Model Reduction Methods

Abstract

This chapter presents an overview of model order reduction – a new paradigm in the field of simulation‐based engineering sciences, and one that can tackle the challenges and leverage the opportunities of modern ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, a number of challenging problems remain intractable. These problems are of different nature, but are common to many branches of science and engineering. Among them are those related to high‐dimensional problems, problems involving very different time scales, models defined in degenerate domains with at least one of the characteristic dimensions much smaller than the others, model requiring real‐time simulation, and parametric models. All these problems represent a challenge for standard mesh‐based discretization techniques; yet the ability to solve these problems efficiently would open unexplored routes for real‐time simulation, inverse analysis, uncertainty quantification and propagation, real‐time optimization, and simulation‐based control – critical needs in many branches of science and engineering. Model order reduction offers new simulation alternatives by circumventing, or at least alleviating, otherwise intractable computational challenges. In the present chapter, we revisit three of these model reduction techniques: proper orthogonal decomposition, proper generalized decomposition, and reduced basis methodologies.

Virtual, Digital and Hybrid Twins: A New Paradigm in Data-Based Engineering and Engineered Data

“…In order to alleviate the computational cost, model order reduction techniques have been proposed and are nowadays intensively used. When considering POD-based model order reduction [3], a learning stage allows extracting the significant modes / i ðxÞ that best approximate the solution. Very often a reduced number of modes R (R ( N) suffices to approximate the solution of problems similar to the one that served to extract the modes at the learning stage.…”

“…Again, it was at the end of the past century and the beginning of the 21st century, that major scientific accomplishments in theoretical and applied mathematics, applied mechanics, and computer sciences contributed to new modeling and simulation procedures. Model Order Reduction (MOR) techniques were one of these major achievements [3]. These techniques do not proceed by simplifying the model, models continue to be well established and validated descriptions of the physics at hand.…”

Efficient geometrical parametrization for finite‐volume‐based reduced order methods

“…Projection‐based ROMs exploit the fact that, in most of the cases, the solution manifold lies in a low‐dimensional space and can be, therefore, approximated by a linear combination of a reduced number of properly selected global basis functions: where are the parameter independent basis functions and are the parameter dependent coefficients. There exist different approaches to construct the set of basis functions such as the greedy approach, the POD, the proper generalized decomposition, and so on . We decided here to rely on a POD approach.…”

Toward Optimality of Proper Generalised Decomposition Bases

“…Numerical simulations appeal as an attractive augmentation to experiments to design and analyse mechanical structures. Despite the recent developments in computational resources that makes it feasible to solve systems with a substantial number of degrees of freedom efficiently, it is of common interest to reduce the numerical cost of numerical models throughout model order reduction (MOR) strategies [1]. The performance of MOR techniques has been shown in different fields such as their application to nonlinear problems [2,3], real-time computations [4] or for performing cyclic, parametric or probabilistic computations in which the information provided by some queries can be efficiently reused to respond to other queries that exhibit some similarities [5,6].…”

“…The singular vectors corresponding to the highest singular values are used to build the ROB [7]. Then, the problem of interest is confined to this ROB resulting in a drastic reduction in the numerical cost [1,8]. However, since the ROB has been defined as an optimal basis for the training stage, some advanced adaptive approaches are required to enrich the basis to tackle nonlinearities [9].…”