Some new developments on the non invasive reduced basis method

• Yvon Maday (UPMC Paris 6)

Room: Ada Lovelace The efficient implementation of reduced basis methods relying on a high fidelity discretization method to compute the elements of the reduced basis, requires to enter within the code, offline, so that the online solution can be produced very rapidly. Since this is sometimes impossible, in particular for codes used in industrial framework, we have proposed a Non Invasive alternative where the code is used at two stages : a) offline to build the reduced basis, b) online to, first, compute a coarse approximation using the code with few degrees of freedom (thus more rapidly than with the high fidelity requirement) then by processing this coarse solution to improve the accuracy and fulfil the high fidelity requirement at much mower cost. This method that first appeared with its numerical analysis in [1] for finite element approximations has been generalised in various directions (finite volume, truncation of the domain, different new rectifications) and implemented in a library in the frame of a joined project with large and medium size companies [2]. In this paper I will present some of the new concepts in this drection [1] Rachida Chakir, Yvon Maday :A two-grid finite-element/reduced basis scheme for the approximation of the solution of parameter dependent PDE. In 9e Colloque national en calcul des structures. ISO 690 Y. (2009, May). [2] Elise Grosjean : Variations and further developments on the Non-Intrusive Reduced Basis two-grid method, PhD Thesis at Sorbonne Université - 2022 Elise Grosjean and Yvon Maday

ROM Closures and Stabilizations for Under-Resolved Turbulent Flows

• Traian Iliescu (Virginia Tech)

Room: Ada Lovelace In this talk, I will survey reduced order model (ROM) closures and stabilizations for under-resolved turbulent flows. Over the past decade, several closure and stabilization strategies have been developed to tackle the ROM inaccuracy in the convection-dominated, under-resolved regime, i.e., when the number of degrees of freedom is too small to capture the complex underlying dynamics. I will present regularized ROMs, which are stabilizations that employ spatial filtering to alleviate the spurious numerical oscillations generally produced by standard ROMs in the convection-dominated, under-resolved regime. I will also survey three classes of ROM closures, i.e., correction terms that increase the ROM accuracy: (i) functional closures, which are based on physical insight; (ii) structural closures, which are developed by using mathematical arguments; and (iii) data-driven closures, which leverage available data. Throughout my talk, I will highlight the impact made by data on classical numerical methods over the past decade. I will also emphasize the role played by physical constraints in data-driven modeling of ROM closures and stabilizations.

Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction

• Charbel Farhat (Stanford)

Room: Ada Lovelace A quadratic approximation manifold is presented for performing nonlinear, projection-based, model order reduction (PMOR). It constitutes a departure from the traditional affine subspace approximation aimed at mitigating the Kolmogorov barrier for nonlinear PMOR, particularly for convection-dominated transport problems. It builds on the data-driven approach underlying the traditional construction of projection-based reduced-order models (PROMs); is application-independent; is linearization-free; and therefore is robust for highly nonlinear problems. Most importantly, this approximation leads to quadratic PROMs that deliver the same accuracy as their traditional counterparts using however the square root of their dimension. The computational advantages of the proposed high-order approach to nonlinear PMOR over the traditional approach are highlighted for the detached-eddy simulation-based prediction of the Ahmed body turbulent wake flow, which is a popular CFD benchmark problem in the automotive industry. For a fixed accuracy level, these advantages include: a reduction of the total offline computational cost by a factor greater than five; a reduction of its online wall clock time by a factor greater than 32; and a reduction of the wall clock time of the underlying high-dimensional model by a factor greater than two orders of magnitude. co-author: Joshua Barnett

Model order reduction for CFD: state of the art, advances in applications

• Gianlugi Rozza (SISSA)

Room: Ada Lovelace