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On the Curse of Dimensionality in the Ritz Method

Gnecco, Giorgio On the Curse of Dimensionality in the Ritz Method. Journal of Optimization Theory and Applications, 168 (2). pp. 488-509. ISSN 0022-3239 (2016)

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It is shown that the classical Ritz method of the calculus of variations suffers from the “curse of dimensionality,” i.e., an exponential growth, as a function of the number of variables, of the dimension a linear subspace needs in order to achieve a desired relative improvement in the accuracy of approximation of the optimal solution value. The proof is constructive and is obtained by exhibiting a family of infinite-dimensional optimization problems for which this happens, namely those with quadratic functional and spherical constraint. The results provide a theoretical motivation for the search of alternative solution methods, such as the so-called “extended Ritz method,” to deal with the curse of dimensionality.

Item Type: Article
Identification Number: 10.1007/s10957-015-0804-y
Additional Information: First online: 03 September 2015
Uncontrolled Keywords: Ritz method; Curse of dimensionality; Infinite-dimensional optimization; Approximation schemes; Extended Ritz method
Subjects: Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Research Area: Computer Science and Applications
Depositing User: Caterina Tangheroni
Date Deposited: 26 Feb 2016 12:39
Last Modified: 26 Feb 2016 12:39
URI: http://eprints.imtlucca.it/id/eprint/3123

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