• DOI:10.1088/0266-5611/21/2/003
• Corpus ID: 120811890

Runge–Kutta integrators yield optimal regularization schemes

@article{Rieder2005RungeKuttaIY,
  title={Runge–Kutta integrators yield optimal regularization schemes},
  author={Andreas Rieder},
  journal={Inverse Problems},
  year={2005},
  volume={21},
  pages={453 - 471}
}

• A. Rieder
• Published 1 April 2005
• Mathematics
• Inverse Problems

Asymptotic regularization (also called Showalter's method) is a theoretically appealing regularization scheme for an ill-posed problem Tx = y, T acting between Hilbert spaces. Here, Tx = y is stably solved by evaluating the solution of the evolution equation u′(t) = T*(y − Tu(t)), u(0) = 0, at a properly chosen finite time. For a numerical realization, however, we have to apply an integrator to the ODE. Fortunately all properties of asymptotic regularization carry over to its numerical realization: Runge–Kutta integrators yield optimal regularization schemes when stopped by the discrepancy principle. In this way a common analysis is obtained for such different regularization schemes as, for instance, the Landweber iteration and the iterated Tikhonov–Phillips method which are generated by the explicit and implicit Euler integrators, respectively. Furthermore it turns out that inconsistent Runge–Kutta schemes, which are useless for solving ODEs, lead to optimal regularizations as well which can even be more efficient than regularizations from consistent Runge–Kutta integrators. The presented computational examples illustrate the theoretical findings and demonstrate that implicit schemes outperform the explicit ones.