A class of robust numerical methods for solving dynamical systems with multiple time scales

In this paper, we develop a class of robust numerical methods for solving dynamical systems with multiple time scales. We first represent the solution of a multiscale dynamical system as a transformation of a slowly varying solution. Then, under the scale separation assumption, we provide a systematic way to construct the transformation map and derive the dynamic equation for the slowly varying solution. We also provide the convergence analysis of the proposed method. Finally, we present several numerical examples, including ODE system with three and four separated time scales to demonstrate the accuracy and efficiency of the proposed method. Numerical results verify that our method is robust in solving ODE systems with multiple time scale, where the time step does not depend on the multiscale parameters

Tip-enhanced Raman spectroscopy of strained semiconductor nanostructures

Raman spectroscopy can serve as a powerful tool to probe the vibrational modes of solid state materials. By taking advantage of the enhanced electric fields caused by the surface-enhanced plasmon resonance of a noble metal coated atomic force microscopy tip, tip-enhanced Raman spectroscopy can dramatically increase local signal intensity and measurement spatial resolution. In this dissertation, work is presented on conventional and tip-enhanced Raman measurements of various semiconductor nanostructures with a specific focus on analyzing strain and strain related properties in these material systems. We use tip-enhanced Raman to study Ge-Si₀.₅Ge₀.₅ core-shell nanowires where we observe two distinct Ge-Ge mode Raman peaks that are affected by strain in the core-shell structure. Tip-enhanced measurements show dramatically increased sensitivity to the modes at the interface between the core and shell and a shift in position of this mode due to plasmonic field localization at the tip apex and the resulting change in phonon self-energy caused by increased coupling between phonons and intervalence-band carrier transitions. We also use tip-enhanced Raman spectroscopy to characterize unstrained and strained MoS₂ and show spatial resolution of approximately 100 nm in the measurements. The strain dependence of the second order Raman modes in MoS₂ reveals changes in the electronic band structure in strained MoS₂ that are manifested through changes in the Raman peak positions and peak area ratios, which are corroborated through density functional theory calculations. Finally, we use conventional Raman spectroscopy to probe uniaxially strained monolayer and three-layer WSe₂. Using mechanical modeling of strain in atomically thin WSe₂ on a stretched elastic substrate, we confirm complete transfer of strain from the substrate to the WSe₂ flakes and analyze the evolution of the Raman spectra with applied uniaxial strain above 1 percent. These studies enable us to experimentally determine the strain induced Raman shift for various Raman modes and to calculate the Grüneisen parameter and strain deformation potential for the first order in-plane Raman mode, with experimental values confirmed with theoretical values calculated using density functional theory.Electrical and Computer Engineerin

A DeepParticle method for learning and generating aggregation patterns in multi-dimensional Keller-Segel chemotaxis systems

We study a regularized interacting particle method for computing aggregation patterns and near singular solutions of a Keller-Segal (KS) chemotaxis system in two and three space dimensions, then further develop DeepParticle (DP) method to learn and generate solutions under variations of physical parameters. The KS solutions are approximated as empirical measures of particles which self-adapt to the high gradient part of solutions. We utilize the expressiveness of deep neural networks (DNNs) to represent the transform of samples from a given initial (source) distribution to a target distribution at finite time T prior to blowup without assuming invertibility of the transforms. In the training stage, we update the network weights by minimizing a discrete 2-Wasserstein distance between the input and target empirical measures. To reduce computational cost, we develop an iterative divide-and-conquer algorithm to find the optimal transition matrix in the Wasserstein distance. We present numerical results of DP framework for successful learning and generation of KS dynamics in the presence of laminar and chaotic flows. The physical parameter in this work is either the small diffusivity of chemo-attractant or the reciprocal of the flow amplitude in the advection-dominated regime

A Novel Stochastic Interacting Particle-Field Algorithm for 3D Parabolic-Parabolic Keller-Segel Chemotaxis System

We introduce an efficient stochastic interacting particle-field (SIPF) algorithm with no history dependence for computing aggregation patterns and near singular solutions of parabolic-parabolic Keller-Segel (KS) chemotaxis system in three space dimensions (3D). The KS solutions are approximated as empirical measures of particles coupled with a smoother field (concentration of chemo-attractant) variable computed by the spectral method. Instead of using heat kernels causing history dependence and high memory cost, we leverage the implicit Euler discretization to derive a one-step recursion in time for stochastic particle positions and the field variable based on the explicit Green's function of an elliptic operator of the form Laplacian minus a positive constant. In numerical experiments, we observe that the resulting SIPF algorithm is convergent and self-adaptive to the high gradient part of solutions. Despite the lack of analytical knowledge (e.g. a self-similar ansatz) of the blowup, the SIPF algorithm provides a low-cost approach to study the emergence of finite time blowup in 3D by only dozens of Fourier modes and through varying the amount of initial mass and tracking the evolution of the field variable. Notably, the algorithm can handle at ease multi-modal initial data and the subsequent complex evolution involving the merging of particle clusters and formation of a finite time singularity

A class of robust numerical methods for solving dynamical systems with multiple time scales

In this paper, we develop a class of robust numerical methods for solving dynamical systems with multiple time scales. We first represent the solution of a multiscale dynamical system as a transformation of a slowly varying solution. Then, under the scale separation assumption, we provide a systematic way to construct the transformation map and derive the dynamic equation for the slowly varying solution. We also provide the convergence analysis of the proposed method. Finally, we present several numerical examples, including ODE system with three and four separated time scales to demonstrate the accuracy and efficiency of the proposed method. Numerical results verify that our method is robust in solving ODE systems with multiple time scale, where the time step does not depend on the multiscale parameters