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Warianty tytułu
Języki publikacji
Abstrakty
A framework for descent algorithms using shape as well as topological sensitivity information is introduced. The concept of gradient-related descent velocities in shape optimization is defined, a corresponding algorithmic approach is developed, and a convergence analysis is provided. It is shown that for a particular choice of the bilinear form involved in the definition of gradient-related directions a shape Newton method can be obtain. The level set methodology is used for representing and updating the geometry during the iterations. In order to include topological changes in addition to merging and splitting of existing geometries, a descent algorithm based on topological sensitivity is proposed. The overall method utilizes the shape sensitivity and topological sensitivity based methods in a serial fashion. Finally, numerical results are presented.
Czasopismo
Rocznik
Tom
Strony
305--324
Opis fizyczny
Bibliogr. 24 poz., wykr.
Twórcy
autor
- Department of Computational and Applied Mathematics, Rice University Houston, Texas, USA
Bibliografia
- Allaire, G., Jouve, F. and Toader, A.-M. (2004) Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys., 194 (1), 363–393.
- Bardi, M., Crandall, M.G., Evans, L.C., Soner, H.M. and Souganidis, P.E. (1997) Viscosity Solutions and Applications. Lecture Notes inMathematics, 1660, Springer-Verlag, Berlin.
- Burger, M. (2003) A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces Free Bound, 5 (3), 301-329.
- Burger, M. (2004) Levenberg-Marquardt level set methods for inverse problems. Inverse Problems, 20, 259–282.
- Burger, M., Hackl, B. and Ring, W. (2004) Incorporating topological derivatives into level set methods. J. Comput. Phys., 194, 344–362.
- Caselles, V., Catte, F., Coll, T. and Dibos, F. (1993) A geometric model for active contours in image processing. Numer. Math., 66 (1), 1–31.
- Crandall, M.G. and Lions, P.-L. (1986) On existence and uniqueness of solutions of Hamilton-Jacobi equations. Nonlinear Analysis, 10 (1), 353–370.
- Delfour, M. and Zolesio, J-P. (2001) Shapes and Geometries. Analysis, Differential Calculus and Optimization. SIAM Advances in Design and Control. SIAM, Philadelphia.
- Dorn, O., Miller, E.L. and Rappaport, C.M. (2000) A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets. Inverse Problems, 16 (5), 1119–1156.
- Eschenauer, H. and Schumacher, A. (1994) Bubble method for topology and shape optimization of structures. Structural Optimization, 8, 42–51.
- Garreau, S., Guillaume, P. and Masmoudi, M. (2001) The topological asymptotic for pde systems: the elasticity case. SIAM J. Control Optim., 39, 1756–1778.
- Hintermuller, M. and Ring, W. (2003) A second order shape optimization approach for image segmentation. SIAM J. Appl. Math., 64 (2), 442–467.
- Hintermuller, M. and Ring, W. (2004) An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional. J. Math. Imag. Vision, 20, 19–42.
- Litman, A., Lesselier, D. and Santosa, F. (1998) Reconstruction of a twodimensional binary obstacle by controlled evolution of a level-set. Inverse Problems, 14 (3), 685–706.
- Nocedal, J. and Wright, S.J. (1999) Numerical Optimization. Springer Series in Operations Research. Springer-Verlag, New York.
- Osher, S.J. and Fedkiw, R.P. (2002) Level Set Methods and Dynamic Implicit Surfaces. Springer Verlag, New York.
- Osher, S.J. and Santosa, F. (2001) Level set methods for optimization problems involving geometry and constraints. I. Frequencies of a two-density inhomogeneous drum. J. Comput. Phys., 171 (1), 272–288.
- Osher, S.J. and Sethian, J.A. (1988) Fronts propagating with curvaturedependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys., 79 (1), 12–49.
- Santosa, F. (1996) A level-set approach for inverse problems involving obstacles. ESAIM Controle Optim. Calc. Var, 1, 17–33.
- Sethian, J.A. (1999) Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge, second edition.
- Simon, J. (1989) Second variations for domain optimization problems. In: Control and Estimation of Distributed Parameter Systems (Vorau, 1988), F. Kappel et al., eds., International Ser. Numer. Math. 91, Birkhauser Verlag, Basel, 361–378.
- Sokołowski, J. and Żochowski, A. (1999) On the topological derivative in shape optimization. SIAM J. Control Optim., 37 (4), 1251–1272.
- Sokołowski, J. and Zolesio, J-P. (1992) Introduction to Shape Optimization. Springer-Verlag, Berlin.
- Yezzi,A., Kichenassamy, S., Kumar, A., Olver, P. and Tannenbaum, A. (1997) A geometric snake model for segmentation of medical imagery. IEEE Trans. Med. Imaging, 16 (3), 199–209.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-BAT5-0007-0091