Gradient descent and fast artificial time integration

Mathematical Modelling and Numerical Analysis

Volumn 43, Issue 4, 2009, Pages 689-708

  Ascher, Uri M ^a   van den Doel, Kees ^a   Huang, Hui ^a   Svaiter, Benar F ^b  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^b IMPA   (Brazil)

Author keywords

Artificial time; Forward Euler; Gradient descent; Lagged steepest descent; Regularization; Steady state

Indexed keywords

EID: 84996139670     PISSN: 0764583X     EISSN: 12903841     Source Type: Journal    
DOI: 10.1051/m2an/2009025     Document Type: Article

References (36)

1. H. Akaike, On a successive transformation of probability distribution and its application to the analysis of the optimum gradient method. Ann. Inst. Stat. Math. Tokyo 11 (1959) 1-16.
2. U. Ascher, Numerical Methods for Evolutionary Differential Equations. SIAM, Philadelphia, USA (2008).
3. U. Ascher, E. Haber and H. Huang, On effective methods for implicit piecewise smooth surface recovery. SIAM J. Sci. Comput. 28 (2006) 339-358.
4. U. Ascher, H. Huang and K. van den Doel, Artificial time integration. BIT 47 (2007) 3-25.
5. J. Barzilai and J. Borwein, Two point step size gradient methods. IMA J. Num. Anal. 8 (1988) 141-148.
6. M. Cheney, D. Isaacson and J.C. Newell, Electrical impedance tomography. SIAM Review 41 (1999) 85-101.
7. E. Chung, T. Chan and X. Tai, Electrical impedance tomography using level set representations and total variation regularization. J. Comp. Phys. 205 (2005) 357-372.
8. Y. Dai and R. Fletcher, Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming. Numer. Math. 100 (2005) 21-47.
9. Y. Dai, W. Hager, K. Schittkowsky and H. Zhang, A cyclic Barzilai-Borwein method for unconstrained optimization. IMA J. Num. Anal. 26 (2006) 604-627.
10. H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems. Kluwer (1996).
11. M. Figueiredo, R. Nowak and S. Wright, Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 1 (2007) 586-598.
12. G.E. Forsythe, On the asymptotic directions of the s-dimensional optimum gradient method. Numer. Math. 11 (1968) 57-76.
13. A. Friedlander, J. Martinez, B. Molina and M. Raydan, Gradient method with retard and generalizations. SIAM J. Num. Anal. 36 (1999) 275-289.
14. G. Golub and Q. Ye, Inexact preconditioned conjugate gradient method with inner-outer iteration. SIAM J. Sci. Comp. 21 (2000) 1305-1320.
15. A. Greenbaum, Iterative Methods for Solving Linear Systems. SIAM, Philadelphia, USA (1997).
16. E. Haber and U. Ascher, Preconditioned all-at-one methods for large, sparse parameter estimation problems. Inverse Problems 17 (2001) 1847-1864.
17. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Second Edition, Springer (1996).
18. H. Huang, Efficient Reconstruction of 2D Images and 3D Surfaces. Ph.D. Thesis, University of BC, Vancouver, Canada (2008).
19. W. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer (2003).
20. Y. Li and D.W. Oldenburg, Inversion of 3-D DC resistivity data using an approximate inverse mapping. Geophys. J. Int. 116 (1994) 557-569.
21. J. Nagy and K. Palmer, Steepest descent, CG and iterative regularization of ill-posed problems. BIT 43 (2003) 1003-1017.
22. J. Nocedal and S. Wright, Numerical Optimization. Springer, New York (1999).
23. J. Nocedal, A. Sartenar and C. Zhu, On the behavior of the gradient norm in the steepest descent method. Comput. Optim. Appl. 22 (2002) 5-35.
24. S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. Springer (2003).
25. P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12 (1990) 629-639.
26. L. Pronzato, H. Wynn and A. Zhigljavsky, Dynamical Search: Applications of Dynamical Systems in Search and Optimization. Chapman & Hall/CRC, Boca Raton (2000).
27. M. Raydan and B. Svaiter, Relaxed steepest descent and Cauchy-Barzilai-Borwein method. Comput. Optim. Appl. 21 (2002) 155-167.
28. R. Sincovec and N. Madsen, Software for nonlinear partial differential equations. ACM Trans. Math. Software 1 (1975) 232-260.
29. N. C. Smith and K. Vozoff, Two dimensional DC resistivity inversion for dipole dipole data. IEEE Trans. Geosci. Remote Sens. 22 (1984) 21-28.
30. G. Strang and G. Fix, An Analysis of the Finite Element Method. Prentice-Hall, Engelwood Cliffs, NJ (1973).
31. 2) decompositions. SIAM J. Multiscale Model. Simul. 2 (2004) 554-579.
32. E. van den Berg and M. Friedlander, Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31 (2008) 840-912.
33. K. van den Doel and U. Ascher, On level set regularization for highly ill-posed distributed parameter estimation problems. J. Comp. Phys. 216 (2006) 707-723.
34. K. van den Doel and U. Ascher, Dynamic level set regularization for large distributed parameter estimation problems. Inverse Problems 23 (2007) 1271-1288.
35. C. Vogel, Computational methods for inverse problem. SIAM, Philadelphia, USA (2002).
36. J. Weickert, Anisotropic Diffusion in Image Processing. B.G. Teubner, Stuttgart (1998).