L1-based reduced over collocation and hyper reduction for steady state and time-dependent nonlinear equations

TL;DR

This paper introduces the L1-ROC method, a novel reduced collocation approach that enhances efficiency and stability in solving nonlinear parametrized PDEs by augmenting the empirical interpolation method with L1-norm-based error estimation.

Contribution

It extends the empirical interpolation method as a direct solver for nonlinear pPDEs, improving efficiency and stability over traditional RBM approaches.

Findings

L1-ROC achieves high efficiency in offline and online stages.

The method maintains high accuracy for time-dependent and steady-state nonlinear problems.

L1-ROC demonstrates superior stability compared to existing methods.

Abstract

The task of repeatedly solving parametrized partial differential equations (pPDEs) in, e.g. optimization or interactive applications, makes it imperative to design highly efficient and equally accurate surrogate models. The reduced basis method (RBM) presents as such an option. Enabled by a mathematically rigorous error estimator, RBM constructs a low-dimensional subspace of the parameter-induced high fidelity solution manifold from which an approximate solution is computed. It can improve efficiency by several orders of magnitudes leveraging an offline-online decomposition procedure. However, this decomposition, usually through the empirical interpolation method (EIM) when the PDE is nonlinear or its parameter dependence nonaffine, is either challenging to implement, or severely degrades online efficiency. In this paper, we augment and extend the EIM approach as a direct solver, as opposed to an assistant, for solving nonlinear pPDEs on the reduced level. The resulting method, called Reduced Over-Collocation method (ROC), is stable and capable of avoiding the efficiency degradation inherent to a traditional application of EIM. Two critical ingredients of the scheme are collocation at about twice as many locations as the dimension of the reduced solution space, and an efficient L1-norm-based error indicator for the strategic selection of the parameter values to build the reduced solution space. Together, these two ingredients render the proposed L1-ROC scheme both offline- and online-efficient. A distinctive feature is that the efficiency degradation appearing in alternative 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 L1-ROC and its superior stability performance.