universal space time coding a universal framework is developed for constructing full rate and full diversity coherent space time codes for systems with arbitrary numbers of transmit and receive antennas the proposed framework combines space time layering concepts with algebraic component codes optimized for single input single output siso channels each component code is assigned to a thread in the space time matrix allowing it thus full access to the channel spatial diversity in the absence of the other threads diophantine approximation theory is then used in order to make the different threads transparent to each other within this framework a special class of signals which uses algebraic number theoretic constellations as component codes is thoroughly investigated the lattice structure of the proposed number theoretic codes along with their minimal delay allow for polynomial complexity maximum likelihood ml decoding using algorithms from lattice theory combining the design framework with the cayley transform allows to construct full diversity differential and noncoherent space time codes the proposed framework subsumes many of the existing codes in the literature extends naturally to time selective and frequency selective channels and allows for more flexibility in the tradeoff between power efficiency bandwidth efficiency and receiver complexity simulation results that demonstrate the significant gains offered by the proposed codes are presented in certain representative scenarios

lattice coding and decoding achieve the optimal diversity multiplexing tradeoff of mimo channels this paper considers communication over coherent multiple input multiple output mimo flat fading channels where the channel is only known at the receiver for this setting we introduce the class of lattice space time last codes we show that these codes achieve the optimal diversity multiplexing tradeoff defined by zheng and tse under generalized minimum euclidean distance lattice decoding our scheme is based on a generalization of erez and zamir mod lambda scheme to the mimo case in our construction the scalar scaling of erez zamir and costa gaussian dirty paper schemes is replaced by the minimum mean square error generalized decision feedback equalizer mmse gdfe this result settles the open problem posed by zheng and tse on the construction of explicit coding and decoding schemes that achieve the optimal diversity multiplexing tradeoff moreover our results shed more light on the structure of optimal coding decoding techniques in delay limited mimo channels and hence open the door for novel approaches for space time code constructions in particular 1 we show that mmse gdfe plays a fundamental role in approaching the limits of delay limited mimo channels in the high signal to noise ratio snr regime unlike the additive white gaussian noise awgn channel case and 2 our random coding arguments represent a major departure from traditional space time code designs based on the rank and or mutual information design criteria

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

optimal designs for space time linear precoders and decoders we introduce a new paradigm for the design of transmitter space time coding that we refer to as linear precoding it leads to simple closed form solutions for transmission over frequency selective multiple input multiple output mimo channels which are scalable with respect to the number of antennas size of the coding block and transmit average peak power the scheme operates as a block transmission system in which vectors of symbols are encoded and modulated through a linear mapping operating jointly in the space and time dimension the specific designs target minimization of the symbol mean square error and the approximate maximization of the minimum distance between symbol hypotheses under average and peak power constraints the solutions are shown to convert the mimo channel with memory into a set of parallel flat fading subchannels regardless of the design criterion while appropriate power bits loading on the subchannels is the specific signature of the different designs the proposed designs are compared in terms of various performance measures such as information rate ber and symbol mean square error

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

maximum norm estimates for an immunology model using reaction diffusion equations with stochastic source terms this paper describes numerical methods and the corresponding maximum norm error estimates for a chemotaxis model proposed by kepler upon detecting pathogens immune cells secrete soluble factors that attract other immune cells to the site of the infection the motion of the model cells is stochastic but biased toward the gradient of one or more of the soluble factors the soluble factors are modeled by a system of reaction diffusion equations with sources that are centered on the cells previously i presented a first order splitting in time for solving the reaction diffusion stochastic system numerically the diffusion reaction and stochastic differential equations can be approximated separately to first order in the supremum norm the three dimensional domain is discretized using finite elements and the diffusion is solved using a backward euler scheme combined with multigrid the reaction is solved using a simple semi implicit first order scheme the stochastic differential equations are given by a langevin process which can be simulated exactly the paper concludes by demonstrating first order convergence of the entire simulation and providing a sample simulation of the reaction diffusion stochastic system

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

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

degree theory on oriented infinite dimensional varieties and the morse number of minimal surfaces spanning a curve in ℝn the algebraic number of disc minimal surfaces spanning a wire in 3 is defined and shown to be equal to one

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

on the complexity of sphere decoding in digital communications sphere decoding has been suggested by a number of authors as an efficient algorithm to solve various detection problems in digital communications in some cases the algorithm is referred to as an algorithm of polynomial complexity without clearly specifying what assumptions are made about the problem structure another claim is that although worst case complexity is exponential the expected complexity of the algorithm is polynomial herein we study the expected complexity where the problem size is defined to be the number of symbols jointly detected and our main result is that the expected complexity is exponential for fixed signal to noise ratio snr contrary to previous claims the sphere radius which is a parameter of the algorithm must be chosen to ensure a nonvanishing probability of solving the detection problem this causes the exponential complexity since the squared radius must grow linearly with problem size the rate of linear increase is however dependent on the noise variance and thus the rate of the exponential function is strongly dependent on the snr therefore sphere decoding can be efficient for some snr and problems of moderate size even though the number of operations required by the algorithm strictly speaking always grows as an exponential function of the problem size

limited feedback unitary precoding for spatial multiplexing systems multiple input multiple output mimo wireless systems use antenna arrays at both the transmitter and receiver to provide communication links with substantial diversity and capacity spatial multiplexing is a common space time modulation technique for mimo communication systems where independent information streams are sent over different transmit antennas unfortunately spatial multiplexing is sensitive to ill conditioning of the channel matrix precoding can improve the resilience of spatial multiplexing at the expense of full channel knowledge at the transmitter which is often not realistic this correspondence proposes a quantized precoding system where the optimal precoder is chosen from a finite codebook known to both receiver and transmitter the index of the optimal precoder is conveyed from the receiver to the transmitter over a low delay feedback link criteria are presented for selecting the optimal precoding matrix based on the error rate and mutual information for different receiver designs codebook design criteria are proposed for each selection criterion by minimizing a bound on the average distortion assuming a rayleigh fading matrix channel the design criteria are shown to be equivalent to packing subspaces in the grassmann manifold using the projection two norm and fubini study distances simulation results show that the proposed system outperforms antenna subset selection and performs close to optimal unitary precoding with a minimal amount of feedback

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

sharply local pointwise a posteriori error estimates for parabolic problems we prove pointwise a posteriori error estimates for semi and fully discrete finite element methods for approximating the solution u to a parabolic model problem our estimates may be used to bound the finite element error u u h l d where d is an arbitrary subset of the space time domain of the definition of the given pde in contrast to standard global error estimates these estimators de emphasize spatial error contributions from space time regions removed from d our results are valid on arbitrary shape regular simplicial meshes which may change in time and also provide insight into the contribution of mesh change to local errors when implemented in an adaptive method these estimates require only enough spatial mesh refinement away from d in order to ensure that local solution quality is not polluted by global effects

generalized linear precoder and decoder design for mimo channels using the weighted mmse criterion we address the problem of designing jointly optimum linear precoder and decoder for a mimo channel possibly with delay spread using a weighted minimum mean squared error mmse criterion subject to a transmit power constraint we show that the optimum linear precoder and decoder diagonalize the mimo channel into eigen subchannels for any set of error weights furthermore we derive the optimum linear precoder and decoder as functions of the error weights and consider specialized designs based on specific choices of error weights we show how to obtain 1 the maximum information rate design 2 qos based design we show how to achieve any set of relative snrs across the subchannels and 3 the unweighted mmse and equal error design for fixed rate systems

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

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

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

on maximum likelihood detection and the search for the closest lattice point maximum likelihood ml decoding algorithms for gaussian multiple input multiple output mimo linear channels are considered linearity over the field of real numbers facilitates the design of ml decoders using number theoretic tools for searching the closest lattice point these decoders are collectively referred to as sphere decoders in the literature in this paper a fresh look at this class of decoding algorithms is taken in particular two novel algorithms are developed the first algorithm is inspired by the pohst enumeration strategy and is shown to offer a significant reduction in complexity compared to the viterbo boutros sphere decoder the connection between the proposed algorithm and the stack sequential decoding algorithm is then established this connection is utilized to construct the second algorithm which can also be viewed as an application of the schnorr euchner strategy to ml decoding aided with a detailed study of preprocessing algorithms a variant of the second algorithm is developed and shown to offer significant reductions in the computational complexity compared to all previously proposed sphere decoders with a near ml detection performance this claim is supported by intuitive arguments and simulation results in many relevant scenarios

optimal transmitter eigen beamforming and space time block coding based on channel mean feedback optimal transmitter designs obeying the water filling principle are well documented they are widely applied when the propagation channel is deterministically known and regularly updated at the transmitter because channel state information is impossible to be known perfectly at the transmitter in practical wireless systems we design in this paper an optimal multiantenna transmitter based on the knowledge of mean values of the underlying channels our optimal transmitter design turns out to be an eigen beamformer with multiple beams pointing to orthogonal directions along the eigenvectors of the correlation matrix of the estimated channel at the transmitter and with proper power loading across beams the optimality pertains to minimizing an upper bound on the symbol error rate which leads to better performance than maximizing the expected signal to noise ratio snr at the receiver coupled with orthogonal space time block codes two directional eigen beamforming emerges as a more attractive choice than conventional one directional beamforming with uniformly improved performance without rate reduction and without essential increase in complexity with multiple receive antennas and reasonably good feedback quality the two directional eigen beamformer is also capable of achieving the best possible performance in a large range of transmit power to noise ratios without a rate penalty

low complexity adaptive design for full diversity full rate space time codes full diversity full rate fdfr space time codes for open loop multiple antenna systems achieve both high data rate and good performance but come with very high decoding complexity in this paper we propose a low complexity adaptive fdfr design for closed loop multiple input multiple output systems with only partial channel subspace knowledge at the transmitter we adapt the open loop fdfr code to the channel to maintain the special layer structure of the code at the receiver that special layer structure enables us to decouple the joint detection over dimension cnt2 into nt individual decoders of dimension cnt where nt is the number of transmit antennas this can also be seen as combining the channel diagonalization with signal diversity rotation the performance of the proposed scheme is analyzed and it is shown that the full diversity property is maintained in the adaptive design adaptive power loading is also incorporated to further exploit channel state information in term of the knowledge of the singular values of the channel the optimal loading schemes are derived for systems with linear receivers

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

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

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

morse conley theory for minimal surfaces of varying topological type after the plateau problem had been solved by douglas and rad6 by finding a possibly branched disk type minimal immersion bounded by a given jordan curve in n soon more general problems were proposed on one hand the existence of minimal surfaces of higher topological structure was investigated by douglas i d courant c and shiffman sh 1 on the other hand the problem of finding unstable minimal surfaces and more generally developing a morse theory for minimal surfaces of disk type was attacked by morse tompkins mt1 and shiffman sh2 a natural question then was to develop a morse theory for minimal surfaces of arbitrary topological structure while morse tompkins in their paper mt2 only treated a very special case namely annulus type surfaces of what their title unstable minimal surfaces of higher topological structure claims shiffman confronted the case of genus 0 and arbitrary connectivity sh3 these classical papers were not in every respect satisfactory the investigations of douglas and courant on the higher genus problem were recently critisized by tromba t 1 however luckhaus i l was able to carry out a systematic reworking of the arguments of courant and shiffman the original approaches of morse tompkins shiffman moreover severely suffer from the fact that these authors work in the c instead of the more natural h l 2 topology in the c dirichlet s integral whose critical points parametrize the sought after minimal surfaces is not differentiable and no intrinsic notion i e depending only on the surface of morse index can be defined this makes it impossible to decide in the work of morse tompkins and shiffman whether the morse relations for minimal surfaces reflect a property of the surfaces

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

improved methods for calculating vectors of short length in a lattice including a complexity analysis the standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of z in a suitable box however it suffices to consider those x e z which lie in a suitable ellipsoid having a much smaller volume than the box we show in this paper that searching through that ellipsoid is in many cases much more efficient if combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions several randomly constructed numerical examples illustrate the superiority of our new method over the known ones

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

asymptotic error expansion and richardson extrapolation for linear fine elements the elliptic ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes this justifies the application of richardson extrapolation for increasing the accuracy

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

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

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

unstable solutions of the euler equations of certain variational problems soit cr n un domaine borne avec une frontiere appartenant a c 2 pour 0 1 soit f 0 f 0 x z p appartenant a c loc 2 r r n et soit f 0 x z r n r convexe pour tout x z r alors pour u c 1 on definit i 0 t u f 0 x u u dx on considere le probleme variationnel engendre en restreignant i 0 a l ensemble u c 1 u avec c 2

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

constellations matched to the rayleigh fading channel we introduce a new technique for designing signal sets matched to the rayleigh fading channel in particular we look for n dimensional n spl ges 2 lattices whose structure provides nth order diversity our approach is based on a geometric formulation of the design problem which in turn can be solved by using a number geometric approach specifically a suitable upper bound on the pairwise error probability makes the design problem tantamount to the determination of what is called a critical lattice of the body s x x sub 1 spl middot spl middot spl middot x sub n spl isin r sup n spl pi sub i 1 sup n x sub i spl les 1 the lattices among which we search for an optimal solution are the standard embeddings in r sup n of the number ring of some totally real number field of degree n over q simulation results confirm that this approach yields lattices with considerable coding gains

optimal transmitter eigen beamforming and space time block coding based on channel correlations optimal transmitter designs obeying the water filling principle are well documented and widely applied when the propagation channel is deterministically known and regularly updated at the transmitter because channel state information csi may be costly or impossible to acquire in rapidly varying wireless environments we develop in this paper statistical water filling approaches for stationary random fading channels these approaches require only knowledge of the channel correlations that do not necessitate frequent updates and can be easily acquired applied to a multiple transmit antenna paradigm our optimal transmitter design turns out to be an eigen beamformer with multiple beams pointing to orthogonal directions along the eigenvectors of the channel s correlation matrix and with proper power loading across the beams the optimality pertains to minimizing a tight bound on the symbol error rate the resulting loaded eigen beamforming outperforms not only the equal power allocation across all antennas but also the conventional beamformer that transmits the available power along the strongest direction coupled with orthogonal space time block codes two dimensional 2 d eigen beamforming emerges as a more attractive choice than conventional one dimensional 1 d beamforming with uniformly better performance without rate reduction and without complexity increase

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

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