An EIM-degradation free reduced basis method via over collocation and residual hyper reduction-based error estimation

JOURNAL OF COMPUTATIONAL PHYSICS

卷:444

Article

Chen, Yanlai¹  Gottlieb, Sigal¹  Ji, Lijie²  Maday, Yvon^3,4,5,6,7,8 

[1] Univ Massachusetts Dartmouth, Dept Math, 285 Old Westport Rd, N Dartmouth, MA 02747 USA

[2] Shanghai Jiao Tong Univ, Sch Math Sci, Shanghai 200240, Peoples R China

[3] Sorbonne Univ, Paris, France

[4] Univ Paris Diderot, Paris, France

[5] CNRS, Paris, France

[6] Lab Jacques Louis Lions, F-75005 Paris, France

[7] SPC, Paris, France

[8] Inst Univ France, Paris, France

关键词: Reduced basis method;    Empirical interpolation method;    Generalized empirical interpolation method;    Collocation;    Over Collocation;    Hyper reductions;    Greedy algorithm;   

DOI  :  10.1016/j.jcp.2021.110545

来源: Elsevier

【 摘 要 】

The need for multiple interactive, real-time simulations using different parameter values has driven the design of fast numerical algorithms with certifiable accuracies. The reduced basis method (RBM) presents itself as such an option. RBM features a mathematically rigorous error estimator which drives the construction of a low-dimensional subspace. A surrogate solution is then sought in this low-dimensional space approximating the parameter-induced high fidelity solution manifold. However when the system is nonlinear or its parameter dependence nonaffine, this efficiency gain degrades tremendously, an inherent drawback of the application of the empirical interpolation method (EIM). In this paper, we augment and extend the EIM approach as a direct solver, as opposed to an assistant, for solving nonlinear partial differential equations on the reduced level. The resulting method, called Reduced Over-Collocation method (ROC), is stable and capable of avoiding the efficiency degradation. Two critical ingredients of the scheme are collocation at about twice as many locations as the number of basis elements for the reduced approximation space, and an efficient error indicator for the strategic building of the reduced solution space. The latter, the main contribution of this paper, results from an adaptive hyper reduction of the residuals for the reduced solution. Together, these two ingredients render the proposed R2-ROC scheme both offline- and online-efficient. A distinctive feature is that the efficiency degradation appearing in traditional RBM approaches that utilize EIM for nonlinear and nonaffine problems is circumvented, both in the offline and online stages. Numerical tests on different families of time-dependent and steady-state nonlinear problems demonstrate the high efficiency and accuracy of our R2-ROC and its superior stability performance. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free