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) [D. C. Liu and {ıJ. Nocedal}, Math. Program., Ser. B 45, No. 3, 503–528 (1989; Zbl 0696.90048)], truncated Newton (T-N) method [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 J. L. Morales and 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 [W. W. Hager and 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 [D. F. Shanno and K. H. Phua, ACM Trans. Math. Softw. 15, No. 4, 618–622 (1989)] and CG-descent [W. W. Hager and 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.


90C90 Applications of mathematical programming
90C30 Nonlinear programming
49J20 Existence theories for optimal control problems involving partial differential equations
47A52 Linear operators and ill-posed problems, regularization
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