J. Zhang. Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications. Comput. Methods Appl. Mech. Engrg., 189(3):825--840, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....A fall into a few clusters, say t of them, whose diameters are small enough, then M 1 A behaves numerically like a matrix with t distinct eigenvalues. As a result, we would expect t iterations of a Krylov method to produce reasonably accurate approximations. It has been shown in [26] 38] and [42] that in practice, with the availability of a high quality preconditioner, the choice of the Krylov subspace accelerator is not so critical. 2.2 Preconditioning. A preconditioner M should satisfy the following demands: M is a good approximation of A in some sense; the construction and ....

J. Zhang. Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications. Tech. Rep. 280-98, Department of Computer Science, University of Kentucky, KY, 1998.

....different processors. In the current situation, the sparse linear system is solved by a preconditioned Krylov subspace method, such as the GMRES method [15] It is generally believed that the quality of the preconditioner determines the convergence rate of a preconditioned Krylov subspace solver [18]. The preconditioners that have been examined in the solid Earth simulations are all constructed by some kind of incomplete LU (ILU) factorizations using static matrix (sparsity) patterns based on the original matrix nonzero structure (sparsity pattern) and within each individual processor in a ....

J. Zhang. Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications. Comput. Methods Appl. Mech. Engrg., 189(3):825--840, 2000.

....of iterations is not affected significantly for PBILU2. 5. 2 FIDAP matrices This set of test matrices were extracted from the test problems provided in the FIDAP package [12] 4 As many of these matrices have small or zero diagonals, they are difficult to solve with standard ILU preconditioners [42]. We tested more than 31 FIDAP matrices for both preconditioners. We found that PBILU2 can solve more than twice as many FIDAP matrices as SLU does. In out tests, PBILU2 solved 20 FIDAP matrices and SLU solved 9. These tests show that our parallel two level block ILU preconditioner is more 4 ....

J. Zhang. Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications. Comput. Methods Appl. Mech. Engrg., 2000. to appear.

....the ILUT preconditioner constructed using different values for Test 1 and the third order integration method. The three Krylov subspace accelerators perform similarly when 10 Gamma7 and the accuracy of ILUT is adequate, although GMRES is slightly better. This observation is in agreement with [35]. We point out that, with = 10 Gamma7 , GMRES solved all linear systems, while BiCGSTAB failed to solve 26 (1:11 ) and TFQMR failed to solve 13 (0:59 ) linear systems. Moreover, when the accuracy of the ILUT preconditioner is reduced ( 10 Gamma6 ) GMRES performs much better than both ....

J. Zhang. Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications. Comput. Methods Appl. Mech. Engrg. to appear.

....than ILUM. Various strategies have been proposed to invert or factor the blocks or domains efficiently. We remark that extracting parallelism from ILU factorizations has been the initial motivation behind the development of these multilevel ILU preconditioners [36, 39, 40] In a recent paper [48] BILUM was tested with several popular Krylov subspace accelerators for solving a few nonsymmetric matrices from applications in computational fluid dynamics. The test results show that the quality of the preconditioner determines the convergence rates of preconditioned iterative schemes. ....

J. Zhang. Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications. Technical Report No. 280-98, Department of Computer Science, University of Kentucky, Lexington, KY, 1998.

....R10000 processor, and 1 MB secondary cache. We used Fortran 77 programming language in 64 bit arithmetic computation. Test matrices. Three test matrices were selected from different applications. Table 1 contains simple descriptions of the test matrices. They have been used in several other papers [6, 9, 39, 46]. None of the three matrices has a zero diagonal. Matrix order nonzeros description RAEFSKY4 19 779 1 328 611 buckling problem for container model UTM5940 5 940 83 842 nuclear fusion plasma simulation WIGTO966 3 864 238 252 Euler equation model Table 1: Simple descriptions of the test matrices. 3 ....

....heat transfer in a square cavity FIDAPM37 9 152 765 944 flow of plastic in a profile extrusion die Table 5: Description of the largest 31 FIDAP matrices. our knowledge, this is the first time that so many FIDAP matrices were solved by a single iterative technique. 20 were solved in [40] 18 in [46], 9 in [39] and 8 in [9] In Table 6 the term unstable means that convergence was not reached in 100 iterations and the condition estimate was greater than 10 15 . Similarly the term inaccurate means that convergence was not reached, but the condition estimate did not exceed 10 15 . They ....

J. Zhang. Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications. Technical Report No. 280-98, Department of Computer Science, University of Kentucky, Lexington, KY, 1998.

....[1, 2, 4] In [1] a modified ILU factorization (MILU) 7, 12] is actually applied to a shifted matrix to insure the existence and stability of the ILU factorization. A multi level block ILU preconditioner (BILUM) has also been tested with several different Krylov subspace accelerators in [26] for solving several sparse matrices from computational fluid dynamics applications. The results in [26] show that the quality of the preconditioner determines the convergence rate of a preconditioned iterative method. With a high quality (high accuracy) BILUM preconditioner the choice of a Krylov ....

....matrix to insure the existence and stability of the ILU factorization. A multi level block ILU preconditioner (BILUM) has also been tested with several different Krylov subspace accelerators in [26] for solving several sparse matrices from computational fluid dynamics applications. The results in [26] show that the quality of the preconditioner determines the convergence rate of a preconditioned iterative method. With a high quality (high accuracy) BILUM preconditioner the choice of a Krylov subspace accelerator is not very critical to determine the success or failure of the preconditioned ....

Zhang, J., Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications, Technical Report No. 280-98, Department of Computer Science, University of Kentucky, Lexington, KY, 1998. Yousef Saad Department of Computer Science and Engineering University of Minnesota 4-192 EE/CS Building, 200 Union Street, S.E.

....as the sizes of the linear systems increase, and for difficult problems, they may even not converge at all. The robustness of the Krylov subspace methods can be improved with the help of preconditioning techniques. Based on evidence from analytical investigations [29] and numerical experiments [47], we know that the convergence rate of a preconditioned Krylov subspace method is largely determined by the quality of the preconditioner. Preconditioning techniques bridge direct and iterative methods and provide balance between reliability and scalability. This observation shifts current ....

J. Zhang. Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications. Technical Report No. 280-98, Department of Computer Science, University of Kentucky, Lexington, KY, 1998.

.... due to their ability to handle general sparse matrices, see, e.g. 36] and a recent survey by Golub and van der Vorst [20] Several numerical experiments and analytical investigations have shown that many of these Krylov subspace methods behave similarly for a large number of test problems [32, 43]. On the other hand, it has been recognized that the performance of these Krylov subspace methods can be remarkably enhanced by coupling them with a suitable preconditioner. Thus, a preconditioned iterative solver consists of an accelerator, commonly chosen as a Krylov subspace method, and a ....

....Krylov subspace method, and a preconditioner. Furthermore, The convergence rate of a preconditioned iterative method is usually dictated by the quality of the preconditioner. With the availability of a high quality preconditioner, the choice of the Krylov subspace accelerator is not that critical [20, 36, 43]. It is therefore of considerable interest to design and identify efficient preconditioning techniques. With the advent Technical Report No. 281 98, Department of Computer Science, University of Kentucky, Lexington, KY, 1998. This research was supported in part by the University of Kentucky ....

J. Zhang. Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications. Technical Report No. 280-98, Department of Computer Science, University of Kentucky, Lexington, KY, 1998.

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