Abstract
The set of solutions of a parameter-dependent linear partial differential equation with smooth coefficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold.
We focus on operators showing an affine parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affine expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold—only spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic equations confirming the predicted convergence rates.
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Notes
- 1.
The other option is to consider local or sequential approximations of the manifold, such as tracking a path on the manifold starting from a certain point and proceeding via a continuation method. In such cases we are usually not interested in the global behavior of the manifold.
- 2.
Ito and Ravindran [16] were perhaps the first ones to suggest using a Hermite ROM in a uniform approximation context, rather than in a pure continuation method. The Lagrange and Hermite ROMs were compared on a driven cavity problem, where the Hermite approach was somewhat superior. No stability problems were reported and the Hermite basis with only two basis functions was able to extrapolate solutions to much larger Reynolds numbers.
- 3.
In one of the pioneering works on RBM, Noor [24] used a Taylor ROM to build a local reduced space that was used to trace the post-buckling behavior of a nonlinear structure. The continuation idea was used also by Peterson [25] to compute Navier-Stokes solutions with increasing Reynolds number flow over a forward facing step. Again a Taylor ROM was constructed and used to extrapolate an initial guess for the Newton method at a slightly higher Reynolds number.
- 4.
In the works of Hay et al. [12, 13] sensitivity information was introduced into the proper orthogonal decomposition framework. The parametric sensitivities of the POD modes were derived and computed. The test problems were related with channel flow around a cylindrical obstacle, either by using a simple parametrization as the Reynolds number, or a more involved geometric parametrization of the obstacle. The use of a Hermite ROM considerably improved the validity of the reduced solutions away from the parametric snapshots. However, in the more involved geometrical parametrization case the Hermite ROM failed completely, as it did not converge to the exact solution even when the number of POD modes was increased.
- 5.
Carlberg and Farhat [6] proposed an approach they call “compact POD”, based on goal-oriented Petrov-Galerkin projection to minimize the approximation error subject to a chosen output criteria, and including sensitivity information with proper weighting coming from the Taylor-expansion and including “mollification” of basis functions far away from the snapshot parameter. The application was the optimization of an aeroelastic wing configuration by building local ROMs along the path to the optimal wing configuration.
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This contribution is published in BUMI (Bollettino Unione Matematica Italiana) as well by a special agreement between UMI (Unione Matematica Italiana) and Springer. This work celebrates the memory of Prof. Enrico Magenes (1923–2010) after the conference held in Pavia in November 2011.
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Lassila, T., Manzoni, A., Quarteroni, A., Rozza, G. (2013). Generalized Reduced Basis Methods and n-Width Estimates for the Approximation of the Solution Manifold of Parametric PDEs. In: Brezzi, F., Colli Franzone, P., Gianazza, U., Gilardi, G. (eds) Analysis and Numerics of Partial Differential Equations. Springer INdAM Series, vol 4. Springer, Milano. https://doi.org/10.1007/978-88-470-2592-9_16
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