experimental study of ilu preconditioners for indefinite matrices incomplete lu factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices however their failure rate is still too high for them to be useful as black box library software for general matrices besides fatal breakdowns due to zero pivots the major causes of failure are inaccuracy and instability of the triangular solves when there are small pivots both these problems can occur but these problems can also occur without small pivots through examples from actual problems this paper shows how these problems evince themselves how these problems can be detected and how these problems can sometimes be circumvented through pivoting reordering scaling perturbing diagonal elements and preserving symmetric structure the goal of this paper is to gain a better practical understanding of ilu preconditioners and help improve their reliability

an extension of the conjugate residual method to nonsymmetric linear systems the conjugate gradient cg method and the conjugate residual cr method are krylov subspace methods for solving symmetric positive definite linear systems to solve nonsymmetric linear systems the bi conjugate gradient bi cg method has been proposed as an extension of cg bi cg has attractive short term recurrences and it is the basis for the successful variants such as bi cgstab in this paper we extend cr to nonsymmetric linear systems with the aim of finding an alternative basic solver numerical experiments show that the resulting algorithm with short term recurrences often gives smoother convergence behavior than bi cg hence it may take the place of bi cg for the successful variants

on the support of mse optimal fixed rate scalar quantizers this paper determines how the support regions of optimal and asymptotically optimal fixed rate scalar quantizers with respect to mean squared error depend on the number of quantization points n and the probability density of the variable being quantized it shows that for asymptotic optimality it is necessary and sufficient that the support region grow fast enough that the outer or overload distortion decreases as o 1 n sup 2 formulas are derived for the minimal support of asymptotically optimal quantizers for generalized gamma densities including gaussian and laplacian interestingly these turn out to be essentially the same as for the support of optimal fixed rate uniform scalar quantizers heuristic arguments are then used to find closed form estimates for the support of truly optimal quantizers for generalized gamma densities these are found to be more accurate than the best prior estimates as computed by numerical algorithms they demonstrate that the support of an optimal quantizer is larger than the minimal asymptotically optimal support by a factor depending on the density but not n and that the outer distortion of optimal quantizers decreases as 1 n sup 3

a survey of the theory of source coding with a fidelity criterion the purpose of this paper is threefold 1 to acquaint the reader with the types of problems that have been considered in the area of source coding with a fidelity criterion 2 to survey results that have been obtained on these problems and 3 to outline future research trends in the area

asymptotic analysis of the exponential penalty trajectory in linear programming we consider the linear program min c x ax b and the associated exponential penalty functionfr x c x r exp aix bi r forr close to 0 the unconstrained minimizerx r offr admits an asymptotic expansion of the formx r x rd r wherex is a particular optimal solution of the linear program and the error term r has an exponentially fast decay using duality theory we exhibit an associated dual trajectory r which converges exponentially fast to a particular dual optimal solution these results are completed by an asymptotic analysis whenr tends to the primal trajectory has an asymptotic ray and the dual trajectory converges to an interior dual feasible solution

convergence of a hybrid projection proximal point algorithm coupled with approximation methods in convex optimization in order to minimize a closed convex function that is approximated by a sequence of better behaved functions we investigate the global convergence of a general hybrid iterative algorithm which consists of an inexact relaxed proximal point step followed by a suitable orthogonal projection onto a hyperplane the latter permits to consider a fixed relative error criterion for the proximal step we provide various sets of conditions ensuring the global convergence of this algorithm the analysis is valid for nonsmooth data in infinite dimensional hilbert spaces some examples are presented focusing on penalty barrier methods in convex programming we also show that some results can be adapted to the zero finding problem for a maximal monotone operator

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

the bicor and cors iterative algorithms for solving nonsymmetric linear systems we present two iterative algorithms for solving real nonsymmetric and complex non hermitian linear systems of equations and that were developed from variants of the nonsymmetric lanczos method in this paper we give the theoretical background of the two iterative methods and discuss their main computational aspects using a large number of numerical experiments we analyze their convergence properties and we also compare them with other popular nonsymmetric iterative solvers in use today

a comparative study of iterative solutions to linear systems arising in quantum mechanics this study is mainly focused on iterative solutions with simple diagonal preconditioning to two complex valued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by sherry li of nersc numerical experiments show the feasibility of iterative methods to some extent when applied to the problems and reveal the competitiveness of our recently proposed lanczos biconjugate a orthonormalization methods to other classic and popular iterative methods by the way experiment results also indicate that application specific preconditioners may be mandatory and required for accelerating convergence

proximal point algorithm controlled by a slowly vanishing term applications to hierarchical minimization let phi 0 mathbb r n to mathbb r cup infty be a closed convex function and phi 1 mathbb r n to mathbb r be a finite convex function that are bounded from below our goal is to build an algorithm which first minimizes the map phi 0 and secondly the map phi 1 over the set s 0 rm argmin phi 0 for that purpose we define the following proximal type algorithm x n 1 x n lambda n in partial eta n phi 0 varepsilon n phi 1 x n 1 leqno mathcal a 1 where lambda n is a positive step sequence eta n is a summable error sequence and varepsilon n is a control sequence tending toward 0 partial eta denotes the eta approximate subdifferential when varepsilon n is a slow control i e sum n 0 infty varepsilon n infty we prove that under adequate conditions the sequence x n defined by mathcal a 1 tends toward an element of s 1 rm argmin s 0 phi 1 more generally given finite convex functions phi 2 ldots phi n mathbb r n to mathbb r let us define the sets s i i in 1 ldots n by the recursive relation s i rm argmin s i 1 phi i we introduce an extension of algorithm mathcal a 1 to minimize hierarchically each function phi i on the set s i 1 for i in 1 ldots n

on quantizer distortion and the upper bound for exponential entropy a sharp upper bound is derived for the exponential entropy in the class of absolutely continuous distributions with specific standard deviation and an exact description of the extremal distributions this result is interpreted as determining the least favorable cases for certain methods of quantization of analog sources it is known that for a large class of quantizers both zero memory and vector the rth power distortion as well as some other distortion criteria are bounded below by a constant depending on r multiplied by a certain integral of the source s probability density it is pointed out that this bound can be rewritten in terms of the exponential entropy the exponential entropy measures the quantitative extent or range of the source distribution this fact gives a physical interpretation of the indicated limits of quantizer performance further elucidated by the main result

a note on the computation of optimal minimum mean square error quantizers this paper considers the problems associated with computing optimal minimum mean square error quantizers most computational methods in current use are iterative these iterative schemes are extremely sensitive to initial conditions various methods of obtaining good initial conditions are presented and discussed

a hybridized iterative algorithm of the bicorstab and gpbicor methods for solving non hermitian linear systems in this study we derive a new iterative algorithm including its preconditioned version which is a hybridized variant of the biconjugate a orthogonal residual stabilized bicorstab method and the generalized product type solvers based on bicor gpbicor method the proposed method which is named gpbicor m similarly to the gpbicg m method proposed by fujino 2002 can be regarded as an extension of the bicorstab2 method introduced by zhao and huang 2013 inspired by fujino s idea for improving the bicgstab2 method in the established gpbicor m method the parameters computed by the bicorstab method are chosen at successive m iteration steps and afterwards the parameters of the gpbicor method are utilized in the subsequent iteration steps therefore the proposed method can inherit the low computational cost of bicorstab and the attractive convergence of gpbicor extensive numerical convergence results on selected real and complex matrices are shown to assess the performances of the proposed gpbicor m method also against other popular non hermitian krylov subspace methods

on the support of fixed rate minimum mean squared error scalar quantizers for a laplacian source this correspondence shows that the support growth of a fixed rate optimum minimum mean squared error scalar quantizer for a laplacian density is logarithmic with the number of quantization points specifically it is shown that for a unit variance laplacian density the ratio of the support determining threshold of an optimum quantizer to 3 spl radic 2lnn 2 converges to 1 as the number n of quantization points grows also derived is a limiting upper bound that says that the support determining threshold cannot exceed the logarithmic growth by more than a small constant e g 0 0669 these results confirm the logarithmic growth of the optimum support that has previously been derived heuristically

ml k bicgstab a bicgstab variant based on multiple lanczos starting vectors we present a variant of the popular bicgstab method for solving nonsymmetric linear systems the method which we denote by ml k bicgstab is derived from a variant of the bicg method based on a lanczos process using multiple k 1 starting left lanczos vectors compared with the original bicgstab method our new method produces a residual polynomial which is of lower degree after the same number of steps but which also requires fewer matrix vector products to generate on average requiring only 1 1 k matvecs per step empirically it also seems to be more stable and more quickly convergent the new method can be implemented as a k term recurrence and can be viewed as a bridge connecting the arnoldi based fom gmres methods and the lanczos based bicgstab methods

self consistency and principal component analysis abstract i examine the self consistency of a principal component axis that is when a distribution is centered about a principal component axis a principal component axis of a random vector x is self consistent if each point on the axis corresponds to the mean of x given that x projects orthogonally onto that point a large class of symmetric multivariate distributions are examined in terms of self consistency of principal component subspaces elliptical distributions are characterized by the preservation of self consistency of principal component axes after arbitrary linear transformations a lack of fit test is proposed that tests for self consistency of a principal axis the test is applied to two real datasets

variants of bicgstab for matrices with complex spectrum recently van der vorst siam j sci statist comput 13 1992 pp 631 644 proposed for solving nonsymmetric linear systems az b a biconjugate gradient bicg based krylov space method called bicgstab that like the biconjugate gradient squared bicgs method of sonneveld does not require matrix vector multiplications with the transposed matrix a t and that has typically a much smoother convergence behavior than bicg and bicgs its nth residual polynomial is the product of the one of bicg i e the nth lanczos polynomial with a polynomial of the same degree with real zeros therefore nonreal eigenvalues of a are not approximated well by the second polynomial factor here the author presents for real nonsymmetric matrices a method bicgstab2 in which the second factor may have complex conjugate zeros moreover versions suitable for complex matrices are given for both methods

vanishing distortion and shrinking cells we establish an asymptotic connection between vanishing rth power distortion and shrinking cell diameters for vector quantizers with convex cells as a consequence a number of shrinking cell conditions may be easily verified by showing that the quantizers in question have distortion tending to zero this also plays an important role in the asymptotic analysis of a common greedy growing scheme for tree structured vector quantizers

an implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

asymptotic distribution of the errors in scalar and vector quantizers high rate or asymptotic quantization theory has found formulas for the average squared length more generally the qth moment of the length of the error produced by various scalar and vector quantizers with many quantization points in contrast this paper finds an asymptotic formula for the probability density of the length of the error and in certain special cases for the probability density of the multidimensional error vector itself the latter can be used to analyze the distortion of two stage vector quantization the former permits one to learn about the point density and cell shapes of a quantizer from a histogram of quantization error lengths histograms of the error lengths in simulations agree well with the derived formulas also presented are a number of properties of the error density including the relationship between the error density the point density and the cell shapes the fact that its qth moment equals bennett s integral a formula for the average distortion of a scalar or vector quantizer and the fact that for stationary sources the marginals of the multidimensional error density of an optimal vector quantizer with large dimension are approximately i i d gaussian

trimmed k means an attempt to robustify quantizers a class of procedures based on impartial trimming self determined by the data is introduced with the aim of robustifying k means hence the associated clustering analysis we include a detailed study of optimal regions showing that only nonpathological regions can arise from impartial trimming procedures the asymptotic results provided in the paper focus on strong consistency of the suggested methods under widely general conditions a section is devoted to exploring the performance of the procedure to detect anomalous data in simulated data sets

an analytic function describing the error performance of optimum quantizers corresp for optimizing or analyzing digital source coding systems a model for the mean square quantizing error as a function of the number of levels is necessary an analytic approximation function is presented which is valid for a wide range of important probability density functions

estimating random integrals from noisy observations sampling designs and their performance the problem of estimating a weighted average of a random process from noisy observations at a finite number of sampling points is considered the performance of sampling designs with optimal or suboptimal but easily computable estimator coefficients is studied several examples and special cases are studied including additive independent noise nonlinear distortion with noise and quantization noise

bennett s integral for vector quantizers this paper extends bennett s 1948 integral from scalar to vector quantizers giving a simple formula that expresses the rth power distortion of a many point vector quantizer in terms of the number of points point density function inertial profile and the distribution of the source the inertial profile specifies the normalized moment of inertia of quantization cells as a function of location the extension is formulated in terms of a sequence of quantizers whose point density and inertial profile approach known functions as the number of points increase precise conditions are given for the convergence of distortion suitably normalized to bennett s integral previous extensions did not include the inertial profile and consequently provided only bounds or applied only to quantizers with congruent cells such as lattice and optimal quantizers the new version of bennett s integral provides a framework for the analysis of suboptimal structured vector quantizers it is shown how the loss in performance of such quantizers relative to optimal unstructured ones can be decomposed into point density and cell shape losses as examples these losses are computed for product quantizers and used to gain further understanding of the performance of scalar quantizers applied to stationary memoryless sources and of transform codes applied to gaussian sources with memory it is shown that the short coming of such quantizers is that they must compromise between point density and cell shapes

idr s a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations we present idr s a new family of efficient short recurrence methods for large nonsymmetric systems of linear equations the new methods are based on the induced dimension reduction idr method proposed by sonneveld in 1980 idr s generates residuals that are forced to be in a sequence of nested subspaces although idr s behaves like an iterative method in exact arithmetic it computes the true solution using at most n n s matrix vector products with n the problem size and s the codimension of a fixed subspace we describe the algorithm and the underlying theory and present numerical experiments to illustrate the theoretical properties of the method and its performance for systems arising from different applications our experiments show that idr s is competitive with or superior to most bi cg based methods and outperforms bi cgstab when s 1

penalization in non classical convex programming via variational convergence a penalty method for convex functions which cannot necessarily be extended outside their effective domains by an everywhere finite convex function is proposed and combined with the proximal method proofs of convergence rely on variational convergence theory

alternating proximal algorithms for linearly constrained variational inequalities application to domain decomposition for pde s abstract let x y z be real hilbert spaces let f x r g y r be closed convex functions and let a x z b y z be linear continuous operators let us consider the constrained minimization problem p min f x g y a x b y given a sequence n which tends toward 0 as n we study the following alternating proximal algorithm a x n 1 argmin n 1 f 1 2 a b y n z 2 2 x n x 2 x y n 1 argmin n 1 g 1 2 a x n 1 b z 2 2 y n y 2 y where and are positive parameters it is shown that if the sequence n tends moderately slowly toward 0 then the iterates of a weakly converge toward a solution of p the study is extended to the setting of maximal monotone operators for which a general ergodic convergence result is obtained applications are given in the area of domain decomposition for pde s

high resolution source coding for non difference distortion measures multidimensional companding entropy coded vector quantization is studied using high resolution multidimensional companding over a class of nondifference distortion measures for distortion measures which are locally quadratic a rigorous derivation of the asymptotic distortion and entropy coded rate of multidimensional companders is given along with conditions for the optimal choice of the compressor function this optimum compressor when it exists depends on the distortion measure but not on the source distribution the rate distortion performance of the companding scheme is studied using an asymptotic expression for the rate distortion function which parallels the shannon lower bound for difference distortion measures it is proved that the high resolution performance of the scheme is arbitrarily close to the rate distortion limit for large quantizer dimensions if the compressor function and the lattice quantizer used in the companding scheme are optimal extending an analogous statement for entropy coded lattice quantization and mse distortion the companding approach is applied to obtain a high resolution quantizing scheme for noisy sources

a convergence result for nonautonomous subgradient evolution equations and its application to the steepest descent exponential penalty trajectory in linear programming abstract we present a new result on the asymptotic behavior of nonautonomous subgradient evolution equations of the form u t t u t where t t 0 is a family of closed proper convex functions the result is used to study the flow generated by the family t x f x r t where f x r ctx r exp aix bi r is the exponential penalty approximation of the linear program min ctx ax b and r t is a positive function tending to 0 when t we prove that the trajectory u t converges to an optimal solution u of the linear program and we give conditions for the convergence of an associated dual trajectory t toward an optimal solution of the dual program

gmres with deflated restarting a modification is given of the gmres iterative method for nonsymmetric systems of linear equations the new method deflates eigenvalues using wu and simon s thick restarting approach siam j matrix anal appl 22 2000 pp 602 616 it has the efficiency of implicit restarting but is simpler and does not have the same numerical concerns the deflation of small eigenvalues can greatly improve the convergence of restarted gmres also it is demonstrated that using harmonic ritz vectors is important because then the whole subspace is a krylov subspace that contains certain important smaller subspaces

weighted toeplitz regularized least squares computation for image restoration the main aim of this paper is to develop a fast algorithm for solving weighted toeplitz regularized least squares problems arising from image restoration based on augmented system formulation we develop new hermitian and skew hermitian splitting hss preconditioners for solving such linear systems the advantage of the proposed preconditioner is that the blurring matrix weighting matrix and regularization matrix can be decoupled such that the resulting preconditioner is not expensive to use we show that for a preconditioned system that is derived from a saddle point structure of size m n times m n the preconditioned matrix has an eigenvalue at 1 with multiplicity n and the other m eigenvalues of the form 1 lambda with lambda 1 we also study how to choose the hss parameter to minimize the magnitude of lambda and therefore the krylov subspace method applied to solving the preconditioned system converges very quickly experimental results for image restoration problems are re

a simple class of asymptotically optimal quantizers a simple class of quantizers is introduced which are asymptotically optimal as the number of quantization levels increases to infinity with respect to a mean r th power absolute error distortion measure these asymptotically optimal quantizers are very easy to compute their performance is evaluated for several distributions and compares favorably with the performance of the optimal quantizers in all cases for which the latter have been computed in addition their asymptotic robustness is studied under location scale and shape mismatch for several families of distributions

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

distributed functional scalar quantization simplified distributed functional scalar quantization dfsq theory provides optimality conditions and predicts performance of data acquisition systems in which a computation on acquired data is desired we address two limitations of previous works prohibitively expensive decoder design and a restriction to source distributions with bounded support we show that a much simpler decoder has equivalent asymptotic performance to the conditional expectation estimator studied previously thus reducing decoder design complexity the simpler decoder features decoupled communication and computation blocks moreover we extend the dfsq framework with the simpler decoder to source distributions with unbounded support finally through simulation results we demonstrate that performance at moderate coding rates is well predicted by the asymptotic analysis and we give new insight on the rate of convergence

a transpose free quasi minimal residual variant of the cors method for solving non hermitian linear systems recently the conjugate a orthogonal residual squared cors method is often competitive and superior to other krylov subspace methods for solving complex non hermitian linear systems from many realistic problems however like the conjugate gradient squared cgs method the cors method often exhibits irregular convergence behavior with wild oscillations in the residual norm to try and remedy this problem we develop a transpose free quasi minimal residual tfqmr variant of the cors method which leads to smooth convergence curves with a convergence rate similar to the cors method we also give convergence results for this new method finally numerical examples are reported to illustrate the effectiveness of the proposed method

motivations and realizations of krylov subspace methods for large sparse linear systems we briefly introduce typical and important direct and iterative methods for solving systems of linear equations concretely describe their fundamental characteristics in viewpoints of both theory and applications and clearly clarify the substantial differences among these methods in particular the motivations of searching the solution of a linear system in a krylov subspace are described and the algorithmic realizations of the generalized minimal residual gmres method are shown and several classes of state of the art algebraic preconditioners are briefly reviewed all this is useful for correctly deeply and completely understanding the application scopes theoretical properties and numerical behaviors of these methods and is also helpful in designing new methods for solving systems of linear equations

bicr variants of the hybrid bicg methods for solving linear systems with nonsymmetric matrices we propose bi conjugate residual bicr variants of the hybrid bi conjugate gradient bicg methods referred to as the hybrid bicr variants for solving linear systems with nonsymmetric coefficient matrices the recurrence formulas used to update an approximation and a residual vector are the same as those used in the corresponding hybrid bicg method but the recurrence coefficients are different they are determined so as to compute the coefficients of the residual polynomial of bicr from our experience it appears that the hybrid bicr variants often converge faster than their bicg counterpart numerical experiments show that our proposed hybrid bicr variants are more effective and less affected by rounding errors the factor in the loss of convergence speed is analyzed to clarify the difference of the convergence between our proposed hybrid bicr variants and the hybrid bicg methods

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

coupling general penalty schemes for convex programming with the steepest descent and the proximal point algorithm we study the asymptotic behavior of the continuous flow generated by coupling the steepest descent method with a general class of penalty schemes for convex programming we establish convergence of the trajectories towards primal optimal solutions as well as convergence of some naturally associated dual paths the results are extended to the sequences generated by an implicit discretization scheme which corresponds to the coupling of an inexact proximal point iteration with the penalty schemes

a generalized product type bicor method and its application in signal deconvolution for solving nonsymmetric linear systems we attempt to establish symmetric structures in nonsymmetric systems and handle them through the methods devised for symmetric cases a biconjugate a orthogonal residual method based on biconjugate a orthonormalization procedure has been proposed and nominated as bicor in y f jing t z huang y zhang l li g h cheng z g ren y duan t sogabe b carpentieri lanczos type variants of the cocr method for complex nonsymmetric linear systems j comput phys 228 2009 6376 6394 as many similar characteristics exist between bicor and bicg the strategies of improved variants of bicg such as cgs and bicgstab can be utilized to enhance the algorithm for bicor making use of the product of residual polynomials of bicor and other polynomials cors and bicorstab have been proposed along the same ideas of cgs and bicgstab respectively in the above mentioned paper in this paper a unified generalized framework of product type bicor which is epitomized by the product of residual polynomials and other polynomials is proposed numerical examples are selected from the blurring signal cases and the effect of the generalized product type bicor method is prominent in signal deconvolution

asymptotic expansion of penalty gradient flows in linear programming we establish asymptotic expansions for nonautonomous gradient flows of the form dot u t nabla f u t r t where f x r is a penalty approximation of a linear program and the penalty parameter r t tends to 0 as t to infty under appropriate conditions we show that every integral curve satisfies u t u infty r t d 0 dot r t r t w 0 o dot r t r t for suitable vectors u infty d 0 and w 0 we deduce an asymptotic expansion for a related dual trajectory and we show that the primal dual limit point is a pair of strictly complementary optimal solutions for the linear program

the effect of quantization on the performance of sampling designs the most common form of quantization is rounding off which occurs in all digital systems a general quantizer approximates an observed value by the nearest among a finite number of representative values in estimating weighted integrals of a time series with no quadratic mean derivatives by means of samples at discrete times it is known that the rate of convergence of the mean square error is reduced from n sup 2 to n sup 1 5 when the samples are quantized for smoother time series with k 1 2 quadratic mean derivatives it is now shown that the rate of convergence is reduced from n sup 2k 2 to n sup 2 when the samples are quantized which is a very significant reduction the interplay between sampling and quantization is also studied leading to asymptotically optimal allocation between the number of samples and the number of levels of quantization

primal and dual convergence of a proximal point exponential penalty method for linear programming we consider the diagonal inexact proximal point iteration frac u k u k 1 lambda k in partial varepsilon k f u k r k nu k where f x r ctx r exp a i x b i r is the exponential penalty approximation of the linear program min ctx ax b we prove that under an appropriate choice of the sequences k e k and with some control on the residual k for every r k 0 the sequence u k converges towards an optimal point u of the linear program we also study the convergence of the associated dual sequence i k exp a i u k b i r k towards a dual optimal solution