a penalization method for optimal control of elliptic problems with state constraints in this paper boundary or distributed stationary control problems are studied in relation to an elliptic operator and state and control constraints different kinds of conditions are formulated to prove the existence of a decoupled optimality system and lagrange multipliers

fractional spectral collocation method for optimal control problem governed by space fractional diffusion equation abstract in this paper we mainly investigate the fractional spectral collocation discretization of optimal control problem governed by a space fractional diffusion equation existence and uniqueness of the solution to optimal control problem is proved the continuous first order optimality condition is derived the eigenfunctions of two classes of fractional strum liouville problems are used as basis functions to approximate state variable and adjoint state variable respectively the fractional spectral collocation scheme for the control problem is constructed based on first optimize then discretize approach note that the solutions of fractional differential equations are usually singular near the boundary a generalized fractional spectral collocation scheme for the control problem is proposed based on first optimize then discretize approach a projected gradient algorithm is designed based on the discrete optimality condition numerical experiments are carried out to verify the effectiveness of the proposed numerical schemes and algorithm

centrosymmetric and centro skewsymmetric toeplitz plus hankel matrices and bezoutians abstract in the paper algebraic properties of centrosymmetric and centro skewsymmetric toeplitz plus hankel t h matrices are investigated it is shown that in the nonsingular case the inverse can be constructed with the help of the solutions of four related toeplitz systems with symmetric or skewsymmetric right hand sides in the singular case the related toeplitz matrices determine the kernel of the t h matrices finally centrosymmetric and centro skewsymmetric t h bezoutians are characterized

displacement structure for hankel vandermonde and related derived matrices abstract we introduce some generalized concepts of displacement structure for structured matrices obtained as products and inverses of toeplitz hankel and vandermonde matrices the toeplitz case has already been studied at some length and the corresponding matrices have been called near toeplitz or toeplitz like or toeplitz derived we focus on hankel and vandermonde like matrices and in particular show how the appropriately defined displacement structure yields fast triangular and orthogonal factorization algorithms for such matrices the main contribution of this paper is presenting a unified framework rather than obtaining the fastest algorithm for each special matrix

extended chandrasekhar recursions we extend the discrete time chandrasekhar recursions for least squares estimation in constant parameter state space models to a class of structured time variant state space models special cases of which often arise in adaptive filtering it can be shown that the much studied exponentially weighted recursive least squares filtering problem can be reformulated as an estimation problem for a state space model having this special time variant structure other applications arise in the multichannel and multidimensional adaptive filtering context

the split levinson algorithm the classical levinson algorithm for computing the predictor polynomial relative to a real positive definite toeplitz matrix is shown to be redundant in complexity it can be broken down into two simpler algorithms either of which needs only to be processed this result can be interpreted in the framework of the theory of orthogonal polynomials on the real line as follows the symmetric and antisymmetric parts of the predictors relative to the sequence of toeplitz matrices constitute two families of polynomials orthogonal on the interval 1 1 with respect to some even weight functions it turns out that the recurrence relations for these orthogonal polynomials can be used efficiently to compute the desired predictor the resulting split levinson algorithm requires roughly one half the number of multiplications and the same number of additions as the classical levinson algorithm a simple derivation of cybenko s method for computing the pisarenko frequencies is obtained from the recurrence relations underlying the split levinson algorithm

convex parabolic boundary control problems with pointwise state constraints abstract optimality conditions and duality are studied for convex parabolic boundary control problems with control constraints and pointwise state constraints caused by the presence of state constraints the multipliers in the optimality conditions and the variables in the dual problem are borel measures these measures appear as data in the adjoint partial differential equation it is shown that its solution as well as the restriction of its solution to the boundary is summable

on the solution of the minimal rational interpolation problem it is shown that the reachability and the observability indices of appropriately defined pairs of matrices provide the key for solving the minimal rational bitangen tial interpolation problem

a look ahead levinson algorithm for general toeplitz systems the authors present an extension of levinson s algorithm that is guaranteed to be weakly stable for a large class of general toeplitz matrices namely those that do not have many consecutive ill conditioned leading principal submatrices the new algorithm adapts itself to the given toeplitz matrix by skipping over all the ill conditioned leading principal submatrices encountered during the solution process this is done by a look ahead strategy that monitors the condition of the leading principal submatrices and if necessary switches to a block step of suitable size the overhead of the look ahead algorithm is typically small compared to the classical levinson algorithm and in addition a reliable condition number estimate is produced

the fractional diffusion equation in one space and in one time dimension a diffusion equation is solved where the first time derivative is replaced by the fractional time derivative 0 1 the solution is given in closed form in terms of fox functions

hankel and loewner matrices abstract mutual relations between the hankel toeplitz bezout and loewner matrices as well as further connections to rational interpolation and projective geometry are investigated

bowtie factors of toeplitz matrices by means of split algorithms the bowtie factorization of a toeplitz correlation matrix r and its inverse is introduced the term bowtie describes the pattern of the factoring matrices several mathematical properties of bowtie matrices are described including their close relation with block triangular matrices the bowtie factors of the inverse r can be computed using a vector version of the split levinson algorithm restricted to orders with the same parity as n the dimension of r the bowtie factors of r can be computed using split vector versions of the schur algorithm the orthogonal factorization of the underlying data matrix y can also be computed by the split versions of the lattice algorithm producing a bowtie orthogonal factor of y

fast recursive identification of state space models via exploitation of displacement structure abstract the seemingly computational burden of state space model identification has limited its real time application though it offers some important advantages over methods based on input output transfer functions it has been shown recently that ideas from the theory of displacement structure can be used in state space identification to reduce the computational burden of batch processing from o mn 2 to o mn flops when the data matrix is of size m n where n m however in many on line identification scenarios with slowly time varying systems it is desirable to update the model as time goes on with the minimal computational burden in this paper we extend our results of the batch processing algorithm to allow updating of the identified state space model with o m 2 flops again the theories of displacement structure and of the fast subspace decomposition fsd technique play crucial roles in the realization of the fast updating algorithm some computer simulation results are also presented

hyperbolic householder transformations a class of transformation matrices analogous to the householder matrices is developed with a nonorthogonal property designed to permit the efficient deletion of data from least squares problems these matrices which we term hyperbolic householder are shown to effect deletion or simultaneous addition and deletion of data with much less sensitivity to rounding errors than for techniques based on normal equations when the addition deletion sets are large this numerical robustness is obtained at the expense of only a modest increase in computations and when only a relatively small fraction of the data set is modified there is a decrease in required computations two applications to signal processing problems are considered first these transformations are used to obtain a square root algorithm for windowed recursive least squares filtering second the transformations are employed to implement the rejection of spurious data from the weight vector estimator in an adaptive array

fast tensor product solvers for optimization problems with fractional differential equations as constraints fractional differential equations have recently received much attention within computational mathematics and applied science and their numerical treatment is an important research area as such equations pose substantial challenges to existing algorithms an optimization problem with constraints given by fractional differential equations is considered which in its discretized form leads to a high dimensional tensor equation to reduce the computation time and storage the solution is sought in the tensor train format we compare three types of solution strategies that employ sophisticated iterative techniques using either preconditioned krylov solvers or tailored alternating schemes the competitiveness of these approaches is presented using several examples with constant and variable coefficients

an interpolation theory approach to h controller degree bounds abstract we derive upper bounds for the mcmillan degree of all h optimal controllers associated with design problems which may be embedded in a certain generalized regular configuration our analysis is confined to problems of the first kind which are characterized by the assumption that both p12 s and p21 s are square but not necessarily of the same size this paper which uses interpolation theory complements a previous paper which addresses the same problem through an approach based on approximation theory we demonstrate that the interpolation theory approach is more direct and circumvents a number of the technical difficulties in the previous method the final outcome is a much shorter proof as a by product we achieve a new result on the degree of an optimal solution of the matrix nevanlinna pick problem

controllability of a backward fractional semilinear differential equation abstract in this paper we study the approximate controllability of a fractional semilinear differential equation involving the right fractional caputo derivative more precisely we construct by means of tikhonov type regularization method the controllability operator then under certain condition on this operator we obtain that the associate backward fractional linear system can be steered to an arbitrary small neighborhood of the state at initial time this allows us to prove the approximate controllability of the semilinear system

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

general optimality conditions for constrained convex control problems in this paper we investigate some optimal convex control problems with mixed constraints on the state and the control we give a general condition which allows us to set optimality conditions for nonqualified problems in the slater sense then we give some applications and examples involving generalized bang bang results

the one dimensional inverse problem of reflection seismology an impulsive normal traction is applied uniformly over the surface of a perfectly stratified plane layered earth the ensuing particle velocity at the surface is assumed to be measured the problem of calculating the subsurface characteristic impedance from knowledge of the input pulse and the measured data is the one dimensional inverse problem of reflection seismology the problem is set up mathematically as an inverse problem for a first order 2 times 2 hyperbolic system the role of propagation of singularities is explained and the relation between the solution of the inverse problem and the cholesky factorization of certain matrices constructed from the data is elaborated for both the continuum problem and the related discrete problem it is found that recursive numerical techniques such as the levinson algorithm and certain other fast methods for cholesky factorization are intimately related to explicit finite difference schemes for solving hyperbolic systems the order of approximation between the

fast matrix factorizations via discrete transmission lines abstract fast algorithms associated with the names of schur and levinson are known for the triangular factorization of symmetric positive definite toeplitz matrices and their inverses in this paper we show that these algorithms can be derived from simple arguments involving causality symmetry and energy conservation in discrete lossless transmission lines the results not only provide a nice interpretation of the classical schur and levinson algorithms and a certain toeplitz inversion formula of gohberg and semencul but they also show immediately that the same fast algorithms apply not only to toeplitz matrices but to all matrices with so called displacement inertia 1 1 the results have been helpful in suggesting new digital filter structures and in the study of nonstationary second order processes

stabilized hyperbolic householder transformations a modification of the hyperbolic householder scheme is introduced which is demonstrably stable theoretically according to an established stability criterion and which exhibits superior numerical behavior in simulations the modified transform scheme effects downdating by applying conventional orthonormal rather than hyperbolic householder transformations to the data the latter have preferable numerical properties however the construction of these orthonormal operators itself requires hyperbolic computations thus the proposed method is in some sense half hyperbolic and half orthonormal there is no computational penalty incurred with these stabilized hyperbolic householder transforms they enjoy an operation count identical to their conventional counterparts

superconvergence phenomenon in the finite element method arising from averaging gradients we study a superconvergence phenomenon which can be obtained when solving a 2nd order elliptic problem by the usual linear elements the averaged gradient is a piecewise linear continuous vector field the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point the convergence rate of the averaged gradient to an exact gradient in thel 2 norm can locally be higher even by one than that of the original piecewise constant discrete gradient

remarks on the origin of the displacement rank concept an account is given of how a relationship was found between some equations of chandrasekhar for solving the wiener hopf integral equations and the nonlinear riccati differential equations encountered in linear least squares estimation problems for systems described in state space form a paper by casti kalaba and murthy 1972 provided the inspiration for seeking such a relationship and then extending it and applying it in many other fields using the concept of displacement structure

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

optimal control of a nonhomogeneous dirichlet boundary fractional diffusion equation we study a nonhomogeneous dirichlet boundary fractional diffusion equation in a bounded domain the fractional time derivative is considered in the riemann liouville sense we first prove by transposition the existence and the uniqueness of the solution of the boundary fractional diffusion equation then under some appropriate assumptions on the closed convex set of the admissible controls we obtain a decoupled optimality system

regularized nonlinear acceleration we describe a convergence acceleration technique for generic optimization problems our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method the weights in this average are computed via a simple linear system whose solution can be updated online this acceleration scheme runs in parallel to the base algorithm providing improved estimates of the solution on the fly while the original optimization method is running numerical experiments are detailed on classical classification problems

on the inverses of toeplitz plus hankel matrices abstract it is well known that the inverses of hankel and toeplitz matrices can be represented as bezoutians of polynomials in the present note a bezoutian type formula for the inverses of toeplitz plus hankel matrices and a complete characterization of toeplitz plus hankel matrix inverses are given

the wright functions as solutions of the time fractional diffusion equation we revisit the cauchy problem for the time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order email protected 0 2 by using the fourier laplace transforms the fundamentals solutions green functions are shown to be high transcendental functions of the wright type that can be interpreted as spatial probability density functions evolving in time with similarity properties we provide a general representation of these functions in terms of mellin barnes integrals useful for numerical computation

inverse scattering for discrete transmission line models this paper presents several methods for the identification of the impedance and reflection coefficient profile of a nonuniform discrete transmission line from its response to a given forcing function this problem is the prototype for a wealth of one dimensional inverse scattering problems arising in various fields a unified straightforward approach is presented providing all known inversion procedures and some novel ones the derivations readily follow from causality of signal propagation on the transmission line it is shown that a direct exploitation of the signal propagation model leads to recursive computationally efficient and reliable inversion algorithms in particular the relationships between layer peeling schur type difference equation and layer adjoining levinson type integral equation algorithms are clearly displayed

a schur algorithm and linearly connected processor array for toeplitz plus hankel matrices a levinson durbin type algorithm for solving toeplitz plus hankel t h linear systems of equations is used to induce a schur type algorithm for such systems a schur type algorithm is defined as one which efficiently computes the ldu decomposition of the matrix on the other hand levinson durbin type algorithms are defined as those algorithms which efficiently compute the udl decomposition of the inverse of a matrix it is shown that the schur algorithm so obtained is amenable to efficient implementation on a linearly connected array of processors in a manner which generalizes the results of s y kung and y h ku 1983 for symmetric toeplitz matrices specifically if t h is of order n then the schur algorithm runs on o n processors in o n time

on the superconvergence of the satisfying babuska brezzi method a simple two point boundary value problem written in a mixed form is shown to have superconvergence characteristics using the satisfying babuska brezzi sbb method convergence and accuracy analyses reveal superior performance of the method compared to the usual galerkin method

dft representations of toeplitz plus hankel bezoutians with application to fast matrixvector multiplication abstract representations for inverses of toeplitz plus hankel matrices and more general t h bezoutians with discrete fourier transformations are presented which can be used for fast matrix vector multiplication with the help of some of the formulas multiplication by a toeplitz plus hankel matrix inverse can be carried out with six dft s plus eight dft s for preprocessing which improves the so far best known result of seven dft s plus 10 dft s for preprocessing in e bozzo linear algebra appl 230 1995 127 150

a look ahead bareiss algorithm for general toeplitz matrices the bareiss algorithm is one of the classical fast solvers for systems of linear equations with toeplitz coefficient matrices the method takes advantage of the special structure and it computes the solution of a toeplitz system of order n with only o n 2 arithmetic operations instead of o n 3 operations however the original bareiss algorithm requires that all leading principal submatrices be nonsingular and the algorithm is numerically unstable if singular or ill conditioned submatrices occur in this paper an extension of the bareiss algorithm to general toeplitz systems is presented using look ahead techniques the proposed algorithm can skip over arbitrary blocks of singular or ill conditioned submatrices and at the same time it still fully exploits the toeplitz structure implementation details and operations counts are given and numerical experiments are reported we also discuss special versions of the proposed look ahead bareiss algorithm for hermitian indefinite toeplitz systems and banded toeplitz systems

the hyberbolic singular value decomposition and applications a new generalization of the singular value decomposition svd the hyperbolic svd is advanced and its existence is established under mild restrictions the hyperbolic svd accurately and efficiently finds the eigenstructure of any matrix that is expressed as the difference of two matrix outer products signal processing applications where this task arises include the covariance differencing algorithm for bearing estimation in sensor arrays sliding rectangular windowing and array calibration two algorithms for effecting this decomposition are detailed one is sequential and follows a similar pattern to the sequential bidiagonal based svd algorithm the other is for parallel implementation and mimics hestenes svd algorithm 1958 numerical examples demonstrate that like its conventional counterpart the hyperbolic svd exhibits superior numerical behavior relative to explicit formation and solution of the normal equations furthermore the hyperbolic svd applies in problems where the conventional svd cannot be employed

the lanczos algorithm and hankel matrix factorization abstract we explore the connections between the lanczos algorithm for matrix tridiagonalization and two fast algorithms for hankel matrix factorization we show how the asymmetric lanczos process is related to the berlekamp massey algorithm and how the symmetrized lanczos process is related to the phillips algorithm we also discuss conditions under which the analysis of phillips applies

fast algorithms for toeplitz and hankel matrices abstract the paper gives a self contained survey of fast algorithms for solving linear systems of equations with toeplitz or hankel coefficient matrices it is written in the style of a textbook algorithms of levinson type and schur type are discussed their connections with triangular factorizations pade recursions and lanczos methods are demonstrated in the case in which the matrices possess additional symmetry properties split algorithms are designed and their relations to butterfly factorizations are developed

finite element approximation of optimal control problems governed by time fractional diffusion equation in this paper galerkin finite element approximation of optimal control problems governed by time fractional diffusion equations is investigated piecewise linear polynomials are used to approximate the state and adjoint state while the control is discretized by variational discretization method a priori error estimates for the semi discrete approximations of the state adjoint state and control are derived furthermore we also discuss the fully discrete scheme for the control problems a finite difference method developed in lin and xu 2007 is used to discretize the time fractional derivative fully discrete first order optimality condition is developed based on first discretize then optimize approach the stability and truncation error of the fully discrete scheme are analyzed numerical example is given to illustrate the theoretical findings

conjugation interpolation and model matching in h abstract the notion of conjugation is introduced to give a unified framework of algebraic operations connected to the analytic aspects of transfer functions two classes of conjugations are shown to be important one is the lossless conjugation that performs inner outer factorization and the other is the j lossless conjugation that solves the model matching problem related to h 8 control the solution of the model matching problem obtained by j lossless conjugation is regarded as the state space version of the directional interpolation theory and is much simpler than those obtained via the nehari approximation

optimal control of fractional diffusion equation in this paper we apply the classical control theory to a fractional diffusion equation in a bounded domain the fractional time derivative is considered in a riemann liouville sense we first study the existence and the uniqueness of the solution of the fractional diffusion equation in a hilbert space then we show that the considered optimal control problem has a unique solution interpreting the euler lagrange first order optimality condition with an adjoint problem defined by means of right fractional caputo derivative we obtain an optimality system for the optimal control

inverse scattering and minimal partial realizations we present an inverse scattering interpretation of the classical minimal partial realization problem posed in a slightly generalized context our approach starts by considering a canonical cascade form structure for the realization of arbitrary transfer functions where the cascade structure can be interpreted as the description of a layered wave scattering medium in this context the partial realization problem calls for a recursive process of layer identification from a given input response pair the scattering data the realization algorithm uses a causality principle to progressively determine the parameters of cascaded linear 2 ports that model the successive wave interaction layers this method for approaching the realization problem turns out to fit nicely into a framework that was also used to obtain fast structured linear estimation algorithms and cascade realizations for digital filters

superfast tikhonov regularization of toeplitz systems toeplitz structured linear systems arise often in practical engineering problems correspondingly a number of algorithms have been developed that exploit toeplitz structure to gain computational efficiency when solving these systems the earliest fast algorithms for toeplitz systems required o n 2 operations while more recent superfast algorithms reduce the cost to o n log n 2 or below in this work we present a superfast algorithm for tikhonov regularization of toeplitz systems using an extension and transformation technique our algorithm translates a tikhonov regularized toeplitz system into a type of specialized polynomial problem known as tangential interpolation under this formulation we can compute the solution in only o n log n 2 operations we use numerical simulations to demonstrate our algorithm s complexity and verify that it returns stable solutions

on discrete schrodinger equations and their two component wave equation equivalents an approach to inverse scattering problems for discrete schrodinger equations which are discrete three term recursions is presented by systematically transforming them into discrete two component wave propagation equations the wave propagation equations permit the immediate application of certain computationally efficient and physically insightful layer peeling algorithms for inverse scattering the mapping of three term recursions to two component evolution equations is one to many because the relation between the potential sequence parametrizing schrodinger equations and the reflection coefficient sequence determining local wave interaction is a nonlinear difference equation this mapping is examined in some detail and it is used to study both direct and inverse scattering problems associated with discrete schrodinger equations

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

optimal control of a fractional diffusion equation with state constraints this paper is concerned with the state constrained optimal control problems of a fractional diffusion equation in a bounded domain the fractional time derivative is considered in the riemann liouville sense under a slater type condition we prove the existence a lagrange multiplier and a decoupled optimality system

on the splitting of classical algorithms in linear prediction theory the levinson algorithm has been shown recently to be redundant in complexity it can be broken down into two simpler algorithms either of these split levinson algorithms needs only to be processed in the present paper it is shown that the schur algorithm the lattice algorithm and the normalized lattice algorithm for computing reflection coefficients can be broken down in a similar manner the number of multiplications in each of the resulting split algorithms is reduced with respect to the classical version by a factor lying between 1 5 and 2

generalized chandrasekhar recursions from the generalized schur algorithm presents a new approach to the chandrasekhar recursions and some generalizations thereof the derivation uses the generalized schur recursions which are o n sup 2 recursions for the triangular factorization of n spl times n matrices having a certain toeplitz like displacement structure it is shown that when the extra structure provided by an underlying state space model is properly incorporated into the generalized schur algorithm it reduces to the chandrasekhar recursions which are o nn sup 2 recursions for estimating the n dimensional state of a time invariant or constant parameter system from n measured outputs it is further noted that the generalized schur algorithm factors more general structured matrices and this fact is readily used to extend the chandrasekhar recursions to a class of time variant state space models special cases of which often arise in adaptive filtering

superconvergence phenomena in the finite element method abstract we give a brief survey of superconvergence phenomena in finding a numerical solution of differential equations by finite elements several new results and open problems are introduced