Krylov Subspace Methods

In my thesis and in subsequent work, the effectiveness of the preconditioned conjugate gradient algorithm was demonstrated for discretizations of linear elliptic partial differential equations [C01], nonlinear elliptic equations [J01], and free boundary problems for linear and nonlinear elliptic equations [J08] [J03]. Application of the conjugate gradient algorithm to general quadratic programming problems was considered [T03]. This work was applied to the analysis of torsion on an elasto-plastic bar [J02] and water flow in an excavation site [C02].

Polynomial preconditioners for the conjugate gradient algorithm were studied in [J33].

A block form of the conjugate gradient algorithm, useful for solving multiple linear systems and for linear systems with specialized eigenvalue distributions, was developed and analyzed [J06]. The parallel implementation of the algorithm was studied [43] [J24] [J32].

The quasi-Newton family of algorithm was extended to block form in [J39], and new insights on Broyden's method applied to linear systems were developed [J41]. Efficient use of conjugate gradient algorithms for computing the search directions in interior point methods was studied in [J55].

A literature review of the first 25 years of the conjugate gradient algorithm was published in 1989, jointly with Gene Golub [J28] and an updated overview is given in [H02].

Stagnation of the GMRES algorithm for solving nonsymmetric systems of equations was studied in [J64].

Zdenek Strakoš, Petr Tichý, and I contributed to the understanding of the convergence of Krylov methods when implemented on computers in inexact arithmetic [J81], by exploiting the relation of these methods to Gauss quadrature.

[C01]

(Invited paper) Paul Concus, Gene H. Golub, and Dianne P. O'Leary, ``A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations,''
in Sparse Matrix Computations, James R. Bunch and Donald J. Rose (Eds.) Academic Press, New York (1976) 309-332.
reprinted in Studies in Numerical Analysis, Gene H. Golub (Ed.), Volume 25 of Studies in Mathematics, The Mathematical Association of America (1984) 178-198.

[C02]

Dianne P. O'Leary, ``Linear programming problems arising from partial differential equations,'' in Sparse Matrix Proceedings 1978, Iain S. Duff and G. W. Stewart (Eds.) SIAM Press, Philadelphia (1979) 25-40.

[C03]

(invited, extended abstract) Dianne P. O'Leary, ``Fine and Medium Grained Parallel Algorithms for Matrix QR Factorization,'' Algorithms and Applications on Vector and Parallel Computers, H.J.J. te Riele, Th.J. Dekker and H.A. van der Vorst, eds., Elsevier Science Publishers B.V. (North Holland), (1987) 347-349.

[H02]

Dianne P. O'Leary, ``Conjugate Gradients and Related KMP Algorithms: The Beginnings," in Linear and Nonlinear Conjugate Gradient-Related Methods, Loyce Adams and J. L. Nazareth, eds., SIAM, Philadelphia, 1996, 1-8.

[J01]

Paul Concus, Gene H. Golub, and Dianne P. O'Leary, ``Numerical solution of nonlinear elliptic partial differential equations by a generalized conjugate gradient method,'' Computing 19 (1978) 321-339.

[J02]

Dianne P. O'Leary and Wei H. Yang, ``Elasto-plastic torsion by quadratic programming,'' Computer Methods in Applied Mechanics and Engineering 16 (1978) 361-368.

[J03]

Dianne P. O'Leary, ``Conjugate gradient algorithms in the solution of optimization problems for nonlinear elliptic partial differential equations,'' Computing 22 (1979) 59-77.

[J06]

Dianne P. O'Leary, ``The block conjugate gradient algorithm and related methods,'' Linear Algebra and Its Applications 29 (1980) 293-322.

[J08]

Dianne P. O'Leary, ``A generalized conjugate gradient algorithm for solving a class of quadratic programming problems,'' Linear Algebra and Its Applications Special Issue on Large Scale Matrix Problems 34 (1980) 371-399. Also in Large Scale Matrix Problems, A. Bjorck, R. J. Plemmons and H. Schneider, eds. North Holland Pub. Co. NY (1981) 391-399.

[J24]

Dianne P. O'Leary, ``Parallel implementation of the block conjugate gradient algorithm,'' Parallel Computing 5 (1987) 127-139.

[J28]

Gene H. Golub and Dianne P. O'Leary, ``Some history of the conjugate gradient and Lanczos algorithms: 1948-1976,'' SIAM Review 31 (1989) 50-102.

[J32]

Dianne P. O'Leary and Peter Whitman, ``Parallel QR factorization by Householder and modified Gram-Schmidt algorithms,'' Parallel Computing 16 (1990) 99-112.

[J33]

Dianne P. O'Leary, ``Yet another polynomial preconditioner for the conjugate gradient algorithm,'' Linear Algebra and Its Applications, 154 (1991) 377-388.

[J39]

Dianne P. O'Leary and A. Yeremin, ``The linear algebra of block quasi-Newton algorithms,'' Linear Algebra and Its Applications, 212/213 (1994) 153-168.

[J41]

Dianne P. O'Leary, ``Why Broyden's nonsymmetric method terminates on linear equations,'' SIAM Journal on Optimization, 5 (1995) 231-235.

[J55]

Weichung Wang and Dianne P. O'Leary, ``Adaptive Use of Iterative Methods in Predictor-Corrector Interior Point Methods for Linear Programming," Numerical Algorithms, (special issue honoring Richard Varga), 25 (2000) 387-406.

[J64]

Ilya Zavorin, Dianne P. O'Leary, and Howard Elman, ``Complete Stagnation of GMRES," Linear Algebra and Its Applications, 367 (2003) 165-183.

[J81]

Dianne P. O'Leary, Zdenek Strakoš, and Petr Tichý, ``On Sensitivity of Gauss-Christoffel Quadrature," Numerische Mathematik, 107 (2007) 147-174. DOI:10.1007/s00211-007-0078-x

[T03]

Dianne P. O'Leary, ``Sparse quadratic programming without matrix updating,'' Computer Science Department Report TR-1200, University of Maryland (1982).