TY  - JOUR
VL  - 168
Y1  - 2016///
JF  - Journal of Optimization Theory and Applications
IS  - 2
A1  - Gnecco, Giorgio
PB  - Springer
SP  - 488
N1  - First online: 03 September 2015
ID  - eprints3123
EP  - 509
TI  - On the Curse of Dimensionality in the Ritz Method
AV  - none
KW  - Ritz method; Curse of dimensionality; Infinite-dimensional optimization; Approximation schemes; Extended Ritz method
UR  - http://link.springer.com/article/10.1007%2Fs10957-015-0804-y
SN  - 0022-3239
N2  - 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.
ER  -