Elwood T. Olsen

L/sub 1/ and L/sub infinity / minimization via a variant of Karmarkar's algorithm

Simple iterative algorithms are presented for L/sub 1/ and L/sub infinity / minimization (regression) based on a variant of Karmarkars linear programming algorithm. Although these algorithms are based on entirely different theoretical principles to the popular IRLS (iteratively reweighted least squares) algorithm, they have almost identical matrix operations. Also presented are the results of a Monte Carlo study comparing the numerical convergence properties of the Karmarkar algorithm for L/sub 1/ minimization to those of an IRLS and a simplex algorithm. The test problem involves L/sub 1/ estimation of AR (autoregressive) model parameters. The Karmarkar algorithm outperformed IRLS by achieving higher numerical accuracy in fewer iterations. Techniques for reducing the computational cost per iteration of the Karmarkar L/sub 1/ algorithm are discussed. >

Steady flows of viscoelastic fluids in axisymmetric abrupt contraction geometry: A comparison of numerical results

Recently two groups of researchers have reported numerical results simulating the steady flow of a KBKZ fluid in torsion free axisymmetric abrupt contraction geometry. The fluid model used in both cases was a constitutive equation chosen to match laboratory behavior of LDPE. In both cases, quadratic finite elements were used. Large corner vortices were observed in the simulations, similar to those observed in laboratory experiments. The agreement between the results in the two papers is good. We repeat the experiment, using linear finite elements. Tracking is performed via an artificial time method, and a novel ‘‘reduced velocity’’ variable is used in our finite element simulation. There is good qualitative and quantitative agreement between the results reported here and the results previously reported by others. Quantitative measures used in the comparison—vortex opening angle, Couette correction, and vortex intensity—are analyzed.

Norms for copulas

We consider several norms on the span of the set C of all copulas. Dominance and equivalence relationships among the norms are discussed, and completeness issues are addressed. The motivation for the study is discussed. Applications to the study of one parameter semigroups of copulas are also addressed.

Obtaining error estimates for optimally constrained incompressible finite elements

Abstract Elements which were predicted to be optimal for incompressible media based on their constraint counts often fail to satisfy the discrete LBB condition. Error estimates for finite element solutions to incompressible Stokesian flows are considered. The estimate for the velocities does not require the satisfaction of the discrete LBB condition. Conditions for convergence of the raw pressures in a negative norm are established. A semi-norm inf-sup condition can be used to establish more usual error estimates for pressures projected into an auxiliary trial space. Numerical evidence suggests that such post-processing may only be necessary for problems with sufficiently rough exact solutions. An example is given of pressure measurements in flows over a transverse slot. Small—but numerically significant—pressure differences are extracted from a pressure field with seemingly chaotic checkerboarding. Similar examples indicate that the Stokesian results carry over to problems with convective and non-Newtonian nonlinearities.

Classical mechanics and entropy

This note addresses a problem of nineteenth century applied mathematics—is it possible in the context of Hamiltonian mechanics to define a functionS of the generalized coordinates and momenta which is monotonically increasing along orbits? The question is of interest, because, for a sytem not in thermodynamic equilibrium, entropy should increase strictly monotonically along an orbit, and a negative answer implies that mechanical principles different from those of Hamiltonian mechanics must be introduced to explain thermodynamics. This note answers the question rigorously for Hamiltonian systems confined to an invariant region of finite volume in phase space; it is not possible to define a continuous function which increases monotonically along orbits. An appendix gives a translation of an 1889 paper of Poincaré addressing the same issue.

Strong consistency of the LAD (L/sub 1/) estimator of parameters of stationary autoregressive processes with zero mean

Strong consistency (almost sure convergence to the true parameters) of the LAD (least absolute deviations) AR (autoregressive) parameter estimator has been proven by S. Gross and W.L. Steiger (1979) under the condition that i.i.d. noise driving a stationary autoregressive process has zero median. This work extends their proof to include the case when the driving noise has zero mean. Thus, when the noise PDF (probability density function) is asymmetric with distinct mean and median, the LAD estimator will be strongly consistent with the PDF centered with either mean or median at the origin. The results of this work extend computer simulations which further indicate that under these conditions, the LAD estimator is MS consistent (mean-squared convergence to the true parameters). The importance of these results in LAD signal processing applications is discussed. >

Wavelet phase filter for emission tomography

The presence of a high level of noise is a characteristic in some tomographic imaging techniques such as positron emission tomography (PET). Wavelet methods can smooth out noise while preserving significant features of images. Mallat et al. proposed a wavelet based denoising scheme exploiting wavelet modulus maxima, but the scheme is sensitive to noise. In this study, we explore the properties of wavelet phase, with a focus on reconstruction of emission tomography images. Specifically, we show that the wavelet phase of regular Poisson noise under a Haar-type wavelet transform converges in distribution to a random variable uniformly distributed on (0, 2(pi) ). We then propose three wavelet-phase-based denoising schemes which exploit this property: edge tracking, local phase variance thresholding, and scale phase variation thresholding. Some numerical results are also presented. The numerical experiments indicate that wavelet phase techniques show promise for wavelet based denoising methods.

Remote object recognition by analysis of surface structure

We present a new algorithm for the discrimination of remote objects by their surface structure. Starting from a range-azimuth profile function, we formulate a range-azimuth matrix whose largest eigenvalues are used as discriminating features to separate object classes. A simpler, competing algorithm uses the number of sign changes in the range-azimuth profile function to discriminate among classes. Whereas both algorithms work well on noiseless data, an experiment involving real data shows that the eigenvalue method is far more robust with respect to noise than is the sign-change method. Two well-known methods based on surface structure, variance, and fractal dimension were also tested on real data. Neither method furnished the aspect invariance and the discriminability of the eigenvalue method.

New results in projection-type algorithms of use in image processing and reconstruction

In image restoration the two main goals are noise smoothing and restoration of sharp details. We consider the noise smoothing problem from the point-of-view of convex projections and introduce several normed derivative constraints and their associated projectors.

Characterization of the LAD (L/sub 1/) AR parameter estimator when applied to stationary ARMA, MA, and higher order AR processes

It is shown that least absolute deviation (LAD) autoregressive (AR) estimates converge to the parameters of an nth-order finite-impulse-response (FIR) filter which minimizes the expectation of the absolute value of the prediction error. Examples are presented in which these parameters are calculated and the efficiencies of the LAD estimates are determined from a Monte Carlo simulation. Applications to order selection and PARCOR parameter estimation are discussed. >