Regularization of nonlinear ill-posed problems by exponential integrators

Marlis Hochbruck; Michael Hönig; Alexander Ostermann

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

  • Volume: 43, Issue: 4, page 709-720
  • ISSN: 0764-583X

Abstract

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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 Problems10 (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 Problems10 (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.

How to cite

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Hochbruck, Marlis, Hönig, Michael, and Ostermann, Alexander. "Regularization of nonlinear ill-posed problems by exponential integrators." ESAIM: Mathematical Modelling and Numerical Analysis 43.4 (2009): 709-720. <http://eudml.org/doc/250593>.

@article{Hochbruck2009,
abstract = { 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 Problems10 (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 Problems10 (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. },
author = {Hochbruck, Marlis, Hönig, Michael, Ostermann, Alexander},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nonlinear ill-posed problems; asymptotic regularization; exponential integrators; variable step sizes; convergence; optimal convergence rates.; nonlinear ill-posed problems; variable step sizes; optimal convergence rates},
language = {eng},
month = {7},
number = {4},
pages = {709-720},
publisher = {EDP Sciences},
title = {Regularization of nonlinear ill-posed problems by exponential integrators},
url = {http://eudml.org/doc/250593},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Hochbruck, Marlis
AU - Hönig, Michael
AU - Ostermann, Alexander
TI - Regularization of nonlinear ill-posed problems by exponential integrators
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2009/7//
PB - EDP Sciences
VL - 43
IS - 4
SP - 709
EP - 720
AB - 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 Problems10 (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 Problems10 (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.
LA - eng
KW - Nonlinear ill-posed problems; asymptotic regularization; exponential integrators; variable step sizes; convergence; optimal convergence rates.; nonlinear ill-posed problems; variable step sizes; optimal convergence rates
UR - http://eudml.org/doc/250593
ER -

References

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