TY - JOUR
T1 - Randomized residual-based error estimators for parametrized equations
AU - Smetana, Kathrin
AU - Zahm, Olivier
AU - Patera, Anthony T.
N1 - Publisher Copyright:
© 2019 Societ y for Industrial and Applied Mathematics
PY - 2019
Y1 - 2019
N2 - We propose a randomized a posteriori error estimator for reduced order approximations of parametrized (partial) differential equations. The error estimator has several important properties: the effectivity is close to unity with prescribed lower and upper bounds at specified high probability; the estimator does not require the calculation of stability (coercivity, or inf-sup) constants; the online cost to evaluate the a posteriori error estimator is commensurate with the cost to find the reduced order approximation; and the probabilistic bounds extend to many queries with only modest increase in cost. To build this estimator, we first estimate the norm of the error with a Monte Carlo estimator using Gaussian random vectors whose covariance is chosen according to the desired error measure, e.g., user-defined norms or quantity of interest. Then, we introduce a dual problem with random right-hand side the solution of which allows us to rewrite the error estimator in terms of the residual of the original equation. In order to have a fast-to-evaluate estimator, model order reduction methods can be used to approximate the random dual solutions. Here, we propose a greedy algorithm that is guided by a scalar quantity of interest depending on the error estimator. Numerical experiments on a multiparametric Helmholtz problem demonstrate that this strategy yields rather low-dimensional reduced dual spaces.
AB - We propose a randomized a posteriori error estimator for reduced order approximations of parametrized (partial) differential equations. The error estimator has several important properties: the effectivity is close to unity with prescribed lower and upper bounds at specified high probability; the estimator does not require the calculation of stability (coercivity, or inf-sup) constants; the online cost to evaluate the a posteriori error estimator is commensurate with the cost to find the reduced order approximation; and the probabilistic bounds extend to many queries with only modest increase in cost. To build this estimator, we first estimate the norm of the error with a Monte Carlo estimator using Gaussian random vectors whose covariance is chosen according to the desired error measure, e.g., user-defined norms or quantity of interest. Then, we introduce a dual problem with random right-hand side the solution of which allows us to rewrite the error estimator in terms of the residual of the original equation. In order to have a fast-to-evaluate estimator, model order reduction methods can be used to approximate the random dual solutions. Here, we propose a greedy algorithm that is guided by a scalar quantity of interest depending on the error estimator. Numerical experiments on a multiparametric Helmholtz problem demonstrate that this strategy yields rather low-dimensional reduced dual spaces.
KW - A posteriori error estimation
KW - Concentration phenomenon
KW - Goal-oriented error estimation
KW - Monte Carlo estimator
KW - Parametrized equations
KW - Projection-based model order reduction
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U2 - 10.1137/18M120364X
DO - 10.1137/18M120364X
M3 - Article
AN - SCOPUS:85065553763
SN - 1064-8275
VL - 41
SP - A900-A926
JO - SIAM Journal on Scientific Computing
JF - SIAM Journal on Scientific Computing
IS - 2
ER -