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56 Chapter 3: Development of a numerical tool for combustion noise analysis, AVSP-f and the Incomplete Cholesky or LU factorization. These last are known as ‘incomplete’ since M −1 is allowed only to have nonzeros in the positions where A has nonzeros. This is done in order to keep the sparsity of the original matrix A. A Complete Cholesky or LU factorization would destroy zeros leading to a dense M matrix. These preconditioners are appropiate for illconditioned problems of small size. But for big and ill-conditioned A matrices, these methods are not the best suited. 3.4.2 Dynamic preconditioning: the embedded GMRES The right preconditioning of the Ax = b problem reads Ax = AMc = b (3.57) The traditional way to solve this system would be first to find a matrix M such that AM is well conditioned; second, to solve the system for the vector c; and finally to find x from x = Mc. In this procedure it is imperative to explicitly construct and store the matrix M. A better way to apply preconditioning would be to find a preconditioning matrix of size n × m by solving an second linear system. In other words, applying GMRES to find a good preconditioner. Recalling Eq. (3.18), we add the preconditioning matrix M into the computation ( ) ( ) Ax = A x 0 + ̂Q m y = b − r m → Ax = A x 0 + M ̂Q m c = b − r m (3.58) where x = x 0 + M ̂Q m c. The matrix Ẑ m , which is the projection of M in the Krylov space, is defined as Ẑ m = M ̂Q m . Eq. (3.58) becomes ( ) A x 0 + Ẑ m c = b − r m (3.59) The preconditioning matrix M is let to be A −1 , so that AM −1 = I. The vector z m of the matrix Ẑ m can now be found by solving the linear system z m = Mq m → Az m = q m (3.60) which is clearly a problem of the type Ax = b. Once the vector z m is found, the orthogonalization (finding q m ) is carried out applying the Arnoldi procedure to AẐ m instead of A ̂Q m , resulting in a faster convergence. Figure 3.8 represents the procedure just described. As it can be observed from Fig. 3.8 the system Ax = b is solved by one GMRES algorithm embedded in another GMRES. As a consequence, the present strategy is called the embedded
3.4 GMRES 57 FGMRES x 0 m =1 AẐm z m ̂Q m+1 ˜Hm Az m = q m m = m +1 c x m = x 0 + Ẑmc NO η < tolerance YES Figure 3.8: The embedded GMRES Algorithm GMRES and is the one applied in the present study to solve the linear system ( ∇ · ¯c 2 ∇ + Iω 2) } {{ } A ˆp }{{} x = iω(γ − 1) ˆ˙ω T } {{ } b (3.61) which represents the propagation of noise due to a thermal source. Note that Eq. (3.61) must be solved once for each frequency ω. As a result, in order to build a proper spectrum of ˆp (e.g.: 0-1000 Hz ; ∆ f = 10Hz) this system must be solved around hundred times. AVSP-f uses a CERFACS implementation of the GMRES algorithm for both real and complex, single and double precision arithmetics suitable for serial, shared memory and distributed memory computers [29, 30].
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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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4 CONTENTS 3.4.1 Preconditioning .
Page 16 and 17: List of Figures 1.1 Main sources of
Page 18 and 19: 8 LIST OF FIGURES 5.19 Acoustic ene
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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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130 BIBLIOGRAPHY [30] FRAYSSÉ, V.,
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132 BIBLIOGRAPHY [62] NICOUD, F., B
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134 BIBLIOGRAPHY [93] SENSIAU, C. S
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