# 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} }

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…

## 28 Citations

Regularization of nonlinear ill-posed problems by exponential integrators

- Mathematics
- 2009

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…

Iterative Runge–Kutta-type methods for nonlinear ill-posed problems

- Mathematics
- 2008

We present a regularization method for solving nonlinear ill-posed problems by applying the family of Runge–Kutta methods to an initial value problem, in particular, to the asymptotical…

Second order asymptotical regularization methods for inverse problems in partial differential equations

- MathematicsJ. Comput. Appl. Math.
- 2020

ASYMPTOTICAL REGULARIZATION METHODS FOR INVERSE PROBLEMS IN PARTIAL DIFFERENTIAL EQUATIONS

- Mathematics
- 2018

We develop Second Order Asymptotical Regularization (SOAR) methods for solving inverse source problems in elliptic partial differential equations with both Dirichlet and Neumann boundary data. We…

Stochastic asymptotical regularization for linear inverse problems

- Mathematics
- 2022

We introduce Stochastic Asymptotical Regularization (SAR) methods for the uncertainty quantification of the stable approximate solution of ill-posed linear-operator equations, which are deterministic…

NOVEL METHODS FOR SOLVING SEVERELY ILL-POSED LINEAR EQUATIONS SYSTEM

- Mathematics
- 2009

We treat an ill-posed system of linear equations by trans forming it into a linear system of stiff ordinary differential equations (SODEs), adding a differential term on the left-hand side. In order…

On fractional asymptotical regularization of linear ill-posed problems in hilbert spaces

- MathematicsFractional Calculus and Applied Analysis
- 2019

Abstract In this paper, we study a fractional-order variant of the asymptotical regularization method, called Fractional Asymptotical Regularization (FAR), for solving linear ill-posed operator…

N A ] 1 4 Ju l 2 01 9 ON FRACTIONAL ASYMPTOTICAL REGULARIZATION OF LINEAR ILL-POSED PROBLEMS IN HILBERT SPACES

- Mathematics
- 2019

In this paper, we study a fractional-order variant of the asymptotical regularization method, called Fractional Asymptotical Regularization (FAR), for solving linear ill-posed operator equations in a…

THE LEVENBERG–MARQUARDT REGULARIZATION FOR THE BACKWARD HEAT EQUATION WITH FRACTIONAL DERIVATIVE∗

- Mathematics
- 2022

. The backward heat problem with time-fractional derivative in Caputo’s sense is studied. The inverse problem is severely ill-posed in the case when the fractional order is close to unity. A…

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