Accelerated Discontinuous Galerkin Solvers with the Chebyshev Iterative Method on the Graphics Processing Unit

This work demonstrates implementations of the discontinuous Galerkin (DG) method on graphics processing units (GPU), which deliver improved computational time compared to the conventional central processing unit (CPU). The linear system developed when applying the DG method to an elliptic problem is solved using the GPU. The conjugate gradient (CG) method and the Chebyshev iterative method are the linear system solvers that are compared, to see which is more efficient when computing with the CPU's parallel architecture. When applying both methods, computational times decreased for large problems executed on the GPU compared to CPU; however, CG is the more efficient method compared to the Chebyshev iterative method. In addition, a constant-free upper bound for the DC spectrum applied to the elliptic problem is developed. Few previous works combine the DG method and the GPU. This thesis will provide useful guidelines for the numerical solution of elliptic problems using DG on the GPU

Investigation of general-purpose computing on graphics processing units and its application to the finite element analysis of electromagnetic problems

In this dissertation, the hardware and API architectures of GPUs are investigated, and the corresponding acceleration techniques are applied on the traditional frequency domain finite element method (FEM), the element-level time-domain methods, and the nonlinear discontinuous Galerkin method. First, the assembly and the solution phases of the FEM are parallelized and mapped onto the granular GPU processors. Efficient parallelization strategies for the finite element matrix assembly on a single GPU and on multiple GPUs are proposed. The parallelization strategies for the finite element matrix solution, in conjunction with parallelizable preconditioners are investigated to reduce the total solution time. Second, the element-level dual-field domain decomposition (DFDD-ELD) method is parallelized on GPU. The element-level algorithms treat each finite element as a subdomain, where the elements march the fields in time by exchanging fields and fluxes on the element boundary interfaces with the neighboring elements. The proposed parallelization framework is readily applicable to similar element-level algorithms, where the application to the discontinuous Galerkin time-domain (DGTD) methods show good acceleration results. Third, the element-level parallelization framework is further adapted to the acceleration of nonlinear DGTD algorithm, which has potential applications in the field of optics. The proposed nonlinear DGTD algorithm describes the third-order instantaneous nonlinear effect between the electromagnetic field and the medium permittivity. The Newton-Raphson method is incorporated to reduce the number of nonlinear iterations through its quadratic convergence. Various nonlinear examples are presented to show the different Kerr effects observed through the third-order nonlinearity. With the acceleration using MPI+GPU under large cluster environments, the solution times for the various linear and nonlinear examples are significantly reduced

Fast Radiative-Transfer-Equation-Based Image Reconstruction Algorithms for Non-Contact Diffuse Optical Tomography Systems

It is well known that the radiative transfer equation (RTE) is the most accurate deterministic light propagation model employed in diffuse optical tomography (DOT). RTE-based algorithms provide more accurate tomographic results than codes that rely on the diffusion equation (DE), which is an approximation to the RTE, in scattering dominant media. However, RTE based DOT (RTE-DOT) has limited utility in practice due to its high computational cost and lack of support for general non-contact imaging systems. In this dissertation, I developed fast reconstruction algorithms for RTE-based DOT (RTE-DOT), which consists of three independent components: an efficient linear solver for forward problems, an improved optimization solver for inverse problem, and the first light propagation model in free space that fully considers the angular dependency, which can provide a suitable measurement operator for RTE-DOT. This algorithm is validated and evaluated with numerical experiments and clinical data. According to these studies, the novel reconstruction algorithm is up to 30 times faster than traditional reconstruction techniques, while achieving comparable reconstruction accuracy

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described