Critical Indices and Self-Similar Power Transform

“Odd” factor approximants of the special form suggested by Gluzman and Yukalov (J. Math. Chem. 2006, 39, 47) are amenable to optimization by power transformation and can be successfully applied to critical phenomena. The approach is based on the idea that the critical index by itself should be optimized through the parameters of power transform to be calculated from the minimal sensitivity (derivative) optimization condition. The critical index is a product of the algebraic self-similar renormalization which contributes to the expressions the set of control parameters typical to the algebraic self-similar renormalization, and of the power transform which corrects them even further. The parameter of power transformation is, in a nutshell, the multiplier connecting the critical exponent and the correction-to-scaling exponent. We mostly study the minimal model of critical phenomena based on expansions with only two coefficients and critical points. The optimization appears to bring quite accurate, uniquely defined results given by simple formulas. Many important cases of critical phenomena are covered by the simple formula. For the longer series, the optimization condition possesses multiple solutions, and additional constraints should be applied. In particular, we constrain the sought solution by requiring it to be the best in prediction of the coefficients not employed in its construction. In principle, the error/measure of such prediction can be optimized by itself, with respect to the parameter of power transform. Methods of calculation based on optimized power-transformed factors are applied and results presented for critical indices of several key models of conductivity and viscosity of random media, swelling of polymers, permeability in two-dimensional channels. Several quantum mechanical problems are discussed as well.

Quantifying parametric uncertainty in the Rothermel model

The purpose of the present work is to quantify parametric uncertainty in the Rothermel wildland fire spread model (implemented in software such as BehavePlus3 and FARSITE), which is undoubtedly among the most widely used fire spread models in the United States. This model consists of a non-linear system of equations that relates environmental variables (input parameter groups) such as fuel type, fuel moisture, terrain, and wind to describe the fire environment. This model predicts important fire quantities (output parameters) such as the head rate of spread, spread direction, effective wind speed, and fireline intensity. The proposed method, which we call sensitivity derivative enhanced sampling, exploits sensitivity derivative information to accelerate the convergence of the classical Monte Carlo method. Coupled with traditional variance reduction procedures, it offers up to two orders of magnitude acceleration in convergence, which implies that two orders of magnitude fewer samples are required for a given level of accuracy. Thus, it provides an efficient method to quantify the impact of input uncertainties on the output parameters.

Multilevel Optimization Procedures for Gas Turbine Design: Part II — Solution and Examples

The mathematical formulations of two procedures for breaking down a large optimization energy problem into a hierarchy of separated but coupled subproblems were presented and discussed in the first part of this work (Ghiglino and Massardo, 1994). Here their applicability is illustrated by formulation of realistic analysis and design problems in the field of gas turbine systems, both for open and closed cycle plants. The definition of objective functions, design variables, constraints and subproblem analyses is discussed; the results obtained with the sensitivity, derivative procedure are compared with the results obtained with the linearization procedure.