Parametric architectures for scientific computing in integrable spaces: the convolution operation as a case of study
- Autores: Gregorio de Miguel Casado
- Directores de la Tesis: Juan Manuel García Chamizo (dir. tes.)
- Lectura: En la Universitat d'Alacant / Universidad de Alicante ( España ) en 2010
- Idioma: inglés
- Tribunal Calificador de la Tesis: Elvira Mayordomo (presid.), Higinio Mora Mora (secret.), Salvador Hernández Muñoz (voc.), Ramón Rizo Aldeguer (voc.), Martín Escardó (voc.)
- Materias:
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- Resumen
The development of Computer Science has forcefully impelled the research capacities in almost all fields in Science, As a matter of fact, nowadays computers play a natural role which is necessary in Science.
Within this context, a slight survey in Physics, whose objectives aim to formulate real world models, outlines the fact that computers are intensively used in many of its fields. However, model validation processes in fields such as Cosmology, Particle Physics or Oceanography perform operations with numerical data acquired in the environment which turn out to be crucial.
This way, the limits of the IEEE754 floating point specification, adopted by most of processor manufacturers and now recently reviewed (IEEE754-2008), suggest some improvements so that to guarantee the reliability of the calculations by establishing a trade off between result accuracy and availability of hardware resources.
In this Ph.D. document, a model for developing specialized processors is proposed. Then, it is particularized to the design of a special purpose processor for supporting calculations in integrable spaces Lp and Sobolev. The functions involved in the calculation are characterized by means of convergent sequences of rational step functions or simple functions. The particular case of the convolution operation is studied and a discrete iterative approximation is proposed and then identified with some basic numerical methods for ODE solution.
The prototype is developed in several abstraction levels. Firstly, the characterization of integral transforms in the corresponding mathematical spaces is performed under the scope of the Computable Analysis paradigm Type-2 Theory of Effectivity (TTE), developed by K. Weihrauch and the community Computability and Complexity in Analysis (CCA). The introduction of this paradigm provides a proper function characterization by means of computable representations which, in their lowest expression level, allow for exploiting a conceptual link with the on-line arithmetic approach (based on signed digit numerical representations) developed by T. Lang and M. Ercegovac. The latter approach consists on left-to-right serial-bit arithmetic operations which resemble the dynamics of a Turing machine. For the following specification level of the processor, the conceptual gap between the Turing machine computability model characterization of the calculation and a Von Neumann architecture is taken on board by introducing a paradigm for algebraic specification of processors developed by A.C.J. Fox, N.A. Harman and J.V. Tucker. As a result, a validated algebraic abstract circuit level specification is obtained.
The implementation of the prototype has being done as a software environment and a hardware prototype. The software consists of an specific library for characterizing (partially) the functions involved in the calculation, according to the computability concepts introduced of TTE and uses a .NET link to connect to the Mathematica kernel. Additionally, an on-line arithmetic library for basic operations (addition, subtraction, multiplication, division and comparison) with periodic rational numbers expressed in fractional positional notation allows for a predictive detection of periodic mantissas in the operations. The hardware prototype consists of a synthesis in recofigurable hardware (FPGA) of the arithmetic-logic unit, which endorses the feasibility of the proposal.
Finally, the evaluation of the precision improvements in the calculation has being done by implementing a library with the UNESCO basic procedures for Physical Oceanography. The comparison of the procedure calculation results with those obtained with the IEEE754 arithmetic approach outline the precision improvements achieved.
