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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

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

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Martin Burger Total Variation 3 Cetraro, September 2008 Optimality Condition

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

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

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

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Martin Burger Total Variation 7 Cetraro, September 2008 Computing Subdifferentials

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

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

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

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

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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 ???

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

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

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Martin Burger Total Variation 15 Cetraro, September 2008 TV Duality

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Martin Burger Total Variation 16 Cetraro, September 2008 TV Duality

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Martin Burger Total Variation 17 Cetraro, September 2008 TV Duality

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

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

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

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

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

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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

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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

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

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

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

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

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

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

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

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

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

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

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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 ?

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