TY - JOUR
T1 - Convergence for long-times of a semidiscrete Perona-Malik equation in one dimension
AU - Paolini, Maurizio
AU - Bellettini, Giovanni
AU - Novaga, Matteo
PY - 2011
Y1 - 2011
N2 - We prove that the semidiscrete schemes of a Perona-Malik type equation converge, in a long time scale, to a suitable system of ordinary differential equations defined on piecewise constant functions. The proof is based on a formal asymptotic expansion argument, and on a careful construction of discrete sub and supersolutions. Despite the equation has a region where it is backward parabolic, we prove a discrete comparison principle, which is the key tool for the convergence result.
AB - We prove that the semidiscrete schemes of a Perona-Malik type equation converge, in a long time scale, to a suitable system of ordinary differential equations defined on piecewise constant functions. The proof is based on a formal asymptotic expansion argument, and on a careful construction of discrete sub and supersolutions. Despite the equation has a region where it is backward parabolic, we prove a discrete comparison principle, which is the key tool for the convergence result.
KW - Perona-Malik
KW - ill-posed problems
KW - image segmentation
KW - partial differential equations
KW - Perona-Malik
KW - ill-posed problems
KW - image segmentation
KW - partial differential equations
UR - https://publicatt.unicatt.it/handle/10807/8935
UR - https://www.scopus.com/inward/citedby.uri?partnerID=HzOxMe3b&scp=79952339633&origin=inward
UR - https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=79952339633&origin=inward
U2 - 10.1142/S0218202511005040
DO - 10.1142/S0218202511005040
M3 - Article
SN - 0218-2025
VL - 21
SP - 241
EP - 265
JO - Mathematical Models and Methods in Applied Sciences
JF - Mathematical Models and Methods in Applied Sciences
IS - 2
ER -