New Type Of Method Research Articles

Tensor Methods for Unconstrained Optimization Using Second Derivatives

A new type of method for unconstrained optimization, called a tensor method, is introduced. It is related in its basic philosophy to the tensor methods for nonlinear equations for Schnabel and Frank [SIAM J. Numer. Anal., 21 (1984), pp. 815–843], but beyond that the methods have significant differences. The tensor method for unconstrained optimization bases each iteration upon a fourth order model of the objective function. This model consists of the quadratic portion of the Taylor series, plus low-rank third and fourth order terms that cause the model to interpolate already calculated function and gradient values from one or more previous iterates. This paper also shows that the costs of forming, storing, and solving the tensor model are not significantly more than these costs for a standard method based upon a quadratic Taylor series model. Test results are presented for sets of problems where the Hessian at the minimizer is nonsingular, and where it is singular. On all the test sets, the tensor method solves considerably more problems than a comparable standard method. On problems solved by both methods, the tensor method requires about half as many iterations, and half as many function and derivative evaluations as the standard method, on the average.

Nonlinear Filters for Efficient Shock Computation

A new type of methods for the numerical approximation of hyperbolic conservation laws with discontinuous solution is introduced. The methods are based on standard finite difference schemes. The difference solution is processed with a nonlinear conservation form filter at every time level to eliminate spurious oscillations near shocks. It is proved that the filter can control the total variation of the solution and also produce sharp discrete shocks. The method is simpler and faster than many other high resolution schemes for shock calculations. Numerical examples in one and two space dimensions are presented.

Quick determination of dielectric anisotropy of paper sheets by means of microwaves

AbstractThis paper deals with a new type of method for determining the complex dielectric constant at microwave frequencies in about 30 s. This method determines the dielectric anisotropy and refractive index for optically opaque paper sheets. The dielectric anisotropy ascribable to the local motions of cellulose molecules is supported by electron microscopic observations and mechanical anisotropy. The angular dependence of the complex dielectric constant gives the degree of molecular orientation and the distribution of cellulose molecules in paper sheets. The direction of the average orientation of molecular chains can be inferred from the anisotropic data. It is also suggested that the cellulose molecules should be oriented on an average in the direction of long axis of pulp fibers. The accumulation of data for complex dielectric constant and refractive index at microwave frequencies gives an important chart for manufacturing the functional paper corresponding to needs.

Nonlinear filters for efficient shock computation

A new type of methods for the numerical approximation of hyperbolic conservation laws with discontinuous solution is introduced. The methods are based on standard finite difference schemes. The difference solution is processed with a nonlinear conservation form filter at every time level to eliminate spurious oscillations near shocks. It is proved that the filter can control the total variation of the solution and also produce sharp discrete shocks. The method is simpler and faster than many other high resolution schemes for shock calculations. Numerical examples in one and two space dimensions are presented.

Using parallel function evaluations to improve hessian approximation for unconstrained optimization

This paper presents a new class of methods for solving unconstrained optimization problems on parallel computers. The methods are intended to solve small to moderate dimensional problems where function and derivative evaluation is the dominant cost. They utilize multiple processors to evaluate the function, (finite difference) gradient, and a portion of the finite difference Hessian simultaneously at each iterate. We introduce three types of new methods, which all utilize the new finite difference Hessian information in forming the new Hessian approximation at each iteration; they differ in whether and how they utilize the standard secant information from the current step as well. We present theoretical analyses of the rate of convergence of several of these methods. We also present computational results which illustrate their performance on parallel computers when function evaluation is expensive.

Spectral estimation with rational modeling of the log spectrum

A new type of methods for spectral estimation is presented starting from the Burg's variational formulation. These methods assume as data an accurate set of autocorrelation values and estimate the spectrum by modeling its logarithm with a rational function. They are compared with the Blackman-Tukey and the maximum entropy methods both theoretically and experimentally. Iterative algorithms of descent type making use of Levinson recursions are employed in the experimental investigation. Two kinds of results are reported: (1) when the data are exact and (2) when the data arise from the periodogram. In this latter case, the comparison is based on the mean and the variance of the spectrum estimates. Ein neuer Typ von spektraler Estimation wird vorgestellt, ausgehend von Burg's variationaler Formulation. In diesen Methoden werden Daten als genaue Autokorrelationswerte angenommen, und der logarithmus des Spektrums wird als eine rationale Funktion modeliert. Diese Methoden werden mit der Blackman-Tukey Methode und der Maximum Entropie Methode verglichen, sowie theoretisch als experimentell. Iterative Algorithmen des Levison Typus werden fu¨r die Experimente eingesetzt. Zwei Arten von Resultaten werden gezeigt: (1) wenn die Daten exact sind und (2) wenn die Daten von Periodogrammen stammen. In dem zweiten Fall basiert der Vergleich auf Mittelwert und Varianz der spektrumsestimations Werten. Un nouveau type de me´thodes pour l'estimation spectrale est pre´sente´e en comman¸ant par la formulation variationnelle de Burg. Ces me´thodes admettent comme donne´es un ensemble pre´cis des valeurs d'autocorre´lation et estiment le spectre en mode´lisant son logarithme par une fonction rationelle. Elles sont compare´esa`celles de Blackman-Tukey et de maximum d'entropie en the´orie et en pratique. Des algorithmes ite´ratifs de type descent utilisant les re´cursions de Levinson sont utilise´es dans la recherche expe´rimentale. Deux types de resultats sont de´crits: (1) quand les donne´es ne sont pas exactes et (2) quand les donne´es proviennent du pe´riodogramme. Dans ce dernier cas, la comparaison est base´e sur la valeur moyenne et la variance de l'estimation spectrale.

Exploring a new type of alcoholism survey.

A new type of method for surveying alcoholism and alcohol-abuse was tested in a way designed to cover about 20% of adults in a town of 50 000. The method proved able to identify types of alcohol-abuser which other survey techniques have difficulty with; it made no undue demands on respondents; and it is much more economical than conventional household surveys. However, more work is still needed on it, and for certain types of information the customary self-report is still going to be required.