Abstract
We propose an alternative iterative method to solve rank deficient problems arising in many real applications such as the finite element approximation to the Stokes equation and computational genetics. Our main contribution is to transform the rank deficient problem into a smaller full rank problem, with structure as sparse as possible. The new system improves the condition number greatly. Numerical experiments suggest that the new iterative method works very well for large sparse rank deficient saddle point problems.
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References
M. Arioli, I.S. Duff and P.P.M. de Rijk, On the augmented system approach to sparse least-squares problems, Numer. Math. 55 (1989) 667–684.
M. Benzi, A direct row-projection method for sparse linear systems, Ph.D. thesis, Graduate Faculty of North Carolina State University, Raleigh (1993).
M. Benzi and C.D. Meyer, A direct projection method for sparse linear systems, SIAM J. Sci. Comput. 16 (1995) 1159–1176.
Å. Björck and C.C. Paige, Solution of augmented linear systems using orthogonal factorization, BIT 34 (1994) 1–24.
Å. Björck and J.Y. Yuan, Preconditioner for least squares problems by LU factorization, ETNA 8 (1999) 26–35.
I.S. Duff and J.K. Reid, Exploiting zeros on the diagonal in the direct solution of indefinite sparse symmetric linear systems, ACM Trans. Software 17 (1996).
H. Elman and D. Silvester, Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations, SIAM J. Sci. Comput. 17 (1996) 33–46.
H. Elman, D.J. Silvester and A. Wathen, Iterative methods for problems in computational fluid dynamics, in: Iterative Methods in Scientific Computing, eds. R. Chan, T. Chan and G. Golub (Springer, Singapore, 1997) pp. 271–327.
B. Fischer, A. Ramage, D.J. Silvester and A.J. Wathen, Minimum residual methods for augmented systems, BIT 38 (1998) 527–543.
G.H. Golub and C. Greif, Techniques for solving general KKT systems, Technical Report, SCCM, Stanford University (2000).
G.H. Golub, X. Wu and J.Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001) 71–85.
C. Helmberg, F. Rendl, R.J. Vanderbei and H. Wolkowicz, An interior-point method for semidefinite programming, SIAM J. Optim. 6 (1996) 342–361.
Z. Luo, B.P.B. Silva and J.Y. Yuan, Direct-projection methods, Internat. J. Comput. Math. 76 (2001) 517–535.
S.G. Nash and A. Sofer, Preconditioned reduced matrices, SIAM J. Matrix Anal. Appl. 17 (1996) 47–68.
C.H. Santos, B.P.B. Silva and J.Y. Yuan, Block SOR methods for rank deficient least squares problems, J. Comput. Appl. Math. 100 (1998) 1–9.
M.A. Saunders, Sparse least squares by conjugate gradients: a comparison of preconditioning methods, in: Proc. of Computer Science and Statistics: 12th Annual Conf. on the Interface, Waterloo, Canada, 1979.
B.P.B. Silva, A preconditioner for least squares problems, Master dissertation, UFPR, Brazil (1997).
R.J. Vanderbei, Splitting dense columns in sparse linear systems: Interior point methods for linear programming, Linear Algebra Appl. 152 (1991) 107–117.
R.J. Vanderbei, Symmetric quasidefinite matrices, SIAM J. Optim. 5 (1995) 100–113.
R.S. Varga, Matrix Iterative Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1962).
S. Wright, Stability of augmented system factorizations in interior-point methods, SIAM J. Matrix Anal. Appl. 18 (1997) 191–222.
D.M. Young, Iterative Solutions of Large Linear Systems (Academic Press, New York, 1971).
J.Y. Yuan, Iterative methods for generalized least squares problems, Ph.D. thesis, IMPA, Rio de Janeiro, Brazil (1993).
J.Y. Yuan, Numerical methods for generalized least squares problems, J. Comput. Appl. Math. 66 (1996) 571–584.
J.Y. Yuan and A.N. Iusem, Preconditioned conjugate gradient method for generalized least squares problems, J. Comput. Appl. Math. 71 (1996) 287–297.
J.Y. Yuan and A.N. Iusem, Preconditioned block SOR method for generalized least squares problems, Acta Math. Appl. Sinica 16 (2000) 130–139.
J.Y. Yuan and X.Q. Jin, Direct iterative methods for rank deficient generalized least squares problems, J. Comput. Math. 18 (2000) 439–448.
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Wu, X., Silva, B. & Yuan, J. Conjugate Gradient Method for Rank Deficient Saddle Point Problems. Numerical Algorithms 35, 139–154 (2004). https://doi.org/10.1023/B:NUMA.0000021758.65113.f5
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DOI: https://doi.org/10.1023/B:NUMA.0000021758.65113.f5