exact ltp representation of the generalized periodic reference fxlms algorithm the analysis of the filtered x least mean square fxlms algorithm can be based either on a stochastic or deterministic approach one of the drawbacks of the stochastic analysis is that it relies on several assumptions which are not justified if the reference signal is time periodic to overcome these limitations several deterministic methods have been suggested however useful results have been reported only for setups with certain filter length secondary path model or reference signal in this paper an exact linear time periodic ltp representation is proposed assuming only that the reference is synchronously sampled the representation is derived for a general fxlms system which also covers the multichannel topology of several parallel filters and its common narrowband modification the representation is found to have a linear time invariant lti form if the filter length is suitably chosen the usability of the method is demonstrated in a numerically computed example by comparing it to an approximate lti model

on the finite element method for singularly perturbed reaction diffusion problems in two and one dimensions second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered the behavior of the ordinary galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed

maximum norm estimates for resolvents of elliptic finite element operators let be a convex domain with smooth boundary in rd it has been shown recently that the semigroup generated by the discrete laplacian for quasi uniform families of piecewise linear finite element spaces on is analytic with respect to the maximum norm uniformly in the mesh width this implies a resolvent estimate of standard form in the maximum norm outside some sector in the right halfplane and conversely here we show directly that such a resolvent estimate holds outside any sector around the positive real axis with arbitrarily small angle this is useful in the study of fully discrete approximations based on a stable rational functions with small

shaping regularization in geophysical estimation problems regularization is a required component of geophysical estimation problems that operate with insufficient data the goal of regularization is to impose additional constraints on the estimated model i introduce shaping regularization a general method for imposing constraints by explicit mapping of the estimated model to the space of admissible models shaping regularization is integrated in a conjugate gradient algorithm for iterative least squares estimation it provides the advantage of better control on the estimated model in comparison with traditional regularization methods and in some cases leads to a faster iterative convergence simple data interpolation and seismic velocity estimation examples illustrate the concept

high order local rate of convergence by mesh refinement in the finite element method we seek approximations of the solution u of the neumann problem for the equation lu f in q with special emphasis on high order accuracy at a given point xo e f2 here q is a bounded domain in rn n 2 with smooth boundary and l is a second order uniformly elliptic differential operator with smooth coefficients an approximate solution uh is determined by the standard galerkin method in a space of continuous piecewise polynomials of degree at most r 1 on a partition ah xo a of u here h is a global mesh size parameter and a is the degree of a certain systematic refinement of the mesh around the given point x0 where larger a s mean finer mesh and a 0 corresponds to the quasi uniform case with no refinement it is proved that for suitable sufficiently large a s the high order error estimate u uh x0 o h2 2 holds a corresponding estimate with the same order of convergence is obtained for the first order derivatives of u uh these estimates are sharp in the sense that the required degree of refinement in each case is essentially the same as is needed for the local approximation to this order near xo for the estimates to hold it is sufficient that the exact solution u have derivatives to the rth order which are bounded close to xo and square integrable in the rest of u the proof of this uses high order negative norm estimates of u uh the number of elements in the considered partitions is of the same order as in the corresponding quasi uniform ones applications of the results to other types of boundary value problems are indicated 0 introduction let i2 be a bounded domain in rn n 2 with smooth boundary aq and consider the neumann problem to find u such that n a au n au 0 1 lu ax aija ea3 au f inn au n au 0 2 an i a a nj 0 on3ai2 a i j1 aj xi here l is assumed to be uniformly elliptic with smooth coefficients n nj and n denote the exterior normal and conormal to aq respectively assume also that the bilinear form n3v w n a v w e a a e a a w avw dx received september 22 1983 revised july 23 1984 1980 mathematics subject classification primary 65n30

pointwise error estimates of linear finite element method for neumann boundary value problems in a smooth domain pointwise error analysis of the linear finite element approximation for delta u u f in omega partial n u tau on partial omega where omega is a bounded smooth domain in mathbb r n is presented we establish o h 2 log h and o h error bounds in the l infty and w 1 infty norms respectively by adopting the technique of regularized green s functions combined with local h 1 and l 2 estimates in dyadic annuli since the computational domain omega h is only polyhedral one has to take into account non conformity of the approximation caused by the discrepancy omega h ne omega in particular the so called galerkin orthogonality relation utilized three times in the proof does not exactly hold and involves domain perturbation terms or boundary skin terms which need to be addressed carefully a numerical example is provided to confirm the theoretical result

uniform error estimates in the finite element method for a singularly perturbed reaction diffusion problem consider the problem e 2 u u f with homogeneous neumann boundary condition in a bounded smooth domain in r n the whole range 0 e 1 is treated the galerkin finite element method is used on a globally quasi uniform mesh of size h the mesh is fixed and independent of e a precise analysis of how the error at each point depends on h and e is is presented as an application first order error estimates in h which are uniform with respect to e are given

analysis of a sign algorithm with delayed data an adaptive filter whose weights are adapted using a sign algorithm with a delayed error signal is analyzed for stationary environments it is proved that the excess average absolute estimation error is bounded for all values of the error signal delay and the algorithm step size for the nonstationary case when the optimal filter weights are time varying the optimum step size which minimizes the excess average absolute error is derived it is shown that the optimum step size does not depend on the additive noise power the analytical results are supported by computer simulations

shape calculus and finite element method in smooth domains the use of finite elements in smooth domains leads naturally to polyhedral or piecewise polynomial approximations of the boundary hence the approximation error consists of two parts the geometric part and the finite element part we propose to exploit this decomposition in the error analysis by introducing an auxiliary problem defined in a polygonal domain approximating the original smooth domain the finite element part of the error can be treated in the standard way to estimate the geometric part of the error we need quantitative estimates related to perturbation of the geometry we derive such estimates using the techniques developed for shape sensitivity analysis

adaptive prediction filtering in t x y domain for random noise attenuation using regularized nonstationary autoregression abstractmany natural phenomena including geologic events and geophysical data are fundamentally nonstationary they may exhibit stationarity on a short timescale but eventually alter their behavior in time and space we developed a 2d t x adaptive prediction filter apf and further extended this to a 3d t x y version for random noise attenuation based on regularized nonstationary autoregression rna instead of patching a popular method for handling nonstationarity we obtained smoothly nonstationary apf coefficients by solving a global regularized least squares problem we used shaping regularization to control the smoothness of the coefficients of apf three dimensional space noncausal t x y apf uses neighboring traces around the target traces in the 3d seismic cube to predict noise free signal so it provided more accurate prediction results than the 2d version in comparison with other denoising methods such as frequency space deconvolution time space prediction filter and frequency space rna

an adaptive newton algorithm based on numerical inversion regularization as postconditioner we examine a class of approximate inversion processes satisfying estimates similar to those defined by finite element or truncated spectral approximations these are to be used as approximate right inverses for newton iteration methods when viewed at the operator level these approximations introduce a defect or loss of derivatives of order one or more regularization is introduced as a form of defect correction a superlinearly convergent approximate newton iteration is thereby obtained by using the numerical inversion adaptively i e with spectral or grid parameters correlated to the magnitude of the current residual in an intermediate norm defined by the defect this adaptive choice makes possible ascribing an order to the convergent process and this is identified as essentially optimal for elliptic problems relative to complexity the design of the algorithm involves multi parameter selection thereby opening up interesting avenues for elliptic problems relative to complexity this applies also to the regularization which may be carried out in the fourier transform space and is band limited in the language of whittaker shannon sampling theory the norms employed in the analysis are of holder space type the iteration is an adaptation of nash moser interation and the complexity studies use vituskin s theory of information processing computational experience is described in the final section

damped multichannel singular spectrum analysis for 3d random noise attenuation abstractmultichannel singular spectrum analysis mssa is an effective algorithm for random noise attenuation in seismic data which decomposes the vector space of the hankel matrix of the noisy signal into a signal subspace and a noise subspace by truncated singular value decomposition tsvd however this signal subspace actually still contains residual noise we have derived a new formula of low rank reduction which is more powerful in distinguishing between signal and noise compared with the traditional tsvd by introducing a damping factor into traditional mssa to dampen the singular values we have developed a new algorithm for random noise attenuation we have named our modified mssa as damped mssa the denoising performance is controlled by the damping factor and our approach reverts to the traditional mssa approach when the damping factor is sufficiently large application of the damped mssa algorithm on synthetic and field seismic data demonstrates superior performance compared with the conve

leaky delayed lms algorithm stochastic analysis for gaussian data and delay modeling error this paper presents a stochastic analysis of the delayed least mean square dlms adaptive algorithm with leakage this analysis is obtained taking into account that mismatches between the system delay and its estimate may occur such an approach is not considered in previous models in addition it is shown that the introduction of a leakage factor into the adaptive algorithm keeps the adaptive algorithm stable under an imperfect delay estimate condition recursive difference equations for the first and second moments of the adaptive filter weights are derived an expression for the critical value of the step size is also determined results of monte carlo simulations present excellent agreement with the proposed model for both white and colored gaussian inputs

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

the lms algorithm with delayed coefficient adaptation the behavior of the delayed least mean square dlms algorithm is studied it is found that the step size in the coefficient update plays a key role in the convergence and stability of the algorithm an upper bound for the step size is derived that ensures the stability of the dlms the relationship between the step size and the convergence speed and the effect of the delay on the convergence speed are also studied the analytical results are supported by computer simulations

random noise attenuation using f x regularized nonstationary autoregression we have developed a novel method for random noise attenuation in seismic data by applying regularized nonstationary autoregression rna in the frequency space f x domain the method adaptively predicts the signal with spatial changes in dip or amplitude using f x rna the key idea is to overcome the assumption of linearity and stationarity of the signal in conventional f x domain prediction technique the conventional f x domain prediction technique uses short temporal and spatial analysis windows to cope with the nonstationary of the seismic data the new method does not require windowing strategies in spatial direction we implement the algorithm by an iterated scheme using the conjugate gradient method we constrain the coefficients of nonstationary autoregression na to be smooth along space and frequency in the f x domain the shaping regularization in least square inversion controls the smoothness of the coefficients of f x rna there are two key parameters in the proposed method filter length

analysis of lms and nlms algorithms with delayed coefficient update under the presence of spherically invariant processes certain conditions require a delay in the coefficient update of the least mean square lms and normalized least mean square nlms algorithms this paper presents an in depth analysis of these modificated versions for the important case of spherically invariant random processes sirps which are known as an excellent model for speech signals some derived bounds and the predicted dynamic behavior of the algorithms are found to correspond very well to simulation results and a real time implementation on a fixed point signal processor a modification of the algorithm is proposed to assure the well known properties of the lms and nlms algorithms

error estimates for the multidimensional two phase stefan problem in this paper we derive rates of convergence for regularizations of the multidimen sional two phase stefan problem and use the regularized problems to define backward dif ference in time and c piecewise linear in space galerkin approximations we find an l2 rate of convergence of order in the e regularization and an l rate of convergence of order h2 e ai 6f in the galerkin estimates which leads to the natural choices a4 3 a a4 3 and a resulting 0 i2 3 l2 rate of convergence of the numerical scheme to the solution of the differential equation an essentially 0 h rate is demonstrated when e 0 and a h2 in our galerkin scheme under a boundedness hypothesis on the galerkin approxima tions the latter result is consistent with computational experience 1 introduction given a smoothly bounded domain s c rn we consider the equation in distribution form l h au fiu 0

sharpl error estimates for semilinear elliptic problems with free boundaries a numerical scheme to approximate a semilinear pde involving a singular maximal monotone graph is analyzed inl a preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute the discrete maximum principle is thus avoided sharp pointwise error estimates are derived for both the smoothing and the discretization procedures an optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence these error estimates for solutions in conjunction with nondegeneracy properties of continuous problems provide sharp interface error estimates two model examples are discussed the obstacle problem and a combustion equation

cosine methods for nonlinear second order hyperbolic equations construction et analyse des methodes efficaces avec une precision d ordre eleve pour l approximation des solutions regulieres d une classe d equations hyperboliques non lineaires du second ordre application des discretisations de type galerkin et des schemas a deux pas estimations d erreurs d ordre optimal

random noise attenuation using local signal and noise orthogonalization abstractwe have developed a novel approach to attenuate random noise based on local signal and noise orthogonalization in this approach we first removed from a seismic section using one of the conventional denoising operators and then applied a weighting operator to the initially denoised section to predict the signal leakage energy as well as retrieve it from the initial noise section the weighting operator was obtained by solving a least squares minimization problem via shaping regularization with a smoothness constraint next the initially denoised section and the retrieved signal were combined to form the final denoised section the proposed denoising approach corresponded to orthogonalizing the initially denoised signal and noise in a local manner we evaluated the denoising performance using local similarity to test the orthogonalization property of the estimated signal and noise we calculated the local similarity map between the denoised signal section and removed noise section low values o

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

an embedding approach to frequency domain and subband adaptive filtering frequency domain and subband implementations improve the computational efficiency and the convergence rate of adaptive schemes the well known multidelay adaptive filter mdf belongs to this class of block adaptive structures and is a dft based algorithm we develop adaptive structures that are based on the trigonometric transforms discrete cosine transform dct and discrete sine transform dst and on the discrete hartley transform dht as a result these structures involve only real arithmetic and are attractive alternatives in cases where the traditional dft based scheme exhibits poor performance the filters are derived by first presenting a derivation for the classical dft based filter that allows us to pursue these extensions immediately the approach used in this paper also provides further insights into subband adaptive filtering

postprocessing the linear finite element method we extend the idea of the postprocessing galerkin methods for dissipative evolution equations to the case of the linear finite element method the postprocessing technique has been developed earlier for spectral methods and for higher order finite element methods the analysis shows that this procedure improves the order of convergence of the piecewise linear galerkin finite element approximation in the h1 norm we show by means of numerical experiments that there is no improvement in the order of convergence in the l2 norm

high order sparse radon transform for avo preserving data reconstruction abstractthe sparse radon transform rt represents seismic data by the superposition of a few constant amplitude events and thus it has trouble dealing with amplitude versus offset avo variations we integrated the gradient and curvature parameters of avo into the rt with these additional properties the lateral continuity of the events amplitude was modeled in the transformation and it could be fitted with orthogonal polynomials this resulted in a higher order rt which included avo terms the high order rt is a highly underdetermined problem which was solved by extracting the major model parameters from energy distribution in a high order radon domain and by decreasing the number of inversion parameters thus a high order sparse rt was achieved the proposed method can be used for data interpolation as well as extrapolation the avo preservation performance of the proposed algorithm in data reconstruction was illustrated using both synthetic and field data examples and the results showed the fe

convergence of the delayed normalized lms algorithm with decreasing step size in several practical applications of the lms algorithm including certain vlsi implementations the coefficient adaptation can be performed only after some fixed delay the resulting algorithm is known as the delayed lms dlms algorithm in the literature previous published analyses of this algorithm are based on mean and moment convergence under the independence assumption between successive input vectors these analyses are interesting and give valuable insights into the convergence properties but from a practical viewpoint they do not guarantee the correct performance of the particular realization with which the user must live we consider a normalized version of this algorithm with a decreasing step size spl mu n and prove the almost sure convergence of the nonhomogeneous algorithm assuming a mixing input condition and the satisfaction of a certain law of large numbers

effect of delay on the performance of the leaky lms adaptive algorithm this correspondence studies the effect of a nonzero delay on the performance of the leaky lms algorithm the stability bound on the step size is derived for error convergence it is shown that leakage allows for larger bounds on the step size of the delayed lms algorithm theoretical bounds are verified by simulations

a new numerical method for backward parabolic problems in the maximum norm setting a new method for solving numerically backward parabolic problems is proposed as usual for this kind of ill posed problems it is assumed that an a priori bound for the solution is available the algorithm consists of two basic steps first a standard forward integration is performed in order to approximate the solution at suitable future time levels second a holomorphic recovery procedure is carried out providing the required approximations for the preceding times the analysis is valid in the maximum norm setting and rigorous estimates are derived among other advantages the method can also be applied to nonlinear problems and it produces a continuous output some numerical illustrations are presented

l error estimate for an approximation of a parabolic variational inequality almost optimall convergence of an approximation of a variational inequality of parabolic type is proved under regularity assumptions which are met by the solution of a one phase stefan problem the discretization employs piecewise linear finite elements in space and the backward euler scheme in time by means of a maximum principle the problem is reduced to an error estimate for an auxiliary parabolic equation the latter bound is obtained by using the smoothing property of the galerkin method

lateral prediction for noise attenuation by t x and f x techniques attenuating random noise with a prediction filter in the time space domain generally produces results similar to those of predictions done in the frequency space domain however in the presence of moderate to high amplitude noise time space or t x prediction passes less random noise than does frequency space or f x prediction the f x prediction may also produce false events in the presence of parallel events where t x prediction does not these advantages of t x prediction are the result of its ability to control the length of the prediction filter in time an f x prediction produces an effective t x domain filter that is as long in time as the input data gulunay s f x domain prediction tends to bias the predictions toward the traces nearest the output trace allowing somewhat more noise to be passed but this bias may be overcome by modifying the system of equations used to calculate the filter the 3 d extension to the 2 d t x and f x prediction techniques allows improved noise attenuation because more samples are used in the predictions and the requirement that events be strictly linear is relaxed

f x adaptive seismic trace interpolation we use exponentially weighted recursive least squares to estimate adaptive prediction filters for frequency space f x seismic interpolation adaptive prediction filters can model signals where the dominant wavenumbers vary in space this concept leads to an f x interpolation method that does not require windowing strategies for optimal results in other words adaptive prediction filters can be used to interpolate waveforms that have spatially variant dips the interpolation method s performance depends on two parameters filter length and forgetting factor we pay particular attention to selection of the forgetting factor because it controls the algorithm s adaptability to changes in local dip finally we use synthetic and real data examples to illustrate the performance of the proposed adaptive f x interpolation method

seismic trace interpolation in the f x domain interpolation of seismic traces is an effective means of improving migration when the data set exhibits spatial aliasing a major difficulty of standard interpolation methods is that they depend on the degree of reliability with which the various geological events can be separated in this respect a multichannel interpolation method is described which requires neither a priori knowledge of the directions of lateral coherence of the events nor estimation of these directions the method is based on the fact that linear events present in a section made of equally spaced traces may be interpolated exactly regardless of the original spatial interval without any attempt to determine their true dips the predictability of linear events in the f x domain allows the missing traces to be expressed as the output of a linear system the input of which consists of the recorded traces the interpolation operator is obtained by solving a set of linear equations whose coefficients depend only on the spectrum of the spatial prediction filter defined by the recorded traces synthetic examples show that this method is insensitive to random noise and that it correctly handles curvatures and lateral amplitude variations assessment of the method with a real data set shows that the interpolation yields an improved migrated section

improving the rate of convergence of high order finite elements on polygons and domains with cusps let u and uv v be the solution and respectively the discrete solution of the non homogeneous dirichlet problem u f on u 0 for any m and any bounded polygonal domain we provide a construction of a new sequence of finite dimensional subspaces vn such that where f hm 1 is arbitrary and c is a constant that depends only on and not on n we do not assume u hm 1 the existence of such a sequence of subspaces was first proved in a ground breaking paper by babuska 8 our method is different it is based on the homogeneity properties of sobolev spaces with weights and the well posedness of non homogeneous dirichlet problem in suitable sobolev spaces with weights for which we provide a new proof and which is a substitute of the usual shift theorems for boundary value problems in domains with smooth boundary our results extended right away to domains whose boundaries have conical points we also indicate some of the changes necessary to deal with domains with cusps our numerical computation are in agreement with our theoretical results

error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data on considere les approximations spatialement discretes continues par rapport au temps des problemes mixtes par la methode des elements finis de galerkin pour les equations paraboliques semi lineaires avec des donnees initiales irregulieres ou incompatibles on etablit des estimations d erreur

optimal l error estimates for piecewise linear finite elements in 1 n 3 for 1 n 3 in this paper for two second order elliptic boundary value problems uniform convergence error estimates are derived using piecewise linear conforming and non conforming finite elements the estimates are valid for polyhedral convex regions rn with 1 n 3 the proof of the estimates is mainly based on sobolev s inequality by which the uniform estimates are reduced to estimates in the mean in this way it is possible to obtain the desired uniformo h2 estimate in the plane as a special case thus confirming theoretically natterer s numerical results

quasi optimal estimates for finite element approximations using orlicz norms on considere l approximation par des elements finis lineaires de la solution du probleme de dirichlet u f on etablit une relation entre l erreur dans la norme infinie et l erreur dans quelques espaces d orlicz on en deduit des estimations uniformes quasi optimales lorsque la solution ne possede des derivees secondes dans l espace d orlicz associe a la fonction exponentielle

optimal convergence for the finite element method in campanato spaces we prove a priori estimates and optimal error estimates for linear finite element approximations of elliptic systems in divergence form with continuous coefficients in campanato spaces the proofs rely on discrete analogues of the campanato inequalities for the solution of the system which locally measure the decay of the energy as an application of our results we derive w 1 p estimates and give a new proof of the well known w 1 results of rannacher and scott

avoiding the order reduction of runge kutta methods for linear initial boundary value problems a new strategy to avoid the order reduction of runge kutta methods when integrating linear autonomous nonhomogeneous initial boundary value problems is presented the solution is decomposed into two parts one of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction a numerical illustration is given this idea applies to practical problems where spatial discretization is also required leading to the full order both in space and time

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

on the discretization in time of semilinear parabolic equations with nonsmooth initial data single step discretization methods are considered for equations of the form u au f t u where a is a linear positive definite operator in a hubert space h it is shown that if the method is consistent with the differential equation then the convergence is essentially of first order in the stepsize even if the initial data v are only in h but also that in contrast to the situation in the linear homogeneous case higher order convergence is not possible in general without further assumptions on v

a family of low complexity blind equalizers two important topics in equalizer design are its complexity and its training we present a family of blind equalizers which by incorporating a decomposition finite impulse response filtering technique can reduce the complexity of the convolution operation therein by about one half the prototype algorithm in this equalizer family employs the prevalent godard cost function several simplified algorithms are proposed including a sign algorithm which eliminates multiplications in coefficient adaptation and a few delayed versions we also study the convergence properties of the algorithms for the prototype algorithm we show that in the limit of an infinitely long equalizer and under mild conditions on signal constellations and channel characteristics there are only two sets of local minima on the performance surface one of the sets is undesirable and is characterized by a null equalized channel response the other corresponds to perfect equalization which can be reached with proper equalizer initialization for the simplified algorithms corresponding cost functions may not exist some understanding of their convergence behaviors are obtained via examination of their adaptation equations simulation results are presented to demonstrate the performance of the algorithms

adaptive multiple subtraction using regularized nonstationary regression stationary regression is the backbone of seismic data processing algorithms including match filtering which is commonly applied for adaptive multiple subtraction however the assumption of stationarity is not always adequate for describing seismic signals i have developed a general method of nonstationary regression and that applies to nonstationary match filtering the key idea is the use of shaping regularization to constrain the variability of nonstationary regression coefficients simple computational experiments demonstrate advantages of shaping regularization over classic tikhonov s regularization including a more intuitive selection of parameters and a faster iterative convergence using benchmark synthetic data examples i have successfully applied this method to the problem of adaptive subtraction of multiple reflections

applications of discrete maximal l p regularity for finite element operators in this paper we present applications of discrete maximal lp regularity for finite element operators more precisely we show error estimates of order h2 for linear and certain semilinear problems in various lp norms discrete maximal regularity allows us to prove error estimates in a very easy and efficient way moreover we also develop interpolation theory for fractional powers of finite element operators and extend the results on discrete maximal lp regularity formerly proved by the author

stability and accuracy of adapted finite element methods for singularly perturbed problems the stability and accuracy of a standard finite element method fem and a new streamline diffusion finite element method sdfem are studied in this paper for a one dimensional singularly perturbed connvection diffusion problem discretized on arbitrary grids both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity often in the form of boundary layers of the solution surprisingly the accuracy of the standard fem is shown to depend crucially on the uniformity of the grid away from the singularity in other words the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but in contrast it is robust with respect to perturbation of properly adapted grid inside the boundary layer motivated by this discovery a new sdfem is developed based on a special choice of the stabilization bubble function the new method is shown to have an optimal maximum norm stability and approximation property in the sense that u u n infty leq c inf v n in v n u v n infty where un is the sdfem approximation in linear finite element space vn of the exact solution u finally several optimal convergence results for the standard fem and the new sdfem are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered

a systolic array realization of an lms adaptive filter and the effects of delayed adaptation a systolic array design for an adaptive filter is presented the filter is based on the least mean square algorithm but due to the problems in implementation of the systolic array a modified algorithm a special case of the delayed lms dlms is used the dlms algorithm introduces a delay in the updating of the filter coefficients the convergence and steady state behavior of the systolic array are analyzed it is shown that the performance of the systolic array is similar to that of a conventional lms implementation for a wide range of practical conditions

seismic data decomposition into spectral components using regularized nonstationary autoregression summary seismic data can be decomposed into nonstationary spectral components with smoothly variable frequencies and smoothly variable amplitudes to estimate local frequencies i use a nonstationary version of prony s spectral analysis method defined with the help of regularized nonstationary autoregression rnar to estimate local amplitudes of different components i fit their sum to the data using regularized nonstationary regression rnr shaping regularization ensures stability of the estimation process and provides controls on smoothness potential applications of the proposed technique include noise attenuation seismic data compression and seismic data regularization

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates this part contains new interior pointwise error estimates for the finite element method for second order elliptic problems in mathbb r n global estimates were considered in part i in the sense to be discussed below these sharpen known interior quasi optimal l infty and w 1 infty estimates in that they indicate a more local dependence of the error at a point on the derivatives of the solution near the point the higher the order of the finite element the more local the behavior of the finite element approximation as a consequence of these estimates new types of local error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in mathbb r n and have applications to superconvergence extrapolation and a posteriori estimates for both smooth and nonsmooth problems

antileakage least squares spectral analysis for seismic data regularization and random noise attenuation spatial transformation of irregularly sampled data series to regularly sampled data series is a challenging problem in many areas such as seismology the classical fourier analysis is limited to regularly sampled data series on the other hand the least squares spectral analysis lssa can analyze irregularly sampled data series although the lssa method takes into account the correlation among the sinusoidal basis functions of irregularly spaced series it still suffers from the problem of spectral leakage energy leaks from one spectral peak into another we propose an iterative method called antileakage lssa to attenuate the spectral leakage and consequently regularize irregular data series in this method we first search for a spectral peak with the highest energy and then we remove suppress it from the original data series in the next step we search for a new peak with the highest energy in the residual data series and remove the new and the old components simultaneously from the original data

the behavior of lms and nlms algorithms in the presence of spherically invariant processes the behavior of least mean square lms and normalized least mean square nlms algorithms with spherically invariant random processes sirps as excitations is shown many random processes fall into this category and sirps closely resemble speech signals the most pertinent properties of these random processes are summarized the lms algorithm is introduced and the first and second order moments of the weight error vector between the wiener solution and the estimated solution are shown the behavior of the nlms algorithm is obtained and the first and second order moments of the weight error vector are calculated the results are verified by comparison with known results when a white noise process and a colored gaussian process are used as input sequences some simulation results for a k sub 0 process are then shown

time frequency analysis of seismic data using local attributes time frequency analysis is an important technology in seismic data processing and interpretation to localize frequency content in time we have developed a novel method for computing a time frequency map for nonstationary signals using an iterative inversion framework we calculated time varying fourier coefficients by solving a least squares problem that uses regularized nonstationary regression we defined the time frequency map as the norm of time varying coefficients time varying average frequency of the seismic data can also be estimated from the time frequency map calculated by our method we tested the method on benchmark synthetic signals and compared it with the well known s transform two field data examples showed applications of the proposed method for delineation of sand channels and for detection of low frequency anomalies

holder estimates for green s functions on convex polyhedral domains and their applications to finite element methods a model second order elliptic equation on a general convex polyhedral domain in three dimensions is considered the aim of this paper is twofold first sharp holder estimates for the corresponding green s function are obtained as an applications of these estimates to finite element methods we show the best approximation property of the error in w 1 infty in contrast to previously known results w p 2 regularity for p 3 which does not hold for general convex polyhedral domains is not required furthermore the new green s function estimates allow us to obtain localized error estimates at a point

self whitening algorithms for adaptive equalization and deconvolution in equalization and deconvolution tasks the correlated nature of the input signal slows the convergence speeds of stochastic gradient adaptive filters prewhitening techniques have been proposed to improve the convergence performance but the additional coefficient memory and updates for the prewhitening filter can be prohibitive in some applications we present two simple algorithms that employ the equalizer as a prewhitening filter within the gradient updates these self whitening algorithms provide quasi newton convergence locally about the optimum coefficient solution for deconvolution and equalization tasks multichannel extensions of the techniques are also described

the stability of adaptive minimum mean square error equalizers using delayed adjustment the necessary and sufficient conditions for the stability and convergence of adaptive minimum mean square error equalizers are developed the analysis includes the effect of delays in the adjustment path as occur for instance when the decoder in a data transmission system delays its output decisions

finite precision error analysis of qrd rls and star rls adaptive filters the qr decomposition based recursive least squares rls adaptive filtering qrd rls algorithm is suitable for vlsi implementation since it has good numerical properties and can be mapped onto a systolic array a new fine grain pipelinable star rls algorithm was developed the pipelined star rls algorithm pstar rls is useful for high speed applications the stability of qrd rls star rls and pstar rls has been proved but the performance of these algorithms in finite precision arithmetic has not yet been analyzed the authors determine expressions for the degradation in the performance of these algorithms due to finite precision by exploiting the steady state properties of these algorithms simple expressions are obtained that depend only on known parameters this analysis can be used to compare the algorithms and to decide the wordlength to be used in an implementation since floating or fixed point arithmetic representations may be used in practice both representations are considered the results show that the three algorithms have about the same finite precision performance with pstar rls performing better than star rls which does better than qrd rls these algorithms can be implemented with as few as 8 bits for the fractional part depending on the filter size and the forgetting factor used the theoretical expressions are found to be in good agreement with the simulation results

linear finite elements may be only first order pointwise accurate on anisotropic triangulations we give a counterexample of an anisotropic triangulation on which the exact solution has a second order error of linear interpolation while the computed solution obtained using linear finite elements is only first order pointwise accurate our example is given in the context of a singularly perturbed reaction diffusion equation whose exact solution exhibits a sharp boundary layer furthermore we give a theoretical justification of the observed numerical phenomena using a finite difference representation of the considered finite element methods both standard and lumped mass cases are addressed

model based interferometric interpolation method we have proposed a two stage model based interferometric interpolation method for filling in gaps in marine seismic data the first stage is creating virtual traces and the second stage is utilizing modified virtual traces for interpolation there are three steps for creating a virtual trace first set the virtual trace s source and receiver coordinates and retrieve two common receiver gathers crg from existing data at least one crg should be obtained otherwise our method cannot be performed second if only one crg is obtained then a synthetic crg should be generated using known velocity model e g water layer model third the virtual trace is created by summing correlated trace pairs over all identical source positions in two crgs when all virtual traces are created a least squares matching filter or a nonstationary prediction error filter pef should be used together with virtual traces to do the interpolation if the trace gap is small or the subsurface geology is simple the local match

fast adaptive algorithms for ar parameters estimation using higher order statistics time varying statistics in linear filtering and linear estimation problems necessitate the use of adaptive or time varying filters in the solution with the rapid availability of vast and inexpensive computation power models which are non gaussian even nonstationary are being investigated at increasing intensity statistical tools used in such investigations usually involve higher order statistics hos the classical instrumental variable iv principle has been widely used to develop adaptive algorithms for the estimation of arma processes despite the great number of iv methods developed in the literature the cumulant based procedures for pure autoregressive ar processes are almost nonexistent except lattice versions of iv algorithms this paper deals with the derivation and the properties of fast transversal algorithms hence by establishing a relationship between classical iv methods and cumulant based ar estimation problems new fast adaptive algorithms fast transversal recursive instrumental variable ftriv and generalized least mean squares glms are proposed for the estimation of ar processes the algorithms are seen to have better performance in terms of convergence speed and misadjustment even in low snr the extra computational complexity is negligible the performance of the algorithms as well as some illustrative tracking comparisons with the existing adaptive ones in the literature are verified via simulations the conditions of convergence are investigated for the glms

a control theory approach to the stability and transient analysis of the filtered x lms adaptive notch filter a method is presented for analyzing the stability and transient response of the filtered x lms adaptive notch filter by formulating the problem in the complex weight domain and applying standard control theory examples are given for pure delay and second order low pass cancellation path transfer functions the method is also extended to the multichannel case where the eigenvalues of the equivalent open loop transfer function matrix characterize the behavior