John A. Kapenga

The Feasibility of a VLSI Chip for Ray Tracing Bicublic Patches

In this article we explore the possibility of a VLSI chip for ray tracing bicubic patches in Bezier form. The purpose of the chip is to calculate the intersection point of a ray with the bicubic patch to a specified level of accuracy, returning parameter values (u,v) specifying the location of the intersection on the patch, and a parameter value, t, which specifies the location of the intersection on the ray. The intersection is calculated by succesively subdividing the patch and computing the intersection of the ray with a bounding box of each subpatch until the bounding volume meets theaccuracy requirement. There are two operating modes: another in which all intersections are found. This algorithm (and the chip) correctly handle the difficult cases of the ray tangentially intersecting a planar patch and intersections of the ray at a silhouette edge of the patch. Estimates indicate that such a chip could be implemented in 2-micron NMOS (N-type metal oxide semiconductor) and could computer patch-ray intersections at the rate of one every 15 microseconds for patces that are prescaled and specified to a 12-bit fixed point for each of the x, y, and z components. A version capable of handling 24-bit patches could compute patch/ray intersections at the rate of one every 140 section point could be performed with the addition of nine scalar subtractions and six scalar multiplies. Images drawn using a software version of the algorithm are presented and discussed.

The Integration of the Multivariate Normal Density Function for the Triangular Method

The model involved in the triangular method is presented, which leads to the need for evaluating a multidimensional integral of the multidimensional normal density function over an irregular region. Work done on the numerical evaluation of this integral is discussed.

A parallelization of adaptive task partitioning algorithms

Abstract An adaptive task partitioning scheme for MIMD architectures is investigated. For many serial adaptive procedures this methodology provides a direct translation into reasonably efficient parallel versions. A proto-type of a two-dimensional integration method has been implemented in this manner. This was facilitated by using a set of high-level macros, layered over the Argonne macro package, which provides the primitives in the adaptive partitioning scheme. Because of the portable nature of the Argonne macro package our code should be readily ported to other MIMD machines with shared memory. A version of the macros for other MIMD architectures is also possible.

Parallel Cubature on Loosely Coupled Systems

We give a survey and results of our work on modifying global adaptive cubature algorithms for use on distributed memory systems. This is an application of a more general task pool management system for MIMD machines, which is under development. On loosely coupled systems, the pool of subregions (tasks) is distributed over the processors. The adaptive nature of the type of algorithm involved causes the grid to be progressively refined in order to meet a set error requirement. In view of a potentially different behavior of the integrand function in subregion sets assigned to different processors, the work loads of the processors may become imbalanced, with a few processors carrying out most of the work. This situation can be alleviated via dynamic load balancing. Results obtained with a simple load balancing strategy show its effects in case of a local integrand problem. An extension of the strategy is proposed, based on a model of the underlying work distribution.

A Parallelization of Adaptive Integration Methods

A meta-algorithm for adaptive integration is specified. It includes a class of adaptive integrators which apply extrapolation to deal with boundary singularities. Methods adhering to the meta-algorithm are parallelized in a straightforward way with use of a set of macros layered over the Argonne macro package.

Regularization with numerical extrapolation for finite and UV-divergent multi-loop integrals

Abstract We give numerical integration results for Feynman loop diagrams such as those covered by Laporta (2000) and by Baikov and Chetyrkin (2010), and which may give rise to loop integrals with UV singularities. We explore automatic adaptive integration using multivariate techniques from the ParInt package for multivariate integration, as well as iterated integration with programs from the Quadpack package, and a trapezoidal method based on a double exponential transformation. ParInt is layered over MPI (Message Passing Interface), and incorporates advanced parallel/distributed techniques including load balancing among processes that may be distributed over a cluster or a network/grid of nodes. Results are included for 2-loop vertex and box diagrams and for sets of 2-, 3- and 4-loop self-energy diagrams with or without UV terms. Numerical regularization of integrals with singular terms is achieved by linear and non-linear extrapolation methods.

Scalable Software for Multivariate Integration on Hybrid Platforms

The paper describes the software infrastructure of the PARINT package for multivariate numerical integration, layered over a hybrid parallel environment with distributed memory computations (on MPI). The parallel problem distribution is typically performed on the region level in the adaptive partitioning procedure. Our objective has been to provide the end-user with state of the art problem solving power packaged as portable software. We will give test results of the multivariate ParInt engine, with significant speedups for a set of 3-loop Feynman integrals. An extrapolation with respect to the dimensional regularization parameter (e) is applied to sequences of multivariate ParInt results Q(e) to obtain the leading asymptotic expansion coefficients as e → 0. This paper further introduces a novel method for a parallel computation of the Q(e) sequence as the components of the integral of a vector function.

Monte Carlo Automatic Integration with Dynamic Parallelism in CUDA

The rapidly evolving CUDA environment is well suited for numerical integration of high dimensional integrals, in particular by Monte Carlo or quasi-Monte Carlo methods. With some care, near peak performance can be obtained on important applications. A basis for efficient numerical integration using using CUDA kernels on NVIDIA GPUs is presented, showing several ways to use CUDA features, provide automatic error control and prevention or detection of roundoff errors. This framework allows easy extension to multiple GPUs, clusters and clouds for addressing problems that were impractical to attack in the past, and is the basis for an update to the ParInt numerical integration package.

Parallelization of Toeplitz Solvers

We shall describe parallel algorithms for solving Toeplitz and block Toeplitz systems, based on the Levinson type methods implemented in serial in the Toeplitz Package (by Arushanian et al. [1]). A matrix A, with blocks A i , j , has block Toeplitz structure if the blocks satisfy A i , j = A j − i for all i, j such that 0 ≤ i, j ≤ n − 1 and each block is a general matrix. The implementation and performance of these algorithms on MIMD shared memory machines will be discussed. It will be shown that the contribution of the inner product calculations to the time complexity is negligible for typical applications.

High Performance and Increased Precision Techniques for Feynman Loop Integrals

For the investigation of physics within and beyond the Standard Model, a precise evaluation of higher order corrections in perturbative quantum field theory is required. We have worked on the development of a computational method for Feynman loop integrals with a fully numerical approach. It is based on numerical integration and extrapolation techniques. In this paper, we describe the status and new developments in our techniques for the numerical computation of Feynman loop integrals. Separation of ultra-violet divergences is important for the renormalization procedure. In our analyses, the separation can be done numerically. For 2-loop integrals we have performed the calculations for up to 4-point functions, and for 2-point functions we can handle up to 4- loop integrals. We report the status and accuracy of the computations with detailed numerical comparisons to results in the literature, in order to demonstrate that our method will evolve into an important component of automated systems for the study of higher-order radiative corrections.