Element-by-element preconditioners for large partially separable optimization problems

SIAM Journal on Scientific Computing

Volumn 18, Issue 6, 1997, Pages 1767-1787

  Daydé, Michel J ^a   L'Excellent, Jean Yves ^a,b   Gould, Nicholas I M ^c  

^a UNIVERSITÉ DE TOULOUSE   (France)

^b CERFACS   (France)

^c RUTHERFORD APPLETON LABORATORY   (United Kingdom)

Author keywords

Element by element preconditioners; Large scale optimization; Partial separability; Preconditioned conjugate gradient

Indexed keywords

ELEMENT BY ELEMENT PRECONDITIONERS; LARGE SCALE OPTIMIZATION; PARTIAL SEPARABILITY; PRECONDITIONED CONJUGATE GRADIENT;

COMPUTATIONAL METHODS; FUNCTIONS; ITERATIVE METHODS; LINEARIZATION; PROBLEM SOLVING;

OPTIMIZATION;

EID: 0031261356     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827594274796     Document Type: Article

References (39)

1. P. AMESTOY (1991), Factorization of Large Sparse Matrices Based on a Multifrontal Approach in a Multiprocessor Environment, Ph.D. thesis TH/PA/91/2, CERFACS, Toulouse, France.
2. M. ARIOLI, T. F. CHAN, I. S. DUFF, N. I. M. GOULD, AND J. K. REID (1993), Computing a Search Direction for Large-Scale Linearly Constrained Nonlinear Optimization Calculations, Tech. report RAL-93-066, Rutherford Appleton Laboratory, Chilton, England.
3. O. AXELSSON (1976), Solution of linear systems of equations: Iterative methods, in Sparse Matrix Techniques (Copenhagen 1976), V. A. Baker, ed., Springer-Verlag, Berlin.
4. I. BONGARTZ, A. R. CONN, N. I. M. GOULD, AND PH. L. TOINT (1993), CUTE: Constrained and Unconstrained Testing Environment, Tech. report TR/PA/93/10, CERFACS, Toulouse, France.
5. A. R. CONN, N. I. M. GOULD, AND PH. L. TOINT (1990), An introduction to the structure of large scale nonlinear optimization problems and the LANCELOT project, in Computing Methods in Applied Sciences and Engineering, R. Glowinski and A. Lichnewsky, eds., SIAM, Philadelphia, PA, pp. 42-54.
6. A. R. CONN, N. I. M. GOULD, AND PH. L. TOINT (1992), Large-scale nonlinear constrained optimization, in Proc. 2nd Internat. Conference on Industrial and Applied Mathematics, R. E. O'Malley, Jr., ed., SIAM, Philadelphia, PA, pp. 51-70; also in Linear Algebra for Large-Scale and Real-Time Applications, M. S. Moonen, G. H. Golub, and B. L. R. De-Moor, eds., NATO ASI Series E: Applied Sciences, Vol. 232, Kluwer Academic Publishers, Norwell, MA, 1993.
7. A. R. CONN, N. I. M. GOULD, AND PH. L. TOINT (1992), Large-scale nonlinear constrained optimization, in Proc. 2nd Internat. Conference on Industrial and Applied Mathematics, R. E. O'Malley, Jr., ed., SIAM, Philadelphia, PA, pp. 51-70; also in Linear Algebra for Large-Scale and Real-Time Applications, M. S. Moonen, G. H. Golub, and B. L. R. De-Moor, eds., NATO ASI Series E: Applied Sciences, Vol. 232, Kluwer Academic Publishers, Norwell, MA, 1993.
8. A. R. CONN, N. I. M. GOULD, AND PH. L. TOINT (1994), Improving the decomposition of partially separable functions in the context of large-scale optimization: A first approach, in Large Scale Optimization: State of the Art, W. W. Hager, D. W. Hearn, and P. M. Pardalos, eds., Kluwer Academic Publishers Group, Norwell, MA, and Dordrecht, the Netherlands, pp. 82-94.
9. M. J. DAYDÉ, J.-Y. L'EXCELLENT, AND N. I. M. GOULD (1994), On the Use of Element-by-Element Preconditioners to Solve Large Scale Partially Separable Optimization Problems, Tech. report RT/APO/94/4, Ecole Nationale Supérieure d'Électronique, d'Électrotechnique, d'Informatique et d'Hydraulique de Toulouse, Toulouse, France.
10. I. S. DUFF AND J. K. REID (1983), The multifrontal solution of indefinite sparse symmetric linear equations, ACM Trans. Math. Software, 9, pp. 302-325.
11. I. S. DUFF, R. G. GRIMES, AND J. G. LEWIS (1989), Sparse matrix test problems, ACM Trans. Math. Software, 15, pp. 1-14.
12. J. ERHEL, A. TRAYNARD, AND M. VIDRASCU (1991), An element-by-element preconditioned conjugate gradient method implemented on a vector computer, Parallel Comput., 17, pp. 1051-1065.
13. P. E. GILL AND W. MURRAY (1974), Newton-type methods for linearly constrained optimization, in Numerical Methods for Constrained Optimization, P. E. Gill and W. Murray, eds., Academic Press, London.
14. P. E. GILL, W. MURRAY, AND M. H. WRIGHT (1981), Practical Optimization, Academic Press, London, New York.
15. G. H. GOLUB AND C. F. VAN LOAN (1989), Matrix Computations, 2nd ed., Johns Hopkins University Press, Baltimore, MD.
16. A. GRIEWANK AND PH. L. TOINT (1982), Local convergence analysis for partitioned quasiNewton updates, Numer. Math., 39, pp. 429-448.
17. A. GRIEWANK AND PH. L. TOINT (1982), On the unconstrained optimization of partially separable functions, in Nonlinear Optimization, M. J. D. Powell, ed., Academic Press, London, New York, pp. 301-312,
18. A. GRIEWANK AND PH. L. TOINT (1982), Partitioned variable metric updates for large structured optimization problems, Numer. Math., 39, pp. 429-448.
19. A. GRIEWANK AND PH. L. TOINT (1984), On the existence of convex decomposition of partially separable functions, Math. Programming, 24, pp. 25-49.
20. I. GUSTAFSSON AND G. LINDSKOG (1986), A preconditioning technique based on element matrix factorization, Comput. Methods Appl. Mech., 55, pp. 201-220.
21. M. R. HESTENES (1969), Multiplier and gradient methods, J. Optim. Theory Appl., 4, pp. 303-320.
22. N. J. HIGHAM (1993), The accuracy of floating-point summation, SIAM J. Sci. Statist. Comput., 14, pp. 783-799.
23. T. J. R. HUGHES (1987), The Finite-Element Method: Linear Static and Dynamic Finite-Element Analysis, Prentice-Hall, Englewood Cliffs, NJ.
24. T. J. R. HUGHES AND R. M. FERENCZ (1988), Fully Vectorized EBE Preconditioners for Nonlinear Solid Mechanics: Applications to Large-Scale Three-Dimensional Continuum, Shell and Contact/Impact Problems, in Proc. 1st Internat. Symp. Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. Périaux, eds., SIAM, Philadelphia, PA, pp. 261-280.
25. T. J. R. HUGHES, R. M. FERENCZ, AND J. O. HALLQUITS (1987), Large-scale vectorized implicit calculations in solid mechanics on a CRAY X-MP/48 utilizing EBE preconditioned conjugate gradients, Comput. Methods Appl. Mech. Engrg., 61, pp. 215-248.
26. T. J. R. HUGHES, I. LEVIT, AND J. WINGET (1983), An element-by-element solution algorithm for problems of structural and solid mechanics, Comput. Methods Appl. Mech. Engrg., 36, pp. 241-254.
27. B. M. IRONS (1970), A frontal solution program for finite-element analysis, Internat. J. Numer. Methods Engrg., 2, pp. 5-32.
28. E. F. KAASSCHIETER (1989), A general finite element preconditioning for the conjugate gradient method, BIT, 29, pp. 824-849.
29. S. G. NASH (1984), Newton-type minimization via the Lanczos method, SIAM J. Numer. Anal., 21, pp. 770-788.
30. B. NOUR-UMID AND B. N. PARLETT (1985), Element preconditioning using splitting techniques, SIAM J. Sci. Statist. Comput., 6, pp. 761-770.
31. M. ORTIZ, P. M. PINSKY, AND R. L. TAYLOR (1983), Unconditionally stable element-by-element algorithms for dynamic problems, Comput. Methods Appl. Mech. Engrg., 36, pp. 223-239.
32. M. J. D. POWELL (1969), A method for nonlinear constraints in minimization problems, in Optimization, R. Fletcher, ed., Academic Press, London, New York, pp. 283-298,
33. J. K. REID (1984), Treesolve, A Fortran Package for Solving Large Sets of Linear Finite Element Equations, Tech. report HL 84/796, AERE Harwell Laboratory, Harwell, UK.
34. Y. SAAD (1990), SPARSKIT: A Basic Tool Kit for Sparse Matrix Computations, Tech. report CSRD TR #1029, Center for Supercomputing Research and Development, University of Illinois, Urbana, IL.
35. R. B. SCHNABEL AND E. ESKOW (1991), A new modified Cholesky factorization, SIAM J. Sci. Statist. Comput., 11, pp. 1136-1158.
36. H. A. VAN DER VORST (1985), The Performance of Fortran Implementations for Preconditioned Conjugate Gradients on Vector Computers, Tech. report 85-09, Delft University of Technology, Delft, the Netherlands.
37. A. J. WATHEN (1989), An analysis of some element-by-element techniques, Comput. Methods Appl. Mech. Engrg., 74, pp. 271-287.
38. A. J. WATHEN (1992), Singular element preconditioning for the finite element method, in Iterative Methods in Linear Algebra, R. Beauwens and P. de. Groen, eds., Elsevier/North-Holland, Amsterdam, pp. 531-540.
39. O. C. ZIENKIEWICZ (1977), The Finite Element Method, McGraw-Hill, London.