Titlebook: Reduced Basis Methods for Partial Differential Equations; An Introduction Alfio Quarteroni,Andrea Manzoni,Federico Negri Textbook 2016 Spri

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Introduction,es have gained an ever increasing importance. In several fields, from aerospace and mechanical engineering to life sciences, numerical simulations of partial differential equations (PDEs) currently provide a virtual platform ancillary to material/mechanics testing or . experiments. These are in turn

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RB Methods: Basic Principles, Basic Properties,n projection onto an .-dimensional space . (the RB space) that approximates the high-fidelity (say, finite element) solution of the given PDE, for any choice of the parameter within a prescribed parameter set. We illustrate the main steps needed to set up such methods efficiently. We discuss in deta

TERRA

On the Algebraic and Geometric Structure of RB Methods,s least-squares RB approximation) and the Galerkin high-fidelity approximation (3.11) are highlighted, for the purpose of illustrating, in a more fitting way and from a different perspective, the mathematical structure underpinning RB methods. The key role played by the transformation matrix in defi

Vasodilation

The Theoretical Rationale Behind,anifold that are directly inherited from the parametrized differential operators. We define the Kolmogorov .-width to measure how well suited .-dimensional subspaces are to approximate the solution manifold. At the end we show that a wise selection of snapshots yields exponential convergence when ap

Left-Atrium

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RB Methods in Action: Setting up the Problem, for this operation are given, and examples of parametrized PDEs are discussed that are inspired by the four problems of Chap. 2. The primary purpose is to highlight the different role played by physical and geometric parameters. A further critical issue addressed concerns the possible affine parame

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Extension to Nonaffine Problems,fline/online decomposition relies on that assumption, in case of nonaffine problems we recover an approximate affine expansion by means of the so-called empirical interpolation method (EIM). We provide a detailed description of the EIM, focusing on linear problems for the sake of simplicity. A possi

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Extension to Nonlinear Problems,milinear equation. Both high-fidelity and RB approximations, as well as their interplay with Newton linearization, are analyzed, before considering in detail the case of Navier-Stokes equations. The underlying RB construction is essentially the same as for the previous linear PDEs, hence the focus i