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Comparison of advanced large-scale minimization algorithms for the solution of inverse ill-posed problems. (English) Zbl 1189.90221
Summary: We compare the performance of several robust large-scale minimization algorithms for the unconstrained minimization of an ill-posed inverse problem. The parabolized Navier-Stokes equation model was used for adjoint parameter estimation. The methods compared consist of three versions of nonlinear conjugate-gradient (CG) method, quasi-Newton Broyden-Fletcher-Goldfarb-Shanno (BFGS), the limited-memory quasi-Newton (L-BFGS) [{\it D. C. Liu} and {ı J. Nocedal}, Math. Program., Ser. B 45, No. 3, 503--528 (1989; Zbl 0696.90048)], truncated Newton (T-N) method [{\it S. G. Nash}, SIAM J. Sci. Stat. Comput. 6, 599--616 (1985); SIAM J. Numer. Anal. 21, 770--788 (1984; Zbl 0558.65041)] and a new hybrid algorithm proposed by {\it J. L. Morales} and {\it J. Nocedal} [Comput. Optim. Appl. 21, No. 2, 143--154 (2002; Zbl 0988.90035)]. For all the methods employed and tested, the gradient of the cost function is obtained via an adjoint method. A detailed description of the algorithmic form of minimization algorithms employed in the minimization comparison is provided. For the inviscid case, the CG-descent method of Hager [{\it W. W. Hager} and {\it H. Zhang}, SIAM J. Optim. 16, No. 1, 170--192 (2005; Zbl 1093.90085)] performed the best followed closely by the hybrid method [Morales and Nocedal, loc. cit.], while in the viscous case, the hybrid method emerged as the best performed followed by CG [{\it D. F. Shanno} and {\it K. H. Phua}, ACM Trans. Math. Softw. 15, No. 4, 618--622 (1989)] and CG-descent [{\it W. W. Hager} and {\it H. Zhang}, SIAM J. Optim. 16, No. 1, 170--192 (2005; Zbl 1093.90085)]. This required an adequate choice of parameters in the CG-descent method as well as controlling the number of L-BFGS and T-N iterations to be interlaced in the hybrid method.

90C90Applications of mathematical programming
90C30Nonlinear programming
49J20Optimal control problems with PDE (existence)
47A52Ill-posed problems, regularization
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