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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 infsup) 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., userdefined norms or quantity of interest. Then, we introduce a dual problem with random righthand 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 fasttoevaluate 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 lowdimensional reduced dual spaces.
Original language  English 

Pages (fromto)  A900A926 
Journal  SIAM journal on scientific computing 
Volume  41 
Issue number  2 
DOIs  
Publication status  Published  28 Mar 2019 
Keywords
 A posteriori error estimation
 parametrized equations
 projectionbased model order reduction
 MonteCarlo estimator
 concentration phenomenon
 goaloriented error estimation
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Activities
 1 Invited talk

Randomized Model Order Reduction
Kathrin Smetana (Speaker)
11 Apr 2018Activity: Talk or presentation › Invited talk