unicity in piecewise polynomial l 1 approximation via an algorithm our main result shows that certain generalized convex functions on a real interval possess a unique best l 1 approximation from the family of piecewise polynomial functions of fixed degree with varying knots this result was anticipated by kioustelidis in 11 however the proof given there is nonconstructive and uses topological degree as the primary tool in a fashion similar to the proof the comparable result for the l 2 case in 5 by contrast the proof given here proceeds by demonstrating the global convergence of an algorithm to calculate a best approximation over the domain of all possible knot vectors the proof uses the contraction mapping theorem to simultaneously establish convergence and uniqueness this algorithm was suggested by kioustelidis 10 in addition an asymptotic uniqueness result and a nonuniqueness result are indicated which analogize known results in the l 2 case

algorithms for optimal discontinuous piecewise linear and constant l 2 fits to continuous functions with adjustable nodes in one and two dimensions in this paper a direct variational approach with nonstandard variations is used to generate algorithms to determine optimal discontinuous piecewise linear and piecewise constant l2 fits to a continuous function of one or two variables with adjustable nodes in the one variable case the algorithm is fast and robust the mesh cannot tangle and the resulting fits are continuous a e in the two variable case on an adjustable triangular grid the algorithm is less robust but gives good results for particular functions possessing a single steep feature the extension to higher dimensions is straightforward

comparisons of gaussian elimination algorithms on a cray y mp abstract the results of various implementations of gaussian elimination on full matrices on a single processor cray y mp are presented and discussed it is shown that when the manufacturer supplied blas kernels are used the difference between thebest versions of level 2 blas and level 3 blas blocked implementations is almost negligible it is suggested that to improve the performance of blocked gaussian elimination it is possible to utilize strassen s matrix multiplication algorithm

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

high performance solution of partial differential equations discretized using a chebyshev spectral collocation method when a chebyshev spectral collocation method is applied to a flow problem in a rectangularly decomposable domain it leads to the solution of a structured linear system since the linear system is solved at each step of a newton type iterative process an efficient method to solve it needs to be provided a level 3 blas based algorithm is presented and its efficiency is studied

quadratic wavefront and traveltime approximations in inhomogeneous layered media with curved interfaces a quadratic approximation for the square of the traveltime from a source region to a receiver region is given for a three dimensional 3 d medium consisting of inhomogeneous layers with curved interfaces the square of the traveltime being a function of source and receiver coordinates is developed in a taylor series around a reference source and receiver point the relationships of the traveltime parameters to the ray parameters and the wavefront curvature matrices are shown using midpoint half offset coordinates gives a simplified traveltime function compared to using source receiver coordinates only in the case that the reference source point and the reference receiver point coincide zero offset approximation for a medium consisting of homogeneous layers with plane dipping interfaces the traveltime approximation is further simplified the derived traveltime approximation is shown to be exact for a reflection from a plane dipping interface in a homogeneous medium explicit expressions for the tr

normal moveout revisited inhomogeneous media and curved interfaces the equation of normal moveout t2 t02 x2 va2 is valid for a reflection from the base of a single homogeneous and isotropic bed but is only an approximation in the real world of multilayered inhomogeneous media and curved interfaces using the theory of geometrical optics we can find another second order equation which represents hyperbolas that are also symmetrical about the time axis however the centers of these hyperbolas do not coincide with the center of coordinates but are shifted along the time axis the equation describing this second type of hyperbola is t tp t0 2 tp2 x2 v12 where tp is the time of focusing depth and v1 the velocity of the input medium this equation is not only more accurate than the usual normal moveout but its use is more economical on a vector computer because the traditional dynamic correction is a static correction in the tp analysis this procedure makes it possible to compute velocities for all the samples of all the stacked traces and produces a velocity secti

signal and traveltime parameter estimation using singular value decomposition abstractsignal detection and traveltime parameter estimation can be performed by computing a coherence function in a data window centered around a traveltime function defined by its parameters we used singular value decomposition of the data matrix not eigendecomposition of a covariance matrix to review the most commonly used coherence measures this resulted in a new reduced semblance coefficient defined from the first eigenimage assuming that the signal amplitude was the same on all data channels as in classical semblance in a second signal model the time signal was constant on each channel but the amplitude changed then the semblance coefficient is the square of the first singular value divided by the energy of the data two normalized crosscorrelation coefficients derived from the first eigenimage can also be used as a coherence measure the normalized crosscorrelation of the spatial singular vector with a vector with all elements equal to one and the normalized crosscorrelation of the tem

using strassen s matrix multiplication in high performance solution of linear systems abstract performance characteristics of dense and structured blocked linear system solvers are studied when strassen s matrix multiplication is used in the update step results of experiments on a multiprocessor cray y mp are presented and discussed

the two faces of nmo abstract in many present day applications in seismic processing the assumption of a homogeneous model leads to simple yet powerful approximations which also work well when heterogeneity is not negligible while the classical normal moveout nmo hyperbola assumes an effective constant velocity medium de bazelaire based on optical projections introduced an alternative way to account for heterogeneity by shifting the reference time rather than the velocity although modern multiparameter nmo descriptions usually are based on the same set of parameters we show through reparameterization that they can be divided into the same two types of approximations one assuming an effective medium the other describing the optical analogue in a medium of constant near surface velocity we provide new insights into the auxiliary medium concept and introduce a generalized osculating equation which allows for the forward and inverse transformation between the effective and the optical domain thereby providing a uni

polynomial splines and their tensor products in extended linear modeling 1994 wald memorial lecture analysis of variance type models are considered for a regression function or for the logarithm of a probability function conditional probability function density function conditional density function hazard function conditional hazard function or spectral density function polynomial splines are used to model the main effects and their tensor products are used to model any interaction components that are included in the special context of survival analysis the baseline hazard function is modeled and nonproportionality is allowed in general the theory involves the l 2 rate of convergence for the fitted model and its components the methodology involves least squares and maximum likelihood estimation stepwise addition of basis functions using rao statistics stepwise deletion using wald statistics and model selection using the bayesian information criterion cross validation or an independent test set publicly available software written in c and interfaced to s s plus is used to apply this methodology to real data

density functions for navigation function based systems in this paper we present a scheme for constructing density functions for systems that are almost globally asymptotically stable i e systems for which all trajectories converge to an equilibrium except for a set of measure zero using navigation functions nfs although recently proven converse theorems guarantee the existence of density functions for such systems such results are only existential and the construction of a density function for almost globally asymptotically stable systems remains a challenging task we show that for a specific class of dynamical systems that are defined based on an nf a density function can be easily derived from the system s underlying nf

parallel solution of almost block diagonal systems on the cray y mp using level 3 blas abstract in a recent publication 1992 the authors showed how efficient a new level 3 blas algorithm for almost block diagonal systems could be using just one processor of a cray y mp here they compare the corresponding results for up to eight processors using standard cray library parallel implementations of the level 3 blas

coordination and collision avoidance for lagrangian systems with disturbances in this paper we address the problem of cooperation and collision avoidance for lagrangian systems with input disturbances we design control laws that guarantee cooperation as well as collision free maneuvers we show using a two step proof that the avoidance part of the control laws guarantees safety of the agents independently of the coordinating part then we establish an ultimate bound on the region to which all the agents converge to the obtained theoretical results are illustrated through numerical examples

amplitude fresnel zone and nmo velocity for pp and ss normal incidence reflections inspired by recent ray theoretical developments the theory of normal incidence rays is generalized to accommodate p and s waves in layered isotropic and anisotropic media the calculation of the three main factors contributing to the two way amplitude i e geometric spreading phase shift from caustics and accumulated reflection transmission coefficients is formulated as a recursive process in the upward direction of the normal incidence rays this step by step approach makes it possible to implement zero offset amplitude modeling as an efficient one way wavefront construction process for the purpose of upward dynamic ray tracing the one way eigensolution matrix is introduced having as minors the paraxial ray tracing matrices for the wavefronts of two hypothetical waves referred to by hubral as the normal incidence point nip wave and the normal wave dynamic ray tracing expressed in terms of the one way eigensolution matrix has two advantages the formulas for geometric spreading phase shif

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

fast diffeomorphic matching to learn globally asymptotically stable nonlinear dynamical systems abstract we propose a new diffeomorphic matching algorithm and use it to learn nonlinear dynamical systems with the guarantee that the learned systems have global asymptotic stability for a given set of demonstration trajectories and a reference globally asymptotically stable time invariant system we compute a diffeomorphism that maps forward orbits of the reference system onto the demonstrations the same diffeomorphism deforms the whole reference system into one that reproduces the demonstrations and is still globally asymptotically stable

a new moving mesh algorithm for the finite element solution of variational problems the paper is devoted to the description and application of a new iterative mesh optimization algorithm for the finite element solution of variational problems set in infinite dimensional spaces the optimality criterion is that the mesh should be such that the variational energy functional evaluated at the finite element approximation be minimized such a criterion has a relatively long history in the finite element literature the chief merit of the procedure presented in this paper is that each node of the mesh and the corresponding nodal value of the discrete approximation are updated by solving sequentially local minimization problems with very few degrees of freedom it is shown that this procedure reduces the energy functional monotonically without the need to solve the global discrete problem at intermediate stages applications to partial differential equations are considered numerical results in two dimensions are obtained by incorporating the algorithm into bank s well known pltmg elliptic solver

stable parallel elimination for boundary value odes a parallelizable and vectorizable algorithm for solving linear algebraic systems arising from two point boundary value odes is described the method is equivalent to gaussian elimination with row partial pivoting applied to a certain column reordered version of the usual almost block diagonal coefficient matrix we present analytical and numerical evidence to show that the algorithm is stable in most circumstances results from implementation on a shared memory multiprocessor and a vector processor are given the approach can be extended to handle problems with multipoint and integral conditions or with algebraic parameters

analysis of an algorithm for generating locally optimal meshes for l 2 approximation by discontinuous piecewise polynomials this paper discusses the problem of constructing a locally optimal mesh for the best l 2 approximation of a given function by discontinuous piecewise polynomials in the one dimensional case it is shown that under certain assumptions on the approximated function baines algorithm m j baines math comp 62 1994 pp 645 669 for piecewise linear or piecewise constant polynomials produces a mesh sequence which converges to an optimal mesh the rate of convergence is investigated a two dimensional modification of this algorithm is proposed in which both the nodes and the connection between the nodes are self adjusting numerical results in one and two dimensions are presented

a local convergence proof for the minvar algorithm for computing continuous piecewise linear approximations the class of continuous piecewise linear pl functions represents a useful family of approximants because invertibility can be readily imposed and if a pl function is invertible then it can be inverted in closed form many applications arising for example in control systems and robotics involve the simultaneous construction of a forward and inverse system model from data most approximation techniques require that separate forward and inverse models be trained whereas an invertible continuous pl affords simultaneously the forward and inverse system model in a single representation the minvar algorithm computes a continuous pl approximation to data local convergence of minvar is proven for the case when the data generating function is itself a pl function and available directly rather than through data

passive velocity field control pvfc part ii application to contour following when the contour following task is represented by a velocity field on the configuration manifold of the system the coordination aspect of the problem is made explicit the pvfc scheme developed in the part i ibid vol 29 9 2001 can then be applied to track the defined velocity field however for some contours an encoding velocity held on the configuration manifold does not exist or is difficult to define and as a consequence the pvfc cannot be directly applied for systems whose configuration manifolds are compact lie groups and the desired contour is represented by a parameterized trajectory a general methodology is developed using a suspension technique to define a velocity field on a manifold related to the configuration manifold of the system for which pvfc can be applied with this strategy timing along the contour can be naturally varied online by a self pacing scheme so that the contour tracking performance can be improved the experimental results for a 2 dof robot following a lissajous contour illustrates and verifies the convergence and robustness properties of the pvfc methodology

two point paraxial traveltime formula for inhomogeneous isotropic and anisotropic media tests of accuracy on several simple models of isotropic and anisotropic media we have studied the accuracy of the two point paraxial traveltime formula designed for the approximate calculation of the traveltime between points s 0 and r 0 located in the vicinity of points s and r on a reference ray the reference ray may be situated in a 3d inhomogeneous isotropic or anisotropic medium with or without smooth curved interfaces the twopoint paraxial traveltime formula has the form of the taylor expansion of the two point traveltime with respect to spatial cartesian coordinates up to quadratic terms at points s and r on the reference ray the constant term and the coefficients of the linear and quadratic terms are determined from quantities obtained from ray tracing and linear dynamic ray tracing along the reference ray the use of linear dynamic ray tracing allows the evaluation of the quadratic terms in arbitrarily inhomogeneous media and as shown by examples it extends the region of accurate results around the reference ray between s and r and even outside this interval obtained with the linear terms only although the formula may be used for very general 3d models we concentrated on simple 2d models of smoothly inhomogeneous isotropic and anisotropic 8 and 20 anisotropy media only on tests in which we estimated twopoint traveltimes between a shifted source and a system of shifted receivers we found that the formula may yield more accurate results than the numerical solution of an eikonal based differential equation the tests also indicated that the accuracy of the formula depends primarily on the length and the curvature of the reference ray and only weakly depends on anisotropy the greater is the curvature of the reference ray the narrower its vicinity in which the formula yields accurate results

kirchhoff migration using eikonal based computation of traveltime source derivatives abstractthe computational efficiency of kirchhoff type migration can be enhanced by using accurate traveltime interpolation algorithms we addressed the problem of interpolating between a sparse source sampling by using the derivative of traveltime with respect to the source location we adopted a first order partial differential equation that originates from differentiating the eikonal equation to compute the traveltime source derivatives efficiently and conveniently unlike methods that rely on finite difference estimations the accuracy of the eikonal based derivative did not depend on input source sampling for smooth velocity models the first order traveltime source derivatives enabled a cubic hermite traveltime interpolation that took into consideration the curvatures of local wavefronts and can be straightforwardly incorporated into kirchhoff antialiasing schemes we provided an implementation of the proposed method to first arrival traveltimes by modifying the fast marching eikonal solver severa