Dalcidio Moraes Claudio

A Coherence Space of Rational Intervals for a Construction of IR

A constructive computational representation of the space of real intervals IR is introduced, in a way that makes it possible to capture both its information structure relevant from a computational standpoint, and its application features as a mathematical structure. The representation consists of the Coherence Space of Rational Intervals IIQ, introduced by defining the web (IQ, ≈) of rational intervals, which is obtained from the set IQ of rational intervals on which a suitable reflexive and symmetric relation ≈ is defined. A two-fold construction of IIQ is performed, such that the internal construction of its domain-like structure leads the transformation of a suitable external algebraic structure defined on IQ into a certain one on IIQ which, when restricted to the set tot(IIQ) of total objects, becomes a structure which is order and algebraically isomorphic to the complete ordered field of the real numbers, R, and such that the family of quasi-total objects when extended with R is order and algebraically isomorphic to the set of the real intervals R.

Dense linear system: a parallel self-verified solver

This article presents a parallel self-verified solver for dense linear systems of equations. This kind of solver is commonly used in many different kinds of real applications which deal with large matrices. Nevertheless, two key problems appear to limit the use of linear system solvers to a more extensive range of real applications: solution correctness and high computational cost. In order to solve the first one, verified computing would be an interesting choice. An algorithm that uses this concept is able to find a highly accurate and automatically verified result providing more reliability. However, the performance of these algorithms quickly becomes a drawback. Aiming at a better performance, parallel computing techniques were employed. Two main parts of this method were parallelized: the computation of the approximate inverse of matrix A and the preconditioning step. The results obtained show that these optimizations increase significantly the overall performance.

Improving the Performance of a Verified Linear System Solver Using Optimized Libraries and Parallel Computation

A parallel version of the self-verified method for solving linear systems was presented in [20, 21]. In this research we propose improvements aiming at a better performance. The idea is to implement an algorithm that uses technologies as MPI communication primitives associated to libraries as LAPACK, BLAS and C-XSC, aiming to provide both self-verification and speed-up at the same time. The algorithms should find an enclosure even for ill-conditioned problems. In this scenario, a parallel version of a self-verified solver for dense linear systems appears to be essential in order to solve bigger problems. Moreover, the major goal of this research is to provide a free, fast, reliable and accurate solver for dense linear systems.

Optimizing a parallel self-verified method for solving linear systems

Solvers for linear equation systems are commonly used in many different kinds of real applications, which deal with large matrices. Nevertheless, two key problems appear to limit the use of linear system solvers to a more extensive range of real applications: computing power and solution correctness. In a previous work, we proposed a method that employs high performance computing techniques together with verified computing techniques in order to eliminate the problems mentioned above. This paper presents an optimization of a previously proposed parallel self-verified method for solving dense linear systems of equations. Basically, improvements are related to the way communication primitives were employed and to the identification of the points in the algorithm in which mathematical accuracy is needed to achieve reliable results.

Optimization Problems Categories

This work presents a categorical approach to cope with some questions originally studied within Computational Complexity Theory. It proceeds a research with theoretical emphasis, aiming at characterising the structural properties of optimization problems, related to the approximative issue, by means of Category Theory. In order to achieve it, two new categories are defined: the OPT category, which objects are optimization problems and the morphisms are the reductions between them, and the APX category, that has approximation problems as objects and approximation-preserving reductions as morphisms. Following the basic idea of categorical shape theory, a comparison mechanism between these two categories is defined and a hierarchical structure of approximation to each optimization problem can be modelled.

Parallel Environment with High Accuracy for Resolution of Numerical Problems

In this paper we describe a high performance environment, like cluster computers, with high accuracy obtained by use of C-XSC library. The C-XSC library is a (free) C++ class library for scientific computing for the development of numerical algorithms delivering highly accurate and automatically verified results by use of the interval arithmetic. These calculus in high accuracy must be available for some basic arithmetic operations, mainly the operations that accomplish the summation and dot product. Because of these aspects, we wish to use the high performance through a cluster environment where we have several nodes executing tasks or calculus. The communication will be done by message passing using the MPI communication library. To obtain the high accuracy in this environment extensions or changes in the parallel programs had done to guarantee that the quality of final result done on cluster, where several nodes collaborate for the final result of the calculus, maintain the same result quality obtained in one sequential high accuracy environment. To validate the environment developed in this work we done basic tests about the dot product, the matrix multiplications, the implementation of interval solvers for banded and dense matrices and the implementation of some numeric methods to solve linear systems with the high accuracy characteristic (some of the methods implemented are used in real life applications like hydrodynamic, agriculture and power electric systems). With these tests we done analysis and comparisons about the performance and accuracy obtained with and without the use of C-XSC library in sequential and parallel programs. With the implementation of these routines and methods will be open a large research field about the study of real life applications that need during their resolution (or in part of their resolution) to calculate arithmetic operations with more accuracy than the accuracy obtained by the traditional computational tools. Our software run on labtec (UFRGS) and Colorado (UPF) clusters. Nowadays we are working in the implementation of parallel versions of programs to solve linear systems (without and with high accuracy) and the optimization of C-XSC library on cluster computers.

On the Approximation of Centered Zonotopes in the Plane

The present work is devoted to computation with zonotopes in the plane. Using ideas from the theory of quasivector spaces we formulate an approximation problem for zonotopes and propose an algorithm for its solution.

A Categorical Approach to NP-Hard Optimization Problems

Aiming at developing a theoretical framework for the formal study of NP-hard optimization problems, which is built on precise mathematical foundations, we have focused on structural properties of optimization problems related to approximative issue. From the observation that, intuitively, there are many connections among categorical concepts and structural complexity notions, in this work we present a categorical approach to cope with some questions originally studied within Computational Complexity Theory. After defining the polynomial time soluble optimization problems category OPTS and the optimization problems category OPT, a comparison mechanism between them and an approximation system to each optimization problem have been introduced, following the basic idea of categorical shape theory. In this direction, we consider new insights and a deeper understanding of some basic questions inside the Structural Complexity field, by an universal language.