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## Resulting Hierarchy (resulting + hierarchy)
## Selected Abstracts## Shallow Bounding Volume Hierarchies for Fast SIMD Ray Tracing of Incoherent Rays COMPUTER GRAPHICS FORUM, Issue 4 2008H. DammertzAbstract Photorealistic image synthesis is a computationally demanding task that relies on ray tracing for the evaluation of integrals. Rendering time is dominated by tracing long paths that are very incoherent by construction. We therefore investigate the use of SIMD instructions to accelerate incoherent rays. SIMD is used in the hierarchy construction, the tree traversal and the leaf intersection. This is achieved by increasing the arity of acceleration structures, which also reduces memory requirements. We show that the resulting hierarchies can be built quickly and are smaller than acceleration structures known so far while at the same time outperforming them for incoherent rays. Our new acceleration structure speeds up ray tracing by a factor of 1.6 to 2.0 compared to a highly optimized bounding interval hierarchy implementation, and 1.3 to 1.6 compared to an efficient kd-tree. At the same time, the memory requirements are reduced by 10,50%. Additionally we show how a caching mechanism in conjunction with this memory efficient hierarchy can be used to speed up shadow rays in a global illumination algorithm without increasing the memory footprint. This optimization decreased the number of traversal steps up to 50%. [source] ## Effective Borel measurability and reducibility of functions MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 1 2005Vasco BrattkaAbstract The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single-valued as well as for multi-valued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and corresponding complete functions. We use this classification and an effective version of a Selection Theorem of Bhattacharya-Srivastava in order to prove a generalization of the Representation Theorem of Kreitz-Weihrauch for Borel measurable functions on computable metric spaces: such functions are Borel measurable on a certain finite level, if and only if they admit a realizer on Baire space of the same quality. This Representation Theorem enables us to introduce a realizer reducibility for functions on metric spaces and we can extend the completeness result to this reducibility. Besides being very useful by itself, this reducibility leads to a new and effective proof of the Banach-Hausdorff-Lebesgue Theorem which connects Borel measurable functions with the Baire functions. Hence, for certain metric spaces the class of Borel computable functions on a certain level is exactly the class of functions which can be expressed as a limit of a pointwise convergent and computable sequence of functions of the next lower level. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Computation and presentation of graphs displaying closure hierarchies of Jordan and Kronecker structures NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6-7 2001Erik ElmrothAbstract StratiGraph, a Java-based tool for computation and presentation of closure hierarchies of Jordan and Kronecker structures is presented. The tool is based on recent theoretical results on stratifications of orbits and bundles of matrices and matrix pencils. A stratification reveals the complete hierarchy of nearby structures, information critical for explaining the qualitative behaviour of linear systems under perturbations. StratiGraph facilitates the application of these theories and visualizes the resulting hierarchy as a graph. Nodes in the graph represent orbits or bundles of matrices or matrix pencils. Edges represent covering relations in the closure hierarchy. Given a Jordan or Kronecker structure, a user can obtain the complete information of nearby structures simply by mouse clicks on nodes of interest. This contribution gives an overview of the StratiGraph tool, presents its main functionalities and other features, and illustrates its use by sample applications. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## ,-matrices for the convection-diffusion equation PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003S. Le Borne Dr.Hierarchical matrices (,-matrices) provide a technique for the sparse approximation of large, fully populated matrices. This technique has been shown to be applicable to stiffness matrices arising in boundary element method applications where the kernel function displays certain smoothness properties. The error estimates for an approximation of the kernel function by a separable function can be carried over directly to error estimates for an approximation of the stiffness matrix by an ,-matrix, using a certain standard partitioning and admissibility condition for matrix blocks. Similarly, ,-matrix techniques can be applied in the finite element context where it is the inverse of the stiffness matrix that is fully populated. Here one needs a separable approximation of Green's function of the underlying boundary value problem in order to prove approximability by matrix blocks of low rank. Unfortunately, Green's function for the convection-diffusion equation does not satisfy the required smoothness properties, hence prohibiting a straightforward generalization of the separable approximation through Taylor polynomials. We will use Green's function to motivate a modification in the (hierarchical) partitioning of the index set and as a consequence the resulting hierarchy of block partitionings as well as the admissibility condition. We will illustrate the effect of the proposed modifications by numerical results. [source] |