@article{eprints3123,
       publisher = {Springer},
            note = {First online: 03 September 2015},
         journal = {Journal of Optimization Theory and Applications},
          author = {Giorgio Gnecco},
           title = {On the Curse of Dimensionality in the Ritz Method},
            year = {2016},
          volume = {168},
          number = {2},
           pages = {488--509},
        keywords = {Ritz method; Curse of dimensionality; Infinite-dimensional optimization; Approximation schemes; Extended Ritz method},
             url = {http://eprints.imtlucca.it/3123/},
        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.}
}