There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. We are concerned with the solution of nonlinear ill-posed operator equations. The first convergence rates results for such problems have been developed by Engl, Kunisch and Neubauer in 1989. While these results apply for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004, more generally, apply to operators formulated in Banach spaces. Recently, Resmerita et al. presented a modification of the convergence rates result of Burger and Osher which turns out a complete generalization of the rates result of Engl et. al. In all these papers relatively strong regularity assumptions are made. However, it has been observed numerically, that violations of the smoothness assumptions of the operator do not necessarily affect the convergence rate negatively. We take this observation and weaken the smoothness assumptions on the operator and prove a novel convergence rate result. The most significant difference in this result to the previous ones is that the source condition is formulated as a variational inequality and not as an equation as before. As examples we present a phase retrieval problem and a specific inverse option pricing problem, both studied in the literature before. For the inverse finance problem, the new approach allows us to bridge the gap to a singular case, where the operator smoothness degenerates just when the degree of ill-posedness is minimal.

Consider a nonlinear ill-posed operator equation F (u) = y where F is defined on a Banach space X. In general, for solving this equation numerically, a finite dimensional approximation of X and an approximation of F are required. Moreover, in general the given data y δ of y are noisy. In this paper we analyze finite dimensional variational regularization, which takes into account operator approximations and noisy data: We show (semi-)convergence of the regularized solution of the finite dimensional problems and establish convergence rates in terms of Bregman distances under appropriate sourcewise representation of a solution of the equation. The more involved case of regularization in nonseparable Banach spaces is discussed in detail. In particular we consider the space of finite total variation functions, the space of functions of finite bounded deformation, and the L ∞ -space.

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...

There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. The first convergence rates results for non-linear problems have been developed by Engl, Kunisch and Neubauer in 1989 [3]. While these results apply for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004 [1], more generally, apply to operators formulated in Banach spaces. Recently, Resmerita and Scherzer [6] presented a modification of the convergence rates result of Burger and Osher which turns out a complete generalization of the result of Engl et. al. In all these papers relatively strong regularity assumptions are made. However, it has been observed numerically, that violations of the smoothness assumptions on the operator do not necessarily affect the convergence rate negatively. We have taken this observation and weakened the smoothness assumptions on the operator and have proved a novel convergence rate result published in [4]. The most significant difference of this result to the previous ones is that the source condition is formulated as a variational inequality and not as an equation as before, which is necessary due to the lack of smoothness assumptions on F .

We study variational methods of bounded variation type for the data analysis. Y. Meyer characterized minimizers of the Rudin-Osher-Fatemi functional in dependence of the G-norm of the data. These results and the follow up work on this topic are generalized to functionals defined on spaces of functions with derivatives of finite bounded variation. In order to derive a characterization of minimizers of convex regularization functionals we use the concept of generalized directional derivatives and duality. Finally we present some examples where the minimizers of convex regularization functionals are calculated analytically, repeating some recent results from the literature and adding some novel results with penalization of higher order derivatives of bounded variation.