Martin Burger Total Variation 1 Cetraro, September 2008 Variational Methods and their Analysis...
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Transcript of Martin Burger Total Variation 1 Cetraro, September 2008 Variational Methods and their Analysis...
![Page 1: Martin Burger Total Variation 1 Cetraro, September 2008 Variational Methods and their Analysis Questions: - Existence - Uniqueness - Optimality conditions.](https://reader038.fdocuments.us/reader038/viewer/2022110320/56649caf5503460f94972a86/html5/thumbnails/1.jpg)
Total Variation 1
Cetraro, September 2008Martin Burger
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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Total Variation 2
Cetraro, September 2008Martin Burger
Basics of Convex Analysis Subgradients:
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Total Variation 3
Cetraro, September 2008Martin Burger
Optimality Condition
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Total Variation 4
Cetraro, September 2008Martin Burger
Computing Subdifferentials Differentiable functionals
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Total Variation 5
Cetraro, September 2008Martin Burger
Computing Subdifferentials Sum of Functionals
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Total Variation 6
Cetraro, September 2008Martin Burger
Computing Subdifferentials Nondifferentiable functionals
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Total Variation 7
Cetraro, September 2008Martin Burger
Computing Subdifferentials
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Total Variation 8
Cetraro, September 2008Martin Burger
Computing Subdifferentials Total Variation
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Total Variation 9
Cetraro, September 2008Martin Burger
Computing Subdifferentials Total Variation
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Total Variation 10
Cetraro, September 2008Martin Burger
Computing Subdifferentials Total variation
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Total Variation 11
Cetraro, September 2008Martin Burger
Optimality Condition Subdifferential of
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Total Variation 12
Cetraro, September 2008Martin Burger
Optimality condition System of equations / variational inequalities for u and gBasis of primal-dual and dual formulation
Can we exchange inf and sup ???
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Total Variation 13
Cetraro, September 2008Martin Burger
Duality Restrict our attention to ROF-Model
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Total Variation 14
Cetraro, September 2008Martin Burger
Duality Cf. Book by Ekeland-Temam for general results
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Total Variation 15
Cetraro, September 2008Martin Burger
TV Duality
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Total Variation 16
Cetraro, September 2008Martin Burger
TV Duality
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Total Variation 17
Cetraro, September 2008Martin Burger
TV Duality
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Total Variation 18
Cetraro, September 2008Martin Burger
Dual Variational Inequality From dual problem it is easy to see
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Total Variation 19
Cetraro, September 2008Martin Burger
Structural Properties of Minimizers Staircasing
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Total Variation 20
Cetraro, September 2008Martin Burger
Meyer Example An exact solution
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Total Variation 21
Cetraro, September 2008Martin Burger
Meyer Example An exact solution
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Total Variation 22
Cetraro, September 2008Martin Burger
Meyer Example Need to find subgradient proportional to f
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Total Variation 23
Cetraro, September 2008Martin Burger
Meyer Example Do 1D integration for g in the radial variable and choose the parameter such that
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Total Variation 24
Cetraro, September 2008Martin Burger
Meyer Example Do 1D integration for g in the radial variable and choose the parameter such that
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Total Variation 25
Cetraro, September 2008Martin Burger
Meyer Example Determine
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Total Variation 26
Cetraro, September 2008Martin Burger
Asymptotic Behaviour of Minimizers Oversmoothing
Leading order
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Total Variation 27
Cetraro, September 2008Martin Burger
Asymptotic Behaviour of Minimizers Oversmoothing
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Total Variation 28
Cetraro, September 2008Martin Burger
Asymptotic Behaviour of Minimizers Close to data
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Total Variation 29
Cetraro, September 2008Martin Burger
Asymptotic Behaviour of MinimizersROF
Formally
Linear analogue
No strong convergence !!
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Total Variation 30
Cetraro, September 2008Martin Burger
Asymptotic Behaviour of MinimizersData Regularity
Exact data
Noisy data
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Total Variation 31
Cetraro, September 2008Martin Burger
Asymptotic Behaviour of MinimizersEnergy estimate
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Total Variation 32
Cetraro, September 2008Martin Burger
Asymptotic Behaviour of MinimizersR-minimal solution in case of nullspace
Multiple Solutions of
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Total Variation 33
Cetraro, September 2008Martin Burger
Asymptotic Behaviour of MinimizersNoisy data
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Total Variation 34
Cetraro, September 2008Martin Burger
Asymptotic Behaviour of MinimizersCoupled limit, regularization parameter depends on noise level
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Total Variation 35
Cetraro, September 2008Martin Burger
Asymptotic Behaviour of MinimizersCan we get quantitative estimates of the error ?
In general only weak* convergence – what is the right error measure ?