On the Curse of Dimensionality in the Ritz Method

Giorgio Gnecco ()
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Giorgio Gnecco: IMT Institute for Advanced Studies

Journal of Optimization Theory and Applications, 2016, vol. 168, issue 2, No 6, 488-509

Abstract: Abstract 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.

Keywords: Ritz method; Curse of dimensionality; Infinite-dimensional optimization; Approximation schemes; Extended Ritz method; 90C06; 90C26; 90C48 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10957-015-0804-y

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