Error Control for the Localized Reduced Basis Multiscale Method with Adaptive On-Line Enrichment

Abstract

Abstract. In this contribution we consider localized, robust and efficient a-posteriori error estimation of the localized reduced basis multi-scale (LRBMS) method for parametric elliptic problems with possibly heterogeneous diffusion coefficient. The numerical treatment of such parametric multi-scale problems are characterized by a high computational complexity, arising from the multi-scale character of the underlying differential equation and the additional parameter dependence. The LRBMS method can be seen as a combination of numerical multi-scale methods and model reduction using reduced basis (RB) methods to efficiently reduce the computational complexity with respect to the multi-scale as well as the parametric aspect of the problem, simultaneously. In contrast to the classical residual based error estimators currently used in RB methods, we are considering error estimators that are based on conservative flux reconstruction and provide an efficient and rigorous bound on the full error with respect to the weak solution. In addition, the resulting error estimator is localized and can thus be used in the on-line phase to adaptively enrich the solution space locally where needed. The resulting certified LRBMS method with adaptive on-line enrichment thus guarantees the quality of the reduced solution during the on-line phase, given any (possibly insufficient) reduced basis that was generated during the offline phase. Numerical experiments are given to demonstrate the applicability of the resulting algorithm with online enrichment to single phase flow in heterogeneous media.

On error controlled numerical model reduction in FE2‐analysis of transient heat flow

“…Error estimators for POD type of techniques have been presented by Abdulle et al for heterogeneous multiscale methods, and by Boyaval for numerical homogenization. Ohlberger and Schindler proposed a procedure for error control in the context of the localized reduced basis multiscale method. Kerfriden et al and Chamoin and Legoll derived error estimation based on the constitutive relation error.…”

A locally conservative reduced flux reconstruction for elliptic problems

“…However, this is not straightforward for RB solutions (5), even if the underlying scheme (2) yields locally conservative approximations. We will not give individual references from here on but refer to [1] and the references therein. The present work closes this gap.…”

“…In the context of model order reduction of elliptic problems (1) by the reduced Basis (RB) method, it is sometimes desirable to obtain a locally conservative flux (4), for instance for a posteriori error estimation [1] or in the context of flow problems [2]. However, this is not straightforward for RB solutions (5), even if the underlying scheme (2) yields locally conservative approximations.…”

“…The present work closes this gap. We will not give individual references from here on but refer to [1] and the references therein.…”

Dynamic data-driven model reduction: adapting reduced models from incomplete data

“…The online adaptive approach [43] adapts the approximation of nonlinear terms from sparse data of the full model. There is also a body of work that rebuilds reduced models from scratch, e.g., in optimization [27,32,50], inverse problems [17,25], and multiscale methods [38]. We also mention that reduced models have been used in the context of dynamical data-driven application systems (DDDAS), which dynamically incorporate data into an executing application, and, in reverse, dynamically steer the measurement process.…”