Parallel Marching Hypercubes Algorithm with Applications to Rigorous Proofs

• Grant number: 24/06833-9
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Start date: July 01, 2024
• End date: August 31, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Antonio Castelo Filho
• Grantee: Antonio Castelo Filho
• Visiting researcher: Marcio Fuzeto Gameiro
• Visiting researcher institution: Rutgers The State University of New Jersey, New Brunswick, United States
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

The Generalized Combinatorial Marching Hypercubes (GCMH) algorithm was developed to calculate an approximation by a cellular complex of a manifold of any dimension and co-dimension, that is, a manifold of dimension n-k immersed in an n-dimensional space. This algorithm uses combinatorial and topological methods to avoid using expensive lookup tables and is therefore efficient in higher dimensions. Another advantage of this algorithm is that it was designed so that each hypercube can be processed independently, making it have great potential for parallelization. One of the objectives of this project is the parallelism of this algorithm with both shared and distributed memory.On the other hand, rigorous computer-assisted proofs are one of the areas that benefit from this type of algorithms. As an application, the objective is to solve the problem of finding periodic orbits of Systems of Ordinary Differential Equations using rigorous computer-assisted proof techniques that use decomposition of solutions into series and with the technique of radial base polynomials, transforming it into the problem of approximating varieties implicitly defined. (AU)