Eine Verallgemeinerung des Ritz-Prozesses (Q1067363)

Let X be a real Hilbert space with a (not necessarily orthogonal) basis \((x_ i)\), M be a closed subspace with basis \((y_ j)\), \(z\in X\setminus M\). The problem is to determine the closest element in M to z. The author describes an algorithm for solving this problem (which would be easily if M and z were known explicitly) using the inner products \((x_ i,z)\) and \((x_ i,y_ j)\) only. The algorithm is based on approximating M by finite-dimensional subspaces. The author proves convergence and gives a condition (on the Gramians of \((x_ 1,...,x_ n))\) that guarantees stability. The algorithm is useful for the following problem: Given a basic force distribution and controllable additional forces; under which linear combination of these forces acting on a plate is the mean stress minimal? On a suitable reproducing kernel Hilbert space, this problem can be reformulated exactly in the above form with M and z known only implicitly.