Resilience represents one of the many challenges to be anticipated and addressed before viable exascale systems can emerge. In this context, I will present an approach to tackle resilience for the solving of deterministic and stochastic partial differential equations (PDEs). The approach is based on a domain decomposition framework, together with a sampling and optimization approach. In particular, the occurence of silent data corruption (SDC), modeled by random bit flips in the 64-bits representation of double precision variables, is overcome by means of robust regression techniques involving l1-minimization. In the case of an elliptic PDE with random coefficients, the stochastic quantities are approximated by a polynomial chaos (PC) surrogate and we then resort to a hybrid approach involving both sampling and Galerkin projection. Finally, I will present discrete a priori bounds on the solution of elliptic PDEs, used to enhance the resilient capabilities of the approach. This a priori knowledge is used to check the admissibility of local PDE solutions and to filter out corrupted data, thus making the subsequent regression more robust.