Reduced basis (RB) methods represent a very efficient approach for the numerical approximation of problems involving the repeated solution of differential equations arising from engineering and applied sciences. Noteworthy examples include partial differential equations (PDEs) depending on several parameters, PDE-constrained optimization, and optimal control and inverse problems.In all these cases, reducing the severe computational complexity is crucial. With this in mind, over the past four decades, reduced-order models (ROMs) have been developed aiming at replacing the original large-dimension numerical problem (typically called high-fidelity approximation) by a reduced problem of substantially smaller dimension. Strategies to generate the reduced problem from the high-fidelity one can be manifold, depending on the context.The strategy adopted in RB methods consists in the projection of the high-fidelity problem upon a subspace made of specially selected basis functions, representing a set of high-fidelity solutions corresponding to suitably chosen parameters. Pioneering works in this area date back to the late 1970s (e.g., B.O. Almroth et al. [5, 6], D. Nagy [193], A.K. Noor and J.M. Peters [201, 202, 203, 204] and address linear and nonlinear structural analysis problems. The first theoretical analysis of RB methods in connection with the use of the continuation method for parametrized equations was presented by J.P. Fink and W.C. Rheinboldt [109, 110] in the mid 1980s. Extensions to problems in fluid dynamics are primarily due to the contributions of Peterson [210] and Gunzburger [124] in the late 1980s.The method was set on a more general and sound mathematical ground in the early 2000s thanks to the seminal work of A.T. Patera, Y. Maday and coauthors [214, 255]. Their work has led to a decisive improvement in the computational aspects of RB methods owing to an efficient criterion for the selection of the basis functions, a systematic splitting of the computational procedure into an offline (parameterindependent) and an online (parameter-dependent) phase, and the use of a posteriori error estimates that guarantee certified numerical solutions for the reduced problem. These have become the essential constituents of the RB methods now most widely used. Often, they are also embedded into more general reduced-order models.

In this work, we present a stable proper orthogonal decomposition-Galerkin approximation for parametrized steady incompressible Navier-Stokes equations with low Reynolds number. Supremizer solutions are added to the reduced velocity space in order to obtain a stable reduced-order system, considering in particular the fulfillment of an inf-sup condition. The stability analysis is first carried out from a theoretical standpoint and then confirmed by numerical tests performed on a parametrized two-dimensional backward-facing step flow. 1137 ological developments were provided in more recent years [15][16][17][18], including the application to new unconventional fields [19,20].In recent years, growing attention has been devoted to the combination of Galerkin strategies with POD [21-23] and to stabilization techniques both for POD [9,[24][25][26] and RB methods [27][28][29][30].The aim of this work is to exploit some features of the RB framework in its state-of-the-art formulation for the stable and accurate approximation of parametrized flows in a POD setting, in order to improve the performance of the latter. In particular, we aim at investigating possible sources of pressure instabilities, in order to avoid spurious pressure modes in the POD approximation of parametrized flows. In fact, it is very well known that the FE approximation of Navier-Stokes (NS) equations presents two main difficulties that can make their numerical approximations meaningless: (i) the indefinite nature of the system and (ii) the stability loss due to convection dominant regimes for high Reynolds numbers. The first issue can be cured by using suitable velocitypressure FE spaces (also known as inf-sup stable elements), satisfying a discrete version of the so-called Ladyzhenskaya-Babuška-Brezzi (LBB) condition (see Section 4). Instead, in order to overcome the velocity stability loss for convection dominant regimes, we can rely on FE stabilization techniques, such as the streamline upwind Petrov-Galerkin (SUPG) method. We underline that stabilization techniques can also be employed to overcome pressure instability, for instance, through the pressure stabilizing Petrov-Galerkin (PSPG) method. Concerning pressure instability, a possible way to cure it in the RB context is the enrichment of velocity spaces through the so-called supremizers.Here, we want to analyze this supremizer stabilization technique to exploit it within a POD context and to develop a stability analysis based on the introduction of an LBB inf-sup condition at the reduced level, too. In this way, POD could benefit from a previously developed robust framework for the RB stabilization of viscous flows that can also manage with a correct pressure recovery. In this work, we test such a method on nonlinear steady viscous flows modeled by NS equations and characterized by both physical and geometrical parametrization.In the offline stage of the resulting strategy, several NS truth solutions are computed, and a POD is performed to extract a low-dimensional representation of both velo...

Reduction strategies, such as model order reduction (MOR) or reduced basis (RB) methods, in scientific computing may become crucial in applications of increasing complexity. In this paper we review the reduced basis methods (built upon a high-fidelity 'truth' finite element approximation) for a rapid and reliable approximation of parametrized partial differential equations, and comment on their potential impact on applications of industrial interest. The essential ingredients of RB methodology are: a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform a competitive Offline-Online splitting in the computational procedure, and a rigorous a posteriori error estimation used for both the basis selection and the certification of the solution. The combination of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for realtime simulation and many-query contexts (for example, optimization, control or parameter identification). After a brief excursus on the methodology, we focus on linear elliptic and parabolic problems, discussing some extensions to more general classes of problems and several perspectives of the ongoing research. We present some re-A Quarteroni · G Rozza ( ) · A Manzoni

This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities -which are mainly related either to nonlinear convection terms and/or some geometric variability -that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and -in the unsteady case -long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

We propose a suitable model reduction paradigm---the certified reduced basis method (RB)---for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations. In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as a constraint and infinite-dimensional control variable. First, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling one to perform competitive offline-online splitting in the computational procedure; and an efficient and rigorous a posteriori error estimate on the state, control, and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique