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On the computational complexity of the Dirichlet Problem for Poisson's Equation


The last years have seen an increasing interest in classifying (existence claims in) classical mathematical theorems according to their strength. We pursue this goal from the refined perspective of computational complexity. Specifically, we establish that rigorously solving the Dirichlet Problem for Poisson's Equation is in a precise sense ‘complete’ for the complexity class ${\#\mathcal{P}}$ and thus as hard or easy as parametric Riemann integration (Friedman 1984; Ko 1991. Complexity Theory of Real Functions).

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BlumensonL.E. (1960). Classroom notes: A derivation of n-Dimensional spherical coordinates. The American Mathematical Monthly 67 (1) 6366.
BournezO., GraçaD.S., PoulyA. and ZhongN. (2013). Computability and computational complexity of the evolution of nonlinear dynamical systems. In: Proc. 9th Conference on Computability in Europe (CiE 2013), Springer LNCS vol. 7921, doi 10.1007/978-3-642-39053-1_2.
BrattkaV. and YoshikawaA. (2006). Towards computability of elliptic boundary value problems in variational formulation. Journal of Complexity, 22 858880.
BravermanM. (2005). On the complexity of real functions. In: Proc. 6th Annual IEEE Symposium on Foundations of Computer Science, doi 10.1109/SFCS.2005.58.
FriedmanH. (1984). The computational complexity of maximization and integration. Advances in Mathematics 53 (1) 8098.
GareyM.R. and JohnsonD.S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, New York.
GilbargD. and TrudingerN.S. (2001). Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin. Reprint of the 1998 edition.
GrzegorczykA. (1957). On the definitions of computable real continuous functions. Fundamenta Mathematicae 44 6171.
KawamuraA. (2010). Lipschitz continuous ordinary differential equations are polynomial-space complete. Computational Complexity 19 (2) 305332.
KawamuraA. (2011). Computational Complexity in Analysis and Geometry. PhD thesis, University of Toronto.
KawamuraA. and CookS. (2012). Complexity theory for operators in analysis. ACM Transactions in Computation Theory 4 (2) Article 5.
KawamuraA., OtaH., RösnickC. and ZieglerM. (2014). Computational complexity of smooth differential equations. Logical Methods in Computer Science 10 (1) 6.
KoK.-I. (1982). The maximum value problem and NP real numbers. Journal of Computer and System Sciences 24 (1) 1535.
KoK.-I. (1990). Inverting a one-to-one real function is inherently sequential. In: Feasible Mathematics (Ithaca, NY, 1989), Progress in Computer Science and Applied Logic, volume 9, Birkhäuser Boston, Boston, MA 239257.
KoK.-I. (1991). Complexity Theory of Real Functions. Progress in Theoretical Computer Science. Birkhäuser Boston Inc., Boston, MA.
KoK.-I. (1992). On the computational complexity of integral equations. Annals of Pure and Applied Logic 53 (3) 201228.
KoK.-I. (1998). Polynomial-time computability in analysis. In: Handbook of Recursive Mathematics, Vol. 2, Studies in Logic and the Foundations of Mathematics, volume 139, North-Holland, Amsterdam 12711317.
KoK.-I. and FriedmanH. (1982). Computational complexity of real functions. Theoretical Computer Sciences 20 (3), 323352.
KoK.-I. and YuF. (2008). On the complexity of convex hulls of subsets of the two-dimensional plane. In: Proceedings of the 4th International Conference on Computability and Complexity in Analysis (CCA 2007). Electron. Notes Theor. Comput. Sci. 202 Elsevier Sci. B. V., Amsterdam 121135.
LabhallaS., LombardiH. and MoutaiE. (2001). Espaces métriques rationnellement présentés et complexité. Theoretical Computer Science 250 265332. le cas de l'espace des fonctions réelles uniformément, continues sur un intervalle compact.
LacombeD. (1958). Sur les possibilités d'extension de la notion de fonction récursive aux fonctions d'une ou plusieurs variables réelles. In: Le raisonnement en mathématiques et en sciences expérimentales, Colloques Internationaux du Centre National de la Recherche Scientifique, LXX, Editions du Centre National de la Recherche Scientifique, Paris 67–75.
MoreraG. (1887). Sulle derivate seconde della funzione potenziale di spazio. Rendiconti 20 302310.
MüllerN.T. (1987). Uniform computational complexity of Taylor series. In: Proceeding 14th International Colloquium on Automata, Languages, and Programming. Springer-Verlag Lecture Notes in Computer Science 267 435444.
MüllerN.T. (1995). Constructive aspects of analytic functions. In: Proc. Workshop on Computability and Complexity in Analysis, volume 190 of InformatikBerichte, FernUniversität Hagen 105–114.
ParberryI. and SchnitgerG. (1988). Parallel computation with threshold functions. Journal of Computer and System Sciences 36 278302.
Pour-ElM.B. and RichardsJ.I. (1989). Computability in Analysis and Physics, Perspectives in Mathematical Logic, Springer-Verlag, Berlin.
RösnickC. (2013). Closed sets and operators thereon: Representations, computability and complexity. In: Proceedings of the 10th International Conference on Computability and Complexity in Analysis (CCA), 8-10 July 2013, Nancy, France.
SunS., ZhongN. and ZieglerM. (2015). Computability of navier-stokes' equation. In: Proceeding of the 11th Conference on Computability in Europe, Lecture Notes in Computer Science 9136 334342.
TodaS. (1991). PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20 (5) 865877.
TuringA. (1936). On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society 42 (2) 230265.
WeihrauchK. (2000). Computable Analysis . An Introduction. Springer, Berlin/Heidelberg.
WeihrauchK. and ZhongN. (2002). Is wave propagation computable or can wave computers beat the Turing machine?. Proceedings of London Mathematical Society 85 (2) 312332.
WeihrauchK. and ZhongN. (2005). Computing the solution of the Korteweg–de Vries equation with arbitrary precision on turing. Theoretical Computer Science 332 (1–3) 337366.
WeihrauchK. and ZhongN. (2006). An algorithm for computing fundamental solutions. SIAM Journal on Computing 35 (6) 12831294.
WeihrauchK. and ZhongN. (2007). Computable analysis of the abstract Cauchy problem in a Banach space and its applications I. Mathematical Logic Quarterly 53 (4–5) 511531.
WienholtzE., KalfH. and KriecherbauerT. (2009). Elliptische Differentialgleichungen zweiter Ordnung. Springer, Dordrecht. Eine Einführung mit historischen Bemerkungen. [An introduction with historical remarks].
ZhongN. (1998). Derivatives of computable functions. Mathematical Logic Quarterly 44 304316.
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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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