High-order adaptive finite element-singly implicit Runge-Kutta methods for parabolic differential equations

We describe an adaptive mesh refinement finite element method-of-lines procedure for solving one-dimensional parabolic partial differential equations. Solutions are calculated using Galerkin's method with a piecewise hierarchical polynomial basis in space and singly implicit Runge-Kutta (SIRK) methods in time. A modified SIRK formulation eliminates a linear systems solution that is required by the traditional SIRK formulation and leads to a new reduced-order interpolation formula. Stability and temporal error estimation techniques allow acceptance of approximate solutions at intermediate stages, yielding increased efficiency when solving partial differential equations. A priori energy estimates of the local discretization error are obtained for a nonlinear scalar problem. A posteriori estimates of local spatial discretization errors, obtained by order variation, are used with the a priori error estimates to control the adaptive mesh refinement strategy. Computational results suggest convergence of the a posteriori error estimate to the exact discretization error and verify the utility of the adaptive technique.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-90-0194; the U.S. Army Research Office under Contract Number DAAL 03-91-G-0215; by the National Science Foundation under Grant Number CDA-8805910; and by a grant from the Committee on Research, Tulane University.     

A high order method for the numerical solution of two-point boundary value problems

A one step finite difference scheme of order 4 for the numerical solution of the general two-point boundary value problemy=f(t,y),a t b, withg(y(a),y(b))=0 is presented. The global discretization error of the scheme is shown, in sufficiently smooth cases, to have an asymptotic expansion containing even powers of the mesh size only. This justifies the use of Richardson extrapolation (or deferred correction) to obtain high orders of accuracy. A theoretical examination of the new scheme for large systems of equations shows that for a given mesh size it generally requires about twice as much work as the Keller box scheme. However, the expectation of higher accuracy usually justifies this extra computational effort. Some numerical results are given which confirm these expectations and show that the new scheme can be generally competitive with the box scheme.     

The impact of integrated STEM professional development on teacher quality

This study of a state‐funded, 3‐year implementation of an integrated STEM professional development (PD) program for teachers from two middle schools in the midwestern U.S. examined if participants in the PD were enabled to transform their practice and perceptions of STEM. An integrated STEM approach includes a focus on the STEM disciplines, along with leveraging social studies/history and English/language arts as important context and tools for solving society’s biggest challenges. Findings in this study indicated that teachers implemented more effective STEM teaching strategies and had more positive perceptions regarding STEM overall. Further, participants became more aware of their personal needs for resources and support to teach through integrated STEM. Implications for research and practice are discussed.     

An advanced tabu search for solving the mixed payload airlift loading problem

This article describes a new, two-dimensional bin packing algorithm that feasibly loads a set of cargo items on a minimal set of airlift aircraft. The problem under consideration is called the Mixed Payload Airlift Loading Problem (MPALP). The heuristic algorithm, called the Mixed Payload Airlift Loading Problem Tabu Search (MPALPTS), surpasses previous research conducted in this area because, in addition to pure pallet cargo loads, MPALPTS can accommodate rolling stock cargo (ie tanks, trucks, HMMMVs, etc) while still maintaining feasibility. To demonstrate its effectiveness, the load plans generated by MPALPTS are directly compared to those generated by the Automated Air Load Planning Software (AALPS) for a given cargo set; AALPS is the load planning software currently mandated for use in all Department of Defense load planning. While more time consuming than AALPS, MPALPTS required the same or fewer aircraft than AALPS in all test scenarios.     

Devising a quick-running heuristic for an unmanned aerial vehicle (UAV) routing system

UAVs provide reconnaissance support for the US military and often need operational routes immediately; current practice involves manual route calculation that can involve hundreds of targets and a complex set of operational restrictions. Our research focused on providing an operational UAV routing system. This system required development of a reasonably effective, quick running routing heuristic. We present the statistical methodology used to devise a quick-running routing heuristic that provides reasonable solutions. We consider three candidate local search heuristic approaches, conduct an empirical analysis to parameterize each heuristic, competitively test each candidate heuristic, and provide statistical analysis on the performance of each candidate heuristic to include comparison of the results of the best candidate heuristic against a compilation of the best-known solutions for standard test problems. Our heuristic is a component of the final UAV routing system and provides the UAV operators a tool to perform their route development tasks quickly and efficiently.     

Laguerre approximation of stable manifolds with application to connecting orbits

We present an algorithm, based on approximation by Laguerre polynomials, for computing a point on the stable manifold of a stationary solution of an autonomous system. A superconvergence phenomenon means that the accuracy of our results is much higher than the usual spectral accuracy. Both the theory and the implementation of the method are considered. Finally, as an application of the algorithm, we describe a fully spectral approximation of homo- and heteroclinic orbits.

     

Multiple levels of metacognition and their elicitation through complex problem-solving tasks

Building on prior efforts, we re-conceptualize metacognition on multiple levels, looking at the sources that trigger metacognition at the individual level, the social level, and the environmental level. This helps resolve the paradox of metacognition: metacognition is personal, but it cannot be explained exclusively by individualistic conceptions. We develop a theoretical model of metacognition in collaborative problem solving based on models and modeling perspectives. The theoretical model addresses several challenges previously found in the research of metacognition. This paper illustrates how metacognition was elicited, at the environmental level, through problems requiring different problem-solving processes (definition building and operationalizing definitions), and how metacognition operated at both the individual level and the social level during complex problem solving. The re-conceptualization of metacognition has the potential to guide the development of metacognitive activities and effective instructional methods to integrate them into existing curricula that are necessary to engage students in active, higher-order learning.     

Robust Dynamics and Control of a Partially Observed Markov Chain

In a seminal paper, Martin Clark (Communications Systems and Random Process Theory, Darlington, 1977, pp. 721–734, 1978) showed how the filtered dynamics giving the optimal estimate of a Markov chain observed in Gaussian noise can be expressed using an ordinary differential equation. These results offer substantial benefits in filtering and in control, often simplifying the analysis and an in some settings providing numerical benefits, see, for example Malcolm et al. (J. Appl. Math. Stoch. Anal., 2007, to appear). Clark’s method uses a gauge transformation and, in effect, solves the Wonham-Zakai equation using variation of constants. In this article, we consider the optimal control of a partially observed Markov chain. This problem is discussed in Elliott et al. (Hidden Markov Models Estimation and Control, Applications of Mathematics Series, vol. 29, 1995). The innovation in our results is that the robust dynamics of Clark are used to compute forward in time dynamics for a simplified adjoint process. A stochastic minimum principle is established.     

The importance for speech intelligibility of random fluctuations in "steady" background noise

Spectrally shaped steady noise is commonly used as a masker of speech. The effects of inherent random fluctuations in amplitude of such a noise are typically ignored. Here, the importance of these random fluctuations was assessed by comparing two cases. For one, speech was mixed with steady speech-shaped noise and N-channel tone vocoded, a process referred to as signal-domain mixing (SDM); this preserved the random fluctuations of the noise. For the second, the envelope of speech alone was extracted for each vocoder channel and a constant was added corresponding to the root-mean-square value of the noise envelope for that channel. This is referred to as envelope-domain mixing (EDM); it removed the random fluctuations of the noise. Sinusoidally modulated noise and a single talker were also used as backgrounds, with both SDM and EDM. Speech intelligibility was measured for N?=?12, 19, and 30, with the target-to-background ratio fixed at -7 dB. For SDM, performance was best for the speech background and worst for the steady noise. For EDM, this pattern was reversed. Intelligibility with steady noise was consistently very poor for SDM, but near-ceiling for EDM, demonstrating that the random fluctuations in steady noise have a large effect.     

The dynamic range of useful temporal fine structure cues for speech in the presence of a competing talker

Within an auditory channel, the speech waveform contains both temporal envelope (E(O)) and temporal fine structure (TFS) information. Vocoder processing extracts a modified version of the temporal envelope (E') within each channel and uses it to modulate a channel carrier. The resulting signal, E'(Carr), has reduced information content compared to the original "E(O)?+ TFS" signal. The dynamic range over which listeners make additional use of E(O)?+ TFS over E'(Carr) cues was investigated in a competing-speech task. The target-and-background mixture was processed using a 30-channel vocoder. In each channel, E(O)?+ TFS replaced E'(Carr) at either the peaks or the valleys of the signal. The replacement decision was based on comparing the short-term channel level to a parametrically varied "switching threshold," expressed relative to the long-term channel level. Intelligibility was measured as a function of switching threshold, carrier type, target-to-background ratio, and replacement method. Scores showed a dependence on all four parameters. Derived intensity-importance functions (IIFs) showed that E(O)?+ TFS information from 8-13 dB below to 10 dB above the channel long-term level was important. When E(O)?+ TFS information was added at the peaks, IIFs peaked around -2 dB, but when E(O)?+ TFS information was added at the valleys, the peaks lay around +1 dB.