SCIENTIFIC COMPUTING AND IMAGING INSTITUTE
at the University of Utah

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

Particle-based simulation methods are used to model a wide range of complex phenomena and to solve time-dependent problems of various scales. Effective visualizations of the resulting state will communicate subtle changes in the three-dimensional structure, spatial organization, and qualitative trends within a simulation as it evolves. We present two algorithms targeting upcoming, highly parallel multicore desktop systems to enable interactive navigation and exploration of large particle data sets with global illumination effects. Monte Carlo path tracing and texture mapping are used to capture computationally expensive illumination effects such as soft shadows and diffuse interreflection. The first approach is based on precomputation of luminance textures and removes expensive illumination calculations from the interactive rendering pipeline. The second approach is based on dynamic luminance texture generation and decouples interactive rendering from the computation of global illumination effects. These algorithms provide visual cues that enhance the ability to perform analysis and feature detection tasks while interrogating the data at interactive rates. We explore the performance of these algorithms and demonstrate their effectiveness using several large data sets.

Keywords: Interactive particle visualization, Global illumination, Ray tracing

In this thesis, adjoint error estimation techniques are applied to complex elastohydrodynamic lubrication (EHL) problems. A functional is introduced, namely the friction, and justification is provided as to why this quantity, and hence its accuracy, is important. An iterative approach has been taken to develop understanding of the mechanisms at work. A series of successively complex cases are proposed, each with adjoint error estimation techniques applied to them. The first step is built up from a model free boundary problem, where the cavitation condition is captured correctly using a sliding mesh. The next problem tackled is a hydrodynamic problem, where non-linear viscosity and density are introduced. Finally, a full EHL line contact problem is introduced, where the surface deforms elastically under pressure. For each case presented, an estimate of a finer mesh friction, calculated from solutions obtained only on a coarse mesh, is corrected according to the adjoint error estimation technique. At each stage, care is taken to ensure that the error estimate is computed accurately when compared against the measured error in the friction.

Non-uniform meshes are introduced for the model free boundary problem. These nonuniform meshes are shown to give the same excellent predictions of the error as uniform meshes. Adaptive refinement is undertaken, with the mesh being refined using the adjoint error estimate. Results for this are presented for both the model free-boundary problem and the full EHL problem. This is shown to enable the accurate calculation of friction values using an order of magnitude fewer mesh points than with a uniform mesh.

Throughout this thesis, standard numerical techniques for calculating EHL solutions have been used. That is, regular mesh finite difference approximations have been used to discretise the problem, with multigrid used to efficiently solve the equations, and spatial adaptivity added through multigrid patches. The adjoint problems have been solved using standard linear algebra packages.

One of the most fundamental concepts in computer graphics is binary space subdivision. In its purest form, this concept leads to binary space partitioning trees (BSP trees) with arbitrarily oriented space partitioning planes. In practice, however, most algorithms use kd-trees-a special case of BSP trees that restrict themselves to axis-aligned planes-since BSP trees are believed to be numerically unstable, costly to traverse, and intractable to build well. In this paper, we show that this is not true. Furthermore, after optimizing our general BSP traversal to also have a fast kd-tree style traversal path for axis-aligned splitting planes, we show it is indeed possible to build a general BSP based ray tracer that is highly competitive with state of the art BVH and kd-tree based systems. We demonstrate our ray tracer on a variety of scenes, and show that it is always competitive with-and often superior to-state of the art BVH and kd-tree based ray tracers.

Keywords: rt, ray tracing, bsp tree

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.