Methods using krylov subspaces as trial spaces are

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Unformatted text preview: MRES iterate from (9.4.8a) as c(M ) = (KT AT AKM );1 KT AT r(0) : M We see that: 1. The GMRES procedure is the extension of the MR procedure to higher-dimensional spaces. 2. The dimension of the Krylov space increases by one after each iteration. 3. As may be expected from our earlier work, x(M ) minimizes kr(M )k2 = kb ; Ax(M ) k2 Unfortunately, the procedure just outlined is unstable (sensitive to round-o error accumulation). It will be necessary to select a more well conditioned basis for KM and we'll do this by constructing a set of mutually orthogonal vectors. This orthogonal projection onto KM is usually done by Arnoldi's method and a pseudocode procedure for its construction appears in Figure 9.4.2. The procedure begins with an initial vector v1 60 Solution Techniques for Elliptic Problems which is the initial residual r(0) normalized to have a unit Euclidean length. Successive vectors vj are chosen to be orthogonal to the previous ones and to have unit lengths. Indeed, the algorithm of Figure 9.4.2 is just classical Gram-Schmidt orthogonalization modi ed by the presence of A to produce an orthonormal basis for the Krylov subspace KM (A v1) = spanfv1 Av1 A2v1 : : : AM ;1v1 g: (9.4.11) The algorithm will halt prematurely if any unnormalized vector wj has a zero Euclidean length. The Boolean variable quit is set to true should this happen. Proving these assertions may be done with an induction argument ( 4], Section 6.3), but we'll proceed less formally. At the rst (j = 1) step of the algorithm, we obtain procedure arnoldi quit = (kr(0) k2 = 0) if not quit then v1 = r(0) =kr(0) k2 end if j=1 while (j M ) and (not quit) do wj = Avj for i = 1 to j do hij = viT Avj wj = wj ; hij vi end for hj+1 j = kwj k2 quit = (hj+1 j = 0) if not quit then vj+1 = wj =hj+1 j end if j =j+1 end while Figure 9.4.2: Arnoldi Gram-Schmidt orthogonal basis construction for KM . h21 v2 = w1 = Av1 ; h11 v1 : Taking an inner product with v1 T T T h21 v1 v2 = v1 Av1 ; h11 v1 v1: 9.4. Krylov Subspace Methods 61 T T...
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