Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference (CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size $h$ and time step $\tau$ as well as the small parameter $0<\varepsilon\le 1$. Based on the error bound, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the CNFD method requests the $\varepsilon$-scalability: $\tau=O(\varepsilon^3)$ and $h=O(\sqrt{\varepsilon})$. Then we propose and analyze two numerical methods for the discretization of the nonlinear Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability is improved to $\tau=O(\varepsilon^2)$ and $h=O(1)$ when $0<\varepsilon\ll 1$ compared with the CNFD method. Extensive numerical results are reported to confirm our error estimates.Comment: 35 pages. 1 figure. arXiv admin note: substantial text overlap with arXiv:1504.0288

A uniformly accurate (UA) multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime

We propose and rigourously analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation with a dimensionless parameter $\varepsilon\in(0,1]$ which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the solution exhibits highly oscillatory propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. Due to the rapid temporal oscillation, it is quite challenging in designing and analyzing numerical methods with uniform error bounds in $\varepsilon\in(0,1]$. We present the MTI-FP method based on properly adopting a multiscale decomposition of the solution of the Dirac equation and applying the exponential wave integrator with appropriate numerical quadratures. By a careful study of the error propagation and using the energy method, we establish two independent error estimates via two different mathematical approaches as $h^{m_0}+\frac{\tau^2}{\varepsilon^2}$ and $h^{m_0}+\tau^2+\varepsilon^2$, where $h$ is the mesh size, $\tau$ is the time step and $m_0$ depends on the regularity of the solution. These two error bounds immediately imply that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $O(\tau)$ for all $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regimes when either $\varepsilon=O(1)$ or $0<\varepsilon\lesssim \tau$. Numerical results are reported to demonstrate that our error estimates are optimal and sharp. Finally, the MTI-FP method is applied to study numerically the convergence rates of the solution of the Dirac equation to those of its limiting models when $\varepsilon\to0^+$.Comment: 25 pages, 1 figur

Thermal analysis of high-bandwidth and energy-efficient 980 nm VCSELs with optimized quantum well gain peak-to-cavity resonance wavelength offset

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Appl. Phys. Lett. 111, 243508 (2017) and may be found at https://doi.org/10.1063/1.5003288.The static and dynamic performance of vertical-cavity surface-emitting lasers (VCSELs) used as light-sources for optical interconnects is highly influenced by temperature. We study the effect of temperature on the performance of high-speed energy-efficient 980 nm VCSELs with a peak wavelength of the quantum well offset to the wavelength of the fundamental longitudinal device cavity mode so that they are aligned at around 60 °C. A simple method to obtain the thermal resistance of the VCSELs as a function of ambient temperature is described, allowing us to extract the active region temperature and the temperature dependence of the dynamic and static parameters. At low bias currents, we can see an increase of the −3 dB modulation bandwidth f−3dB with increasing active region temperature, which is different from the classically known situation. From the detailed analysis of f−3dB versus the active region temperature, we obtain a better understanding of the thermal limitations of VCSELs, giving a basis for next generation device designs with improved temperature stability

Studying Spread Patterns of COVID-19 based on Spatiotemporal Data

The current COVID-19 epidemic have transformed every aspect of our lives, especially our behavior and routines. These changes have been drastically impacting the economy in each region, such as local restaurants and transportation systems. With massive amounts of ambient data being collected everywhere, we now can develop innovative algorithms to have a much greater understanding of epidemic spread patterns of COVID-19 based on spatiotemporal data. The findings will open up the possibility to design adaptive planning or scheduling systems that will help preventing the spread of COVID-19 and other infectious diseases. In this tutorial, we will review the trending state-of-theart machine learning techniques to model epidemic spread patterns with spatiotemporal data. These techniques are organized from two aspects: (1) providing a comprehensive review of recent studies about human routine behavior modeling, such as inverse reinforcement learning and graph neural network, and the impacts of behaviors on the spread patterns of infectious diseases based on GPS data; (2) introducing the existing literature on using remote sensing data to monitor the spatiotemporal pattern of the epidemic spread. Under current epidemic with unknown lasting time, we believe that modeling the spread patterns of COVID-19 epidemic is an important topic that will benefit to researchers and practitioners from both academia and industry

NUMERICAL METHODS AND COMPARISONS FOR THE DIRAC EQUATION IN THE NONRELATIVISTIC LIMIT REGIME

Ph.DDOCTOR OF PHILOSOPH

Physics-guided Machine Learning for Scientific Knowledge Discovery

Machine learning (ML) has found immense success in commercial applications such as computer vision and natural language processing. Given the success of ML in commercial domains, there is an increasing interest in using ML models for advancing scientific discovery. However, direct application of ``black-box" ML models has met with limited success in scientific domains given that the data available for many scientific problems is far smaller than what is needed to effectively train advanced ML models. Additional challenge arises due to the data non-stationarity in space and time. In the absence of adequate information about the physical mechanisms of real-world processes, ML approaches are prone to false discoveries of patterns which look deceptively good on training data but cannot generalize to unseen scenarios

Ultra-bright Photoactivatable Fluorophores Created by Reductive Caging

Sub-diffraction-limit imaging can be achieved by sequential localization of photoactivatable fluorophores, where the image resolution depends on the number of photons detected per localization. Here, we report a strategy for fluorophore caging that creates photoactivatable probes with high photon yields. Upon photoactivation, these probes can provide 104–106 photons per localization and allow imaging of fixed samples with resolutions of several nanometers. This strategy can be applied to many fluorophores across the visible spectrum