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This article is concerned with a regularity analysis of parametric operatorequations with a perspective on uncertainty quantification. We study theregularity of mappings between Banach spaces near branches of isolatedsolutions that are implicitly defined by a residual equation. Under $s$-Gevreyassumptions on on the residual equation, we establish $s$-Gevrey bounds on theFr\'echet derivatives of the local data-to-solution mapping. This abstractframework is illustrated in a proof of regularity bounds for a semilinearelliptic partial differential equation with parametric and random field input.

The Gevrey class implicit mapping theorem with application to UQ of semilinear elliptic PDEs