A mixed-variable continuously deforming finite element method for parabolic evolution problems. Part III: Numerical implementation and computational results

In Part I of this paper,¹ the conceptual framework of a rate variational least squares formulation of a continuously deforming mixed-variable finite element method was presented for solving a single evolution equation. In Part II² a system of ordinary differential equations with respect to time was derived for solving a system of three coupled evolution equations by the deforming grid mixed-variable least squares rate variational finite element method. The system of evolution equations describes the coupled heat flow, fluid flow and trace species transport in porous media under conditions when the flow velocities and constituent phase transitions induce sharp fronts in the solution domain. In this paper, we present the method we have adopted to integrate with respect to time the resulting spatially discretized system of non-linear ordinary differential equations. Next, we present computational results obtained using the code in which this deforming mixed finite element method was implemented. Because several features of the formulation are novel and have not been previously attempted, the problems were selected to exercise these features with the objective of demonstrating that the formulation is correct and that the numerical procedures adopted converge to the correct solutions.     

Designing mechanical components to a specified reliability and confidence level

A methodology for determining the lower one-sided confidence limit on the reliability, R_L1, of mechanical components and structural members is presented. To accomplish this, charts for R_L1 values at 50%, 60%, 70%, 80%, 90%, 95%, 99% and 99.9% confidence levels are given. A methodology for designing components and structural members to a desired reliability with greater than 50% assurance is proposed. All of these methodologies are illustrated by examples.     

Mixed-Weibull parameter estimation for burn-in data using the Bayesian approach

A Bayesian approach is presented for both grouped and ungrouped burn-in test data, to come up with the posterior parameters of the bimodal mixed-Weibull distribution. The concepts of belonging probabilities and fractional ranks are introduced for this approach. A numerical comparison is made by conducting the Kolmogorov-Smirnov goodness-of-fit test on the parameter estimates obtained by Jensen's graphical method, by the Bayesian approach using the conventional separation plotting method and by the Bayesian approach using the fractional rank plotting method. It turns out that the Bayesian approach with the fractional rank plotting method yields the best results.     

Fitting the Weibull log-linear model to accelerated life-test data

The Weibull log-linear model is a widely-used accelerated life-test model in reliability engineering. The standard deviation of log(life), s, was often assumed to be a constant or stress-independent. However, theoretical and experimental research results suggest that, in many cases, s is stress-dependent. The data analysis via the MLE method must be performed numerically, because of the complexity of the model and many unknown parameters being involved. The commonly-used methods often fail to converge when the starting point is not close to the solution, especially for censored data. Generally, no easy-to-use software is available for the Weibull log-linear model. To facilitate this process, an efficient algorithm is presented in this paper, to obtain the MLE of the model parameters from test data (with or without censoring) for both stress-independent and stress-dependent models. The validity and effectiveness of this procedure are illustrated with numerical examples. The method is numerically stable, and easy to implement and program     

EMP: execution time measurement protocol for compute‐bound programs

Measuring execution time is one of the most used performance evaluation techniques in computer science research. Inaccurate measurements cannot be used for a fair performance comparison between programs. Despite the prevalence of its use, the intrinsic variability in the time measurement makes it hard to obtain repeatable and accurate timing results of a program running on an operating system. We propose a novel execution time measurement protocol (termed EMP) for measuring the execution time of a compute‐bound program on Linux, while minimizing that measurement's variability. During the development of execution time measurement protocol, we identified several factors that disturb execution time measurement. We introduce successive refinements to the protocol by addressing each of these factors, in concert, reducing variability by more than an order of magnitude. We also introduce a new visualization technique, what we term ‘dual‐execution scatter plot’ that highlights infrequent, long‐running daemons, differentiating them from frequent and/or short‐running daemons. Our empirical results show that the proposed protocol successfully achieves three major aspects—precision, accuracy, and scalability—in execution time measurement that can work for open‐source and proprietary software. Copyright © 2017 John Wiley & Sons, Ltd.     

Maximum likelihood estimates, from censored data, for mixed-Weibulldistributions

An algorithm for estimating the parameters of mixed-Weibull distributions from censored data is presented. The algorithm follows the principle of the MLE (maximum likelihood estimate) through the EM (expectation and maximization) algorithm, and it is derived for both postmortem and non-postmortem time-to-failure data. The MLEs of the nonpostmortem data are obtained for mixed-Weibull distributions with up to 14 parameters in a five-subpopulation mixed-Weibull distribution. Numerical examples indicate that some of the log-likelihood functions of the mixed-Weibull distributions have multiple local maxima; therefore the algorithm should start at several initial guesses of the parameters set. It is shown that the EM algorithm is very efficient. On the average for two-Weibull mixtures with a sample size of 200, the CPU time (on a VAX 8650) is 0.13 s/iteration. The number of iterations depends on the characteristics of the mixture. The number of iterations is small if the subpopulations in the mixture are well separated. Generally, the algorithm is not sensitive to the initial guesses of the parameters     

A mixed-variable continuously deforming finite element method for parabolic evolution problems. Part II: The coupled problem of phase-change in porous media

The conceptual framework of a least squares rate variational approach to the formulation of continuously deforming mixed-variable finite element computational scheme for a single evolution equation was presented in Part I.¹ In this paper (Part II), we extend these concepts and present an adaptively deforming mixed variable finite element method for solving general two-dimensional transport problems governed by a system of coupled non-linear partial differential evolution equations. In particular, we consider porous media problems that involve coupled heat and mass transport processes that yield steep continuous moving fronts, and abrupt, discontinuous, moving phase-change interfaces. In this method, the potentials, such as the temperature, pressure and species concentration, and the corresponding fluxes, are permitted to jump in value across the phase-change interfaces. The equations, and the jump conditions, governing the physical phenomena, which were specialized from a general multiphase, multiconstituent mixture theory, provided the basis for the development and implementation of a two-dimensional numerical simulator. This simulator can effectively resolve steep continuous fronts (i.e. shock capturing) without oscillations or numerical dispersion, and can accurately represent and track discontinuous fronts (i.e. shock fitting) through adaptive grid deformation and redistribution. The numerical implementation of this simulator and numerical examples that demonstrate the performance of the computational method are presented in Part III² of this paper.     

Combinatorial algorithms for DNA sequence assembly

The trend toward very large DNA sequencing projects, such as those being undertaken as part of the Human Genome Program, necessitates the development of efficient and precise algorithms for assembling a long DNA sequence from the fragments obtained by shotgun sequencing or other methods. The sequence reconstruction problem that we take as our formulation of DNA sequence assembly is a variation of the shortest common superstring problem, complicated by the presence of sequencing errors and reverse complements of fragments. Since the simpler superstring problem is NP-hard, any efficient reconstruction procedure must resort to heuristics. In this paper, however, a four-phase approach based on rigorous design criteria is presented, and has been found to be very accurate in practice. Our method is robust in the sense that it can accommodate high sequencing error rates, and list a series of alternate solutions in the event that several appear equally good. Moreover, it uses a limited form of multiple sequence alignment to detect, and often correct, errors in the data. Our combined algorithm has successfully reconstructed nonrepetitive sequences of length 50,000 sampled at error rates of as high as 10%.This research was supported by the National Library of Medicine under Grant R01-LM4960, by a postdoctoral fellowship from the Program in Mathematics and Molecular Biology of the University of California at Berkeley under National Science Foundation Grant DMS-8720208, and by a fellowship from the Centre de recherches mathématiques of the Université de Montréal.