TY - JOUR
T1 - Randomized local model order reduction
AU - Buhr, Andreas
AU - Smetana, Kathrin
N1 - Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.
PY - 2018
Y1 - 2018
N2 - 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. Babu\v ska and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373-406; K. Smetana and A. T. Patera, SIAM J. Sci. Comput., 38 (2016), pp. A3318-A3356]. 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, P. G. Martinsson, and J. A. Tropp, SIAM Rev., 53 (2011), pp. 217-288] which constructs a local reduced space approximating the range of the transfer operator and thus the optimal local approximation spaces. 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.
AB - 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. Babu\v ska and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373-406; K. Smetana and A. T. Patera, SIAM J. Sci. Comput., 38 (2016), pp. A3318-A3356]. 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, P. G. Martinsson, and J. A. Tropp, SIAM Rev., 53 (2011), pp. 217-288] which constructs a local reduced space approximating the range of the transfer operator and thus the optimal local approximation spaces. 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.
KW - A posteriori error estimation
KW - A priori error bound
KW - Domain decomposition methods
KW - Localized model order reduction
KW - Multiscale methods
KW - Randomized linear algebra
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U2 - 10.1137/17M1138480
DO - 10.1137/17M1138480
M3 - Article
AN - SCOPUS:85053800106
SN - 1064-8275
VL - 40
SP - A2120-A2151
JO - SIAM Journal on Scientific Computing
JF - SIAM Journal on Scientific Computing
IS - 4
ER -