Visualization

Polynomial chaos methods (and generalized polynomial chaos methods) have been extensively applied to analyze PDEs that contain uncertainties. However this approach is rarely applied to hyperbolic systems. In this paper we analyze the properties of the resulting deterministic system of equations obtained by stochastic Galerkin projection. We consider a simple model of a scalar wave equation with random wave speed. We show that when uncertainty causes the change of characteristic directions, the resulting deterministic system of equations is a symmetric hyperbolic system with both positive and negative eigenvalues. A consistent method of imposing the boundary conditions is proposed and its convergence is established. Numerical examples are presented to support the analysis.

Keywords: Generalized polynomial chaos, stochastic PDE, Galerkin method, hyperbolic equation, uncertainty quantification

The method of Moving Least Squares (MLS) is a popular framework for reconstructing continuous functions from scattered data due to its rich mathematical properties and well-understood theoretical foundations. This paper applies MLS to volume rendering, providing a unified mathematical framework for ray casting of scalar data stored over regular as well as irregular grids. We use the MLS reconstruction to render smooth isosurfaces and to compute accurate derivatives for high-quality shading effects. We also present a novel, adaptive preintegration scheme to improve the efficiency of the ray casting algorithm by reducing the overall number of function evaluations, and an efficient implementation of our framework exploiting modern graphics hardware. The resulting system enables high-quality volume integration and shaded isosurface rendering for regular and irregular volume data.

A ubiquitous requirement in many mathematical and computational problems is a set of well-placed point samples. For producing very even distributions of samples across complex surfaces, a dynamic particle system is a controllable mechanism that naturally accommodates strict sampling requirements. The systemfirst constrains particles to a surface, and then moves the particles across the surface until they are arranged in minimal energy configurations. Adaptivity is added into the system by scaling the distance between particles, causing higher densities of points around surface features. In this dissertation we explore and refine the dynamics of particle systems for generating efficient and adaptive point samples of implicit surfaces.

Throughout this dissertation, we apply the adaptive particle system framework to several application areas. First, efficient visualizations of high-order finite element datasets are generated by developing adaptivity metrics of surfaces that exist in the presence of curvilinear coordinate transformation. Second, a framework is proposed that meets fundamental sampling constraints of Delaunay-based surface reconstruction algorithms. In meeting these constraints, the particle distributions produce nearly-regular, efficient isosurface tessellation that are geometrically and topologically accurate. And third, a novel analytic representation of material boundaries in multimaterial volume datasets is developed, as well as a set of projection operators, that allow for explicit sampling of nonmanifold material intersections. Using a tetrahedral labeling algorithm, the material intersections are extracted as watertight, nonmanifold meshes that are well-suited for simulations.

With the continual increase in computing power, volumetric datasets with sizes ranging from only a few megabytes to petascale are generated thousands of times per day. Such data may come from an ordinary source such as simple everyday medical imaging procedures, while larger datasets may be generated from cluster-based scientific simulations or measurements of large scale experiments. In computer science an incredible amount of work worldwide is put into the efficient visualization of these datasets. As researchers in the field of scientific visualization, we often have to face the task of handling very large data from various sources. This data usually comes in many different data formats. In medical imaging, the DICOM standard is well established, however, most research labs use their own data formats to store and process data. To simplify the task of reading the many different formats used with all of the different visualization programs, we present a system for the efficient handling of many types of large scientific datasets (see Figure 1 for just a few examples). While primarily targeted at structured volumetric data, UVF can store just about any type of structured and unstructured data. The system is composed of a file format specification with a reference implementation of a reader. It is not only a common, easy to implement format but also allows for efficient rendering of most datasets without the need to convert the data in memory.

Visualization is often employed as part of the simulation science pipeline, it's the window through which scientists examine their data for deriving new science, and the lens used to view modeling and discretization interactions within their simulations. We advocate that as a component of the simulation science pipeline, visualization must be explicitly considered as part of the validation and verification (V&V) process. In this article, the authors define V&V in the context of computational science, discuss the role of V&V in the scientific process, and present arguments for the need for verifiable visualization.