Optimal Iterative Refinement Method
Optimal Iterative Refinement Method
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4. Some Additive Algorithms. We recall, that in the so called additive algoritlmis. We solve equation (3) by the conjugate gradient or some other standard iterative algorithm. The different algorithms can simply be defined by specifying the subspaces V, or alterna- tively the projections P, . Before we discuss specific algorithms, we make some remarks on estimating the eigenvalues of P from above and below.
The upper bound on the spectrum is obtained by bounding a{Pvh, Vh) = a{PiVh, Vh) + a(P2V...h, Vh) + ••• + a[PkVh, Vh), from above in terms of a{vh, Vh). We can use Schwarz' inequality and the fact that P^ is a projection to prove that each term is bounded by a(vh, Vh) and thus the spectrmn of P is bounded from above by k. Our goal, however, is to establish a uniform boimd on the condition number. This can be done if the terms are orthogonal or almost orthogonal; cf. Discussion below.
A lemma from Lions [9] provides a method for obtaining lower bounds. Since the proof of his result is quite short, we include it in this paper.
What to read after Optimal Iterative Refinement Method?
You can find similar books in the "Read Also" column, or choose other free books by Olof B Widlund to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
4. Some Additive Algorithms. We recall, that in the so called additive algoritlmis. We solve equation (3) by the conjugate gradient or some other standard iterative algorithm. The different algorithms can simply be defined by specifying the subspaces V, or alterna- tively the projections P, . Before we discuss specific algorithms, we make some remarks on estimating the eigenvalues of P from above and below.
The upper bound on the spectrum is obtained by bounding a{Pvh, Vh) = a{PiVh, Vh) + a(P2V...h, Vh) + ••• + a[PkVh, Vh), from above in terms of a{vh, Vh). We can use Schwarz' inequality and the fact that P^ is a projection to prove that each term is bounded by a(vh, Vh) and thus the spectrmn of P is bounded from above by k. Our goal, however, is to establish a uniform boimd on the condition number. This can be done if the terms are orthogonal or almost orthogonal; cf. Discussion below.
A lemma from Lions [9] provides a method for obtaining lower bounds. Since the proof of his result is quite short, we include it in this paper.
What to read after Optimal Iterative Refinement Method?
You can find similar books in the "Read Also" column, or choose other free books by Olof B Widlund to read onlineMoreLess
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