On the approximate minimization of functionals

Abstract

This paper considers in general the problem of finding the minimum of a given functional f ( u ) f(u) over a set B B by approximately minimizing a sequence of functionals f n ( u n ) {f_n}({u_n}) over a "discretized" set B n {B_n} ; theorems are given proving the convergence of the approximating points u n {u_n} in B n {B_n} to the desired point u u in B B . Applications are given to the Rayleigh-Ritz method, regularization, Chebyshev solution of differential equations, and the calculus of variations.