Short Course on Reduced Basis Methods

Course Title

Reduced Basis Methods: Theory, Implementation and Applications


Yanlai Chen, Department of Mathematics, University of Massachusetts Dartmouth


Room 602, Pao Yue-Kong library, Shanghai Jiao Tong University

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Models of reduced computational complexity are indispensable in scenarios where a large number of numerical solutions to a parametrized problem are desired in a fast/real-time fashion. Thanks to an offline-online procedure and the recognition that the parameter-induced solution manifolds can be well approximated by finite-dimensional spaces, reduced basis method (RBM) and reduced collocation method (RCM) can improve the simulation efficiency in such scenarios by several orders of magnitudes. The accuracy of the RBM solution is maintained through a rigorous a posteriori error estimator whose efficient development is critical and involves fast eigensolves.

This short course will present an introduction of the RBM, their theoretical underpinning, and all ingredients of the algorithm in its classical form. We will showcase some of our recent work on novel approaches for speeding up the offline portion of the RBM/RCM, and new residual-based and residual-free strategies for circumventing error stagnation that is traditional of the classical RBM. A Matlab-based software system and PDE Apps jointly designed by Akselos S.A. and MIT (https://atpwui.akselos.com/acoustics/pdeappsdata.html) will be demonstrated in the course. Lastly, the course will feature a data compression algorithm inspired by RBM, a Parametrized-Background Data-Weak Formulation for Real-Time Data Assimilation, and the integration of RBM into the generalized Polynomial Chaos (gPC) framework for uncertainty quantification significantly delaying the curse of dimensionality.


Date Time Lecture
December 13, 2017 9:00-11:40 Lecture 1
December 14, 2017 9:00-11:40 Lecture 2
December 15, 2017 9:00-11:40 Lecture 3

Tentative outline

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Zhenli Xu, SJTU