Regularization of nonlinear ill-posed problems by exponential integrators
ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 4, pp. 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 : 10.1051/m2an/2009021
Classification : 65J20, 65N21, 65L05
Mots clés : nonlinear ill-posed problems, asymptotic regularization, exponential integrators, variable step sizes, convergence, optimal convergence rates
Hochbruck, Marlis  ; Hönig, Michael  ; Ostermann, Alexander 1

1 Institut für Mathematik, Universität Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria.
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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] C. Böckmann and P. Pornsawad, Iterative Runge-Kutta-type methods for nonlinear ill-posed problems. Inverse Problems 24 (2008) 025002. | MR | Zbl

[2] 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 | Zbl

[3] 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 | Zbl

[4] 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 | Zbl

[5] 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).

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

[7] 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 | Zbl

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

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

[10] 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

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

[12] 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 | Zbl

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

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

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

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

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

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

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

[20] 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 | Zbl

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

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

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

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