Accurate solution of near-colliding Prony systems via decimation and homotopy continuation
This work provides a more robust numerical method for solving Prony systems, which are important in various mathematical applications, particularly in near-colliding scenarios where existing methods struggle.
The paper addresses the challenge of solving near-colliding Prony systems, which are difficult to solve numerically. By transforming the system into a Hankel-type polynomial system and combining it with a decimation technique and homotopy continuation, the authors achieve high-accuracy solutions for the nonlinear variables under data perturbation.
We consider polynomial systems of Prony type, appearing in many areas of mathematics. Their robust numerical solution is considered to be difficult, especially in "near-colliding" situations. We consider a case when the structure of the system is a-priori fixed. We transform the nonlinear part of the Prony system into a Hankel-type polynomial system. Combining this representation with a recently discovered "decimation" technique, we present an algorithm which applies homotopy continuation to an appropriately chosen Hankel-type system as above. In this way, we are able to solve for the nonlinear variables of the original system with high accuracy when the data is perturbed.