Quantifier elimination for approximate Beals-Kartashova factorization

TL;DR

This paper explores the use of quantifier elimination via cylindrical algebraic decomposition to achieve approximate factorization of linear partial differential operators, potentially simplifying numerical simulations.

Contribution

It introduces a novel approach applying quantifier elimination to approximate BK factorization for LPDOs, enabling numerical computations with lower-order operators.

Findings

Feasibility of using quantifier elimination for approximate factorization.

Potential simplification of numerical simulations with LPDEs.

New methodology for approximate factorization of LPDOs.

Abstract

The only known constructive factorization algorithm for linear partial differential operators (LPDOs) is Beals-Kartashova (BK) factorization \cite{bk2005}. One of the most interesting features of BK-factorization: at the beginning all the first-order factors are constructed and afterwards the factorization condition(s) should be checked. This leads to the important application area - namely, numerical simulations which could be simplified substantially if instead of computation with one LPDE of order $n$ we will be able to proceed computations with $n$ LPDEs all of order 1. In numerical simulations it is not necessary to fulfill factorization conditions exactly but with some given accuracy, which we call approximate factorization. The idea of the present paper is to look into the feasibility of solving problems of this kind using quantifier elinination by cylindrical algebraic decomposition.