83 research outputs found
Discontinuous Galerkin Methods For The Boltzmann-Poisson Systems In Semiconductor Device Simulations
No full textWe are interested in the deterministic computation of the transients for the Boltzmann-Poisson system describing electron transport in semiconductor devices. The main difficulty of such computation arises from the very high dimensions of the model, making it necessary to use relatively coarse meshes and hence requiring the numerical solver to be stable and to have good resolution under coarse meshes. In this paper we consider the discontinuous Galerkin (DG) method, which is a finite element method using discontinuous piecewise polynomials as basis functions and numerical fluxes based on upwinding for stability, for solving the Boltzmann-Poisson system. In many situations, the deterministic DG solver can produce accurate solutions with equal or less CPU time than the traditional DSMC (Direct Simulation Monte Carlo) solvers. Numerical simulation results on a diode and a 2D double-gate MOSFET are given.Mathematic
Superconvergence and accuracy enhancement of discontinuous Galerkin solutions for Vlasov-Maxwell equations and Numerical Analysis of a Hybrid Method for Radiation Transport
No full textIn this thesis we will analyze and enhance two schemes for kinetic equations. Namely the discontinous Galerkin (DG) methods for solving the the Vlasov-Maxwell (VM) system and a hybrid method for solving the time-dependent radiation transport equation (RTE). In Chapter 2 we will consider the DG methods for solving the VM system, a fundamental model for collisionless magnetized plasma. The DG methods provide accurate numerical description with conservation and stability properties. However, to resolve the high dimensional probability distribution function, the computational cost is the main bottleneck even for modern-day supercomputers. The first part of this thesis studies the applicability of a post-processing technique to the DG solution to enhance its accuracy and resolution for the VM system. This postprocessor is applied at the final time of the simulation, and its cost is negligible, it succeeds by producing a high-resolution solution with the same cost of computing a low-resolution one, thus saving computational time in the process. In particular, we prove the superconvergence of order in the negative order norm for the probability distribution function and the electromagnetic fields when piecewise polynomial degree is used. Numerical tests including Landau damping, two-stream instability and streaming Weibel instabilities are considered showing the performance of the post-processor. This is based on joint work with Yingda Cheng, Juntao Huang and Jennyfer Ryan [1] In Chapter 3, we prove rigorous error estimates for a hybrid method introduced in [2] for solving the time-dependent RTE. The method relies on a splitting of the kinetic distribution function for the radiation into uncollided and collided components. A high-resolution method (in angle) is used to approximate the uncollided components and a low-resolution method is used to approximate the the collided component. After each time step, the kinetic distribution is reinitialized to be entirely uncollided. For this analysis, we consider a mono-energetic problem on a periodic domains, with constant material cross-sections of arbitrary size. We assume the uncollided equation is solved exactly and the collided part is approximated in angle via a spherical harmonic expansion (\Peqn method). Using a non-standard set of semi-norms, we obtain estimates of the form C(\e,\sigma,\dt)N^{-s} where denotes the regularity of the solution in angle, \e and are scattering parameters, \dt is the time-step before reinitialization, and is a complicated function of \e, , and \dt. These estimates involve analysis of the multiscale RTE that includes, but necessarily goes beyond, usual spectral analysis. We also compute error estimates for the monolithic \Peqn method with the same resolution as the collided part in the hybrid. Our results highlight the benefits of the hybrid approach over the monolithic discretization in both highly scattering and streaming regimes. This is based in a joint work with Cory D. Hauck and Victor Decaria [3]Thesis (Ph.D.)--Michigan State University. Applied Mathematics - Doctor of Philosophy, 2023Includes bibliographical reference
A discontinuous Galerkin solver for front propagation
No full textInternational audienceWe propose a new discontinuous Galerkin (DG) method based on [Cheng and Shu, JCP, 2007] to solve a class of Hamilton-Jacobi equations that arises from optimal control problems. These equations are connected to front propagation problems or minimal time problems with non isotropic dynamics. Several numerical experiments show the relevance of our method, in particular for front propagation
A discontinuous Galerkin solver for front propagation
No full textInternational audienceWe propose a new discontinuous Galerkin (DG) method based on [Cheng and Shu, JCP, 2007] to solve a class of Hamilton-Jacobi equations that arises from optimal control problems. These equations are connected to front propagation problems or minimal time problems with non isotropic dynamics. Several numerical experiments show the relevance of our method, in particular for front propagation
A discontinuous Galerkin scheme for front propagation with obstacles
No full textInternational audienceWe are interested in front propagation problems in the presence of obstacles. We extend a previous work (Bokanowski, Cheng and Shu, SIAM J. Scient. Comput., 2011), to propose a simple and direct discontinuous Galerkin (DG) method adapted to such front propagation problems. We follow the formulation of (Bokanowski, Forcadel and Zidani, SIAM J. Control Optim. 2010), leading to a level set formulation driven by , where is an obstacle function. The DG scheme is motivated by the variational formulation when the Hamiltonian is a linear function of , corresponding to linear convection problems in presence of obstacles. The scheme is then generalized to nonlinear equations, written in an explicit form. Stability analysis are performed for the linear case with Euler forward, a Heun scheme and a Runge-Kutta third order time discretization using the technique proposed in (Zhang and Shu, SIAM J. Control and Optim., 2010). Several numerical examples are provided to demonstrate the robustness of the method. Finally, a narrow band approach is considered in order to reduce the computational cost
STATISTICAL SIGNAL PROCESSING APPROACHES FOR MULTI-REFERENCE ALIGNMENT AND NEURAL TEXTURE SYNTHESIS
No full textStatistical signal processing plays a crucial role in numerous fields of modern technology and science. Some of the important applications include extracting signals from noisy data, processing images and videos for tasks like compression and enhancement, and analyzing time-varying data, such as climate data and asset prices.In this dissertation, we address two problems related to statistical signal/image processing.The first issue involves a generalized version of the multi-reference alignment problem in one dimension, inspired by modern data applications such as cryo-electron microscopy. The objective is to recover an unknown signal f : R \u2192 R from multiple observations that have been translated, dilated, and corrupted by additive noise. In the presence of large dilations and corruptions, observational data do not resemble the underlying signal. Although current approaches in the field have shown empirical success in the absence of dilations, no approach has successfully provided convergence guarantees for signal inversion while dilating, translating, and corrupting observational data all at once. Thus, we propose an unbiased estimator for the bispectrum of the unknown signal which depends on the corrupted samples and knowledge of the dilation distribution. To validate our proposed estimator, we use it for bispectrum recovery, and invert the recovered bispectrum to achieve full signal inversion. The second problem concerns neural texture synthesis, which is important for understanding how humans perceive texture. Current approaches require regularization terms or some type of supervision to capture long range constraints, such as the alignment of bricks, in images. To remedy this issue, we propose a new set of statistics for examplar-based neural texture synthesis based on Sliced Wasserstein Loss, and augment our proposed algorithm with a multi-scale synthesis process. Based on qualitative and quantitative experiments, our results are comparable or better than current state of the art methodsThesis (Ph.D.)--Michigan State University. Applied Mathematics - Doctor of Philosophy, 2023Includes bibliographical reference
An adaptive multiresolution ultra-weak discontinuous Galerkin method for nonlinear Schrödinger equations
No full textPreprint from available from arXiv. Also available from publisher.This paper develops a high-order adaptive scheme for solving nonlinear Schrödinger equations. The solutions to such equations often exhibit solitary wave and local structures, which make adaptivity essential in improving the simulation efficiency. Our scheme uses the ultra-weak discontinuous Galerkin (DG) formulation and belongs to the framework of adaptive multiresolution schemes. Various numerical experiments are presented to demonstrate the excellent capability of capturing the soliton waves and the blow-up phenomenon.Y. Liu: Research supported in part by a grant from the Simons Foundation (426993, Yuan Liu). W. Guo: Research is supported by NSF grant DMS-1830838. Y. Cheng: Research is supported by NSF grants DMS-1453661 and DMS-1720023. Z. Tao: Research is supported by NSFC Grant 12001231
Discontinuous Galerkin methods for Vlasov-type systems
No full textNon UBCUnreviewedAuthor affiliation: Michigan State UniversityFacult
High frequency computation in wave equations and optimal design for a cavity
No full textTwo types of problems are studied in this thesis.One part of the thesis is devoted to high frequency computation. Motivated by fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beam methods which were originally designed for initial value problems of wave equations in the high frequency regime, we develop fast multiscale Gaussian beam methods for wave equations in bounded convex domains in the high frequency regime. To compute the wave propagation in bounded convex domains, we have to take into account reflecting multiscale Gaussian beams, which are accomplished by enforcing reflecting boundary conditions during beam propagation and carrying out suitable reflecting beam summation. To propagate multiscale beams efficiently, we prove that the ratio of the squared magnitude of beam amplitude and the beam width is roughly conserved, and accordingly we propose an effective indicator to identify significant beams. We also prove that the resulting multiscale Gaussian beam methods converge asymptotically. Numerical examples demonstrate the accuracy and efficiency of the method.The second part of the thesis studies the reduction of backscatter radar cross section (RCS) for a cavity embedded in the ground plane. One approach for RCS reduction is through the coating material. Assume the bottom of the cavity is coated by a thin, multilayered radar absorbing material (RAM) with possibly different permittivities. The objective is to minimize the backscatter RCS by the incidence of a plane wave over a single or a set of incident angles and frequencies. By formulating the scattering problem as a Helmholtz equation with artificial boundary condition, the gradient with respect to the material permittivities is determined efficiently by the adjoint state method, which is integrated into a nonlinear optimization scheme. Numerical example shows the RCS may be significantly reduced.Another approach is through shape optimization. By introducing a transparent boundary condition, the unbounded scattering problem from a cavity is reduced to a bounded domain problem. RCS reduction for the cavity is formulated as a shape optimization problem involving the Helmholtz equation. The existence of the minimizer is proved under an appropriate constraint. Descent directions of the objective function with respect to the boundary may be found via the domain derivative. It is used in a gradient-based optimization scheme to find the optimal shape of the cavity. Numerical examples show that the RCS is effectively reduced at different incident frequencies.Thesis (Ph. D.)--Michigan State University. Applied Mathematics, 2013Includes bibliographical references (pages 145-149
ALGEBRAIC TOPOLOGY AND GRAPH THEORY BASED APPROACHES FOR PROTEIN FLEXIBILITY ANALYSIS AND B FACTOR PREDICTION
No full textProtein fluctuation, measured by B factors, has been shown to highly correlate to protein flexibility and function. Several methods have been developed to predict protein B factoras well as related applications such as docking pose ranking, domain separation, entropycalculation, hinge detection, hot spot detection, stability analysis, etc. While many B factormethods exist, reliable B factor prediction continues to be an ongoing challenge and there ismuch room for improvement.This work introduces a paradigm shifting geometric graph based model called the multi-scale weighted colored graph (MWCG) model. The MWCG model is a new generation of computational algorithms that signicantly improves the current landscape of protein struc-tural fluctuation analysis. The MWCG model treats each protein as a colored graph where colored nodes correspond to atomic element types and edges are weighted by a generalized centrality metric. Each graph contains multiple subgraphs based on interaction typesbetween graphic nodes, then protein rigidity is represented by generalized centralities of subgraphs. MWCGs predict the B factors of protein residues and accurately analyze the flexibility of all atoms in a protein simultaneously. The MWCG model presented in thiswork captures element specific interactions across multiple scales and is a novel visual tool for identifying various protein secondary structures. This work also demonstrates MWCG protein hinge detection using a variety of proteins.Cross protein prediction of protein B factors has previously been an unsolved problem in terms of B factor prediction methods. Since many proteins are dicult to crystallize, and for some it is likely impossible, models that can cross predict protein B factor are absolutelynecessary. By integrating machine learning and the advanced graph theory MWCG method, this work provides a robust cross protein B factor prediction solution using a set of known proteins to predict the B factors of a protein previously unseen to the algorithm. Thealgorithm connects different proteins using global protein features such as the resolution of the X-ray crystallography data. The combination of global and local features results in successful cross protein B factor prediction. To test and validate these results this work considers several machine learning approaches such as random forest, gradient boosted trees, and deep convolutional neural networks.Recently, persistent homology has had tremendous success in biomolecular data analysis. It works by examining the topological relationship or connectivity of a group of atoms in a molecule at a variety of scales, then rendering a family of topological representations of the molecule. However, persistent homology is rarely employed for the analysis of atomic properties, such as biomolecular flexibility analysis or B factor prediction. This work introduces atom specific persistent homology (ASPH) to provide a local atomic level representation of a molecule via a global topological tool. This is achieved through the construction of a pair of conjugated sets of atoms and corresponding conjugated simplicial complexes, as well as conjugated topological spaces. The difference between the topological invariants of the pair of conjugated sets is measured by Bottleneck and Wasserstein metrics and leads to anatom specic topological representation of individual atomic properties in a molecule. Atom specific topological features are integrated with various machine learning algorithms, including gradient boosting trees and convolutional neural network for protein thermal fluctuation analysis and blind cross protein B factor prediction.Extensive numerical testing indicates the proposed methods provide novel and powerful graph theory and algebraic topology based tools for analyzing and predicting atom specific, localized protein flexibility information.Thesis (Ph.D.)--Michigan State University. Mathematics - Doctor of Philosophy, 2019Includes bibliographical reference
- …
