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

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

Martin Burger Total Variation 3 Cetraro, September 2008 Optimality Condition

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

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

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

Martin Burger Total Variation 7 Cetraro, September 2008 Computing Subdifferentials

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

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

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

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

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 ???

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

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

Martin Burger Total Variation 15 Cetraro, September 2008 TV Duality

Martin Burger Total Variation 16 Cetraro, September 2008 TV Duality

Martin Burger Total Variation 17 Cetraro, September 2008 TV Duality

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 ?