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Perona-Malik equation error estimates for numerical finite volume scheme Angela 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
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Perona-Malik equation - properties of data g(s) is Lipschitz continuous decreasing function, g(0)=1, 0< g(s) 0 for s is a smoothing kernel with compact support u 0 (x) is initial condition with and for 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 x p N(p) …neighbors of p e pq …common edge between volumes p and q of measure m(e pq ) d pq :=|x p -x q |, T pq :=m(e pq )/d pq u p n …numerical solution
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Explicit scheme
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Semi - implicit scheme
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Regularity of the solution
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Error estimates For explicit scheme For semi -implicit scheme
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Thank you Angela Handlovičová Zuzana Krivá Department of Mathematics Slovak University of Technology, Bratislava, Slovakia
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