numerical estimation of coercivity constants for boundary integral operators in acoustic scattering coercivity is an important concept for proving existence and uniqueness of solutions to variational problems in hilbert spaces but while coercivity estimates are well known for many variational problems arising from partial differential equations they are still an open problem in the context of boundary integral operators arising from acoustic scattering problems where rigorous coercivity results have so far only been established for combined integral operators on the unit circle and sphere the fact that coercivity holds even in these special cases is perhaps surprising as formulations of helmholtz problems are generally thought to be indefinite the main motivation for investigating coercivity in this context is that it has the potential to give error estimates for the galerkin method which are both explicit in the wavenumber k and valid regardless of the approximation space used thus they apply to hybrid asymptotic numerical methods recently developed for the high frequency case one way to interpret coercivity is by considering the numerical range of the operator the numerical range is a well established tool in spectral theory and algorithms exist to approximate the numerical range of finite dimensional matrices we can therefore use galerkin projections of the boundary integral operators to approximate the numerical range of the original operator we prove convergence estimates for the numerical range of galerkin projections of a general bounded linear operator on a hilbert space to justify this approach by computing the numerical range of the combined integral operator in acoustic scattering for several interesting convex nonconvex smooth and polygonal domains we numerically study coercivity estimates for varying wavenumbers we find that coercivity holds uniformly in the wavenumber k for a wide variety of domains finally we consider a trapping domain for which there exist resonances also called scattering poles very close to the real line to demonstrate that coercivity for a certain wavenumber k seems to be strongly dependent on the distance to the nearest resonance

card shuffling and the polynomial numerical hull of degree emk the polynomial numerical hull of degree k is a set in the complex plane associated with a matrix a that is designed to give more information than the spectrum can provide about the behavior of a under the action of polynomials we give an example of the application of the polynomial numerical hull of degree k to explain the cutoff phenomenon that is often observed in markov processes in particular we consider the much publicized result that it takes seven riffle shuffles to randomize a deck of cards d bayer and p diaconis ann appl probab 2 1992 pp 294 313 while this result cannot be explained by eigenvalues it is shown to be very well predicted by the 1 norm polynomial numerical hulls of various degrees associated with a matrix derived from the probability transition matrix

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

upper and lower bounds on norms of functions of matrices abstract given an n by n matrix a we look for a set s in the complex plane and positive scalars m and m such that for all functions p bounded and analytic on s and throughout a neighborhood of each eigenvalue of a the inequalities m inf f l s f a p a p a m inf f l s f a p a hold we show that for 2 by 2 matrices if s is the field of values then one can take m 1 and m 2 we show that for a perturbed jordan block a matrix a that is an n by n jordan block with eigenvalue 0 except that its n 1 entry is with 0 1 if s is the unit disk then m m 1 we argue however that in general due to the behavior of minimal norm interpolating functions it may be very difficult or impossible to find such a set s for which the ratio m m is of moderate size

stochastic programming approach to optimization under uncertainty in this paper we discuss computational complexity and risk averse approaches to two and multistage stochastic programming problems we argue that two stage say linear stochastic programming problems can be solved with a reasonable accuracy by monte carlo sampling techniques while there are indications that complexity of multistage programs grows fast with increase of the number of stages we discuss an extension of coherent risk measures to a multistage setting and in particular dynamic programming equations for such problems

numerical ranges of large toeplitz matrices abstract a bananch algebraic approach is proposed to study the asymptotic behaviour of the numerical ranges of certain finite approximation matrices of infinite operators the approach works for large classes of approximation methods it is examined in detail here for the finite sections of toeplitz operators and of operators which are generated by toeplitz operators the basic ingredient is a precise knowledge of the finite section method for toeplitz and related operators

new conditions for non stagnation of minimal residual methods in the solution of large linear systems a condition guaranteeing that a minimal residual krylov subspace method makes some progress i e that it does not stagnate is that the symmetric part of the coefficient matrix be positive definite this condition results in a well established worst case bound for the convergence rate of the iterative method due to elman this bound has been extensively used e g when the linear system comes from discretized partial differential equations to show that the convergence of gmres is independent of the underlying mesh size in this paper we introduce more general non stagnation conditions which do not require the symmetric part of the coefficient matrix to be positive definite and that guarantee for example the non stagnation of restarted gmres for certain values of the restarting parameter

spectral zero crossings localization properties and applications we present an analysis of the rate of sign changes in the discrete fourier spectrum of a sequence the sign changes of either the real or imaginary parts of the spectrum are considered and the rate of sign changes is termed as the spectral zero crossing rate szcr we show that szcr carries information pertaining to the locations of transients within the temporal observation window we show duality with temporal zero crossing rate analysis by expressing the spectrum of a signal as a sum of sinusoids with random phases this extension leads to spectral domain iterative filtering approaches to stabilize the spectral zero crossing rate and to improve upon the location estimates the localization properties are compared with group delay based localization metrics in a stylized signal setting well known in speech processing literature we show applications to epoch estimation in voiced speech signals using the szcr on the integrated linear prediction residue the performance of the szcr based epoch localization technique is competitive with the state of the art epoch estimation techniques that are based on average pitch period

linear preservers of polynomial numerical hulls of matrices abstract let m n be the algebra of all n n complex matrices 1 k n 1 be an integer and m n m n be a linear operator in this paper it is shown that preserves the polynomial numerical hull of order k if and only if there exists a unitary matrix u m n such that either a u a u for all a m n or a u a t u for all a m n

polynomial numerical hulls of the direct sum of a normal matrix and a jordan block abstract let j k be the k k jordan block with eigenvalue and let n be an m m normal matrix in this paper we study the polynomial numerical hulls of order 2 and n 1 for a j k n where n m k we obtain a necessary and sufficient condition such that v 2 a has an interior point also we analytically characterize v 2 j 2 n and we show that if n is co linear then v 2 j 2 n a n v 2 j 2 a finally we study v n 1 a and we show that if n is neither co linear nor co circular then v n 1 a has at most one point more than a

on the limiting set of singular values of toeplitz matrices abstract this paper is devoted to the description of the limiting set of teoplitz matrices generated by functions being locally normal over qc

worst case distribution analysis of stochastic programs we show that for even quasi concave objective functions the worst case distribution with respect to a family of unimodal distributions of a stochastic programming problem is a uniform distribution this extends the so called uniformity principle of barmish and lagoa 1997 where the objective function is the indicator function of a convex symmetric set

higher order crossings spectral analysis of an almost periodic random sequence in noise the precise effect of noise on hoc higher order crossing sequences is examined settling some issues not dealt with previously it is shown that it is possible to construct hoc sequences that converge to true discrete frequencies in a mixed spectrum even for very low signal to noise ratios the availability of the correlation function is then assumed when the expected hocs from repeated differences are known the correlation function is automatically known as well however other quantities not just the expected hoc can serve the same purpose for example under the gaussian assumption the correlation function can be obtained from that of the binary process derived by clipping the original process at any level

asymptotic analysis of a fast algorithm for efficient multiple frequency estimation based on an asymptotic analysis of the contraction mapping cm method of li and kedem 1993 a bandwidth shrinkage rule is proposed for fast and accurate estimation of the frequencies of multiple sinusoids from noisy measurements the cm frequency estimates are defined as the fixed points of a contractive mapping formed by the lag one autocorrelation coefficient calculated from the output of a parametric filter applied to the observed time series with bandwidth parameters judiciously chosen according to the asymptotic analysis the algorithm is shown to be able to accommodate possibly poor initial values of precision o n sup 1 3 and converge to a final estimate whose accuracy is arbitrarily close to o n sup 3 2 the optimal error rate for frequency estimation under the gaussian assumption the total computational complexity of the algorithm is shown to be o n log n which is comparable to that of n point fast fourier transform fft a novelty in the asymptotic analysis is that it accommodates closely spaced frequencies by allowing not only the filter bandwidth but also the frequency separation to be functions of the sample size n this enables an assessment of the accuracy of the frequency estimates for given bandwidths and initial values in situations where some or all of the frequencies are close to each other

the polynomial numerical hulls of jordan blocks and related matrices abstract the polynomial numerical hull of degree k for a square matrix a is a set designed to give useful information about the norms of polynomial functions of the matrix it is defined as z c p a p z for all p of degree k or less whilethese sets have been computed numerically for a number of matrices the computations have not been verified analytically in most cases in this paper we show analytically that the 2 norm polynomial numerical hulls of degrees 1 through n 1 for an n by n jordan block are disks about the eigenvalue with radii approaching 1 as n and we prove a theorem characterizing these radii rk n in the special case where k n 1 this theorem leads to a known result in complex approximation theory for n even rn 1 n is the positive root of 2rn r 1 0 and for n odd it satisfies a similar formula for large n this means that rn 1 n 1 log 2n n log log 2n n these results are used to obtain bounds on the polynomial numerical hulls of certain degrees for banded triangular toeplitz matrices and for block diagonal matrices with triangular toeplitz blocks

generalizations of the field of values useful in the study of polynomial functions of a matrix for a given square matrix a and positive integer k we consider sets in the complex plane satisfying p a maxz p z for all polynomials p of degree k or less the largest such set referred to as the polynomial numerical hull of degree k was introduced by o nevanlinna convergence of iterations for linear equations birkhauser basel 1993 and a number of properties of this set were derived for both matrices and linear operators we give several equivalent characterizations of the polynomial numerical hull of degree k and we actually compute these sets for several matrices for k 1 this set is just the field of values of a and for k m where m is the degree of the minimal polynomial of a it is the spectrum of a for 1 k m these sets are intermediate between the field of values and the spectrum and sometimes resemble pseudospectra

iterative filtering for multiple frequency estimation it is well known that prony s least squares estimator gives inconsistent estimates for multiple frequency estimation in a recent attempt to diminish this problem dragosevic and stankovic 1989 couple the least squares method of autoregressive ar estimation with an iterative filtering scheme discussed by kay 1988 using an all pole filter but the inconsistency still persists this paper attacks the chronic inconsistency with a general approach of parametric filtering that unifies and extends the previous work it is shown that the inconsistency can be eliminated with an appropriately parametrized filter the clue for the correct parametrization comes from a formula for the bias of the least squares ar estimator the fact of the matter is that as long as a filter satisfies the parametrization requirement consistent estimates can be obtained from the least squares ar estimator on the basis of the filtered data in particular the all pole filter considered by dragosevic and stankovic can be easily reparametrized so that it too satisfies the parametrization requirement and thus leads to a consistent estimator experimental results show that the modified method has a higher resolution than the discrete fourier transform and that its overall performance is quite remarkable

polynomial numerical hulls of the direct sum of two jordan blocks abstract let j n be the n n jordan block with a positive real eigenvalue and a j n j n in this paper we study the polynomial numerical hull of degree k h k a when k 2 and k 2 n 1 if 0 3 2 we show that r 2 n r 2 n h 2 a where r 2 n represents the radius of the circular disk h 2 j n 0 and if 2 then there exists 0 such that h 2 a x i y c x 2 y 2 also we give explicit formulas for computing the sets h 3 a and h 2 a r when n 2 although the spectrum and numerical range are continuous set valued functions with respect to the hausdorff metric by an example we show that polynomial numerical hulls may not be even lower semicontinuous

some contributions to a frequency location method due to he and kedem the author derives a useful information theoretical technology from the tunable filter that automatically detects the frequency of an unknown signal in white gaussian noise by adjusting its parameter to the correlation coefficient of its output the tunable filter earlier proposed by s he and b kedem see ibid vol 35 no 2 p 360 9 1989 is known as the hk filter the methodology sidesteps the white noise and normality assumptions more importantly it is shown that broad classes of linear filters can be made to implement the detection plan in particular bandpass filters can be expected to lock in even more quickly and accurately than the hk filter this analysis yields a formula for the error contraction at each recursion

eigenvalues and pseudo eigenvalues of toeplitz matrices the eigenvalues of a nonhermitian toeplitz matrix a are usually highly sensitive to perturbations having condition numbers that increase exponentially with the dimension n an equivalent statement is that the resolvent zz a of a toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest exponentially large as a function of n even when z is far from the spectrum because of these facts the meaningfulness of the eigenvalues of nonhermitian toeplitz matrices for any but the most theoretical purposes should be considered suspect in many applications it is more meaningful to investigate the e pseudo eigenvalues the complex numbers z with ll zz a 11 l this paper analyzes the pseudospectra of toeplitz matrices and in particular relates them to the symbols of the matrices and thereby to the spectra of the associated toeplitz and laurent operators our results are reasonably complete in the triangular case and preliminary in the cases of nontriangular toeplitz matrices block toeplitz matrices and toeplitz like matrices with smoothly varying coefficients computed examples of pseudospectra are presented throughout and applications in numerical analysis are mentioned

a distributional interpretation of robust optimization motivated by data driven decision making and sampling problems we investigate probabilistic interpretations of robust optimization ro we establish a connection between ro and distributionally robust stochastic programming drsp showing that the solution to any ro problem is also a solution to a drsp problem specifically we consider the case where multiple uncertain parameters belong to the same fixed dimensional space and find the set of distributions of the equivalent drsp problem the equivalence we derive enables us to construct ro formulations for sampled problems as in stochastic programming and machine learning that are statistically consistent even when the original sampled problem is not in the process this provides a systematic approach for tuning the uncertainty set the equivalence further provides a probabilistic explanation for the common shrinkage heuristic where the uncertainty set used in an ro problem is a shrunken version of the original uncertainty set

characterizations of the polynomial numerical hull of degree k six characterizations of the polynomial numerical hull of degree k are established for bounded linear operators on a hilbert space it is shown how these characterizations provide a natural distinction between interior and boundary points one of the characterizations is used to prove that the polynomial numerical hull of any fixed degree k for a toeplitz matrix whose symbol is piecewise continuous approaches all or most of that of the infinite dimensional toeplitz operator as the matrix size goes to infinity

price of correlations in stochastic optimization when decisions are made in the presence of high dimensional stochastic data handling joint distribution of correlated random variables can present a formidable task both in terms of sampling and estimation as well as algorithmic complexity a common heuristic is to estimate only marginal distributions and substitute joint distribution by independent product distribution in this paper we study possible loss incurred on ignoring correlations through a distributionally robust stochastic programming model and we quantify that loss as price of correlations poc using techniques of cost sharing from game theory we identify a wide class of problems for which poc has a small upper bound to our interest this class will include many stochastic convex programs uncapacitated facility location steiner tree and submodular functions suggesting that the intuitive approach of assuming independent distribution acceptably approximates the robust model for these stochastic optimization problems additionally we demonstrate hardness of bounding poc via examples of subadditive and supermodular functions that have large poc we find that our results are also useful for solving many deterministic optimization problems like welfare maximization k dimensional matching and transportation problems under certain conditions

strong consistency of the contraction mapping method for frequency estimation some statistical properties in regard to the contraction mapping cm method are discussed one of the requirements in this method is that the filter be parameterized to satisfy a certain fundamental property the parameterization clearly depends on the normalized noise spectrum which theoretically has to be known or estimated a priori if this information is available one can first whiten the noise with a linear filter and then apply the cm method to the filtered data in this way the parameterization only needs to be done under the white noise assumption and filters like the ar 2 can be used by the cm method in applications however prewhitening may not always be necessary

iterative methods for cyclically reduced non self adjoint linear systems we study iterative methods for solving linear systems of the type arising from two cyclic discretizations of non self adjoint two dimensional ellip tic partial differential equations a prototype is the convection diffusion equa tion the methods consist of applying one step of cyclic reduction resulting in a reduced system of half the order of the original discrete problem com bined with a reordering and a block iterative technique for solving the reduced system for constant coefficient problems we present analytic bounds on the spectral radii of the iteration matrices in terms of cell reynolds numbers that show the methods to be rapidly convergent in addition we describe numerical experiments that supplement the analysis and that indicate that the methods compare favorably with methods for solving the unreduced system

asymptotic theory for a fast frequency detector a fast technique for efficient online signal detection and tracking is extended by analyzing the asymptotic distribution of the detector a version of the technique is seen to have potentially greater precision than other fast estimators the upper bound being o 1 n sup 2 3 coupled with another device there is potential for achieving the convergence precision of the maximum likelihood estimate which by conventional methods entails considerably computational expense

distributionally robust markov decision processes we consider markov decision processes where the values of the parameters are uncertain this uncertainty is described by a sequence of nested sets that is each set contains the previous one each of which corresponds to a probabilistic guarantee for a different confidence level consequently a set of admissible probability distributions of the unknown parameters is specified this formulation models the case where the decision maker is aware of and wants to exploit some yet imprecise a priori information of the distribution of parameters and it arises naturally in practice where methods for estimating the confidence region of parameters abound we propose a decision criterion based on distributional robustness the optimal strategy maximizes the expected total reward under the most adversarial admissible probability distributions we show that finding the optimal distributionally robust strategy can be reduced to the standard robust mdp where parameters are known to belong to a single uncertainty set hence it can be computed in polynomial time under mild technical conditions

linear maps preserving the polynomial numerical radius of matrices abstract let n 2 be a fixed integer and denote by m n the algebra of all n n complex matrices fix also an integer k such that 1 k n we prove that if m n m n is a linear map which preserves the polynomial numerical radius of order k there exist then a unitary n n complex matrix u and a complex number of modulus one such that either t u t u for all t m n or t u t t u for all t m n

ambiguous chance constrained problems and robust optimization in this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain we focus primarily on the special case where the uncertainty set of the distributions is of the form where p denotes the prohorov metric the ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure the main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability this result is established using the strassen dudley representation theorem that states that when the distributions of two random variables are close in the prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability we also show that the robust sampled problem can be solved efficiently both in theory and in practice

fields of values and iterative methods abstract the performance of an iterative scheme to solve x t x c t c n n c c n is often judged by spectral properties of t if t is not normal it is however well known that only conclusions about the assymptotic behavior of an iterative method can be drawn from spectral information to anticipate the progress of the iteration after a finite number of steps the knowledge of the eigenvalues alone is often useless in addition the spectrum of t may be highly sensitive to perturbations if t is not normal an iterative method which on the basis of some spectral information is predicted to converge rapidly for t may well diverge if t is slightly perturbed in practice the convergence of the iteration x m t x m 1 c is therefore frequently measured by some norm verbar t verbar rather than by the spectral radius t but apart from the fact that norms lead to error estimates which are often too pessimistic they cannot be used to analyze more general schemes such as e g the chebyshev iterative methods here we discuss another tool to analyze the behavior of an iterative method namely the field of values w t the collection of all rayleigh quotients of t w t contains the eigenvalues of t and the numerical radius t max z w t z defines a norm on c n n the field of values represents therefore an intermediate concept to judge an iterative scheme by it is related to the spectral approach but has also certain norm properties