Formally Verified Computation of Enclosures of Solutions of Ordinary Differential Equations

Ordinary differential equations (ODEs) are ubiquitous when modeling continuous dynamics. Classical numerical methods compute approximations of the solution, however without any guarantees on the quality of the approximation. Nevertheless, methods have been developed that are supposed to compute enclosures of the solution. In this paper, we demonstrate that enclosures of the solution can be verified with a high level of rigor: We implement a functional algorithm that computes enclosures of solutions of ODEs in the interactive theorem prover Isabelle/HOL, where we formally verify (and have mechanically checked) the safety of the enclosures against the existing theory of ODEs in Isabelle/HOL. Our algorithm works with dyadic rational numbers with statically fixed precision and is based on the well-known Euler method. We abstract discretization and round-off errors in the domain of affine forms. Code can be extracted from the verified algorithm and experiments indicate that the extracted code exhibits reasonable efficiency. 1 Relations to the paper Here we relate the contents of our NFM 2014 paper [2] with the sources you find here. In the following list we show which notions and theorems in the paper correspond to which parts of the source code. If you are (still) interested in the relations to our ITP 2012 paper [3], you should take a look at the document of older releases (before Isabelle 2013-1) of this AFP entry.