Abstract
We devise and analyze the bit-cost of solvers for linear evolutionary systems of Partial Differential Equations (PDEs) with given analytic initial conditions. Our algorithms are rigorous in that they produce approximations to the solution up to guaranteed absolute error \(1/2^n\) for any desired number n of output bits. Previous work has shown that smooth (i.e. infinitely differentiable but non-analytic) initial data does not yield polynomial-time computable solutions unless it holds P=NP (or stronger complexity hypotheses). We first resume earlier complexity investigations of the Cauchy-Kovalevskaya Theorem about linear PDEs with analytic matrix coefficients: from qualitative polynomial-time solutions for any fixed polynomial-time computable analytic initial conditions, to quantitative parameterized bit-cost analyses for any given analytic initial data, as well as turn devised algorithms into computational practice. We secondly devise a parameterized polynomial-time solver for the Heat and the Schrödinger equation with given analytic initial data: PDEs not covered by Cauchy-Kovalevskaya. Reliable implementations and empirical performance evaluation (including testing on the Elasticity and Acoustic systems examples) in the Exact Real Computation (ERC) paradigm confirm the theoretical predictions and practical applicability of our algorithms. These involve new continuous abstract data types operating on power and Fourier series without rounding error.
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Notes
- 1.
The source code for the implementation can be found on https://github.com/holgerthies/irram-pde.
- 2.
For space reasons, we omit a detailed analysis of our experiments but some more information can be found at the github repository of our implementation.
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Acknowledgments
This work was supported by the National Research Foundation of Korea (grant 2017R1E1A1A03071032), by the International Research & Development Program of the Korean Ministry of Science and ICT (grant 2016K1A3A7A03950702), by the NRF Brain Pool program (grant 2019H1D3A2A02102240) and by JSPS KAKENHI Grant Number JP20K19744.
We thank Filippo Morabito for his lectures about PDEs on manifolds and Pieter Collins and Norbert Müller for helpful discussions on possibilities of implementing differential equations in ERC packages.
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Selivanova, S., Steinberg, F., Thies, H., Ziegler, M. (2021). Exact Real Computation of Solution Operators for Linear Analytic Systems of Partial Differential Equations. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2021. Lecture Notes in Computer Science(), vol 12865. Springer, Cham. https://doi.org/10.1007/978-3-030-85165-1_21
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