DSC Publications

Core Information
Title: Comparison of advanced large-scale minimization algorithms for the solution of inverse ill-posed problems
Abstract: 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 J. Nocedal, On the limited memory BFGS method for large scale minimization, Math. Program. 45 (1989), pp. 503–528], truncated Newton (T-N) method [S.G. Nash, Preconditioning of truncated Newton methods, SIAM J. Sci. Stat. Comput. 6 (1985), pp. 599–616, S.G. Nash, Newton-type minimization via the Lanczos method, SIAM J. Numer. Anal. 21 (1984), pp. 770–788] and a new hybrid algorithm proposed by Morales and Nocedal [J.L. Morales and J. Nocedal, Enriched methods for large-scale unconstrained optimization, Comput. Optim. Appl. 21 (2002), pp. 143–154]. 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, A new conjugate gradient method with guaranteed descent and efficient line search, SIAM J. Optim. 16 (1) (2005), pp. 170– 192] performed the best followed closely by the hybrid method [J.L. Morales and J. Nocedal, Enriched methods for large-scale unconstrained optimization, Comput. Optim. Appl. 21 (2002), pp. 143–154], while in the viscous case, the hybrid method emerged as the best performed followed by CG [D.F. Shanno and K.H. Phua, Remark on algorithm 500. Minimization of unconstrained multivariate functions,ACM Trans. Math. Softw. 6 (1980), pp. 618–622] and CG-descent [W.W. Hager and H. Zhang,A new conjugate gradient method with guaranteed descent and efficient line search, SIAM J. Optim. 16 (1) (2005), pp. 170–192]. 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.
Keywords: large-scale minimization methods, inverse problems, adjoint parameter estimation, ill-posed problems
Author Information
1. Alekseev, Aleksey K[ Department of Aerodynamics and Heat Transfer, RSC, ENERGIA, Korolev (Kaliningrad), Moscow Region, Russian Federation ]Neither
2. Navon, Ionel M[ FSU/SCS ]Neither
3. Steward, Jeffrey L.[ FSU/SCS ]Graduate
Detailed Scientific Article Information
Journal Name: Optimization Methods and Software
Volume: Early View
Page Range: 1–25
Article Number: Not Provided
Number of Pages: 25
Year of Publication: 2008
Refereed: Yes
Digital Object Identifier (DOI), if available: 10.1080/10556780802370746
Official Url: http://www.informaworld.com
ISSN: 1055-6788 print ; 1029-4937 online
Subjects Information
1. Mathematics
2. Fluid Dynamics

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