A function space framework for structural total variation regularization with applications in inverse problems

Hintermüller, Michael; Höller, Martin; Papafitsoros, Kostas

doi:10.1088/1361-6420/aab586

Cited by 27 publications

(31 citation statements)

References 61 publications

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“…Classical regularisers such as TV, Huber-TV and TGV of any order, satisfy the above properties. Note that this is also true for a large class of structural TV-type functionals that are commonly used in inverse problems, see [31].…”

Section: Asymptotic Behaviourmentioning

confidence: 86%

Analysis and automatic parameter selection of a variational model for mixed Gaussian and salt-and-pepper noise removal

Calatroni¹,

Papafitsoros²

2019

Inverse Problems

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We analyse a variational regularisation problem for mixed noise removal that was recently proposed in [14]. The data discrepancy term of the model combines L 1 and L 2 terms in an infimal convolution fashion and it is appropriate for the joint removal of Gaussian and Salt & Pepper noise. In this work we perform a finer analysis of the model which emphasises on the balancing effect of the two parameters appearing in the discrepancy term. Namely, we study the asymptotic behaviour of the model for large and small values of these parameters and we compare it to the corresponding variational models with L 1 and L 2 data fidelity. Furthermore, we compute exact solutions for simple data functions taking the total variation as regulariser. Using these theoretical results, we then analytically study a bilevel optimisation strategy for automatically selecting the parameters of the model by means of a training set. Finally, we report some numerical results on the selection of the optimal noise model via such strategy which confirm the validity of our analysis and the use of popular data models in the case of "blind" model selection.

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Section: Asymptotic Behaviourmentioning

confidence: 86%

Analysis and automatic parameter selection of a variational model for mixed Gaussian and salt-and-pepper noise removal

Calatroni¹,

Papafitsoros²

2019

Inverse Problems

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“…was first shown by Strang 38 for continuous and extended to piece-wise continuous by Hintermüller et al 62 This minimum cut-maximum flow duality immediately gives rise to simple bounds on the effective crack resistance. Indeed, inserting = into the primal problem, and v = (min ) into the dual problem provides the two-sided bound…”

Section: A Fully Periodic Cell Formula and The Maximum Flow Formulationmentioning

confidence: 86%

“…Actually, Strang only considers the case of continuous γ. However, Hintermüller et al provide a corresponding result for crack resistances with jumps using a Fenchel predualization technique.…”

Section: Homogenization Of Brittle Fracture and Cell Formulaementioning

confidence: 99%

“…For fixed

\overline{ξ} \in R^{d}

, introduce the (convex) set

𝒦_{\overline{ξ}}

of kinematic constraints

𝒦_{\overline{ξ}} = \{ξ : Q \to R^{d} periodic | ξ = \overline{ξ} + \nabla ϕ for some periodic ϕ : Q \to R\}

and denote by ⟨·⟩ Q the averaging operator on Q . Then, for a convex function f , the formal dual of the problem

f (ξ) \to \min_{ξ \in 𝒦_{\overline{ξ}}}

is given by

f^{*} (v) - \overline{ξ} \cdot {⟨ v ⟩}_{Q} \to \min_{div v = 0},

see Hintermüller et al, where f ∗ denotes the Legendre‐Fenchel dual of f , that is,

f^{*} (v) = \sup_{ξ} {⟨ ξ \cdot v ⟩}_{Q} - f (ξ) .

…”

Section: Homogenization Of Brittle Fracture and Cell Formulaementioning

confidence: 99%

“…Put differently, the formal dual of the minimum cut problem is the maximumflow problem , which seeks to maximize the flow ⟨ v ⟩ Q in direction

\overline{ξ}

subject to an incompressibility constraint and a bound constraint on the flow v . The strong duality of these problems, that is,

γ^{eff} (\overline{ξ}) \equiv \min_{ξ \in 𝒦_{\overline{ξ}}} {⟨ γ ‖ ξ ‖ ⟩}_{Q} = \max_{div v = 0, ‖ v ‖ \leq γ} \overline{ξ} \cdot {⟨ v ⟩}_{Q}

was first shown by Strang for continuous γ and extended to piece‐wise continuous γ by Hintermüller et al This minimum cut—maximum flow duality immediately gives rise to simple bounds on the effective crack resistance. Indeed, inserting

ξ = \overline{ξ}

into the primal problem, and

v = (\min γ) \overline{ξ}

into the dual problem provides the two‐sided bound

\min_{x \in Q} γ (x) \leq γ^{eff} (\overline{ξ}) \leq {⟨ γ ⟩}_{Q}

on the effective crack resistance.…”

Section: Homogenization Of Brittle Fracture and Cell Formulaementioning

confidence: 99%

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An FFT‐based method for computing weighted minimal surfaces in microstructures with applications to the computational homogenization of brittle fracture

Schneider

2019

Numerical Meth Engineering

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Summary Cell formulae for the effective crack resistance of a heterogeneous medium obeying Francfort‐Marigo's formulation of linear elastic fracture mechanics have been proved recently, both in the context of periodic and stochastic homogenization. This work proposes a numerical strategy for computing the effective, possibly anisotropic, crack resistance of voxelized microstructures using the fast Fourier transform (FFT). Based on Strang's continuous minimum cut—maximum flow duality, we explore a primal‐dual hybrid gradient method for computing the effective crack resistance, which may be readily integrated into an existing FFT‐based code for homogenizing thermal conductivity. We close with demonstrative numerical experiments.

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Reduced modelling and optimal control of epidemiological individual‐based models with contact heterogeneity

Courtès

Franck

Killian

et al. 2023

Optim Control Appl Methods

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Modelling epidemics using classical population-based models suffers from shortcomings that so-called individual-based models are able to overcome, as they are able to take into account heterogeneity features, such as super-spreaders, and describe the dynamics involved in small clusters. In return, such models often involve large graphs which are expensive to simulate and difficult to optimize, both in theory and in practice.By combining the reinforcement learning philosophy with reduced models, we propose a numerical approach to determine optimal health policies for a stochastic individual-based model taking into account heterogeneity in the population. More precisely, we introduce a deterministic reduced population-based model involving a neural network, designed to faithfully mimic the local dynamics of the more complex individual-based model. Then the optimal control is determined by sequentially training the network until an optimal strategy for the population-based model succeeds in also containing the epidemic when simulated on the individual-based model.After describing the practical implementation of the method, several numerical tests are proposed to demonstrate its ability to determine controls for models with contact heterogeneity.

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A function space framework for structural total variation regularization with applications in inverse problems

Cited by 27 publications

References 61 publications

Analysis and automatic parameter selection of a variational model for mixed Gaussian and salt-and-pepper noise removal

Analysis and automatic parameter selection of a variational model for mixed Gaussian and salt-and-pepper noise removal

An FFT‐based method for computing weighted minimal surfaces in microstructures with applications to the computational homogenization of brittle fracture

Reduced modelling and optimal control of epidemiological individual‐based models with contact heterogeneity

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