Regularization of nonlinear ill-posed problems by exponential integrators
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 4, p. 709-720

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.

DOI : https://doi.org/10.1051/m2an/2009021
Classification:  65J20,  65N21,  65L05
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {43},
number = {4},
year = {2009},
pages = {709-720},
doi = {10.1051/m2an/2009021},
zbl = {1167.65369},
mrnumber = {2542873},
language = {en},
url = {http://www.numdam.org/item/M2AN_2009__43_4_709_0}
}

Hochbruck, Marlis; Hönig, Michael; Ostermann, Alexander. Regularization of nonlinear ill-posed problems by exponential integrators. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 4, pp. 709-720. doi : 10.1051/m2an/2009021. http://www.numdam.org/item/M2AN_2009__43_4_709_0/

 C. Böckmann and P. Pornsawad, Iterative Runge-Kutta-type methods for nonlinear ill-posed problems. Inverse Problems 24 (2008) 025002. | MR 2408539 | Zbl 1151.35097

 J. Daniel, W.B. Gragg, L. Kaufman and G.W. Stewart, Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization. Math. Comp. 30 (1976) 772-795. | MR 431641 | Zbl 0345.65021

 V.L. Druskin and L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2 (1995) 205-217. | MR 1332710 | Zbl 0831.65042

 H.W. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems. Inverse Problems 5 (1989) 523-540. | MR 1009037 | Zbl 0695.65037

 B. Hackl, Geometry Variations, Level Set and Phase-field Methods for Perimeter Regularized Geometric Inverse Problems. Ph.D. Thesis, Johannes Keppler Universität Linz, Austria (2006).

 M. Hanke, A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Problems 13 (1997) 79-95. | MR 1435869 | Zbl 0873.65057

 M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math. 72 (1995) 21-37. | MR 1359706 | Zbl 0840.65049

 M. Hochbruck and C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34 (1997) 1911-1925. | MR 1472203 | Zbl 0888.65032

 M. Hochbruck and A. Ostermann, Explicit exponential Runge-Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43 (2005) 1069-1090. | MR 2177796 | Zbl 1093.65052

 M. Hochbruck, M. Hönig and A. Ostermann, A convergence analysis of the exponential Euler iteration for nonlinear ill-posed problems. Inv. Prob. 25 (2009) 075009. | MR 2519861 | Zbl pre05588142

 M. Hochbruck, A. Ostermann and J. Schweitzer, Exponential Rosenbrock-type methods. SIAM J. Numer. Anal. 47 (2009) 786-803. | MR 2475962

 T. Hohage and S. Langer, 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 2337589 | Zbl 1129.65043

 M. Hönig, Asymptotische Regularisierung schlecht gestellter Probleme mittels steifer Integratoren. Diplomarbeit, Universität Karlsruhe, Germany (2004).

 B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems. De Gruyter, Berlin, New York (2008). | MR 2459012 | Zbl 1145.65037

 A. Neubauer, Tikhonov regularization for non-linear ill-posed problems: optimal convergence rates and finite-dimensional approximation. Inverse Problems 5 (1989) 541-557. | MR 1009038 | Zbl 0695.65038

 A. Rieder, On the regularization of nonlinear ill-posed problems via inexact Newton iterations. Inverse Problems 15 (1999) 309-327. | MR 1675352 | Zbl 0969.65049

 A. Rieder, On convergence rates of inexact Newton regularizations. Numer. Math. 88 (2001) 347-365. | MR 1826857 | Zbl 0990.65061

 A. Rieder, Inexact Newton regularization using conjugate gradients as inner iteration. SIAM J. Numer. Anal. 43 (2005) 604-622. | MR 2177882 | Zbl 1092.65047

 A. Rieder, Runge-Kutta integrators yield optimal regularization schemes. Inverse Problems 21 (2005) 453-471. | MR 2146271 | Zbl 1075.65078

 T.I. Seidman and C.R. Vogel, Well-posedness and convergence of some regularization methods for nonlinear ill-posed problems. Inverse Problems 5 (1989) 227-238. | MR 991919 | Zbl 0691.35090

 D. Showalter, Representation and computation of the pseudoinverse. Proc. Amer. Math. Soc. 18 (1967) 584-586. | MR 212594 | Zbl 0148.38205

 U. Tautenhahn, On the asymptotical regularization of nonlinear ill-posed problems. Inverse Problems 10 (1994) 1405-1418. | MR 1306812 | Zbl 0828.65055

 J. Van Den Eshof and M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comp. 27 (2006) 1438-1457. | MR 2199756 | Zbl 1105.65051