We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a parameter. This model can be seen as a reaction-diffusion problem with a polynomial nonlinearity in the reaction term. The efficiency of various numerical approximations across the entire parameter space is closely related to the regularity of the solution with respect to the parameter. We show that if the coefficients and the right-hand side are analytic or Gevrey class regular with respect to the parameter, the same type of parametric regularity is valid for the solution. The key ingredient of the proof is the combination of the alternative-to-factorial technique from our previous work [1] with a novel argument for the treatment of the power-type nonlinearity in the reaction term. As an application of this abstract result, we obtain rigorous convergence estimates for numerical integration of semilinear reaction-diffusion problems with random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Our theoretical findings are confirmed in numerical experiments.

翻译：我们研究一类具有齐次本质边界条件的二阶参数椭圆半线性偏微分方程，其中系数和右端项（因此解也）可能依赖于参数。该模型可视为反应项具有多项式非线性的反应扩散问题。在整个参数空间上各种数值逼近的效率与解关于参数的正则性密切相关。我们证明：若系数和右端项关于参数具有解析性或Gevrey类正则性，则解也具有相同类型的参数正则性。证明的关键步骤是将前期工作[1]中的非阶乘技巧与处理反应项中幂型非线性的新论证相结合。作为该抽象结果的应用，我们利用高斯拟蒙特卡罗求积法，获得了随机系数半线性反应扩散问题数值积分的严格收敛估计。数值实验验证了理论结果。