A local estimate for vectorial total variation minimization in one dimension

Giacomelli, Lorenzo; Łasica, Michał

doi:10.1016/j.na.2018.11.009

Cited by 5 publications

(4 citation statements)

References 22 publications

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“…We stress that Theorem 1 is, to our knowledge, the first result of this type which admits vectorial problems. An exception is [14], where a stronger result is obtained in the case m = 1, Φ = | • |. We note however, that there is also a paper [2], where existence of W 1,1 solutions is obtained in vectorial setting for functionals of linear growth with a regular enough source term instead of fidelity term.…”

Section: Introductionmentioning

confidence: 92%

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Existence of $W^{1,1}$ solutions to a class of variational problems with linear growth on convex domains

Łasica¹,

Rybka²

2020

Preprint

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We consider a class of convex integral functionals composed of a term of linear growth in the gradient of the argument, and a fidelity term involving L 2 distance from a datum. Such functionals are known to attain their infima in the BV space. Under the assumption that the domain of integration is convex, we prove that if the datum is in W 1,1 , then the functional has a minimizer in W 1,1 . In fact, the minimizer inherits W 1,p regularity from the datum for any p ∈ [1, +∞]. We also obtain a quantitative bound on the singular part of the gradient of the minimizer in the case that the datum is in BV . We infer analogous results for the gradient flow of the underlying functional of linear growth. We admit any convex integrand of linear growth, possibly defined on vector-valued maps.The functional E λ f is weakly lower semicontinuous on W. However, this space is not reflexive. Hence, without additional assumptions E λ f may fail to attain its infimum. In order to resolve this issue, one may opt to consider instead its lower semicontinuous envelope E λ f in L 2 (Ω, R n ).

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Section: Introductionmentioning

confidence: 92%

“…In [7] and [4], it has been established for m = n = 1 and Φ = | • | that |∇u| ≤ |∇f | in the sense of measures. This was later generalized to the vectorial case n > 1 in [14]. Such an estimate is known to fail if m > 1.…”

Section: Introductionmentioning

confidence: 99%

Existence of $W^{1,1}$ solutions to a class of variational problems with linear growth on convex domains

Łasica¹,

Rybka²

2020

Preprint

Self Cite

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“…In all papers listed above, only the scalar case n = 1 is considered. The only generalization of the mentioned results to the vector-valued case that we know of is [20], where inequality (1.6) is obtained in the case m = 1, n > 1, F = | • | using integral estimates. In this paper, our goal is to obtain estimates on u x or u s x in the case m = 1, n > 1 for possibly general F .…”

Section: Introductionmentioning

confidence: 99%

Local estimates for vectorial Rudin–Osher–Fatemi type problems in one dimension

Grochulska,

Łasica

2024

ESAIM: COCV

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We consider the Rudin–Osher–Fatemi variational denoising model with general regularizing term and L2 fidelity in one-dimensional, vector-valued setting. We obtain local estimates on the singular part of the variation measure of the minimizer in terms of the singular part of the variation measure of the datum. In the case of homogeneous regularizer, we prove local estimates on the whole variation measure of the minimizer and deduce an analogous result for the gradient flow of the regularizer. We also discuss the question of extending our results to other fidelities.

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“…We point out that similar results have already been reported for the case of the total variation flow. In particular, in the one-dimensional case, the total variation of the solution can be controlled by that of the initial data in a completely local way (see [20]) while in the multidimensional case, this has only been shown for the jump part (see [17] for dimension less than or equal to 7 or [23] for a related result in any dimension).…”

Section: Introductionmentioning

confidence: 99%

The Neumann problem for one-dimensional parabolic equations with linear growth Lagrangian: evolution of singularities

Moll

Smarrazzo

2022

J. Evol. Equ.

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In this paper, we obtain existence and uniqueness of strong solutions to the inhomogenous Neumann initial-boundary problem for a parabolic PDE which arises as a generalization of the time-dependent minimal surface equation. Existence and regularity in time of the solution are proved by means of a suitable pseudoparabolic relaxed approximation of the equation and the corresponding passage to the limit. Our main result is monotonicity in time of the positive and negative singular parts of the distributional space derivative for bounded variation initial data. Sufficient conditions for instantaneous $$L^1$$ L 1 -$$W^{1,1}_{{\mathrm{loc}}}$$ W loc 1 , 1 or BV-$$W^{1,1}_{{\mathrm{loc}}}$$ W loc 1 , 1 regularizing effects are also discussed.

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A local estimate for vectorial total variation minimization in one dimension

Cited by 5 publications

References 22 publications

Existence of $W^{1,1}$ solutions to a class of variational problems with linear growth on convex domains

Existence of $W^{1,1}$ solutions to a class of variational problems with linear growth on convex domains

Local estimates for vectorial Rudin–Osher–Fatemi type problems in one dimension

The Neumann problem for one-dimensional parabolic equations with linear growth Lagrangian: evolution of singularities

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