Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution

AbstractA singularly perturbed two-point boundary value problem with an exponential boundary layer is solved numerically by using an adaptive grid method. The mesh is constructed adaptively by equidistributing a monitor function based on the arc-length of the exact solution. The error analysis for this approach was carried out by Qiu et al. (J. Comput. Appl. Math. 101 (1999) 1–25). In this work, their error bound will be improved to the optimal order which is independent of the perturbation parameter. The main ingredient used to obtain the improved result is the theory of the discrete Green's function

Efficient Estimation of Copula-based Semiparametric Markov Models

This paper considers efficient estimation of copula-based semiparametric strictly stationary Markov models. These models are characterized by nonparametric invariant (one-dimensional marginal) distributions and parametric bivariate copula functions; where the copulas capture temporal dependence and tail dependence of the processes. The Markov processes generated via tail dependent copulas may look highly persistent and are useful for financial and economic applications. We first show that Markov processes generated via Clayton, Gumbel and Student's $t$ copulas and their survival copulas are all geometrically ergodic. We then propose a sieve maximum likelihood estimation (MLE) for the copula parameter, the invariant distribution and the conditional quantiles. We show that the sieve MLEs of any smooth functionals are root-$n$ consistent, asymptotically normal and efficient; and that their sieve likelihood ratio statistics are asymptotically chi-square distributed. We present Monte Carlo studies to compare the finite sample performance of the sieve MLE, the two-step estimator of Chen and Fan (2006), the correctly specified parametric MLE and the incorrectly specified parametric MLE. The simulation results indicate that our sieve MLEs perform very well; having much smaller biases and smaller variances than the two-step estimator for Markov models generated via Clayton, Gumbel and other tail dependent copulas.Copula, Tail dependence, Nonlinear Markov models, Geometric ergodicity, Sieve MLE, Semiparametric efficiency, Sieve likelihood ratio statistics, Value-at-Risk

Spiral Deployment of Optical Fiber Sensors for Distributed Strain Measurement in Seven-Wire Twisted Steel Cables, Post-Tensioned Against Precast Concrete Bars

On-time monitoring and condition assessments of steel cables provide mission-critical data for informed decision making, ensuring the structural safety of post-tensioned concrete structures. This study aimed to develop a spiral deployment scheme of distributed fiber optic sensors (DFOS) and to monitor/assess the post-tensioned force in seven-wire twisted steel cables, based on the pulse-pre-pump Brillouin optical time domain analysis. Each DFOS was placed in a spiral shape between two surface wires of a steel cable and glued to the steel cable by epoxy. Image observations were conducted to investigate the entireness and bonding condition between the optical fiber and the steel wires. Eight concrete bar specimens were cast, each with a pre-embedded plastic or metal duct at its center and each was post-tensioned by a steel strand through the duct once they were instrumented with two strain and two temperature sensors. The strand was loaded/unloaded and monitored by measuring the Brillouin frequency shifts and correlating them with the applied strains and the resulting cable force after temperature compensation. The maximum, minimum, and average cable forces integrated from the measured stain data were compared and validated with those from a load cell. The maximum (or average) cable force was linearly related to the ground truth data with a less than 10% error between them, after any initial slack had been removed from the test setup. The post-tensioned force loss was bounded by approximately 30%, using the test setup designed in this study