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Arithmetic Complexity of Solutions of the Dirichlet Problem

Holger Boche, Volker Pohl, H. Vincent Poor

TL;DR

This work examines the computability of the Dirichlet problem on the unit disk $\mathbb{D}$ under Turing-machine computation, focusing on two canonical solution methods: Poisson integral evaluation and Dirichlet-energy minimization. Using computable-analysis tools and the Zheng--Weihrauch hierarchy, it proves that even with computable boundary data, the resulting solutions are generally not Turing computable, and it provides precise upper and lower bounds for the non-computability of each method. Specifically, the minimal energy $\underline{\mathrm{E}}(f)$ associated with the variational approach lies in $\Sigma_1$ (recursively approximable) and can be noncomputable, while Poisson-integral boundary values $f(e^{i\theta})$ reside in $\Delta_2$ (recursively approximable) with lower bounds in $\mathbf{C}_2$, and there exist boundary data yielding noncomputable values at computable points. The results offer a rigorous, hierarchy-based account of the intrinsic limits of digital computation for classical boundary-value problems and guide which computability framework is applicable given a boundary-data class.

Abstract

The classical Dirichlet problem on the unit disk can be solved by different numerical approaches. The two most common and popular approaches are the integration of the associated Poisson integral and, by applying Dirichlet's principle, solving a particular minimization problem. For practical use, these procedures need to be implemented on concrete computing platforms. This paper studies the realization of these procedures on Turing machines, the fundamental model for any digital computer. We show that on this computing platform both approaches to solve Dirichlet's problem yield generally a solution that is not Turing computable, even if the boundary function is computable. Then the paper provides a precise characterization of this non-computability in terms of the Zheng--Weihrauch hierarchy. For both approaches, we derive a lower and an upper bound on the degree of non-computability in the Zheng--Weihrauch hierarchy.

Arithmetic Complexity of Solutions of the Dirichlet Problem

TL;DR

This work examines the computability of the Dirichlet problem on the unit disk under Turing-machine computation, focusing on two canonical solution methods: Poisson integral evaluation and Dirichlet-energy minimization. Using computable-analysis tools and the Zheng--Weihrauch hierarchy, it proves that even with computable boundary data, the resulting solutions are generally not Turing computable, and it provides precise upper and lower bounds for the non-computability of each method. Specifically, the minimal energy associated with the variational approach lies in (recursively approximable) and can be noncomputable, while Poisson-integral boundary values reside in (recursively approximable) with lower bounds in , and there exist boundary data yielding noncomputable values at computable points. The results offer a rigorous, hierarchy-based account of the intrinsic limits of digital computation for classical boundary-value problems and guide which computability framework is applicable given a boundary-data class.

Abstract

The classical Dirichlet problem on the unit disk can be solved by different numerical approaches. The two most common and popular approaches are the integration of the associated Poisson integral and, by applying Dirichlet's principle, solving a particular minimization problem. For practical use, these procedures need to be implemented on concrete computing platforms. This paper studies the realization of these procedures on Turing machines, the fundamental model for any digital computer. We show that on this computing platform both approaches to solve Dirichlet's problem yield generally a solution that is not Turing computable, even if the boundary function is computable. Then the paper provides a precise characterization of this non-computability in terms of the Zheng--Weihrauch hierarchy. For both approaches, we derive a lower and an upper bound on the degree of non-computability in the Zheng--Weihrauch hierarchy.
Paper Structure (14 sections, 9 theorems, 82 equations)

This paper contains 14 sections, 9 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. General Notation and Function Spaces
  3. Computability Analysis
  4. The Zheng--Weihrauch Hierarchy of Non-Computability
  5. The first layer in the Zheng--Weihrauch hierarchy
  6. The second layer in the Zheng--Weihrauch hierarchy
  7. The Classical Dirichlet Problem on $\mathbb{D}$ -- Problem Statement
  8. Solution by Poisson integral
  9. Solution by Dirichlet's principle
  10. Problem statement
  11. Computation via the Minimization of Dirichlet energy
  12. Computation via Integrating the Poisson Integral
  13. Conclusions and Discussions
  14. The complete Zheng--Weihrauch hierarchy

Key Result

Proposition 1

A number $x \in \mathbb{R}$ belongs to $\Delta_{1}$ if and only if $x \in \Sigma_{1} \cap \Pi_{1}$, that is $\Delta_{1} = \Sigma_{1} \cap \Pi_{1}$.

Theorems & Definitions (37)

  • Definition 1: Computable rational sequence
  • Definition 2: Effective convergence
  • Definition 3: Computable number
  • Remark 1
  • Definition 4: Computable sequence
  • Definition 5: Banach--Mazur computable function
  • Remark 2
  • Remark 3
  • Definition 6: Effectively separable
  • Definition 7: Computable functions in Banach spaces
  • ...and 27 more