1. Perona-Malik equation error estimates for numerical finite volume scheme Angela Handlovičová Zuzana Krivá Department of Mathematics Slovak University of Technology, Bratislava, Slovakia

2. Initial noisy image Smooth initial noisy image and preserve edges

3. Perona-Malik Equation

4. 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

5. 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)

6. 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

7. Explicit scheme

8. Semi - implicit scheme

9. Regularity of the solution

10. Error estimates For explicit scheme For semi -implicit scheme

16. Thank you Angela Handlovičová Zuzana Krivá Department of Mathematics Slovak University of Technology, Bratislava, Slovakia