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.
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Gnecco, G. On the Curse of Dimensionality in the Ritz Method. J Optim Theory Appl 168, 488–509 (2016). https://doi.org/10.1007/s10957-015-0804-y
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DOI: https://doi.org/10.1007/s10957-015-0804-y
Keywords
- Ritz method
- Curse of dimensionality
- Infinite-dimensional optimization
- Approximation schemes
- Extended Ritz method