This article is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under $s$-Gevrey assumptions on on the residual equation, we establish $s$-Gevrey bounds on the Fr\'echet derivatives of the local data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input.

翻译：本文关注参数化算子方程的正则性分析，着眼于不确定性量化。我们研究了由残差方程隐式定义的孤立解分支附近Banach空间之间映射的正则性。在残差方程满足$s$-Gevrey假设的条件下，我们建立了局部数据到解映射的Fr\'echet导数的$s$-Gevrey界。这一抽象框架通过证明具有参数化和随机场输入的半线性椭圆型偏微分方程的正则性界得到了具体说明。