### Abstract

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.

Original language | English |
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Pages (from-to) | A900-A926 |

Journal | SIAM journal on scientific computing |

Volume | 41 |

Issue number | 2 |

DOIs | |

Publication status | Published - 28 Mar 2019 |

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### Keywords

- A posteriori error estimation
- parametrized equations
- projection-based model order reduction
- Monte-Carlo estimator
- concentration phenomenon
- goal-oriented error estimation

### Cite this

*SIAM journal on scientific computing*,

*41*(2), A900-A926. https://doi.org/10.1137/18M120364X