of historical data in order to yield accurate results in a real world situation and in financial markets specifically the environment is full of uncertainties and changes occur rapidly thus future situations must be usually forecasted using the scant data made available over a short span of time therefore forecasting in these situations requires methods that work efficiently with incomplete data although fuzzy forecasting methods are suitable for incomplete data situations their performance is not always satisfactory in this paper based on the basic concepts of anns and fuzzy regression models a new hybrid method is proposed that yields more accurate results with incomplete data sets in our proposed model the advantages of anns and fuzzy regression are combined to overcome the limitations in both anns and fuzzy regression models the empirical results of financial market forecasting indicate that the proposed model can be an effective way of improving forecasting accuracy

generalized choi maps in three dimensional matrix algebra we consider a class of positive linear maps in the three dimensional matrix algebra which are generalizations of the positive linear map constructed by choi in the relation with positive semidefinite biquadratic forms we find conditions for which such maps are completely positive completely copositive decomposable and two positive

a priori error analysis for discretization of sparse elliptic optimal control problems in measure space in this paper an optimal control problem is considered where the control variable lies in a measure space and the state variable fulfills an elliptic equation this formulation leads to a sparse structure of the optimal control in this setting we prove a new regularity result for the optimal state and the optimal control moreover a finite element discretization based on e casas c clason and k kunisch siam j control optim 50 2012 pp 1735 1752 is discussed and a priori error estimates are derived which significantly improve the estimates from that paper numerical examples for problems in two and three space dimensions illustrate our results

on sparse optimal control for general linear systems in this paper we investigate an l 0 optimization problem with constraints in a form of volterra integral equation and the l infty norm in particular the equivalence theorem among the l p optimizations with p in 0 1 is derived which provides the following twofold extension of the existing results first these theoretical results enable us to solve sparse optimal control problems without imposing the finite dimensionality of the system to be controlled which was the crucial assumption for the derivation of the existing results second the relationship between the partial state constrained problem and output controllability is newly characterized

optimal control of semilinear elliptic equations in measure spaces optimal control problems in measure spaces governed by semilinear elliptic equations are considered first order optimality conditions are derived and structural properties of their solutions in particular sparsity are discussed necessary and sufficient second order optimality conditions are obtained as well on the basis of the sufficient conditions stability of the solutions is analyzed highly nonlinear terms can be incorporated by utilizing an l infty omega regularity result for solutions of the first order necessary optimality conditions

adaptive finite element method for parabolic equations with dirac measure abstract in this paper we study the adaptive finite element method for parabolic equations with dirac measure two kinds of problems with separate measure data in time and measure data in space are considered it is well known that the solutions of such kind of problems may exhibit lower regularity due to the existence of the dirac measure and thus fit to adaptive fem for space discretization and variable time steps for time discretization for both cases we use piecewise linear and continuous finite elements for the space discretization and backward euler scheme or equivalently piecewise constant discontinuous galerkin method for the time discretization the a posteriori error estimates based on energy and l 2 norms for the fully discrete problems are then derived to guide the adaptive procedure numerical results are provided at the end of the paper to support our theoretical findings

a green s formula for quasilinear elliptic operators abstract we prove a theorem of traces which allows us to define the boundary condition of the neumann problem associated to a quasilinear elliptic operator in divergence form later we prove a green s formula that justifies the variational formulation of the problem

optimal control of semilinear parabolic equations by bv functions optimal control problems for semilinear parabolic equations with control costs involving the total bounded variation seminorm are analyzed this choice of control cost favors optimal controls which are piecewise constant and it penalizes the number of jumps it is an appropriate choice if a simple structure of the optimal controls is desired which however is still sufficiently flexible so that good tracking properties can be maintained existence of optimal controls necessary and sufficient optimality conditions and sparsity properties of the derivatives are obtained convergence of a finite element approximation is analyzed and numerical examples illustrating structural properties of the optimal controls are provided

second order sufficient optimality conditions for some state constrained control problems of semilinear elliptic equations this paper deals with a class of optimal control problems governed by elliptic equations with nonlinear boundary condition the case of boundary control is studied pointwise constraints on the control and certain equality and set constraints on the state are considered second order sufficient conditions for local optimality of controls are established

second order analysis for bang bang control problems of pdes in this paper we derive some sufficient second order optimality conditions for control problems of partial differential equations pdes when the cost functional does not involve the usual quadratic term for the control or higher nonlinearities for it though not always in this situation the optimal control is typically bang bang two different control problems are studied the second differs from the first in the presence of the l 1 norm of the control this term leads to optimal controls that are sparse and usually take only three different values we call them bang bang bang controls though the proofs are detailed in the case of a semilinear elliptic state equation the approach can be extended to parabolic control problems some hints are provided in the last section to extend the results

a refined fuzzy time series model for forecasting abstract fuzzy time series models have been used to model observations where each one of them contains multiple values the formulation of fuzzy relationships and the lengths of intervals are considered to be two of the critical factors that affect forecasting results unfortunately the lengths of the intervals were determined during the early stages of forecasting in these models and they thus often failed to reflect the distribution of observations this study therefore proposes a refined fuzzy time series model to further refine the lengths of intervals this model can refine the lengths of intervals during the formulation of fuzzy relationships and hence capture the fuzzy relationships more appropriately as a result the forecasting results can be improved both the stock index and enrollment are used as the targets in the empirical analysis

optimal a priori error estimates for an elliptic problem with dirac right hand side it is well known that finite element solutions for elliptic pdes with dirac measures as source terms converge due to the fact that the solution is not in h 1 suboptimal in classical norms a standard remedy is to use graded meshes then quasioptimality i e optimal up to a log factor for low order finite elements can be recovered e g in the l 2 norm here we show for the lowest order case quasioptimality and for higher order finite elements optimal order a priori estimates on a family of quasi uniform meshes in an l 2 seminorm the seminorm is defined as an l 2 norm on a fixed subdomain which excludes the locations of the delta source terms our motivation in the use of such a norm results from the observation that in many applications the error at the singularity is dominated by the model error e g in dimension reduced settings or is not the quantity of interest e g in optimal control problems the quasi optimal and optimal order a priori bounds are obtained recursively by using aub

stabilization by sparse controls for a class of semilinear parabolic equations stabilization problems for parabolic equations with polynomial nonlinearities are investigated in the context of an optimal control formulation with a sparsity enhancing cost functional this formulation allows that the optimal control completely shuts down once the trajectory is sufficiently close to a stable steady state such a property is not present for commonly chosen control mechanisms to establish these results it is necessary to develop a function space framework for a class of optimal control problems posed on infinite time horizons which is otherwise not available

on maximal parabolic regularity for non autonomous parabolic operators abstract we consider linear inhomogeneous non autonomous parabolic problems associated to sesquilinear forms with discontinuous dependence of time we show that for these problems the property of maximal parabolic regularity can be extrapolated to time integrability exponents r 2 this allows us to prove maximal parabolic l r regularity for discontinuous non autonomous second order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations

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

spike controls for elliptic and parabolic pdes abstract we analyze the use of measures of the minimal norm to control elliptic and parabolic equations we prove the sparsity of the optimal control in the parabolic case we prove that the solution of the optimization problem is a borel measure supported in a set of lebesgue measure zero in both cases the approximate controllability can be achieved efficiently by means of controls that are activated in some finite number of pointwise locations we also analyze the corresponding dual problem

weighted fuzzy time series models for taiex forecasting abstract this study proposes weighted models to tackle two issues in fuzzy time series forecasting namely recurrence and weighting it is argued that recurrent fuzzy relationships which were simply ignored in previous studies should be considered in forecasting it is also recommended that different weights be assigned to various fuzzy relationships in previous studies these fuzzy relationships were treated as if they were equally important which might not have properly reflected the importance of each individual fuzzy relationship in forecasting the weighted models are compared with the local regression models in which weight functions also play an important role both models are different by nature but certain theoretical backgrounds in local regression models are adopted by using the taiwan stock index as the forecasting target the empirical results show that the weighted model outperforms one of the conventional fuzzy time series models

generalising wigner s theorem we analyse linear maps of operator algebras mapping the set of rank k projectors onto the set of rank l projectors surjectively a complete characterisation of such maps for prime is provided a particular case corresponding to is well known as wigner s theorem hence our result may be considered as a generalisation of this celebrated wigner s result

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

fuzzy seasonality forecasting abstract in this paper a fuzzy forecasting technique for seasonality in the time series data is presented using the following procedure first with the fuzzy regression analysis the fuzzy trend of a time series is analyzed then the fuzzy seasonality is defined by realizing the membership grades of the seasons to the fuzzy regression model both making fuzzy forecast and crisp forecast are investigated seasonal fuzziness and trends are analyzed the method is applied to the sales forecasting problem of a food distribution company

control of an elliptic problem with pointwise state constraints this paper deals with a quadratic control problem for elliptic equations with pointwise state constraints existence and uniqueness of the solution is proved optimality conditions are given and regularity of the optimal solution is investigated

the galerkin finite element method for a multi term time fractional diffusion equation we consider the initial boundary value problem for a diffusion equation involving multiple time fractional derivatives on a bounded convex polyhedral domain we analyze a space semidiscrete scheme based on the standard galerkin finite element method using continuous piecewise linear functions nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived which cover both smooth and nonsmooth data further we develop a fully discrete scheme based on a finite difference discretization of the time fractional derivatives and discuss its stability and error estimate extensive numerical experiments for one and two dimensional problems confirm the theoretical convergence rates

positive maps from irreducibly covariant operators in this paper we discuss positive maps induced by irreducibly covariant linear operators for finite groups the application of group theory methods allows deriving some new results of a different kind in particular a family of necessary conditions for positivity for such objects is derived stemming either from the definition of a positive map or the novel method inspired by the inverse reduction map in the low dimensional cases for the permutation group s 3 and the quaternion group q the necessary and sufficient conditions are given together with the discussion on their decomposability in higher dimensions we present positive maps induced by a three dimensional irreducible representation of the permutation group s 4 and d dimensional representation of the monomial unitary group mu d in the latter case we deliver if and only if condition for the positivity and compare the results with the method inspired by the inverse reduction we show that the generalised choi map can be obtained by considered covariant maps induced by the monomial unitary group as an additional result a novel interpretation of the fujiwara algolet conditions for positivity and complete positivity is presented finally in the end a new form of an irreducible representation of the symmetric group s n is constructed allowing us to simplify the form of certain choi jamiolkowski images derived for irreducible representations of the symmetric group

finite element approximations of parabolic optimal control problems with controls acting on a lower dimensional manifold this paper is devoted to the study of finite element approximations to parabolic optimal control problems with controls acting on a lower dimensional manifold the manifold can be a point a curve or a surface which may be independent of time or evolve in the time horizon and is assumed to be strictly contained in the space domain at first we obtain the first order optimality conditions for the control problems and the corresponding regularity results then for the control problems we consider the fully discrete finite element approximations based on the piecewise constant discontinuous galerkin scheme for time discretization and piecewise linear finite elements for space discretization and variational discretization to the control variable a priori error estimates are finally obtained for the fully discretized control problems and supported by numerical examples

optimal control of the two dimensional stationary navier stokes equations with measure valued controls the first author was supported by the spanish ministerio de economia industria y competitividad under research projects mtm2014 57531 p and mtm2017 83185 p the second author was supported by the erc advanced grant 668998 ocloc under the eu s h2020 research program

local error analysis for the stokes equations with a punctual source term the solution of the stokes problem with a punctual force in source term is not h 1 times mathbb l 2 and therefore the approximation by a finite element method is suboptimal in the case of poisson problem with a dirac mass in the right hand side an optimal convergence for the lagrange finite elements has been shown on a subdomain which excludes the singularity in mathbb l 2 norm by koppl and wohlmuth here we show a quasi optimal local convergence in h 1 times mathbb l 2 norm for a mathbb p k mathbb p k 1 finite element method k geqslant 2 and for the mathbb p 1 mathrm b mathbb p 1 the error is still analysed on a subdomain which does not contain the singularity the proof is based on local arnold and liu error estimates a weak version of aubin nitsche duality lemma applied to the stokes problem and discrete inf sup conditions these theoretical results are generalized to a wide class of finite element methods before being illustrated by numerical simulations

fuzzy time series and its models based upon the works by zadeh 1973 1975 in this paper a special dynamic process with linguistic values as its observations is defined and studied this dynamic process is called fuzzy time series to describe fuzzy time series fuzzy relational equations are employed as the models there are two classes of fuzzy time series time variant and time invariant the definition of fuzzy time series is given some properties of fuzzy time series are explored and procedures to develop fuzzy time series models are discussed finally a procedure for using fuzzy time series for foracasting is suggested

spectral conditions for positive maps we provide partial classification of positive linear maps in matrix algebras which is based on a family of spectral conditions this construction generalizes the celebrated choi example of a map which is positive but not completely positive it is shown how the spectral conditions enable one to construct linear maps on tensor products of matrix algebras which are positive but only on a convex subset of separable elements such maps provide basic tools to study quantum entanglement in multipartite systems

optimality conditions and error analysis of semilinear elliptic control problems with l 1 cost functional semilinear elliptic optimal control problems involving the l 1 norm of the control in the objective are considered necessary and sufficient second order optimality conditions are derived a priori finite element error estimates for piecewise constant discretizations for the control and piecewise linear discretizations of the state are shown error estimates for the variational discretization of the problem in the sense of m hinze comput optim appl 30 2005 pp 45 61 are also obtained numerical experiments confirm the convergence rates

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

approximation of elliptic control problems in measure spaces with sparse solutions optimal control problems in measure spaces governed by elliptic equations are considered for distributed and neumann boundary control which are known to promote sparse solutions optimality conditions are derived and some of the structural properties of their solutions in particular sparsity are discussed a framework for their approximation is proposed which is efficient for numerical computations and for which we prove convergence and provide error estimates

mesh independence of semismooth newton methods for lavrentiev regularized state constrained nonlinear optimal control problems a class of nonlinear elliptic optimal control problems with mixed control state constraints arising e g in lavrentiev type regularized state constrained optimal control is considered based on its first order necessary optimality conditions a semismooth newton method is proposed and its fast local convergence in function space as well as a mesh independence principle for appropriate discretizations are proved the paper ends by a numerical verification of the theoretical results including a study of the algorithm in the case of vanishing lavrentiev parameter the latter process is realized numerically by a combination of a nested iteration concept and an extrapolation technique for the state with respect to the lavrentiev parameter

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

measure valued directional sparsity for parabolic optimal control problems a directional sparsity framework allowing for measure valued controls in the spatial direction is proposed for parabolic optimal control problems it allows for controls which are localized in space where the spatial support is independent of time well posedness of the optimal control problems is established and the optimality system is derived it is used to establish structural properties of the minimizer an a priori error analysis for finite element discretization is obtained and numerical results illustrate the effects of the proposed cost functional and the convergence results

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

on positive linear maps between matrix algebras abstract there is a concrete example of a positive linear map from m 2 to m 4 which is not decomposable modification of this map gives an explicit counterexample to a conjecture of woronowicz on the strong kadison inequality

an optimization approach for well targeted transcranial direct current stimulation transcranial direct current stimulation tdcs is a noninvasive brain stimulation technique which modifies neural excitability by providing weak currents through scalp electrodes the aim of this study is to introduce and analyze a novel optimization method for safe and well targeted multiarray tdcs for optimization we consider an optimal control problem for a laplace equation with neumann boundary conditions with control and pointwise gradient state constraints we prove well posedness results for the proposed methods and provide computer simulation results in a highly realistic six compartment geometry adapted hexahedral head model for discretization of the proposed minimization problem the finite element method is employed and the existence of at least one minimizer to the discretized optimization problem is shown for numerical solution of the corresponding discretized problem we employ the alternating direction method of multipliers and comprehensively examine the cortical current flow field with regard to focality target intensity and orientation the numerical results reveal that the optimized current flow fields show significantly higher focality and in most cases higher directional agreement to the target vector in comparison to standard bipolar electrode montages

pontryagin s principle for state constrained boundary control problems of semilinear parabolic equations this paper deals with state constrained optimal control problems governed by semilinear parabolic equations we establish a minimum principle of pontryagin s type to deal with the state constraints we introduce a penalty problem by using ekeland s principle the key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined conditions for normality of optimality conditions are given

an a posteriori error analysis of an elliptic optimal control problem in measure space abstract we propose an a posteriori error estimator for an optimal control problem with sparsity promoting cost the control variable lies in the space of regular borel measures we consider a solution technique that relies on the discretization of the control variable as a linear combination of dirac measures the proposed a posteriori error estimator can be decomposed into the sum of two contributions an error estimator in the maximum norm for the discretization of the adjoint equation and an estimator in the l 2 norm that accounts for the approximation of the state equation we prove that the designed error estimator is locally efficient and we explore its reliability properties the analysis is valid for two and three dimensional domains we illustrate the theory with numerical examples

error estimates for the numerical approximation of dirichlet boundary control for semilinear elliptic equations we study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control the control is the trace of the state on the boundary of the domain which is assumed to be a convex polygonal open set in mathbb r 2 piecewise linear finite elements are used to approximate the control as well as the state we prove that the error estimates are of order o h 1 1 p for some p 2 which is consistent with the w 1 1 p p gamma regularity of the optimal control

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

sparse source identification of linear diffusion advection equations by adjoint methods abstract we present an algorithm for the time inversion of diffusion advection equations based on the adjoint methodology given a final state distribution our main aim is to recover sparse initial conditions constituted by a finite combination of kronecker deltas identifying their location and mass we discuss the strengths of the adjoint machinery and the difficulties that are to be faced in particular when the diffusivity coefficient or the time horizon is large

immersed finite elements for optimal control problems of elliptic pdes with interfaces this paper presents a numerical method and analysis based on the variational discretization concept for optimal control problems governed by elliptic pdes with interfaces the method uses a simple uniform mesh which is independent of the interface due to the jump of the coefficient across the interface the standard linear finite element method cannot achieve optimal convergence when the uniform mesh is used therefore the immersed finite element method ifem developed in li et al 20 is used to discretize the state equation required in the variational discretization approach optimal error estimates for the control state and adjoint state are derived numerical examples are provided to confirm the theoretical results

optimal a priori error estimates of parabolic optimal control problems with pointwise control in this paper we consider a parabolic optimal control problem with a pointwise dirac type control in space but variable in time in two space dimensions to approximate the problem we use the standard continuous piecewise linear approximation in space and the piecewise constant discontinuous galerkin method in time despite low regularity of the state equation we show almost optimal h 2 k convergence rate for the control in l 2 norm this result improves almost twice the previously known estimate in w gong m hinze and z zhou a priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control tech report 2011 07 hamburger beitrage zur angewandten mathematik hamburg germany 2011

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

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

boundary control of semilinear elliptic equations with pointwise state constraints this paper is concerned with state constrained optimal control problems of semilinear elliptic equations the control being on the boundary optimality conditions are derived and regularity of the optimal solution is investigated

characterization of local quadratic growth for strong minima in the optimal control of semi linear elliptic equations in this article we consider an optimal control problem of a semi linear elliptic equation with bound constraints on the control our aim is to characterize local quadratic growth for the cost function j in the sense of strong solutions this means that the function j growths quadratically over all feasible controls whose associated state is close enough to the nominal one in the uniform topology the study of strong solutions classical in the calculus of variations seems to be new in the context of pde optimization our analysis based on a decomposition result for the variation of the cost combines pontryagin s principle and second order conditions while these two ingredients are known we use them in such a way that we do not need to assume that the hessian of lagrangian of the problem is a legendre form or that it is uniformly positive on an extended set of critical directions

proximal schemes for parabolic optimal control problems with sparsity promoting cost functionals abstractfast first order proximal methods that solve linear and bilinear parabolic optimal control problems with a sparsity cost functional are discussed weak convergence of these methods is proved and for benchmarking purposes the proposed inexact proximal schemes are compared to an inexact semi smooth newton method results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes and to validate the theoretical estimates

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