A general Approach to Methods with a Sparse Jacobian for Solving Nonlinear Systems of Equations

2000 Mathematics Subject Classification: 65H10.

Parabolic similaritons in optical fibres

Recent developments in nonlinear optics have brought to the fore of intensive research an interesting class of pulses with a parabolic intensity profile and a linear instantaneous frequency shift or chirp. Parabolic pulses propagate in optical fibres with normal group-velocity dispersion in a self-similar manner, holding certain relations (scaling) between pulse power, duration and chirp parameter, and can tolerate strong nonlinearity without distortion or wave breaking. These solutions, which have been dubbed similaritons, were demonstrated theoretically and experimentally in fibre amplifiers in 2000. Similaritons in fibre amplifiers are, along with solitons in passive fibres, the most well-known classes of nonlinear attractors for pulse propagation in optical fibre, so they take on major fundamental importance. The unique properties of parabolic similaritons have stimulated numerous applications in nonlinear optics, ranging from ultrashort high-power pulse generation to highly coherent continuum sources and to optical nonlinear processing of telecommunication signals. In this work, we review the physics underlying the generation of parabolic similaritons as well as recent results obtained in a wide range of experimental configurations.

Impact of the carrier relaxation paths on two-state operation in quantum dot lasers

We study InGaAs QD laser operating simultaneously at ground (GS) and excited (ES) states under 30ns pulsed-pumping and distinguish three regimes of operation depending on the pump current and the carrier relaxation pathways. An increased current leads to an increase in ES intensity and to a decrease in GS intensity (or saturation) for low pump range, as typical for the cascade-like pathway. Both the GS and ES intensities are steadily increased for high current ranges, which prove the dominance of the direct capture pathway. The relaxation oscillations are not pronounced for these ranges. For the mediate currents, the interplay between the both pathways leads to the damped large amplitude relaxation oscillations with significant deviation of the relaxation oscillation frequency from the initial value during the pulse.

A gender study investigating physics self-efficacy

The underrepresentation of women in physics has been well documented and a source of concern for both policy makers and educators. My dissertation focuses on understanding the role self-efficacy plays in retaining students, particularly women, in introductory physics. I use an explanatory mixed methods approach to first investigate quantitatively the influence of self-efficacy in predicting success and then to qualitatively explore the development of self-efficacy. In the initial quantitative studies, I explore the utility of self-efficacy in predicting the success of introductory physics students, both women and men. Results indicate that self-efficacy is a significant predictor of success for all students. I then disaggregate the data to examine how self-efficacy develops differently for women and men in the introductory physics course. Results show women rely on different sources of self-efficacy than do men, and that a particular instructional environment, Modeling Instruction, has a positive impact on these sources of self-efficacy. In the qualitative phase of the project, this dissertation focuses on the development of self-efficacy. Using the qualitative tool of microanalysis, I introduce a methodology for understanding how self-efficacy develops moment-by-moment using the lens of self-efficacy opportunities. I then use the characterizations of self-efficacy opportunities to focus on a particular course environment and to identify and describe a mechanism by which Modeling Instruction impacts student self-efficacy. Results indicate that the emphasizing the development and deployment of models affords opportunities to impact self-efficacy. The findings of this dissertation indicate that introducing key elements into the classroom, such as cooperative group work, model development and deployment, and interaction with the instructor, create a mechanism by which instructors can impact the self-efficacy of their students. Results from this study indicate that creating a model to impact the retention rates of women in physics should include attending to self-efficacy and designing activities in the classroom that create self-efficacy opportunities.

An Exploratory Qualitative Study of the Proximal Goal Setting of Two Introductory Modeling Physics Students

A qualitative investigation on the impact of goal-setting strategies on self-efficacy of two students taking Introductory Modeling Physics was conducted. The study found that the problem solving process can be divided into two main themes: goal-setting and self-efficacy. Self-efficacy plays a role in the goal setting process of these two students, and may be linked to the retention of students in physics.

Multimode Analysis of Nanoscale Biomolecular Interactions

Biomolecular interactions, including protein-protein, protein-DNA, and protein-ligand interactions, are of special importance in all biological systems. These interactions may occer during the loading of biomolecules to interfaces, the translocation of biomolecules through transmembrane protein pores, and the movement of biomolecules in a crowded intracellular environment. The molecular interaction of a protein with its binding partners is crucial in fundamental biological processes such as electron transfer, intracellular signal transmission and regulation, neuroprotective mechanisms, and regulation of DNA topology. In this dissertation, a customized surface plasmon resonance (SPR) has been optimized and new theoretical and label free experimental methods with related analytical calculations have been developed for the analysis of biomolecular interactions. Human neuroglobin (hNgb) and cytochrome c from equine heart (Cyt c) proteins have been used to optimize the customized SPR instrument. The obtained Kd value (~13 µM), from SPR results, for Cyt c-hNgb molecular interactions is in general agreement with a previously published result. The SPR results also confirmed no significant impact of the internal disulfide bridge between Cys 46 and Cys 55 on hNgb binding to Cyt c. Using SPR, E. coli topoisomerase I enzyme turnover during plasmid DNA relaxation was found to be enhanced in the presence of Mg2+. In addition, a new theoretical approach of analyzing biphasic SPR data has been introduced based on analytical solutions of the biphasic rate equations. In order to develop a new label free method to quantitatively study protein-protein interactions, quartz nanopipettes were chemically modified. The derived Kd (~20 µM) value for the Cyt c-hNgb complex formations matched very well with SPR measurements (Kd ~16 µM). The finite element numerical simulation results were similar to the nanopipette experimental results. These results demonstrate that nanopipettes can potentially be used as a new class of a label-free analytical method to quantitatively characterize protein-protein interactions in attoliter sensing volumes, based on a charge sensing mechanism. Moreover, the molecule-based selective nature of hydrophobic and nanometer sized carbon nanotube (CNT) pores was observed. This result might be helpful to understand the selective nature of cellular transport through transmembrane protein pores.

Heat transport in low dimensions : introduction and phenomenology

Review paper, to appear in the Springer Lecture Notes in Physics volume "Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer" (S. Lepri ed.)

Optical wave turbulence and wave condensation in a nonlinear optical experiment

We present theory, numerical simulations and experimental observations of a 1D optical wave system. We show that this system is of a dual cascade type, namely, the energy cascading directly to small scales, and the photons or wave action cascading to large scales. In the optical context the inverse cascade is particularly interesting because it means the condensation of photons. We show that the cascades are induced by a six-wave resonant interaction process described by weak turbulence theory. We show that by starting with weakly nonlinear randomized waves as an initial condition, there exists an inverse cascade of photons towards the lowest wavenumbers. During the cascade nonlinearity becomes strong at low wavenumbers and, due to the focusing nature of the nonlinearity, it leads to modulational instability resulting in the formation of solitons. Further interaction of the solitons among themselves and with incoherent waves leads to the final condensate state dominated by a single strong soliton. In addition, we show the existence of the direct energy cascade numerically and that it agrees with the wave turbulence prediction.

Cooperative or collaborative learning: Is there a difference in university students’ perceptions?

The hypothesis that the same educational objective, raised as cooperative or collaborative learning in university teaching does not affect students’ perceptions of the learning model, leads this study. It analyses the reflections of two students groups of engineering that shared the same educational goals implemented through two different methodological active learning strategies: Simulation as cooperative learning strategy and Problem-based Learning as a collaborative one. The different number of participants per group (eighty-five and sixty-five, respectively) as well as the use of two active learning strategies, either collaborative or cooperative, did not show differences in the results from a qualitative perspective.

Numerical study of neutron beam divergence in a beam-fusion scenario employing laser driven ions

The most established route to create a laser-based neutron source is by employing laser accelerated, low atomic-number ions in fusion reactions. In addition to the high reaction cross-sections at moderate energies of the projectile ions, the anisotropy in neutron emission is another important feature of beam-fusion reactions. Using a simple numerical model based on neutron generation in a pitcher–catcher scenario, anisotropy in neutron emission was studied for the deuterium–deuterium fusion reaction. Simulation results are consistent with the narrow-divergence ( ∼ 70 ° full width at half maximum) neutron beam recently served in an experiment employing multi-MeV deuteron beams of narrow divergence (up to 30° FWHM, depending on the ion energy) accelerated by a sub-petawatt laser pulse from thin deuterated plastic foils via the Target Normal Sheath Acceleration mechanism. By varying the input ion beam parameters, simulations show that a further improvement in the neutron beam directionality (i.e. reduction in the beam divergence) can be obtained by increasing the projectile ion beam temperature and cut-off energy, as expected from interactions employing higher power lasers at upcoming facilities.

Variationelle Zeitdiskretisierungen höherer Ordnung der Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängigen Gebieten

Wir betrachten zeitabhängige Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängi- gen Gebieten, wobei die Bewegung des Gebietsrandes bekannt ist. Die zeitliche Entwicklung des Gebietes wird durch die ALE-Formulierung behandelt, die die Nachteile der klassischen Euler- und Lagrange-Betrachtungsweisen behebt. Die Position des Randes und seine Geschwindigkeit werden dabei so in das Gebietsinnere fortgesetzt, dass starke Gitterdeformationen verhindert werden. Als Zeitdiskretisierungen höherer Ordnung werden stetige Galerkin-Petrov-Verfahren (cGP) und unstetige Galerkin-Verfahren (dG) auf Probleme in zeitabhängigen Gebieten angewendet. Weiterhin werden das C 1 -stetige Galerkin-Petrov-Verfahren und das C 0 -stetige Galerkin- Verfahren vorgestellt. Deren Lösungen lassen sich auch in zeitabhängigen Gebieten durch ein einfaches einheitliches Postprocessing aus der Lösung des cGP-Problems bzw. dG-Problems erhalten. Für Problemstellungen in festen Gebieten und mit zeitlich konstanten Konvektions- und Reaktionstermen werden Stabilitätsresultate sowie optimale Fehlerabschätzungen für die nachbereiteten Lösungen der cGP-Verfahren und der dG-Verfahren angegeben. Für zeitabhängige Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängigen Gebieten präsentieren wir konservative und nicht-konservative Formulierungen, wobei eine besondere Aufmerksamkeit der Behandlung der Zeitableitung und der Gittergeschwindigkeit gilt. Stabilität und optimale Fehlerschätzungen für die in der Zeit semi-diskretisierten konservativen und nicht-konservativen Formulierungen werden vorgestellt. Abschließend wird das volldiskretisierte Problem betrachtet, wobei eine Finite-Elemente-Methode zur Ortsdiskretisierung der Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängigen Gebieten im ALE-Rahmen einbezogen wurde. Darüber hinaus wird eine lokale Projektionsstabilisierung (LPS) eingesetzt, um der Konvektionsdominanz Rechnung zu tragen. Weiterhin wird numerisch untersucht, wie sich die Approximation der Gebietsgeschwindigkeit auf die Genauigkeit der Zeitdiskretisierungsverfahren auswirkt.

Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations I: Method Formulation

In this article we consider the development of discontinuous Galerkin finite element methods for the numerical approximation of the compressible Navier-Stokes equations. For the discretization of the leading order terms, we propose employing the generalization of the symmetric version of the interior penalty method, originally developed for the numerical approximation of linear self-adjoint second-order elliptic partial differential equations. In order to solve the resulting system of nonlinear equations, we exploit a (damped) Newton-GMRES algorithm. Numerical experiments demonstrating the practical performance of the proposed discontinuous Galerkin method with higher-order polynomials are presented.

Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations II: Goal--Oriented A Posteriori Error Estimation

In this article we consider the application of the generalization of the symmetric version of the interior penalty discontinuous Galerkin finite element method to the numerical approximation of the compressible Navier--Stokes equations. In particular, we consider the a posteriori error analysis and adaptive mesh design for the underlying discretization method. Indeed, by employing a duality argument (weighted) Type I a posteriori bounds are derived for the estimation of the error measured in terms of general target functionals of the solution; these error estimates involve the product of the finite element residuals with local weighting terms involving the solution of a certain dual problem that must be numerically approximated. This general approach leads to the design of economical finite element meshes specifically tailored to the computation of the target functional of interest, as well as providing efficient error estimation. Numerical experiments demonstrating the performance of the proposed approach will be presented.