In this article, an iterative method suitable for inverting semilinear problems is presented. The method employs a two-way parametrization that is able to reduce semilinear boundary-value problems into simpler and linear systems governed by the Laplacian. In contrast to Newton–Picard iterations, the method prescribes a well-defined procedure for constructing initial iterates, and thus, does not suffer from lack of global convergence. With rectangular domains and periodic boundary conditions, the technique also allows fast Poisson solvers to be engaged. Theoretical justifications are provided and supported by several experiments. Numerical results show that the method yields approximations which are comparable to reference solutions.

在本文中，提出了一种适用于反半线性问题的迭代方法。该方法采用双向参数化，能够将半线性边界值问题简化为由拉普拉斯算子控制的简单线性系统。与Newton-Picard迭代相反，该方法规定了用于定义初始迭代的定义明确的过程，因此不会遭受缺乏全局收敛性的困扰。利用矩形域和周期性边界条件，该技术还允许使用快速泊松求解器。理论证明是由几个实验提供和支持的。数值结果表明，该方法得出的近似值可与参考溶液相比。