Special Issue on Model Reduction

Abstract

Model reduction, in the sense of the approximation of a numerical solution into a subspace of much smaller dimension than that associated with usual semi-discretization or discretization methods, is rapidly becoming an indispensable tool for many areas of computational sciences and engineering. These include, to name only a few, computational-based design and optimization, statistical analysis, optimal control, and embedded computing. It is also essential for all scenarios where real-time simulation responses are desired. Linear model reduction has deep roots in many engineering fields such as structural dynamics and electronic circuit analysis. For applications with a single or no parameter, it has matured to the point where it is embraced today by many practitioners. By comparison, parametric and nonlinear model reduction methods are still in their infancy. Nevertheless, giant strides have been recently made in these two areas. Consequently, these two topics have gained importance for many applications where high-dimensional, nonlinear, parametric simulations remain computationally prohibitive. For this reason, the main objective of this Special Issue is to disseminate some of the recent advances made in parametric and nonlinear model reduction methods and highlight their potential for making a strong impact on the broad field of computational engineering. To this effect, this issue contains 13 papers 1-13 covering several important aspects of the mathematical theory, numerical algorithms, and computational strategies underlying parametric and nonlinear model reduction approaches. Nine of these papers focus on the parametric aspect of the problem in the linear case. They emphasize two main strategies for constructing a globally accurate parametric reduced-order model (ROM): the offline/online strategy where a ROM is built during a computationally intensive offline phase and exploited in real-time during an online phase and an alternative strategy where a parametric ROM is iteratively constructed, exploited, and updated as needed. The four other papers address more specifically the topic of nonlinear model reduction using the notion of a hierarchy of approximations. Three papers focus also on data assimilation and system monitoring—a rapidly growing field where model reduction is essential—three on numerical stability, and three on accuracy analysis and error estimators. In a field where advances can often give, rightfully or wrongfully, the impression of ‘more of the same’, the editors hope that this Special Issue will bring to the readers a fresh perspective on an important direction where computational engineering should go to continue shining in the future.