In a recent paper by A. Chambolle et al. [11] it was proven that if the subgradient of the total variation at the noise free data is not empty, the level-sets of the total variation denoised solutions converge to the level-sets of the noise free data with respect to the Hausdorff distance. The condition on the subgradient corresponds to the source condition introduced by Burger and Osher [10], who proved convergence rates results with respect to the Bregman distance under this condition. We generalize the result of Chambolle et al. to total variation regularization of general linear inverse problems under such a source condition. As particular applications we present denoising in bounded and unbounded, convex and non convex domains, deblurring and inversion of the circular Radon transform. In all these examples the convergence result applies. Moreover, we illustrate the convergence behavior through numerical examples.

We consider the fluid mechanical problem of identifying the critical yield number Yc of a dense solid inclusion (particle) settling under gravity within a bounded domain of Bingham fluid, i.e., the critical ratio of yield stress to buoyancy stress that is sufficient to prevent motion. We restrict ourselves to a two-dimensional planar configuration with a single antiplane component of velocity. Thus, both particle and fluid domains are infinite cylinders of fixed cross-section. We then show that such yield numbers arise from an eigenvalue problem for a constrained total variation. We construct particular solutions to this problem by consecutively solving two Cheeger-type set optimization problems. Finally, we present a number of example geometries in which these geometric solutions can be found explicitly and discuss general features of the solutions. 639 of the fluid. Plug regions may occur either within the interior of a flow or may be attached to the wall. In general, as the applied forcing decreases, the plug regions increase in size and the velocity decreases in magnitude. It is natural that at some critical ratio of the driving stresses to the resistive yield stress of the fluid, the flow stops altogether. This critical yield ratio or yield number is the topic of this paper.Critical yield numbers are found for even the simplest one-dimensional (1D) flows, such as Poiseuille flows in pipes and plane channels or uniform film flows, e.g., paint on a vertical wall. These limits have been estimated and calculated exactly for flows around isolated particles, such as the sphere [8] (axisymmetric flow) and the circular disc [46, 48] (two-dimensional (2D) flow). Such flows have practical application in industrial non-Newtonian suspensions, e.g., mined tailings transport, cuttings removal in drilling of wells, etc.The first systematic study of critical yield numbers was carried out by Mosolov and Miasnikov [40,41], who considered antiplane shear flows, i.e., flows with velocity u = (0, 0, w(x 1 , x 2 )) in the x 3 -direction along ducts (infinite cylinders) of arbitrary cross-section Ω. These flows driven by a constant pressure gradient only admit the static solution (w(x 1 , x 2 ) = 0) if the yield stress is sufficiently large. Amongst the many interesting results in [40,41] the key contributions relate to exposing the strongly geometric nature of calculating the critical yield number Y c . First, they show that Y c can be related to the maximal ratio of area to perimeter of subsets of Ω. Second, they develop an algorithmic methodology for calculating Y c for specific symmetric Ω, e.g., rectangular ducts. This methodology is extended further by [29].Critical yield numbers have been studied for many other flows, using analytical estimates, computational approximations, and experimentation. Critical yield numbers to prevent bubble motion are considered in [18,50]. Settling of shaped particles is considered in [31,45]. Natural convection is studied in [32,33]. The onset of landslides is studied in [28,30,26] (where the...

We extend some recent results on the Hausdorff convergence of level-sets for total variation regularized linear inverse problems. Dimensions higher than two and measurements in Banach spaces are considered. We investigate the relation between the dimension and the assumed integrability of the solution that makes such an extension possible. We also give some counterexamples of practical application scenarios where the natural choice of fidelity term makes such a convergence fail.

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