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In this paper we propose local approximation spaces for localized model order reduction procedures such as domain decomposition and multiscale methods. Those spaces are constructed from local solutions of the partial differential equation (PDE) with random boundary conditions, yield an approximation that converges provably at a nearly optimal rate, and can be generated at close to optimal computational complexity. In many localized model order reduction approaches like the generalized finite element method, static condensation procedures, and the multiscale finite element method local approximation spaces can be constructed by approximating the range of a suitably defined transfer operator that acts on the space of local solutions of the PDE. Optimal local approximation spaces that yield in general an exponentially convergent approximation are given by the left singular vectors of this transfer operator [I Babuska and R Lipton 2011, K Smetana and AT Patera 2016]. However, the direct calculation of these singular vectors is computationally very expensive. In this paper, we propose an adaptive randomized algorithm based on methods from randomized linear algebra [N Halko et al. 2011], which constructs a local reduced space approximating the range of the transfer operator and thus the optimal local approximation spaces. The number of local solutions of the PDE required by the algorithm equals approximately the dimension of the desired local reduced spaces and the algorithm thus realizes the construction of the local approximation spaces at nearly optimal computational complexity. Starting from results in randomized linear algebra [N Halko et al. 2011] we prove an a priori error bound showing that the local spaces constructed by the algorithm presented in this article result in an approximation that converges at a nearly optimal rate. Moreover, the adaptive algorithm relies on a probabilistic a posteriori error estimator for which we prove that it is both efficient and reliable with high probability. Several numerical experiments confirm the theoretical findings.
- Randomized linear algebra
- Domain decomposition methods
- Multiscale methods
- A priori error bound
- A posteriori error estimation
- Localized model order reduction