Abstract
This paper provides a brief survey of recent achievements in characterizing computational complexity of partial differential equations (PDEs), as well as computing solutions with guaranteed precision within the exact real computation approach. The emphasis is on classical solutions and linear PDE systems, since these are the cases where most of the progress has been achieved so far. Complexity, as it turns out, heavily depends on the smoothness of the initial data, which has similarities with the situation for ordinary differential equations (ODEs).
Supported by the National Research Foundation of Korea (grant 2017R1E1A1A03071032) and by the International Research & Development Program of the Korean Ministry of Science and ICT (grant 2016K1A3A7A03950702) and by the NRF Brain Pool program (grant 2019H1D3A2A02102240) and by the RFBR-JSPS Grant 20-51-50001. The author is thankful to Victor Selivanov, Holger Thies and Martin Ziegler for valuable discussions.
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References
Bournez, O., Graça, D.S., Pouly, A.: Solving analytic differential equations in polynomial time over unbounded domains. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 170–181. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22993-0_18
Brattka, V., Hertling, P., Weihrauch, K.: A tutorial on computable analysis. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) New Computational Paradigms: Changing Conceptions of What Is Computable, pp. 425–491. Springer, New York (2008). https://doi.org/10.1007/978-0-387-68546-5_18
Braverman, M., Cook, S.A.: Computing over the reals: foundations for scientific computing. Notices AMS 53(3), 318–329 (2006)
Ershov, Y., Goncharov, S.: Constructive Models. Novosibirsk, Scientific Book (in Russian, there is an English Translation) (1999)
Gay, W., Zhang, B.Y., Zhong, N.: Computability of solutions of the Korteweg-de Vries equation. Math. Log. Q. 47(1), 93–110 (2001)
Graça, D., Zhong, N.: Computability of differential equations. In: Handbook of Computability and Complexity in Analysis (Editors: Vasco Brattka and Peter Hertling), pp. 71–99. Theory and Applications of Computability (2021)
Kawamura, A.: Lipschitz continuous ordinary differential equations are polynomial-space complete. Comput. Complex. 19(2), 305–332 (2010). https://doi.org/10.1007/s00037-010-0286-0
Kawamura, A., Cook, S.: Complexity theory for operators in analysis. In: Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, pp. 495–502. ACM, New York (2010). https://doi.org/10.1145/1806689.1806758
Kawamura, A., Ota, H., Rösnick, C., Ziegler, M.: Computational complexity of smooth differential equations. Log. Methods Comput. Sci. 10(1:6), 15 (2014). https://doi.org/10.2168/LMCS-10(1:6)2014
Kawamura, A., Steinberg, F., Thies, H.: Parameterized complexity for uniform operators on multidimensional analytic functions and ODE solving. In: Proc 25th International Workshop on Logic, Language, Information, and Computation (WOLLIC), pp. 223–236 (2018). https://doi.org/10.1007/978-3-662-57669-4_13
Kawamura, A., Steinberg, F., Ziegler, M.: On the computational complexity of the Dirichlet problem for Poisson’s equation. Math. Struct. Comput. Sci. 27(8), 1437–1465 (2017). https://doi.org/10.1017/S096012951600013X
Kawamura, A., Thies, H., Ziegler, M.: Average-case polynomial-time computability of hamiltonian dynamics. In: Potapov, I., Spirakis, P.G., Worrell, J. (eds.) 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018, 27–31 August 2018, Liverpool, UK. LIPIcs, vol. 117, pp. 30:1–30:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
Kawamura, A., Ziegler, M.: Invitation to real complexity theory: algorithmic foundations to reliable numerics with bit-costs (2018). https://arxiv.org/abs/1801.07108
Ko, K., Friedman, H.: Computational complexity of real functions. Theoret. Comput. Sci. 20(3), 323–352 (1982). https://doi.org/10.1016/S0304-3975(82)80003-0
Ko, K.I.: Complexity Theory of Real Functions. Progress in Theoretical Computer Science, Birkhäuser, Boston (1991)
Koswara, I., Pogudin, G., Selivanova, S., Ziegler, M.: Bit-complexity of solving systems of linear evolutionary partial differential equations. In: Santhanam, R., Musatov, D. (eds.) CSR 2021. LNCS, vol. 12730, pp. 223–241. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-79416-3_13
Koswara, I., Pogudin, G., Selivanova, S., Ziegler, M.: Bit-complexity of classical solutions of linear evolutionary systems of partial differential equations. J. Complex. (2022, submitted)
Koswara, I., Selivanova, S., Ziegler, M.: Computational complexity of real powering and improved solving linear differential equations. In: van Bevern, R., Kucherov, G. (eds.) CSR 2019. LNCS, vol. 11532, pp. 215–227. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-19955-5_19
Lim, D., Selivanova, S., Ziegler, M.: Complexity and coding theory of hilbert spaces: what is a polynomial-time computable \(l_2\) function? In: Proceedings of Computability and Complexity in Analysis (CCA 2020), pp. 41–42 (2020)
Müller, N.T.: The iRRAM: exact arithmetic in C++. In: Blanck, J., Brattka, V., Hertling, P. (eds.) CCA 2000. LNCS, vol. 2064, pp. 222–252. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45335-0_14
Pan, V., Reif, J.: The bit-complexity of discrete solutions of partial differential equations: compact multigrid. Comput. Math. Appl. 20, 9–16 (1990)
Park, S., et al.: Foundation of computer (algebra) analysis systems: semantics, logic, programming, verification (2020). https://arxiv.org/abs/1608.05787
Plum, M.: Computer-assisted proofs for semilinear elliptic boundary value problems. Japan J. Indust. Appl. Math. 26(2–3), 419–442 (2009)
Pouly, A., Graça, D.S.: Computational complexity of solving polynomial differential equations over unbounded domains. Theoret. Comput. Sci. 626, 67–82 (2016). https://doi.org/10.1016/j.tcs.2016.02.002
Pour-El, M., Richards, J.: Computability in Analysis and Physics. Cambridge University Press, Cambridge (2017)
Selivanova, S., Selivanov, V.: Computing solution operators of boundary-value problems for some linear hyperbolic systems of PDEs. Log. Methods Comput. Sci. 13(4:13), 1–31 (2017)
Selivanov, V., Selivanova, S.: Primitive recursive ordered fields and some applications. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds.) CASC 2021. LNCS, vol. 12865, pp. 353–369. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-85165-1_20
Selivanova, S., Selivanov, V.: Primitive recursive ordered fields and some applications (2021). arXiv:2010.10189
Selivanova, S., Steinberg, F., Thies, H., Ziegler, M.: 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.) CASC 2021. LNCS, vol. 12865, pp. 370–390. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-85165-1_21
Selivanova, S., Selivanov, V.: Bit complexity of computing solutions for symmetric hyperbolic systems of PDEs with guaranteed precision. Computability 10(2), 123–140 (2021). https://doi.org/10.3233/COM-180215
Selivanova, S., Selivanov, V.L.: Computing solution operators of boundary-value problems for some linear hyperbolic systems of PDEs. Log. Methods Comput. Sci. 13(4) (2017). https://doi.org/10.23638/LMCS-13(4:13)2017
Steinberg, F.: Complexity theory for spaces of integrable functions. Log. Methods Comput. Sci. 13(3), Paper No. 21, 39 (2017). https://doi.org/10.23638/LMCS-13(3:21)2017
Sun, S.-M., Zhong, N., Ziegler, M.: Computability of the solutions to navier-stokes equations via effective approximation. In: Du, D.-Z., Wang, J. (eds.) Complexity and Approximation. LNCS, vol. 12000, pp. 80–112. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-41672-0_7
Weihrauch, K.: Computable Analysis. Springer, Berlin (2000). https://doi.org/10.1007/978-3-642-56999-9
Weihrauch, K.: Computational complexity on computable metric spaces. Math. Log. Q. 49(1), 3–21 (2003)
Ziegler, M., Brattka, V.: Computability in linear algebra. Theoret. Comput. Sci. 326(1–3), 187–211 (2004)
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Selivanova, S. (2022). Computational Complexity of Classical Solutions of Partial Differential Equations. In: Berger, U., Franklin, J.N.Y., Manea, F., Pauly, A. (eds) Revolutions and Revelations in Computability. CiE 2022. Lecture Notes in Computer Science, vol 13359. Springer, Cham. https://doi.org/10.1007/978-3-031-08740-0_25
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