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On the Curse of Dimensionality in the Ritz Method

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  • Giorgio Gnecco

    (IMT Institute for Advanced Studies)

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.

Suggested Citation

  • Giorgio Gnecco, 2016. "On the Curse of Dimensionality in the Ritz Method," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 488-509, February.
    • Handle: RePEc:spr:joptap:v:168:y:2016:i:2:d:10.1007_s10957-015-0804-y
    DOI: 10.1007/s10957-015-0804-y
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    References listed on IDEAS

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    1. G. Gnecco & M. Sanguineti, 2010. "Estimates of Variation with Respect to a Set and Applications to Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 145(1), pages 53-75, April.
    2. A. Alessandri & C. Cervellera & M. Sanguineti, 2007. "Functional Optimal Estimation Problems and Their Solution by Nonlinear Approximation Schemes," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 445-466, September.
    3. R. Zoppoli & M. Sanguineti & T. Parisini, 2002. "Approximating Networks and Extended Ritz Method for the Solution of Functional Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 112(2), pages 403-440, February.
    4. S. Giulini & M. Sanguineti, 2009. "Approximation Schemes for Functional Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 140(1), pages 33-54, January.
    5. G. Gnecco & M. Sanguineti, 2010. "Suboptimal Solutions to Dynamic Optimization Problems via Approximations of the Policy Functions," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 764-794, September.
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