Bit complexity of computing solutions for symmetric hyperbolic systems of PDEs with guaranteed precision

Abstract

We establish upper bounds on bit complexity of computing solution operators for symmetric hyperbolic systems of PDEs, combining symbolic and approximate algorithms to obtain the solutions with guaranteed prescribed precision. Restricting to algebraic real inputs allows us to use the classical (“discrete”) bit complexity concept.

Primitive Recursive Ordered Fields and Some Applications

“…In particular, we find a PR-version (for the archimedean case) of the Ershov-Madison theorem on the computable real closure [8,18], relate the PR ordered fields of reals to the field of PR reals, give a sufficient condition for PR root-finding, propose (apparently, new) notions of PR-computability in analysis and apply them to obtain new results on computations of PDE-solutions. These results complement the results in [1] about root-finding in the field R alg of algebraic reals and in [28] about the complexity of PDE-solutions. The class of PR real closed fields of reals is shown to be richer than the class of PTIME-presentable fields.…”

“…Here we search for a PR-analogue of the following fact from [26,27,28]: for any finite set F of computable reals there is a computable real closed ordered subfield (B, β) of the computable reals such that F ⊆ B (see also a more general Theorem 4.1 in [22] obtained independently). This implies R c = {A | α ∈ c(R)} = {A | α ∈ cs(R)} where R c is the set of computable reals and c(R), cs(R) are the computable analogues of pra(R), pras(R).…”

“…Multilinear interpolations of rational grid functions are taken as approximations to the solutions of Cauchy problem below. We refer the reader to [26,28] and references therein for additional information related to the mentioned spaces, multilinear interpolations and Godunov's difference scheme discussed below.…”

“…In [26,27] computable ordered fields of reals were related to the field of computable reals and used to prove computability of some problems in algebra and analysis in the rigorous sense of computable analysis [32]. In [28], PTIME-computability of some problems on algebraic numbers established in [1] (based on previous results from computer algebra, see e.g. [17]) was applied to find upper complexity bounds for some problems in algebra and analysis.…”

Computability of Differential Equations

“…, B m to the solution of the PDE is uniformly computable, provided the matrices satisfy some algebraic conditions. More recently, the computational complexity of solving PDEs with the format (0.12) was studied in [76], [72]. There the (uniform) complexity of this problem is shown to be in EXPTIME in general and in PTIME under certain conditions.…”