Gradient descent and fast artificial time integration
Uri M. Ascher; Kees van den Doel; Hui Huang; Benar F. Svaiter
ESAIM: Mathematical Modelling and Numerical Analysis (2009)
- Volume: 43, Issue: 4, page 689-708
- ISSN: 0764-583X
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top- H. Akaike, On a successive transformation of probability distribution and its application to the analysis of the optimum gradient method. Ann. Inst. Stat. Math. Tokyo11 (1959) 1–16.
- U. Ascher, Numerical Methods for Evolutionary Differential Equations. SIAM, Philadelphia, USA (2008).
- U. Ascher, E. Haber and H. Huang, On effective methods for implicit piecewise smooth surface recovery. SIAM J. Sci. Comput.28 (2006) 339–358.
- U. Ascher, H. Huang and K. van den Doel, Artificial time integration. BIT47 (2007) 3–25.
- J. Barzilai and J. Borwein, Two point step size gradient methods. IMA J. Num. Anal.8 (1988) 141–148.
- M. Cheney, D. Isaacson and J.C. Newell, Electrical impedance tomography. SIAM Review41 (1999) 85–101.
- 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.
- Y. Dai and R. Fletcher, Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming. Numer. Math.100 (2005) 21–47.
- 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.
- H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems. Kluwer (1996).
- 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.
- G.E. Forsythe, On the asymptotic directions of the s-dimensional optimum gradient method. Numer. Math.11 (1968) 57–76.
- A. Friedlander, J. Martinez, B. Molina and M. Raydan, Gradient method with retard and generalizations. SIAM J. Num. Anal.36 (1999) 275–289.
- G. Golub and Q. Ye, Inexact preconditioned conjugate gradient method with inner-outer iteration. SIAM J. Sci. Comp.21 (2000) 1305–1320.
- A. Greenbaum, Iterative Methods for Solving Linear Systems. SIAM, Philadelphia, USA (1997).
- E. Haber and U. Ascher, Preconditioned all-at-one methods for large, sparse parameter estimation problems. Inverse Problems17 (2001) 1847–1864.
- E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Second Edition, Springer (1996).
- H. Huang, Efficient Reconstruction of 2D Images and 3D Surfaces. Ph.D. Thesis, University of BC, Vancouver, Canada (2008).
- W. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer (2003).
- 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.
- J. Nagy and K. Palmer, Steepest descent, CG and iterative regularization of ill-posed problems. BIT43 (2003) 1003–1017.
- J. Nocedal and S. Wright, Numerical Optimization. Springer, New York (1999).
- 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.
- S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. Springer (2003).
- P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell.12 (1990) 629–639.
- L. Pronzato, H. Wynn and A. Zhigljavsky, Dynamical Search: Applications of Dynamical Systems in Search and Optimization. Chapman & Hall/CRC, Boca Raton (2000).
- M. Raydan and B. Svaiter, Relaxed steepest descent and Cauchy-Barzilai-Borwein method. Comput. Optim. Appl.21 (2002) 155–167.
- R. Sincovec and N. Madsen, Software for nonlinear partial differential equations. ACM Trans. Math. Software1 (1975) 232–260.
- N.C. Smith and K. Vozoff, Two dimensional DC resistivity inversion for dipole dipole data. IEEE Trans. Geosci. Remote Sens.22 (1984) 21–28.
- G. Strang and G. Fix, An Analysis of the Finite Element Method. Prentice-Hall, Engelwood Cliffs, NJ (1973).
- E. Tadmor, S. Nezzar and L. Vese, A multiscale image representation using hierarchical (BV, L2) decompositions. SIAM J. Multiscale Model. Simul.2 (2004) 554–579.
- E. van den Berg and M. Friedlander, Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput.31 (2008) 840–912.
- 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.
- K. van den Doel and U. Ascher, Dynamic level set regularization for large distributed parameter estimation problems. Inverse Problems23 (2007) 1271–1288.
- C. Vogel, Computational methods for inverse problem. SIAM, Philadelphia, USA (2002).
- J. Weickert, Anisotropic Diffusion in Image Processing. B.G. Teubner, Stuttgart (1998).