Approach for the reduced basis method - Convergence rates
Sciences_Maths_Paris
2010-06-22
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Ron DeVore (Université Texas A&M, lauréat 2009 de la Chaire d'Excellence de la Fondation SMP) et Albert Cohen (Professeur à l'UPMC) organisent une conférence autour des Problèmes en grande dimension.---
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Convergence Rates for Greedy Algorithms in Reduced Basis Methods
Wolfgang Dahmen, RWTH Aachen joint work with Peter Binev, Albert Cohen, Ronald DeVore, Guergana Petrova, and Przemyslaw Wojtaszczyk
The reduced basis method was introduced for the accurate online evaluation of solutions to a parameter dependent family of elliptic partial differential equations. Abstractly, it can be viewed as determining a “good” n dimensional space Hn to be used in approx- imating the elements of a compact set F in a Hilbert space H. One, by now popular, computational approach is to find Hn through a greedy strategy. It is natural to com- pare the approximation performance of the Hn generated by this strategy with that of the Kolmogorov widths dn(F) since the latter gives the smallest error that can be achieved by subspaces of fixed dimension n. The first such comparisons show that the approximation error, σn(F) := dist(F,Hn), obtained by the greedy strategy satisfies σn(F) ≤ Cn2ndn(F). In this talk various improvements on this result will be given for a greedy strategy that is computationally practicable and efficient in the above context of parameter dependent PDEs. The main objective is to obtain convergence rates for σn(F) also in a regime where the known bounds no longer provide any information, namely when the decay of the widths dn(F) is slower than n−12−n. While the above direct comparison between σn(F) and dn(F) can generally not be improved by much it can be shown that certain convergence rates for dn(F) imply certain convergence rates for σn(F). For instance, whenever dn(F) ≤ Mn−α for all n > 0, for some M,α > 0, one also have σn(F) ≤ CαMn−α for all n > 0, where Cα depends only on α. Similar results are derived for generalized exponential rates of the form Me−anα.
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Convergence Rates for Greedy Algorithms in Reduced Basis Methods
Wolfgang Dahmen, RWTH Aachen joint work with Peter Binev, Albert Cohen, Ronald DeVore, Guergana Petrova, and Przemyslaw Wojtaszczyk
The reduced basis method was introduced for the accurate online evaluation of solutions to a parameter dependent family of elliptic partial differential equations. Abstractly, it can be viewed as determining a “good” n dimensional space Hn to be used in approx- imating the elements of a compact set F in a Hilbert space H. One, by now popular, computational approach is to find Hn through a greedy strategy. It is natural to com- pare the approximation performance of the Hn generated by this strategy with that of the Kolmogorov widths dn(F) since the latter gives the smallest error that can be achieved by subspaces of fixed dimension n. The first such comparisons show that the approximation error, σn(F) := dist(F,Hn), obtained by the greedy strategy satisfies σn(F) ≤ Cn2ndn(F). In this talk various improvements on this result will be given for a greedy strategy that is computationally practicable and efficient in the above context of parameter dependent PDEs. The main objective is to obtain convergence rates for σn(F) also in a regime where the known bounds no longer provide any information, namely when the decay of the widths dn(F) is slower than n−12−n. While the above direct comparison between σn(F) and dn(F) can generally not be improved by much it can be shown that certain convergence rates for dn(F) imply certain convergence rates for σn(F). For instance, whenever dn(F) ≤ Mn−α for all n > 0, for some M,α > 0, one also have σn(F) ≤ CαMn−α for all n > 0, where Cα depends only on α. Similar results are derived for generalized exponential rates of the form Me−anα.
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