Doctor of Philosophy

dissertationSmoothness-increasing accuracy-conserving (SIAC) filters were introduced as a class of postprocessing techniques to ameliorate the quality of numerical solutions of discontinuous Galerkin (DG) simulations. SIAC filtering works to eliminate the oscillations in the error by introducing smoothness back to the DG field and raises the accuracy in the L2-n o rm up to its natural superconvergent accuracy in the negative-order norm. The increased smoothness in the filtered DG solutions can then be exploited by simulation postprocessing tools such as streamline integrators where the absence of continuity in the data can lead to erroneous visualizations. However, lack of extension of this filtering technique, both theoretically and computationally, to nontrivial mesh structures along with the expensive core operators have been a hindrance towards the application of the SIAC filters to more realistic simulations. In this dissertation, we focus on the numerical solutions of linear hyperbolic equations solved with the discontinuous Galerkin scheme and provide a thorough analysis of SIAC filtering applied to such solution data. In particular, we investigate how the use of different quadrature techniques could mitigate the extensive processing required when filtering over the whole computational field. Moreover, we provide detailed and efficient algorithms that a numerical practitioner requires to know in order to implement this filtering technique effectively. In our first attempt to expand the application scope of this filtering technique, we demonstrate both mathematically and through numerical examples that it is indeed possible to observe SIAC filtering characteristics when applied to numerical solutions obtained over structured triangular meshes. We further provide a thorough investigation of the interplay between mesh geometry and filtering. Building upon these promising results, we present how SIAC filtering could be applied to gain higher accuracy and smoothness when dealing with totally unstructured triangular meshes. Lastly, we provide the extension of our filtering scheme to structured tetrahedral meshes. Guidelines and future work regarding the application of the SIAC filter in the visualization domain are also presented. We further note that throughout this document, the terms postprocessing and filtering will be used interchangeably

Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering

Discontinuous Galerkin (DG) methods are a popular class of numerical techniques to solve partial differential equations due to their higher order of accuracy. However, the inter-element discontinuity of a DG solution hinders its utility in various applications, including visualization and feature extraction. This shortcoming can be alleviated by postprocessing of DG solutions to increase the inter-element smoothness. A class of postprocessing techniques proposed to increase the inter-element smoothness is SIAC filtering. In addition to increasing the inter-element continuity, SIAC filtering also raises the convergence rate from order k+1k+1 to order 2k+12k+1 . Since the introduction of SIAC filtering for univariate hyperbolic equations by Cockburn et al. (Math Comput 72(242):577–606, 2003), many generalizations of SIAC filtering have been proposed. Recently, the idea of dimensionality reduction through rotation has been the focus of studies in which a univariate SIAC kernel has been used to postprocess a two-dimensional DG solution (Docampo-Sánchez et al. in Multi-dimensional filtering: reducing the dimension through rotation, 2016. arXiv preprint arXiv:1610.02317). However, the scope of theoretical development of multidimensional SIAC filters has never gone beyond the usage of tensor product multidimensional B-splines or the reduction of the filter dimension. In this paper, we define a new SIAC filter called hexagonal SIAC (HSIAC) that uses a nonseparable class of two-dimensional spline functions called hex splines. In addition to relaxing the separability assumption, the proposed HSIAC filter provides more symmetry to its tensor-product counterpart. We prove that the superconvergence property holds for a specific class of structured triangular meshes using HSIAC filtering and provide numerical results to demonstrate and validate our theoretical results

Smoothness-Increasing Accuracy-Conserving (SIAC) filtering and quasi interpolation: A unified view

Filtering plays a crucial role in postprocessing and analyzing data in scientific and engineering applications. Various application-specific filtering schemes have been proposed based on particular design criteria. In this paper, we focus on establishing the theoretical connection between quasi-interpolation and a class of kernels (based on B-splines) that are specifically designed for the postprocessing of the discontinuous Galerkin (DG) method called Smoothness-Increasing Accuracy-Conserving (SIAC) filtering. SIAC filtering, as the name suggests, aims to increase the smoothness of the DG approximation while conserving the inherent accuracy of the DG solution (superconvergence). Superconvergence properties of SIAC filtering has been studied in the literature. In this paper, we present the theoretical results that establish the connection between SIAC filtering to long-standing concepts in approximation theory such as quasi-interpolation and polynomial reproduction. This connection bridges the gap between the two related disciplines and provides a decisive advancement in designing new filters and mathematical analysis of their properties. In particular, we derive a closed formulation for convolution of SIAC kernels with polynomials. We also compare and contrast cardinal spline functions as an example of filters designed for image processing applications with SIAC filters of the same order, and study their properties

Multi-dimensional filtering: Reducing the dimension through rotation

Over the past few decades there has been a strong effort towards the development of Smoothness-Increasing Accuracy-Conserving (SIAC) filters for Discontinuous Galerkin (DG) methods, designed to increase the smoothness and improve the convergence rate of the DG solution through this post-processor. These advantages can be exploited during flow visualization, for example by applying the SIAC filter to the DG data before streamline computations [Steffan et al., IEEE-TVCG 14(3): 680-692]. However, introducing these filters in engineering applications can be challenging since a tensor product filter grows in support size as the field dimension increases, becoming computationally expensive. As an alternative, [Walfisch et al., JOMP 38(2);164-184] proposed a univariate filter implemented along the streamline curves. Until now, this technique remained a numerical experiment. In this paper we introduce the line SIAC filter and explore how the orientation, structure and filter size affect the order of accuracy and global errors. We present theoretical error estimates showing how line filtering preserves the properties of traditional tensor product filtering, including smoothness and improvement in the convergence rate. Furthermore, numerical experiments are included, exhibiting how these filters achieve the same accuracy at significantly lower computational costs, becoming an attractive tool for the scientific visualization community

On the Treatment of Field Quantities and Elemental Continuity in FEM Solutions

As the finite element method (FEM) and the finite volume method (FVM), both traditional and high-order variants, continue their proliferation into various applied engineering disciplines, it is important that the visualization techniques and corresponding data analysis tools that act on the results produced by these methods faithfully represent the underlying data. To state this in another way: the interpretation of data generated by simulation needs to be consistent with the numerical schemes that underpin the specific solver technology. As the verifiable visualization literature has demonstrated: visual artifacts produced by the introduction of either explicit or implicit data transformations, such as data resampling, can sometimes distort or even obfuscate key scientific features in the data. In this paper, we focus on the handling of elemental continuity, which is often only C0 continuous or piecewise discontinuous, when visualizing primary or derived fields from FEM or FVM simulations. We demonstrate that traditional data handling and visualization of these fields introduce visual errors. In addition, we show how the use of the recently proposed line-SIAC filter provides a way of handling elemental continuity issues in an accuracy-conserving manner with the added benefit of casting the data in a smooth context even if the representation is element discontinuous

Divided difference estimates and accuracy enhancement of discontinuous Galerkin methods for nonlinear symmetric systems of hyperbolic conservation laws

In this paper, we investigate the accuracy-enhancement for the discontinuous Galerkin (DG) method for solving one-dimensional nonlinear symmetric systems of hyperbolic conservation laws. For nonlinear equations, the divided difference estimate is an important tool that allows for superconvergence of the post-processed solutions in the local L2-norm. Therefore, we first prove that the L2-norm of the α-th order (1≤ α≤ k+1) divided difference of the DG error with upwind fluxes is of order k+(3-α)/2, provided that the flux Jacobian matrix, f'(u), is symmetric positive definite. Furthermore, using the duality argument, we are able to derive superconvergence estimates of order 2k+(3-α)/2 for the negative-order norm, indicating that some particular compact kernels can be used to extract at least (3k/2+1)-th order superconvergence for nonlinear systems of conservation laws. Numerical experiments are shown to demonstrate the theoretical results

Doctor of Philosophy

dissertationMemory access irregularities are a major bottleneck for bandwidth limited problems on Graphics Processing Unit (GPU) architectures. GPU memory systems are designed to allow consecutive memory accesses to be coalesced into a single memory access. Noncontiguous accesses within a parallel group of threads working in lock step may cause serialized memory transfers. Irregular algorithms may have data-dependent control flow and memory access, which requires runtime information to be evaluated. Compile time methods for evaluating parallelism, such as static dependence graphs, are not capable of evaluating irregular algorithms. The goals of this dissertation are to study irregularities within the context of unstructured mesh and sparse matrix problems, analyze the impact of vectorization widths on irregularities, and present data-centric methods that improve control flow and memory access irregularity within those contexts. Reordering associative operations has often been exploited for performance gains in parallel algorithms. This dissertation presents a method for associative reordering of stencil computations over unstructured meshes that increases data reuse through caching. This novel parallelization scheme offers considerable speedups over standard methods. Vectorization widths can have significant impact on performance in vectorized computations. Although the hardware vector width is generally fixed, the logical vector width used within a computation can range from one up to the width of the computation. Significant performance differences can occur due to thread scheduling and resource limitations. This dissertation analyzes the impact of vectorization widths on dense numerical computations such as 3D dG postprocessing. It is difficult to efficiently perform dynamic updates on traditional sparse matrix formats. Explicitly controlling memory segmentation allows for in-place dynamic updates in sparse matrices. Dynamically updating the matrix without rebuilding or sorting greatly improves processing time and overall throughput. This dissertation presents a new sparse matrix format, dynamic compressed sparse row (DCSR), which allows for dynamic streaming updates to a sparse matrix. A new method for parallel sparse matrix-matrix multiplication (SpMM) that uses dynamic updates is also presented