TY - JOUR

T1 - Convergence of discrete schemes for the Perona-Malik equation

AU - Paolini, Maurizio

AU - Bellettini, Giovanni

AU - Novaga, Matteo

AU - Tornese, C.

PY - 2008

Y1 - 2008

N2 - We prove the convergence, up to a subsequence, of the spatial semidiscrete scheme for the one-dimensional Perona-Malik equation $u_t = (\phi'(u_x))_x$, $\phi(p) := \log(1+p^2)/2$, when the initial datum $\bar u$ is 1-Lipschitz out of a finite number of jump points, and we characterize the problem satisfied by the limit solution.
In the more difficult case when $\bar u$ has a whole interval where $\phi''(\bar u_x)$ is negative, we construct a solution by a careful inspection of the behaviour of the approximating solutions in a space-time neighbourhood of the jump points.
The limit solution u we obtain is the same as the one obtained by replacing $\phi(\cdot)$ with the truncated function $\min(\phi(\cdot),1)$, and it turns out that $u$ solves a free boundary problem.
The free boundary consists of the points dividing the region where $|u_x| > 1$ from the region where $|u_x| \leq 1$.
Finally, we consider the full space-time discretization (implicit in time) of the Perona-Malik equation, and we show that, if the time step is small with respect to the spatial grid h, then the limit is the same as the one obtained with the spatial semidiscrete scheme. On the other hand, if the time step is large with respect to h, then the limit solution equals $\bar u$, i.e., the standing solution of the convexified problem.

AB - We prove the convergence, up to a subsequence, of the spatial semidiscrete scheme for the one-dimensional Perona-Malik equation $u_t = (\phi'(u_x))_x$, $\phi(p) := \log(1+p^2)/2$, when the initial datum $\bar u$ is 1-Lipschitz out of a finite number of jump points, and we characterize the problem satisfied by the limit solution.
In the more difficult case when $\bar u$ has a whole interval where $\phi''(\bar u_x)$ is negative, we construct a solution by a careful inspection of the behaviour of the approximating solutions in a space-time neighbourhood of the jump points.
The limit solution u we obtain is the same as the one obtained by replacing $\phi(\cdot)$ with the truncated function $\min(\phi(\cdot),1)$, and it turns out that $u$ solves a free boundary problem.
The free boundary consists of the points dividing the region where $|u_x| > 1$ from the region where $|u_x| \leq 1$.
Finally, we consider the full space-time discretization (implicit in time) of the Perona-Malik equation, and we show that, if the time step is small with respect to the spatial grid h, then the limit is the same as the one obtained with the spatial semidiscrete scheme. On the other hand, if the time step is large with respect to h, then the limit solution equals $\bar u$, i.e., the standing solution of the convexified problem.

KW - differential equations

KW - finite difference scheme

KW - illposed problems

KW - differential equations

KW - finite difference scheme

KW - illposed problems

UR - http://hdl.handle.net/10807/19289

M3 - Article

SP - 892

EP - 924

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

ER -