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首页> 外文期刊>Journal of Computational Physics >Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees
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Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees

机译：非线性二维抛物线问题的随机树域分解解

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摘要

A domain decomposition method is developed for the numerical solution of nonlinear parabolic partial differential equations in any space dimension, based on the probabilistic representation of solutions as an average of suitable multiplicative functionals. Such a direct probabilistic representation requires generating a number of random trees, whose role is that of the realizations of stochastic processes used in the linear problems. First, only few values of the sought solution inside the space-time domain are computed (by a Monte Carlo method on the trees). An interpolation is then carried out, in order to approximate interfacial values of the solution inside the domain. Thus, a fully decoupled set of sub-problems is obtained. The algorithm is suited to massively parallel implementation, enjoying arbitrary scalability and fault tolerance properties. Pruning the trees is shown to increase appreciably the efficiency of the algorithm. Numerical examples conducted in 2D, including some for the KPP equation, are given.
机译：开发了一种域分解方法，用于以任何适当维数的均值形式表示的解的概率表示，从而在任何空间维度上求解非线性抛物型偏微分方程的数值解。这种直接的概率表示方法需要生成许多随机树，其作用是实现线性问题中使用的随机过程。首先，仅计算了时空域内所需解的很少几个值（通过树上的蒙特卡洛方法）。然后执行插值，以便近似域内溶液的界面值。因此，获得了完全解耦的子问题集。该算法适用于大规模并行实现，具有任意的可伸缩性和容错特性。修剪树木表明可以显着提高算法的效率。给出了在2D模式下进行的数值示例，其中包括一些KPP方程。

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