Approximation of the Constant in a Markov-Type Inequality on a Simplex Using Meta-Heuristics

Grzegorz Sroka,Mariusz Oszust

doi:10.3390/math9030264

Grzegorz Sroka, Mariusz Oszust

Open Access

https://doi.org/10.3390/math9030264

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Journal: Mathematics	Publication Date: Jan 29, 2021
Citations: 1	License type: CC BY 4.0

Affiliation: Rzeszów University of Technology

Abstract

Markov-type inequalities are often used in numerical solutions of differential equations, and their constants improve error bounds. In this paper, the upper approximation of the constant in a Markov-type inequality on a simplex is considered. To determine the constant, the minimal polynomial and pluripotential theories were employed. They include a complex equilibrium measure that solves the extreme problem by minimizing the energy integral. Consequently, examples of polynomials of the second degree are introduced. Then, a challenging bilevel optimization problem that uses the polynomials for the approximation was formulated. Finally, three popular meta-heuristics were applied to the problem, and their results were investigated.

Highlights

Markov-type polynomial inequalities [1,2,3,4,5,6], to Bernstein-type inequalities [1,5,6,7,8,9], are often found in many areas of applied mathematics, including popular numerical solutions of differential equations
They are often used in the error analysis of variational techniques, including the finite element method or the discontinuous Galerkin method used for solving partial differential equations (PDEs) [10]
The results reveal that the Particle Swarm Optimization method (PSO) is better suited to the problem than the Genetic Algorithm (GA), since it can determine a better C, and it finds it with fewer function calls

Summary

Introduction

Markov-type polynomial inequalities (or inverse inequalities) [1,2,3,4,5,6], to Bernstein-type inequalities [1,5,6,7,8,9], are often found in many areas of applied mathematics, including popular numerical solutions of differential equations. Proper estimates of optimal constants in both types of inequalities can help to improve the bounds of numerical errors. Ozisik et al [10] determined the constants in multivariate Markov inequalities on an interval, a triangle, and a tetrahedron with the L2 -norm They derived explicit expressions for the constants on all above-mentioned simplexes using orthonormal polynomials. Vianello [11] used the approximation theory notions of a polynomial mesh and the Dubiner distance in a compact set to determine error estimates for the total degree of polynomial optimization on Chebyshev grids of the hypercube. Vianello [12] constructed norming meshes for polynomial optimization by using a classical Markov inequality on the general convex body in RN. An approach to determining an approximation of the constant in Markov’s inequality using introduced minimal polynomials is proposed.

Minimal Polynomials

Estimation of the Constant C in a Markov-Type Inequality

Optimization Problem in a Markov-Type Inequality

Conclusions