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Regularization of nonlinear ill-posed problems by exponential integrators. (English) Zbl 1167.65369

Summary: 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 Probl. 10, No. 6, 1405–1418 (1994; Zbl 0828.65055)], 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, loc. cit.], and of our discrete version [the authors, Inverse Probl. 25, No. 7, Article ID 075009 (2009; Zbl 1184.65063)]. 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.

MSC:

65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
65L05 Numerical methods for initial value problems involving ordinary differential equations
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References:

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