1. Martin Burger Total Variation 1 Cetraro, September 2008 Variational Methods and their Analysis Questions: - Existence - Uniqueness - Optimality conditions for solutions (-> numerical methods) - Structural properties of solutions - Asymptotic behaviour with respect to

2. Martin Burger Total Variation 2 Cetraro, September 2008 Basics of Convex Analysis Subgradients:

3. Martin Burger Total Variation 3 Cetraro, September 2008 Optimality Condition

4. Martin Burger Total Variation 4 Cetraro, September 2008 Computing Subdifferentials Differentiable functionals

5. Martin Burger Total Variation 5 Cetraro, September 2008 Computing Subdifferentials Sum of Functionals

6. Martin Burger Total Variation 6 Cetraro, September 2008 Computing Subdifferentials Nondifferentiable functionals

7. Martin Burger Total Variation 7 Cetraro, September 2008 Computing Subdifferentials

8. Martin Burger Total Variation 8 Cetraro, September 2008 Computing Subdifferentials Total Variation

9. Martin Burger Total Variation 9 Cetraro, September 2008 Computing Subdifferentials Total Variation

10. Martin Burger Total Variation 10 Cetraro, September 2008 Computing Subdifferentials Total variation

11. Martin Burger Total Variation 11 Cetraro, September 2008 Optimality Condition Subdifferential of

12. Martin Burger Total Variation 12 Cetraro, September 2008 Optimality condition System of equations / variational inequalities for u and g Basis of primal-dual and dual formulation Can we exchange inf and sup ???

13. Martin Burger Total Variation 13 Cetraro, September 2008 Duality Restrict our attention to ROF-Model

14. Martin Burger Total Variation 14 Cetraro, September 2008 Duality Cf. Book by Ekeland-Temam for general results

15. Martin Burger Total Variation 15 Cetraro, September 2008 TV Duality

16. Martin Burger Total Variation 16 Cetraro, September 2008 TV Duality

17. Martin Burger Total Variation 17 Cetraro, September 2008 TV Duality

18. Martin Burger Total Variation 18 Cetraro, September 2008 Dual Variational Inequality From dual problem it is easy to see

19. Martin Burger Total Variation 19 Cetraro, September 2008 Structural Properties of Minimizers Staircasing

20. Martin Burger Total Variation 20 Cetraro, September 2008 Meyer Example An exact solution

21. Martin Burger Total Variation 21 Cetraro, September 2008 Meyer Example An exact solution

22. Martin Burger Total Variation 22 Cetraro, September 2008 Meyer Example Need to find subgradient proportional to f

23. Martin Burger Total Variation 23 Cetraro, September 2008 Meyer Example Do 1D integration for g in the radial variable and choose the parameter such that

24. Martin Burger Total Variation 24 Cetraro, September 2008 Meyer Example Do 1D integration for g in the radial variable and choose the parameter such that

25. Martin Burger Total Variation 25 Cetraro, September 2008 Meyer Example Determine

26. Martin Burger Total Variation 26 Cetraro, September 2008 Asymptotic Behaviour of Minimizers Oversmoothing Leading order

27. Martin Burger Total Variation 27 Cetraro, September 2008 Asymptotic Behaviour of Minimizers Oversmoothing

28. Martin Burger Total Variation 28 Cetraro, September 2008 Asymptotic Behaviour of Minimizers Close to data

29. Martin Burger Total Variation 29 Cetraro, September 2008 Asymptotic Behaviour of Minimizers ROF Formally Linear analogue No strong convergence !!

30. Martin Burger Total Variation 30 Cetraro, September 2008 Asymptotic Behaviour of Minimizers Data Regularity Exact data Noisy data

31. Martin Burger Total Variation 31 Cetraro, September 2008 Asymptotic Behaviour of Minimizers Energy estimate

32. Martin Burger Total Variation 32 Cetraro, September 2008 Asymptotic Behaviour of Minimizers R-minimal solution in case of nullspace Multiple Solutions of

33. Martin Burger Total Variation 33 Cetraro, September 2008 Asymptotic Behaviour of Minimizers Noisy data

34. Martin Burger Total Variation 34 Cetraro, September 2008 Asymptotic Behaviour of Minimizers Coupled limit, regularization parameter depends on noise level

35. Martin Burger Total Variation 35 Cetraro, September 2008 Asymptotic Behaviour of Minimizers Can we get quantitative estimates of the error ? In general only weak* convergence – what is the right error measure ?