In this talk we will present localized model order reduction (LMOR) methods for nonlinear partial differential equations with the following features: 1) In contrast to existing methods such as [ST22], we can fully control the error on the subdomains without assessing the global computational domain. 2) The reduced basis functions on the subdomains can be computed on small oversampling domains that enclose the target subdomain; the methods are thus localizable. 3) We do not require any structural assumptions such as scale separation. These features make the LMOR methods well suited for digital twins. To construct the local reduced or surrogate model we consider a transfer operator that maps arbitrary admissible boundary data on the boundary of an oversampling domain to the respective (local) solution on the target subdomain. Then, we try to approximate the set of all local solutions on the target subdomain. Interpreting the boundary data as some input parameter, we can view this set of local solutions as a set of solutions depending on a parameter. This motivates using methods from model order reduction such as the Greedy algorithm [VPRP03] to approximate this set. However, the Greedy algorithm [VPRP03] only provides certification over the training set of finite cardinality used to construct the reduced model. We will thus present a randomized greedy algorithm that provides with high probability a certification for the whole parameter set rather than only for the parameters in the training set. This then facilitates controlling the local approximation error of the LMOR methods. [ST22] K. Smetana and T. Taddei, Localized Model Reduction for Nonlinear Elliptic Partial Differential Equations: Localized Training, Partition of Unity, and Adaptive Enrichment, SIAM J. Sci. Comput., 45(3), A1300-A1331, 2023. [VPRP03] K. Veroy, C. Prud'homme, D.V. Rovas, and A.T. Patera, A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, in Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, 2003.