%P 488-509
%T On the Curse of Dimensionality in the Ritz Method
%O First online: 03 September 2015
%I Springer
%V 168
%A Giorgio Gnecco
%K Ritz method; Curse of dimensionality; Infinite-dimensional optimization; Approximation schemes; Extended Ritz method
%L eprints3123
%D 2016
%X 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.
%J Journal of Optimization Theory and Applications
%N 2
%R 10.1007/s10957-015-0804-y