Markov Chain Mote Carlo solution of BK equation through Newton-Kantorovich method

TL;DR

This paper introduces a novel Monte Carlo approach combining Newton-Kantorovich and MCMC methods to solve the non-linear BK equation efficiently, especially for complex, high-dimensional cases.

Contribution

The paper presents a new hybrid method that transforms non-linear equations into linear ones for MCMC solution, applied to the BK equation in quantum chromodynamics.

Findings

MCMC method achieves high precision in solving the BK equation.

The approach is efficient and scalable for complex, high-dimensional non-linear equations.

Numerical results demonstrate the method's effectiveness and potential for broader applications.

Abstract

We propose a new method for Monte Carlo solution of non-linear integral equations by combining the Newton-Kantorovich method for solving non-linear equations with the Markov Chain Monte Carlo (MCMC) method for solving linear equations. The Newton-Kantorovich method allows to express the non-linear equation as a system of the linear equations which then can be treated by the MCMC (random walk) algorithm. We apply this method to the Balitsky-Kovchegov (BK) equation describing evolution of gluon density at low x. Results of numerical computations show that the MCMC method is both precise and efficient. The presented algorithm may be particularly suited for solving more complicated and higher-dimensional non-linear integral equation, for which traditional methods become unfeasible.