Hodiex: A Sixth Order Accurate Method for Solving Elliptical PDEs

This paper describes a method for discretizing general linear two dimensional elliptical PDEs with variable coefficients, Lu=g, which achieves high orders of accuracy on an extended range of problems. The method can be viewed as an extension of the ELLPACK6 discretization module HODIE ("High Order Difference Approximation with Identity Expansion"), which achieves high orders of accuracy on a more limited class of problems. We thus call this method HODIEX. An advantage of HODIEX methods, including the one described here, is that they are based on a compact 9-point stencil which yields linear systems with a smaller bandwidth than if a larger stencil were used to achieve higher accuracy

Parallel ELLPACK for Shared Memory Multiprocessors

This paper describes a parallel version of ELLPACK for shared memory multiprocessors. ELLPACK is a system for numerically solving elliptic PDEs. It consists of a very high level language for defining PDE problems and selecting methods of solution, and a library of approximately fifty problem solving modules. Earlier work considered three discretization modules (five point star, hodie, and hermite collocation), two linear system solution modules (linpack spd band and jacobi cg), and a triple module (hodie fft) which includes both discretization and solution, all for rectangular domains and simple boundary conditions. Here we describe parallel versions of six additional modules (hermite collocation, hodie helmholtz, five point star, band ge, sor, symmetric sor cg) for general boundary conditions and domains, and discuss modifications to the ELLPACK preprocessor, the tool that translates an ELLPACK "program" into FORTRAN

Strategies for Parallelizing PDE Software

Three strategies for parallelizing components of the mathematical software package ELLPACK are considered: an explicit approach using compiler directives available only on the target machine, an automatic approach using an optimizing and parallelizing precompiler, and a two-level approach based on extensive use of a set of low level computational kernels. Each approach to parallelization is described in detail, along with a discussion of the effort involved. In connection with the third strategy, a set of computational kernels useful for PDE solving is proposed. We describe our experience in parallelizing six problem solving components of ELLPACK using each of the three strategies and give performance results for a shared memory multiprocessor. Our results suggest that the two-level strategy allows the best balance among programmer effort, portability, and parallel performance

Quantitative assessment of renal perfusion and oxygenation by invasive probes: basic concepts

Renal tissue hypoperfusion and hypoxia are early key elements in the pathophysiology of acute kidney injury of various origins, and may also promote progression from acute injury to chronic kidney disease. Here we describe basic principles of methodology to quantify renal hemodynamics and tissue oxygenation by means of invasive probes in experimental animals. Advantages and disadvantages of the various methods are discussed in the context of the heterogeneity of renal tissue perfusion and oxygenation.This chapter is based upon work from the COST Action PARENCHIMA, a community-driven network funded by the European Cooperation in Science and Technology (COST) program of the European Union, which aims to improve the reproducibility and standardization of renal MRI biomarkers. This introduction chapter is complemented by a separate chapter describing the experimental procedure and data analysis