##
**A multigrid method for a fourth-order diffusion equation with application to image processing.**
*(English)*
Zbl 1096.65105

This paper deals with the initial–boundary value problem for the following biharmonic diffusion equation
\[
\begin{aligned} & \frac{\partial u_{i}(x,y,t)}{\partial t} + \epsilon(t)\Delta^{2}u_{i}(x,y,t) = d(u(x,y,t))\quad on\;\Omega\times (0, T) \end{aligned}
\]
together with some special boundary conditions. These boundary conditions are completely different from the conditions used by B. Fischer and J. Modersitzki [J. Math. Imaging Vis. 18, No.-1, 81–85 (2003; Zbl 1034.68110)], where so-called curvature approach is used, no coupling of the boundary conditions is presented and thus, the biharmonic system can be fully decoupled into two Poisson equations.

In this paper, the curvature approach is not applicable due to the special boundary conditions. Thus, the author introduces a new fast multigrid scheme for solving the biharmonic diffusion problem using a semi-implicit time discretization. The resulting stationary problem is given by a biharmonic operator with higher-order boundary conditions. In the case of spatial discretization, the author uses the finite difference method. In the last section of the paper, the author presents a real image registration example in order to demonstrate the principle and the reliability of used approach.

In this paper, the curvature approach is not applicable due to the special boundary conditions. Thus, the author introduces a new fast multigrid scheme for solving the biharmonic diffusion problem using a semi-implicit time discretization. The resulting stationary problem is given by a biharmonic operator with higher-order boundary conditions. In the case of spatial discretization, the author uses the finite difference method. In the last section of the paper, the author presents a real image registration example in order to demonstrate the principle and the reliability of used approach.

Reviewer: Petr Necesal (Plzen)

### MSC:

65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35K55 | Nonlinear parabolic equations |

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |