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

Published online by Cambridge University Press:  28 July 2016

AKITOSHI KAWAMURA ,
FLORIAN STEINBERG  and
MARTIN ZIEGLER
AKITOSHI KAWAMURA
Affiliation:
Graduate School of Arts and Sciences, University of Tokyo, Tokyo, Japan
FLORIAN STEINBERG
Affiliation:
Graduate School of Arts and Sciences, University of Tokyo, Tokyo, Japan Department of Mathematics, TU Darmstadt, Darmstadt, Germany
MARTIN ZIEGLER
Affiliation:
Department of Mathematics, TU Darmstadt, Darmstadt, Germany
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Abstract

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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