A stochastic homotopy tracking algorithm for parametric systems of nonlinear equations

TL;DR

This paper introduces a stochastic homotopy tracking algorithm that perturbs parametric systems randomly to avoid singularities, improving efficiency in solving nonlinear equations.

Contribution

The paper proposes a novel stochastic homotopy tracking method that effectively avoids singularities during path tracking, with theoretical guarantees and demonstrated efficiency.

Findings

The stochastic method successfully avoids singularities during tracking.

The solution path remains close to the original path theoretically.

Numerical examples show improved efficiency over traditional methods.

Abstract

The homotopy continuation method has been widely used in solving parametric systems of nonlinear equations. But it can be very expensive and inefficient due to singularities during the tracking even though both start and end points are non-singular. The current tracking algorithms focus on the adaptivity of the stepsize by estimating the distance to the singularities but cannot avoid these singularities during the tracking. We present a stochastic homotopy tracking algorithm that perturbs the original parametric system randomly each step to avoid the singularities. We then prove that the stochastic solution path introduced by this new method is still closed to the original solution path theoretically. Moreover, several homotopy examples have been tested to show the efficiency of the stochastic homotopy tracking method.