The numerical solution of ill-posed problems requires suitable regularization techniques. One possible option is to consider time integration methods to solve the Showalter differential equation numerically. The stopping time of the numerical integrator corresponds to the regularization parameter. A number of well-known regularization methods such as the Landweber iteration or the Levenberg-Marquardt method can be interpreted as variants of the Euler method for solving the Showalter differential equation. Motivated by an analysis of the regularization properties of the exact solution of this equation presented by [U. Tautenhahn, Inverse Problems 10 (1994) 1405-1418], we consider a variant of the exponential Euler method for solving the Showalter ordinary differential equation. We discuss a suitable discrepancy principle for selecting the step sizes within the numerical method and we review the convergence properties of [U. Tautenhahn, Inverse Problems 10 (1994) 1405-1418], and of our discrete version [M. Hochbruck et al., Technical Report (2008)]. Finally, we present numerical experiments which show that this method can be efficiently implemented by using Krylov subspace methods to approximate the product of a matrix function with a vector.
Mots clés : nonlinear ill-posed problems, asymptotic regularization, exponential integrators, variable step sizes, convergence, optimal convergence rates
@article{M2AN_2009__43_4_709_0,
author = {Hochbruck, Marlis and H\"onig, Michael and Ostermann, Alexander},
title = {Regularization of nonlinear ill-posed problems by exponential integrators},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {709--720},
publisher = {EDP-Sciences},
volume = {43},
number = {4},
year = {2009},
doi = {10.1051/m2an/2009021},
mrnumber = {2542873},
zbl = {1167.65369},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2009021/}
}
TY - JOUR AU - Hochbruck, Marlis AU - Hönig, Michael AU - Ostermann, Alexander TI - Regularization of nonlinear ill-posed problems by exponential integrators JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2009 SP - 709 EP - 720 VL - 43 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2009021/ DO - 10.1051/m2an/2009021 LA - en ID - M2AN_2009__43_4_709_0 ER -
%0 Journal Article %A Hochbruck, Marlis %A Hönig, Michael %A Ostermann, Alexander %T Regularization of nonlinear ill-posed problems by exponential integrators %J ESAIM: Modélisation mathématique et analyse numérique %D 2009 %P 709-720 %V 43 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2009021/ %R 10.1051/m2an/2009021 %G en %F M2AN_2009__43_4_709_0
Hochbruck, Marlis; Hönig, Michael; Ostermann, Alexander. Regularization of nonlinear ill-posed problems by exponential integrators. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 4, pp. 709-720. doi : 10.1051/m2an/2009021. http://www.numdam.org/articles/10.1051/m2an/2009021/
[1] and , Iterative Runge-Kutta-type methods for nonlinear ill-posed problems. Inverse Problems 24 (2008) 025002. | MR | Zbl
[2] , , and , Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization. Math. Comp. 30 (1976) 772-795. | MR | Zbl
[3] and , Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2 (1995) 205-217. | MR | Zbl
[4] , and , Convergence rates for Tikhonov regularization of nonlinear ill-posed problems. Inverse Problems 5 (1989) 523-540. | MR | Zbl
[5] , Geometry Variations, Level Set and Phase-field Methods for Perimeter Regularized Geometric Inverse Problems. Ph.D. Thesis, Johannes Keppler Universität Linz, Austria (2006).
[6] , A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Problems 13 (1997) 79-95. | MR | Zbl
[7] , and , A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math. 72 (1995) 21-37. | MR | Zbl
[8] and , On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34 (1997) 1911-1925. | MR | Zbl
[9] and , Explicit exponential Runge-Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43 (2005) 1069-1090. | MR | Zbl
[10] , and , A convergence analysis of the exponential Euler iteration for nonlinear ill-posed problems. Inv. Prob. 25 (2009) 075009. | MR
[11] , and , Exponential Rosenbrock-type methods. SIAM J. Numer. Anal. 47 (2009) 786-803. | MR
[12] and , Convergence analysis of an inexact iteratively regularized Gauss-Newton method under general source conditions. Journal of Inverse and Ill-Posed Problems 15 (2007) 19-35. | MR | Zbl
[13] , Asymptotische Regularisierung schlecht gestellter Probleme mittels steifer Integratoren. Diplomarbeit, Universität Karlsruhe, Germany (2004).
[14] , and , Iterative Regularization Methods for Nonlinear Ill-Posed Problems. De Gruyter, Berlin, New York (2008). | MR | Zbl
[15] , Tikhonov regularization for non-linear ill-posed problems: optimal convergence rates and finite-dimensional approximation. Inverse Problems 5 (1989) 541-557. | MR | Zbl
[16] , On the regularization of nonlinear ill-posed problems via inexact Newton iterations. Inverse Problems 15 (1999) 309-327. | MR | Zbl
[17] , On convergence rates of inexact Newton regularizations. Numer. Math. 88 (2001) 347-365. | MR | Zbl
[18] , Inexact Newton regularization using conjugate gradients as inner iteration. SIAM J. Numer. Anal. 43 (2005) 604-622. | MR | Zbl
[19] , Runge-Kutta integrators yield optimal regularization schemes. Inverse Problems 21 (2005) 453-471. | MR | Zbl
[20] and , Well-posedness and convergence of some regularization methods for nonlinear ill-posed problems. Inverse Problems 5 (1989) 227-238. | MR | Zbl
[21] , Representation and computation of the pseudoinverse. Proc. Amer. Math. Soc. 18 (1967) 584-586. | MR | Zbl
[22] , On the asymptotical regularization of nonlinear ill-posed problems. Inverse Problems 10 (1994) 1405-1418. | MR | Zbl
[23] and , Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comp. 27 (2006) 1438-1457. | MR | Zbl
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