An Algebraic Programming Style for Numerical Software and Its Optimization

Abstract

The abstract mathematical theory of partial differential equations (PDEs) is formulated in terms of manifolds, scalar fields, tensors, and the like, but these algebraic structures are hardly recognizable in actual PDE solvers. The general aim of the Sophus programming style is to bridge the gap between theory and practice in the domain of PDE solvers. Its main ingredients are a library of abstract datatypes corresponding to the algebraic structures used in the mathematical theory and an algebraic expression style similar to the expression style used in the mathematical theory. Because of its emphasis on abstract datatypes, Sophus is most naturally combined with object-oriented languages or other languages supporting abstract datatypes. The resulting source code patterns are beyond the scope of current compiler optimizations, but are sufficiently specific for a dedicated source-to-source optimizer. The limited, domain-specific, character of Sophus is the key to success here. This kind of optimization has been tested on computationally intensive Sophus style code with promising results. The general approach may be useful for other styles and in other application domains as well.

Compiling language definitions

“…Program Analysis -Typechecking of Pascal , Chapter 2] -Typechecking and execution of CLaX [Dinesh and Tip 1992;Dinesh and Tip 1997] -Type inference, object identification, and documentation for Cobol ; van Deursen and Moonen 1998;van Deursen and Kuipers 1998;van Deursen and Kuipers 1999] Program Transformation -Interactive program transformation for Clean [van den Brand et al 1995] and Prolog [Brunekreef 1996] -Automatic program transformation for C++ [Dinesh et al 1998] Software Renovation -Description of the multiplicity of languages and dialects encountered in software renovation applications such as Cobol (including embedded languages like SQL and CICS) van den Brand et al 1997;] -Automatic program transformation for restructuring of Cobol programs (including embedded languages like SQL and CICS) [van den Brand et al 1997;van den Brand et al 1998;] -Derivation of language descriptions from compilers and on-line manuals Sellink and Verhoef 2000] Specification and Prototyping of New Applications and Tools -PIM [Field 1992;Bergstra et al 1997] (compiler toolkit) -µCRL [Hillebrand 1996] (proof checking and simulation toolkit) -Components of the ASF+SDF Meta-Environment itself [van den Brand et al 1997] (including a parser generator, a prettyprinter generator, and the ASF+SDF compiler described in this article) Table I. Main application areas of the ASF+SDF Meta-Environment.…”

Coordinate Free Programming of Computational Fluid Dynamics Problems

“…Unfortunately, the resulting efficiency seems sensitive to memory layout and other factors which are difficult to control. However, a source-level transformation tool, such as CodeBoost [9], can be invaluable in this respect. It makes it possible to systemise experiments with various data layouts and other transformations of the code.…”

On the use of an algebraic language interface for waveform definition

“…We then briefly describe the interface to the underlying SDR framework. As SDR is another form of scientific programming, we follow the general models already set up in that domain [34].…”