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Monte-Carlo Solution of the Neumann Problem for Nonlinear Helmholts Equation

机译：非线性亥姆霍兹方程Neumann问题的Monte-Carlo解决方案

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In this paper we consider the Neumann boundary-value problem for the Helmholtz equation with a polynomial non-linearity on the right-hand side. We will assume that a solution of the problem exists, and this permits us to construct an unbiased Monte Carlo estimator using the trajectories of certain branching processes. We utilize Green's formula and an elliptic mean-value theorem to derive a special integral equation which equates the value of the function with its integral over the considered domain. The solution of the problem is then given in the form of a mathematical expectation over some particular random variables. According to this probabilistic representation, a branching stochastic process is constructed and an unbiased estimator of the solution of the nonlinear problem is formed by taking the expectation over these branching processes. It is shown that the unbiased estimator has a finite variance. In addition, the proposed branching process has a finite average number of branches, and hence is easily simulated. Finally, we provide numerical results based on numerical experiments carried out with these algorithms to validate our approach.
机译：在本文中，我们考虑了Helmhholtz方程的Neumann边值问题，右侧的多项式非线性。我们将假设存在问题的解决方案，这允许我们使用某些分支过程的轨迹来构造一个无偏见的蒙特卡罗估计器。我们利用绿色的公式和椭圆形均值定理来得出特殊的整体方程，其等于函数的值与所考虑的域中的整体相同。然后以某种特定随机变量的数学期望的形式给出问题的解决方案。根据这种概率表示，构造了分支随机过程，并且通过在这些分支过程中考虑期望来形成非线性问题的解的不偏的估计器。结果表明，非偏见的估计器具有有限的方差。另外，所提出的分支过程具有有限的平均分支数，因此很容易模拟。最后，我们提供了基于与这些算法进行的数值实验提供的数值结果，以验证我们的方法。

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