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The numerical solution of the Dirichlet generalized and classical harmonic problems for irregular n-sided pyramidal domains by the method of probabilistic solutions

M. Zakradze, Z. Tabagari, N. Koblishvili, T. Davitashvili, J. M. Sanchez-Saez, F., Criado-Aldeanueva

TL;DR

The paper addresses numerical solution of Dirichlet generalized and classical harmonic problems on irregular pyramidal domains $P_n(h)$ by the Method of Probabilistic Solutions (MPS). It uses Wiener-process simulations to approximate $u(x)$ via $u(x)=E_x g(x( au))$, with Monte Carlo estimate $u_N(x)=\tfrac{1}{N}\sum_{i=1}^N g(y^i)$, and develops a geometric boundary-intersection routine for pyramids to obtain the boundary data $g(y)$. An auxiliary classical problem with exact solution $u(x,x^0)=\tfrac{1}{|x-x^0|}$ is used to validate the implementation, and two irregular pyramid examples demonstrate convergence as the quantization number $n_q$ and the path count $N$ increase. The results show that the approach accurately solves 3D Laplace problems on complex polyhedral domains and provides a foundation for extending MPS to broader geometries and infinite domains.

Abstract

This paper describes the application of the method of probabilistic solutions (MPS) to numerically solve the Dirichlet generalized and classical harmonic problems for irregular n sided pyramidal domains. Here, generalized means that the boundary function has a finite number of first kind discontinuity curves, with the pyramid edges acting as these curves. The pyramid base is a convex polygon, and its vertex projection lies within the base. The proposed algorithm for solving boundary problems numerically includes the following steps: a) applying MPS, which relies on computer modeling of the Wiener process; b) determining the intersection point between the simulated Wiener process path and the pyramid surface; c) developing a code for numerical implementation and verifying the accuracy of the results; d) calculating the desired function value at any chosen point. Two examples are provided for illustration, and the results of the numerical experiments are presented and discussed.

The numerical solution of the Dirichlet generalized and classical harmonic problems for irregular n-sided pyramidal domains by the method of probabilistic solutions

TL;DR

The paper addresses numerical solution of Dirichlet generalized and classical harmonic problems on irregular pyramidal domains by the Method of Probabilistic Solutions (MPS). It uses Wiener-process simulations to approximate via , with Monte Carlo estimate , and develops a geometric boundary-intersection routine for pyramids to obtain the boundary data . An auxiliary classical problem with exact solution is used to validate the implementation, and two irregular pyramid examples demonstrate convergence as the quantization number and the path count increase. The results show that the approach accurately solves 3D Laplace problems on complex polyhedral domains and provides a foundation for extending MPS to broader geometries and infinite domains.

Abstract

This paper describes the application of the method of probabilistic solutions (MPS) to numerically solve the Dirichlet generalized and classical harmonic problems for irregular n sided pyramidal domains. Here, generalized means that the boundary function has a finite number of first kind discontinuity curves, with the pyramid edges acting as these curves. The pyramid base is a convex polygon, and its vertex projection lies within the base. The proposed algorithm for solving boundary problems numerically includes the following steps: a) applying MPS, which relies on computer modeling of the Wiener process; b) determining the intersection point between the simulated Wiener process path and the pyramid surface; c) developing a code for numerical implementation and verifying the accuracy of the results; d) calculating the desired function value at any chosen point. Two examples are provided for illustration, and the results of the numerical experiments are presented and discussed.
Paper Structure (7 sections, 1 theorem, 20 equations, 2 figures, 4 tables)

This paper contains 7 sections, 1 theorem, 20 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Formulation of the Generalized Problem
  3. Simulation of the Wiener Process and the Method of Probabilistic Solution
  4. Finding the Intersection Point of the Trajectory of the Simulated Wiener Process and the Surface $S$ of the Pyramid $P_n$
  5. An Auxiliary Classical Problem
  6. Numerical Examples
  7. Concluding remarks

Key Result

Theorem 1

Suppose $g(y)$ is continuous (or discontinuous) bounded function on $S$ and the finite domain $D\subset \mathbb{R}^3$ is bounded by piecewise smooth surface $S$, then the solution of the Dirichlet classical (or generalized) boundary value problem for the Laplace equation at the fixed point $x\in D$

Figures (2)

  • Figure 1: Results for the Problem \ref{['problem:2']} (Example \ref{['example:6.1']}) for starting point $(0, 0, 0.5)$
  • Figure 2: Illustration of single run for the Problem \ref{['problem:2']} (Example \ref{['example:6.1']}) ($13657$ steps)

Theorems & Definitions (6)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Example 1
  • Example 2