Solving ODEs by Obtaining Purely Second Degree Multinomials via Branch and Bound with Admissible Heuristic

Abstract

Probabilistic evolution theory (PREVTH) forms a framework for the solution of explicit ODEs. The purpose of the paper is two-fold: (1) conversion of multinomial right-hand sides of the ODEs to purely second degree multinomial right-hand sides by space extension; (2) decrease the computational burden of probabilistic evolution theory by using the condensed Kronecker product. A first order ODE set with multinomial right-hand side functions may be converted to a first order ODE set with purely second degree multinomial right-hand side functions at the expense of an increase in the number of equations and unknowns. Obtaining purely second degree multinomial right-hand side functions is important because the solution of such equation set may be approximated by probabilistic evolution theory. A recent article by the authors states that the ODE set with the smallest number of unknowns can be found by searching. This paper gives the details of a way to search for the optimal space extension. As for the second purpose of the paper, the computational burden can be reduced by considering the properties of the Kronecker product of vectors and how the Kronecker product appears within the recursion of PREVTH: as a Cauchy product structure.

Dissipative quadratizations of polynomial ODE systems

“…It turns out that every polynomial ODE system can be similarly lifted to an at most quadratic one: this fact has been established at least 100 years ago [2,27] and has been rediscovered several times since then [8,10,11,17,24]. In the recent years quadratization has been used in a number of application areas including model order reduction [5,6,17,25,26], synthetic biology [13,20,21], numerical integration [16,18,19], and reachability analysis [14]. While it has been shown in [21] that the problem of finding the minimal number of extra variables necessary for quadratization is NP-hard, at least two practically useful software packages have been developed for performing quadratization: BioCham [21] and QBee [7].…”