Perona-Malik equation error estimates for numerical finite volume scheme
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Perona-Malik equation error estimates for
numerical finite volume schemeAngela Handlovičová
Zuzana Krivá
Department of Mathematics Slovak University of Technology, Bratislava, Slovakia
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Initial noisy image
Smooth initial noisy image and preserve edges
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Perona-Malik Equation
)()0,( 0 xuxu
,0)),(|)),(*(|(),(
xtuxtuGg
txtu
Itxxtu
,,0),(
Itx ,
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Perona-Malik equation - properties of datag(s) is Lipschitz continuous decreasing function,g(0)=1, 0< g(s) 0 for s
)( dCG
is a smoothing kernel with compact support
u0(x) is initial condition
Kwith
d
dxxG 1)( andxxG )( for 0
Dirac function at point x
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Results obtained for finite volume numerical scheme• Existence of weak solutions and regularity-Catté,
Lions, Morel, Coll (1992)
• Convergence of semi-implicit finite volume numerical scheme – Mikula Ramarossy (2001)
• Convergence of explicit finite volume scheme Krivá (2003)
• Error estimates for semi-implicit FV scheme Handlovičová, Krivá (2005)
• Error estimates for explicit scheme H,K (2005)
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Finite volume
p…volume of measure m(p), with representative point xpN(p) …neighbors of p
epq…common edge between volumes p and q of measure m(epq)
dpq:=|xp-xq|, Tpq :=m(epq)/d pq
upn …numerical solution
maxNT
k
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• Explicit scheme
)()()( 11
)(
1,1
np
nqpq
pNq
npq
np
np uuTgkpmuu
p
p dxxupm
u )()(
10
0
|))(*(|: 1,1pq
nnpq xuGgg
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• Semi - implicit scheme
)()()()(
1,1 np
nqpq
pNq
npq
np
np uuTgkpmuu
p
p dxxupm
u )()(
10
0
|))(*(|: 1,1pq
nnpq xuGgg
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Regularity of the solution
))(,()(
)),(,(
)),(,(
)),(,(
11
11
2,1
2,22
LILu
LILu
WILu
WILu
t
tt
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Error estimates
2Chk
max
0
21,1 |),(),(|
N
n I
nkhn
n
Chxtuxtu
max
0
211
))(
1()(
N
n I Ie e
pqpqpq
np
nq
pqpq
n pq pq
Chnuemd
uudem
For explicit scheme
For semi -implicit scheme Chk
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Thank you
Angela HandlovičováZuzana Krivá
Department of Mathematics Slovak University of Technology, Bratislava, Slovakia