martin burger total variation 1 cetraro, september 2008 variational methods and their analysis...
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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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Cetraro, September 2008Martin Burger
Basics of Convex Analysis Subgradients:
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Cetraro, September 2008Martin Burger
Optimality Condition
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Cetraro, September 2008Martin Burger
Computing Subdifferentials Differentiable functionals
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Cetraro, September 2008Martin Burger
Computing Subdifferentials Sum of Functionals
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Cetraro, September 2008Martin Burger
Computing Subdifferentials Nondifferentiable functionals
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Cetraro, September 2008Martin Burger
Computing Subdifferentials
Total Variation 8
Cetraro, September 2008Martin Burger
Computing Subdifferentials Total Variation
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Cetraro, September 2008Martin Burger
Computing Subdifferentials Total Variation
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Cetraro, September 2008Martin Burger
Computing Subdifferentials Total variation
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Cetraro, September 2008Martin Burger
Optimality Condition Subdifferential of
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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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Cetraro, September 2008Martin Burger
Duality Restrict our attention to ROF-Model
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Cetraro, September 2008Martin Burger
Duality Cf. Book by Ekeland-Temam for general results
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Cetraro, September 2008Martin Burger
TV Duality
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Cetraro, September 2008Martin Burger
TV Duality
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Cetraro, September 2008Martin Burger
TV Duality
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Cetraro, September 2008Martin Burger
Dual Variational Inequality From dual problem it is easy to see
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Cetraro, September 2008Martin Burger
Structural Properties of Minimizers Staircasing
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Cetraro, September 2008Martin Burger
Meyer Example An exact solution
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Cetraro, September 2008Martin Burger
Meyer Example An exact solution
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Cetraro, September 2008Martin Burger
Meyer Example Need to find subgradient proportional to f
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Cetraro, September 2008Martin Burger
Meyer Example Do 1D integration for g in the radial variable and choose the parameter such that
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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Cetraro, September 2008Martin Burger
Meyer Example Determine
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Cetraro, September 2008Martin Burger
Asymptotic Behaviour of Minimizers Oversmoothing
Leading order
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Cetraro, September 2008Martin Burger
Asymptotic Behaviour of Minimizers Oversmoothing
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Cetraro, September 2008Martin Burger
Asymptotic Behaviour of Minimizers Close to data
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Cetraro, September 2008Martin Burger
Asymptotic Behaviour of MinimizersROF
Formally
Linear analogue
No strong convergence !!
Total Variation 30
Cetraro, September 2008Martin Burger
Asymptotic Behaviour of MinimizersData Regularity
Exact data
Noisy data
Total Variation 31
Cetraro, September 2008Martin Burger
Asymptotic Behaviour of MinimizersEnergy estimate
Total Variation 32
Cetraro, September 2008Martin Burger
Asymptotic Behaviour of MinimizersR-minimal solution in case of nullspace
Multiple Solutions of
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Cetraro, September 2008Martin Burger
Asymptotic Behaviour of MinimizersNoisy data
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Cetraro, September 2008Martin Burger
Asymptotic Behaviour of MinimizersCoupled limit, regularization parameter depends on noise level
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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 ?