Bivariational Bounding Methods

Abstract

A problem of interest in optimization theory is that of bounding the inner product ‹g,u› of an arbitrary vector g with u, the solution-vector of an equation Fu = 0 in the inner-product space. In many situations a suitable choice of g can lead to bounds on the solution u itself. We describe here an approach based on a bivariational approximation to ‹g,u›, namely

$$J(V,U)=-<V,FU>+<g,U>.$$

(1)

For simplicity we assume that the space is real and the inner product is symmetric. The square-norm notation ${{\left\| U \right\|}^{2}}=<U,U>$is employed.