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54 Chapter 3: Development of a numerical tool for combustion noise analysis, AVSP-f 3.4 GMRES The Generalized Minimal RESidual (GMRES) method is a very well known algorithm for solving linear systems of considerable size. It was proposed by Saad and Schultz in 1986 [83] in order to solve large, sparse and non Hermitian linear systems. GMRES belongs to the class of Krylov based iterative methods and is composed by the procedures described in the previous sections. Fig. 3.7 shows a scheme in which the GMRES method is summarized. GMRES x 0 m =1 A ̂Q m m = m +1 ̂Q m+1 ˜Hm y x m = x 0 + ̂Q m y NO η < tolerance YES Figure 3.7: The GMRES Algorithm GMRES has been chosen to solve the problem of Eq. (3.9) due to its portability, simplicity, flexibility and efficiency. It is important to emphasize that the fact that GMRES belongs to the class of Krylov based iterative methods and allows to solve the linear system by using only matrix-vector products instead of storing the full matrix A.
3.4 GMRES 55 Stopping criteria of GMRES In order to analyze the convergence of GMRES, it is important to know which mathematical properties of A determine the size of ‖r‖. There are two main observations. The first is that GMRES converges monotonically ‖r m+1 ‖ ≤ ‖r m ‖. The reason is that ‖r m ‖ is as small as possible for the subspace K m . By enlarging K m to the space K m+1 , it is only possible to decrease the residual norm, or at worst leave it unchanged. The second is that after at most n steps the process must converge, at least in the absence of round errors ‖r n ‖ → 0. Many algorithms of numerical linear algebra satisfy a condition that is both stronger and simpler than stability. We say that an algorithm ˜f for a problem f is backward stable if ˜f (x) = f ( ˜x) for some approximated solution ˜x = x m . Here ˜x = x + δx and ˜f = f + δ f are a perturbed solution and a perturbed function respectively. The normwise backward error η is then defined as: η = ‖x m − x‖ ‖x‖ = ‖A−1 (b − r m ) − A −1 b‖ ‖A −1 b‖ = ‖r m‖ 2 ‖b‖ 2 (3.56) The backward error measures the distance between the data of the initial problem and those of the perturbed problem. The best η one can require from an algorithm is a backward error of the order of the machine precision. In practice, the approximation of the solution is acceptable when its backward error is lower than the uncertainty on the data. 3.4.1 Preconditioning The convergence of a matrix iteration depends on the properties of the matrix, i.e., the eigenvalues, the singular values, or sometimes other information. It is interesting to comprehend that in many cases, the problem of interest can be transformed so that the properties of the matrix are improved drastically. This process of ‘preconditioning’ is essential to most successful applications of iterative methods. Suppose we wish to solve an n × n nonsingular system Ax = b. For any nonsingular n × n matrix M, the system MAx = Mb has the same solution. If we solve this system iteratively, the convergence will depend on the properties of MA instead of those of A. If this preconditioner M is well chosen, the problem may be solved much more rapidly. The best choice ever for a preconditioner is the inverse of the system matrix A, i.e., M = A −1 . Of course it would be ridiculous to use A −1 as a preconditioner since this is already the solution of the problem. The idea is then to make M −1 as close as A as possible. In other words, to make the eigenvalues of M −1 A be close to 1 and ‖M −1 A − I‖ 2 small where I is the identity matrix. In doing so, a quick convergence of the solution is expected. Two well known preconditioning methods are the Diagonal scaling or Jacobi (M −1 = diag(A))
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UNIVERSITE MONTPELLIER II SCIENCES
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An expert is a man who has made all
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Y gracias familia mía!!, que así
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Abstract Today, much of the current
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7 Computation of the Reflection Coe
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Chapter 7: Computation of the Refle
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Bibliography [1] AGARWAL, A., MORRI
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