Hyperspherical path tracking methodology as correction step in homotopic continuation methods

• The homotopic mapping is a strategy for finding solutions in nonlinear systems.
• Hyperspherical path tracking method is a versatile tool of homotopic continuation.
• Hyperspherical path tracking is a robust technique for searching global optima.
• SEHPE code was validated successfully using previously reported problems.

Homotopic trajectories are constructed through the calculation of the discrete points that constitute a curve that is known as the homotopic path, which is mathematically the same algebraic system that is mapped by the homotopy function at every point but exhibits an arbitrary variation in the homotopic parameter. Thus, to calculate the homotopic parameter, complex strategies are commonly used to define a step size that favors convergence. However, we demonstrate that these strategies cannot guarantee the numerical stability of the path tracking process and are also quite complicated to understand and implement in numerical procedures. In this work, an N+1 dimensional version of the canonical equation of the sphere was solved in conjunction with the homotopic system. Thus, through the use of N+1 variables and N+1 equations, the problem is defined and geometrically closed. This method was named “hyperspherical path tracking”. In a combined methodology and results section, we present some heuristic observations in the construction of a novel convergence criterion for homotopic methods. In addition, numerical evidence of the stability and good behavior of the tracking hyperspheres is presented. In all the solved example systems, the solution vectors that have been previously reported by other authors were localized using our method. In some cases, additional solution vectors were found. In addition, our method was able to circumvent the numerical challenge that is presented in the construction of homotopic paths that are deformed by bounded homotopies. The solution of a system derived from one benchmark function of two variables with multiple minima is presented, and some conclusions were obtained for this application. Finally, our method found 15 solution vectors for a large and highly nonlinear algebraic system of equations, which was obtained through the discretization of the set of elliptic partial differential equations (PDEs) that govern the natural convection in a differentially heated square cavity; these solutions had not been previously reported.