A PRIORI CONVERGENCE OF THE GREEDY ALGORITHM FOR THE PARAMETRIZED REDUCED BASIS METHOD

By Annalisa Buffa, Yvon Maday, Anthony T. Patera, Christophe Prud'Homme, Gabriel Turinici. 2012

325 citations recorded by the source corpus.

Subfields: Computational Mechanics, Mechanics of Materials, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5669ef026480bdb5f62f17ca96fe2795ad08b694

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In Computational Mechanics

In Mechanics of Materials

In Statistical and Nonlinear Physics

Citations in the corpus

Decay of the Kolmogorov $N$-width for wave problems (2019, unknown): “It has been proven that for certain linear, coercive parameterized problems, the Kolmogorov N -width decays exponentially fast, i.e., with some constants C < ∞ and β > 0 , see e.g. strategy [2, 8].”
Model Order Reduction in Fluid Dynamics: Challenges and Perspectives (2014, result): “Results regarding the connection between n-width decay rates and greedy algorithm converges rates can be found in [18, 22] for parametric problems, in [55] for time-dependent problems, and results regarding the n-width decay rates for parameter-dependent elliptic PDEs in [77, 86].”
Low-rank methods for high-dimensional approximation and model order reduction (2015, unknown): “as presented in section 5.1 (replacing kw(y) − u(y)kV by kB(y)w(y) − f(y)kC(y)). This is the so-called oﬄine phase of Reduced Basis methods, and convergence results for this algorithm can be found in [10, 12, 32], where the error ku−urk∞,K = supy∈Ξ K ku(y)− PVru(y)kV is compared with the best rank-r”
Non-intrusive reduced order modeling of nonlinear problems using neural networks (2018, background): “Once a reduced order framework has been properly set up, an approximation to the truth solution for a new parameter value is sought online as a linear combination of the RB functions, with the expansion coefficients determined via a projection of the full-order system onto the reduced space [7].”
A parameterized‐background data‐weak approach to variational data assimilation: formulation, analysis, and application to acoustics (2015, methodology): “The algorithm has been proven to generate an optimal sequence of spaces with respect to the Kolmogorov width of Mbk in [5, 25, 26].”
Reduced basis methods for partial differential equations with stochastic influences (2013, methodology): “The selection is usually based upon a Greedy algorithm [14, 73, 98].”
Certified reduced basis approximation for parametrized partial differential equations and applications (2011, background): “Not only, an a priori theory for RB approximations is also available, dealing with a class of single parameter coercive problems [19] and more recently extended also to the multi-parameter case [20].”
Greedy algorithms for high-dimensional non-symmetric linear problems (2012, methodology): “f dimensionality [2]. Several approaches have recently been proposed in order to circumvent this signiﬁcant diﬃculty. Let us mention among others sparse grids [21], tensor formats [11], reduced bases [4] and adaptive polynomial approximations [6]. In this paper, we will focus on a particular kind of”