feasible and noninterior path following in constrained minimization with low multiplier regularity primal dual path following methods for constrained minimization problems in function space with low multiplier regularity are introduced and analyzed regularity properties of the path are proved the path structure allows us to define approximating models which are used for controlling the path parameter in an iterative process for computing a solution of the original problem the moreau yosida regularized subproblems of the new path following technique are solved efficiently by semismooth newton methods the overall algorithmic concept is provided and numerical tests including a comparison with primal dual path following interior point methods for state constrained optimal control problems show the efficiency of the new concept

the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

absolute equal distribution of families of finite sets let u uin i 1dn n n0 and v vin i 1dn n n0 where u1n u2n udn n v1n v2n vdn n n n0 and limn dn let f be a set of continuous real valued functions on r then u and v are equally distributed with respect to f if i 1dn f uin f vin o dn f f or absolutely equally distributed with respect to f if i 1dn f uin f vin o dn f f we show that these definitions are equivalent if f f c r limx f x andlimx f x exist finite and we give sufficient conditions for u and v to be absolutely equally distributed with respect to f f fisboundedanduniformlycontinuousonr and f f c r fisbounded

unbounded solutions of the equation asymptotic behavior of solutions of first order differential equation with deviating arguments in the email protected t email protected email protected i t y email protected i y email protected i is discussed for t a criterion for representing solutions in exponential form is proved inequalities for solution estimation are given sufficient conditions for the existence of unbounded solutions are derived a relevant illustrative example is given as well known results are discussed and compared

the spectral approximation of multiplication operators via asymptotic structured linear algebra abstract a multiplication operator on a hilbert space may be approximated with finite sections by choosing an orthonormal basis of the hilbert space multiplication operators with nonzero symbols defined on l 2 spaces of functions are never compact and then such approximations cannot converge in the norm topology instead we consider how well the spectra of the finite sections approximate the spectrum of the multiplication operator whose expression is simply given by the essential range of the symbol i e the multiplier we discuss the case of real orthogonal polynomial bases and the relations with the classical fourier basis whose choice leads to the well studied toeplitz case indeed the asymptotic approximation of the spectrum by the spectra of the associated toeplitz sections is possible only under precise geometric assumptions on the range of the symbol conversely the use of circulant approximations leads to constructive algorithms with o n log n complexity n number of sections working in general and generalizable to the separable multivariate and matrix valued cases as well

spectral clustering properties of block multilevel hankel matrices abstract by means of recent results concerning spectral distributions of toeplitz matrices we show that the singular values of a sequence of block p level hankel matrices h n generated by a p variate matrix valued measure whose singular part is finitely supported are always clustered at zero thus extending a result known when p 1 and is real valued and lipschitz continuous the theorems hold for both eigenvalues and singular values in the case of singular values we allow the involved matrices to be rectangular

barrier methods for optimal control problems with state constraints we study barrier methods for state constrained optimal control problems with pdes the focus of our analysis is the path of minimizers of the barrier subproblems with the aim to provide a solid theoretical basis for function space oriented path following algorithms we establish results on existence continuity and convergence of this path moreover we consider the structure of barrier subdifferentials which play the role of dual variables

multidimensional esprit for damped and undamped signals algorithm computations and perturbation analysis in this paper we present and analyze the performance of multidimensional esprit n d esprit method for estimating parameters of n d superimposed damped and or undamped exponentials n d esprit algorithm is based on low rank decomposition of multilevel hankel matrices formed by the n d data in order to reduce the computational complexity for large signals we propose a fast n d esprit using truncated singular value decomposition svd then through a first order perturbation analysis we derive simple expressions of the variance of the estimates in n d multiple tones case these expressions do not involve the factors of the svd we also derive closed form expressions of the variances of the complex modes frequencies and damping factors estimates in the n d single tone case computer results are presented to show effectiveness of the fast version of n d esprit and verify theoretical expressions

spectral properties of hankel matrices and numerical solutions of finite moment problems abstract after proving that any hankel matrix generated by moments of positive functions is conditioned essentially the same as the hilbert matrix of the same size we show a preconditioning technique i e a congruence transform of the original hankel matrix that drastically reduces its ill conditioning applications of this result to classical orthogonal polynomial sequences and to modified moment problems are given also we outline an efficient algorithm for the computation of the function f x w x exp p x where w x is positive and p x is a polynomial of degree n 1 from the knowledge of its first n moments

stability and asymptotic properties of a system of functional differential equations with nonconstant delays the asymptotic properties of a real two dimensional differential system x t a t x t k 1 mb k t x q k t h t x t x q 1 t x q m t with unbounded nonconstant delays t q k t 0 satisfying lim t q k t are studied here a b k and h are supposed to be matrix functions and a vector function the conditions for the stability and asymptotic stability of solutions and the conditions under which all solutions tend to zero are given the methods are based on the transformation of the considered real system to one equation with complex valued coefficients asymptotic properties are studied by means of a lyapunov krasovskii functional the results generalize some previous ones where the asymptotic properties for two dimensional systems with one or more constant delays or one nonconstant delay were studied

an interior point algorithm with inexact step computation in function space for state constrained optimal control we consider an interior point method in function space for pde constrained optimal control problems with state constraints our emphasis is on the construction and analysis of an algorithm that integrates a newton path following method with adaptive grid refinement this is done in the framework of inexact newton methods in function space where the discretization error of each newton step is controlled by adaptive grid refinement in the innermost loop this allows to perform most of the required newton steps on coarse grids such that the overall computational time is dominated by the last few steps for this purpose we propose an a posteriori error estimator for a problem suited norm

several path following methods for a class of gradient constrained variational inequalities path following splitting and semismooth newton methods for solving a class of problems related to elasto plastic material deformations are proposed analyzed and tested numerically while the splitting techniques result in alternating minimization schemes which are typically linearly convergent the proposed moreau yosida regularization based semismooth newton technique and an associated lifting step yield local superlinear convergence in function space the lifting step accounts for the fact that the operator associated with the linear system in the newton iteration need not be boundedly invertible uniformly along the iterates for devising an efficient update strategy for the path following parameter regularity properties of the path are studied and utilized within an inexact path following scheme for all approaches the paper ends by a report on numerical tests of the different approaches

asymptotic convergence criteria of solutions of delayed functional differential equations abstract the convergence of all solutions of the scalar equation x f t x t is studied under the main assumption that every constant is its solution the criterion and the sufficient conditions for convergence are proved for the considered classes of equations it is among others proved that all solutions are convergent if and only if there exists at least one increasing and convergent solution as applications criteria for some classes of linear equations as well as for nonlinear equations are obtained some comparisons with already known results are given too

pushing the envelope of the test functions in the szego and avram parter theorems the szego and avram parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of hermitian toeplitz matrices and the singular values of arbitrary toeplitz matrices respectively as the matrix dimension goes to infinity the question on whether these theorems are true whenever they make sense is essentially the question on whether they are valid for all continuous nonnegative and monotonously increasing test functions we show that surprisingly the answer to this question is negative on the other hand we prove the two theorems in a general form which includes all versions known so far

convergence of the solutions of the equation y t β t y t δ y t τ in the critical case we study the asymptotic behavior of the solutions of the first order differential equation containing two delays y t t y t y t with t0 r 0 the convergence of all solutions is characterized by the existence of a strictly increasing bounded solution a critical case is found for the coefficient function for coefficients below the critical function a strictly increasing and bounded solution is constructed and thus the convergence of all solutions is shown relations with known results are discussed too

asymptotic representation of solutions of equationẏ t β t y t y t τ t abstract in this paper the equation t t y t y t t is considered where c i 1 r i 1 t 1 r 0 t t is an increasing function on i 1 t t i 1 0 c i 1 r results about asymptotic structure and behavior of the solutions of this equation are proved a connection with the equation t c t x t t and the asymptotic representation of its solutions is obtained

unbounded increasing solutions of a system of difference equations with delays we consider a homogeneous system of difference equations with deviating arguments in the form y n k 1 q k n y n p k y n r k where n n 0 n 0 z p k r k are integers r k p k 0 q is a positive integer y y 1 y s t y n 0 r n 0 r 1 r s is an unknown discrete vector function s 1 is an integer r max r 1 r q y n y n 1 y n and k n ij k n i j 1 s are real matrices such that ij k n 0 n 0 1 0 and k 1 q j 1 s ij k n 0 for each admissible i and all n n 0 the behavior of solutions of this system is discussed for n the existence of unbounded increasing solutions in an exponential form is proved and estimates of solutions are given the scalar case is discussed as well

exponential solutions of equation ẏ t β t y t δ y t τ abstract asymptotic behaviour of solutions of first order differential equation with two deviating arguments of the form y t t y t y t is discussed for t a criterion for representing solutions in exponential form is proved as consequences inequalities for such solutions are given connections with known results are discussed and a sufficient condition for existence of unbounded solutions generalizing previous ones is derived an illustrative example is considered too

asymptotic convergence of the solutions of a discrete system with delays abstract a system of s discrete equations y n n y n j y n k is considered where k and j are integers k j 0 n is a real s s square matrix defined for n n 0 k n 0 z with non negative elements ij n i j 1 s such that j 1 s ij n 0 y y 1 y 2 y s t n 0 k n 0 k 1 r s and y n y n 1 y n for n n 0 a method of auxiliary inequalities is used to prove that every solution of the given system is asymptotically convergent under some conditions i e for every solution y n defined for all sufficiently large n there exists a finite limit lim n y n moreover it is proved that the asymptotic convergence of all solutions is equivalent to the existence of one asymptotically convergent solution with increasing coordinates some discussion related to the so called critical case known for scalar equations is given as well

approximation of nonnegative functions by means of exponentiated trigonometric polynomials we consider the problem of approximating a nonnegative function from the knowledge of its first fourier coefficients here we analyze a method introduced heuristically in a paper by borwein and huang siam j opt 5 1995 68 99 where it is shown how to construct cheaply a trigonometric or algebraic polynomial whose exponential is close in some sense to the considered function in this note we prove that approximations given by borwein and huang s method in the trigonometric case can be related to a nonlinear constrained optimization problem and their convergence can be easily proved under mild hypotheses as a consequence of known results in approximation theory and spectral properties of toeplitz matrices moreover they allow to obtain an improved convergence theorem for best entropy approximations

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

a unifying approach to some old and new theorems on distribution and clustering abstract a unifying approach is proposed to studying the distributions of eigenvalues and singular values of toeplitz matrices associated with a fourier series and multilevel toeplitz matrices associated with a multidimensional fourier series obtained are the extensions of the szego and avram parter theorems where the generating function is now required to belong to l 2 and not necessarily to l analogous extensions are given for multilevel toeplitz matrices in particular it is proved that if f x 1 x p l 2 then the p level complex toeplitz matrices allied with f have their singular values distributed as f x 1 x p the distribution results for the cesaro optimal circulants hold even if f l 1 also suggested are new theorems on clustering that have to do with the preconditioning of multilevel toeplitz matrices by multilevel circulants

the caratheodory fejer method for real rational approximation a caratheodory fejer method is presented for near best real rational approximation on intervals based on the eigenvalue or singular value analysis of a hankel matrix of chebyshev coefficients in approximation of a smooth function f the cf approximant r cf frequently differs from the best approximation r by only one part in millions or billions to account for this we show here under weak assumptions that if f is approximated on varepsilon varepsilon then as varepsilon to 0 f r o varepsilon m n 1 while r cf r o varepsilon 3m 2n 3 in contrast the latter figure would be o varepsilon m n 2 for the chebyshev economization approximant of maehly or the chebyshev pade approximant of gragg it follows that as varepsilon to 0 best approximation error curves approach the real parts of m n 1 winding rational functions of constant modulus to within o varepsilon 3m 2n 3 numerical examples are given

asymptotic expansions for interior penalty solutions of control constrained linear quadratic problems we consider a quadratic optimal control problem governed by a nonautonomous affine differential equation subject to nonnegativity control constraints for a general class of interior penalty functions we show how to compute the principal term of the pointwise expansion of the state and the adjoint state our main argument relies on the following fact if the control of the initial problem satisfies strict complementarity conditions for its hamiltonian except for a finite number of times the estimates for the penalized optimal control problem can be derived from the estimations of a related stationary problem our results provide several types of efficiency measures of the penalization technique error estimations of the control for l s norms s in 1 infty error estimations of the state and the adjoint state in sobolev spaces w 1 s s in 1 infty and also error estimates for the value function for the l 1 norm and the logarithmic penalty the optimal results are given in this case we indeed establish that the penalized control and the value function errors are of order o eps log eps

adjusted least squares fitting of algebraic hypersurfaces we consider the problem of fitting a set of points in euclidean space by an algebraic hypersurface we assume that points on a true hypersurface described by a polynomial equation are corrupted by zero mean independent gaussian noise and we estimate the coefficients of the true polynomial equation the adjusted least squares estimator accounts for the bias present in the ordinary least squares estimator the adjusted least squares estimator is based on constructing a quasi hankel matrix which is a bias corrected matrix of moments for the case of unknown noise variance the estimator is defined as a solution of a polynomial eigenvalue problem in this paper we present new results on invariance properties of the adjusted least squares estimator and an improved algorithm for computing the estimator for an arbitrary set of monomials in the polynomial equation

spectra of multilevel toeplitz matrices advanced theory via simple matrix relationships we consider the eigenvalue and singular value distributions for m level toeplitz matrices generated by a complex valued periodic function of m real variables we show that familiar formulations for l due to szeg and others can be preserved so long as f l1 and what is more with g weyl s definitions just a bit changed in contrast to other approaches the one we follow is based on simple matrix relationships

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

asymptotic convergence of the solutions of a discrete equation with several delays abstract a discrete equation y n i 1 s i n y n j i y n k i with integer delays k i and j i k i j i 0 is considered it is assumed that i z n 0 k 0 n 0 z k max k 1 k 2 k s z p p p 1 p z i 1 s i n 0 s z 1 is a fixed integer and n z n 0 criteria of asymptotic convergence of solutions are expressed in terms of inequalities for the functions i i 1 s some general properties of solutions are derived as well it is e g proved that for the asymptotical convergence of all solutions the existence of a strictly monotone and asymptotically convergent solution is sufficient a crucial role in the analysis of convergence is played by an auxiliary inequality derived from the form of a given discrete equation

an interior point method in function space for the efficient solution of state constrained optimal control problems we propose and analyze an interior point path following method in function space for state constrained optimal control our emphasis is on proving convergence in function space and on constructing a practical path following algorithm in particular the introduction of a pointwise damping step leads to a very efficient method as verified by numerical experiments

a note on the spectral distribution of symmetrized toeplitz sequences abstract the singular value and spectral distribution of toeplitz matrix sequences with lebesgue integrable generating functions is well studied early results were provided in the classical szeg theorem and the avram parter theorem in which the singular value symbol coincides with the generating function more general versions of the theorem were later proved by zamarashkin and tyrtyshnikov and tilli considering real nonsymmetric toeplitz matrix sequences we first symmetrize them via a simple permutation matrix and then we show that the singular value and spectral distribution of the symmetrized matrix sequence can be obtained analytically by using the notion of approximating class of sequences in particular under the assumption that the symbol is sparsely vanishing we show that roughly half of the eigenvalues of the symmetrized toeplitz matrix i e a hankel matrix are negative positive for sufficiently large dimension i e the matrix sequence is symmetric asymptotically indefinite

the asymptotic bounds of solutions of linear delay systems abstract our aim is to obtain conditions under which every solutionyof the vector functional differential equation formula can be estimated in terms of a solution of a scalar functional equation formula wherea x is a suitable real positive function andbis a suitable real negative constant moreover ifb x is diagonal matrix we derive more precise estimates ofyin terms of a solution of a vector functional equation formula where the elements ofa x are formed by the absolute values of the corresponding elements ofa x these asymptotic results enable us to obtain sharp estimates of solutions of certain linear delay systems

spectral analysis of the anti reflective algebra abstract anti reflective boundary conditions have been studied in connection with fast deblurring algorithms in the case of d dimensional objects signals for d 1 images for d 2 here we study how under the assumption of strong symmetry of the point spread functions and under mild degree conditions the associated matrices depend on a symbol and define an algebra homomorphism furthermore the eigenvalues can be exhaustively described in terms of samplings of the symbol and other related functions and appropriate o n d log n arithmetic operations algorithms can be derived for the related computations these results in connection with the use of the anti reflective transform are of interest when employing filtering type procedures for the reconstruction of noisy and blurred objects

absolute equal distribution of the spectra of hermitian matrices abstract if x if x x if x if x let a n b n p n where b n and p n are n n hermitian matrices we show that if p n f 2 o n then for any a i 1 n f mid i a n f mid i b n o n if f c eigenvalues numbered in nondecreasing order we consider the special case where p n are real hankel matrices we also show that if rank p n o n then a holds for every and f c combining these results yields a result concerning c n b n e n r n where e n 2 f o n and rank r n o n we also consider the case where the conditions on e n are stated in terms of schatten p norms finally we show that if t n are hermitian toeplitz matrices generated by f c with minimum m f and maximum m f 2 i 1 n n in 2 i n n 1 i n and n is a permutation of 1 2 n such that f n 1 n f n 2 n f n n n then i 1 n f i t n f f n i n o n if f c m f m f

an admm numerical approach to linear parabolic state constrained optimal control problems optimal control problems arising from systems modeled by linear parabolic equations may be difficult for both theoretical analysis and algorithmic design for the case where there are additional constraints on the state variables restrictive regularity assumptions are usually required to guarantee the existence of the associated lagrange multiplier and thus some regularization type methods such as the moreau yosida and lavrentiev methods have been discussed in the literature in this article we study the application of the alternating direction method of multipliers admm to linear parabolic state constrained optimal control problems and propose an admm numerical approach we prove the convergence of the admm algorithm without any existence or regularity assumption on the lagrange multiplier and estimate its worst case convergence rate in both the ergodic and nonergodic senses an important feature of the admm approach is that it decouples the state constraints and the parabolic optimal control problems inside each iteration we show the efficiency of the admm approach by testing some control problems in two space dimensions

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

affine invariant convergence analysis for inexact augmented lagrangian sqp methods an affine invariant convergence analysis for inexact augmented lagrangian sqp methods is presented the theory is used for the construction of an accuracy matching between iteration errors and truncation errors which arise from the inexact linear system solvers the theoretical investigations are illustrated numerically by an optimal control problem for the burgers equation

convergence in asymptotically autonomous functional differential equations in this paper we consider linear and nonlinear perturbations of a linear autonomous functional differential equation which has infinitely many equilibria we give sufficient conditions under which the solutions of the perturbed equation tend to the equilibria of the unperturbed equation at infinity as a consequence we obtain sufficient conditions for systems of delay differential equations to have asymptotic equilibrium