The regularization of a convex program is exact if all solutions of the regularized problem are also solutions of the original problem for all values of the regularization parameter below some positive threshold. For a general convex program, we show that the regularization is exact if and only if a certain selection problem has a Lagrange multiplier. Moreover, the regularization parameter thre...

Regularization is a required component of geophysical estimation problems that operate with insufficient data. The goal of regularization is to impose additional constraints on the estimated model. I introduce shaping regularization, a general method for imposing constraints by explicit mapping of the estimated model to the space of admissible models. Shaping regularization is integrated in a c...

We generalize the notion of Bregman distance using concepts from abstract convexity in order to derive convergence rates for Tikhonov regularization with non-convex regularization terms. In particular, we study the non-convex regularization of linear operator equations on Hilbert spaces, showing that the conditions required for the application of the convergence rates results are strongly relat...

We illustrate the dimensional regularization technique using a simple example from electrostatics. This example illustrates the virtues of dimensional regularization without the complications of a full quantum field theory calculation. We contrast the dimensional regularization approach with the cutoff regularization approach, and demonstrate that dimensional regularization preserves translatio...

The inverse problem in the diffuse optical tomography is known to be nonlinear, ill-posed, and sometimes under-determined, requiring regularization to obtain meaningful results, with Tikhonov-type regularization being the most popular one. The choice of this regularization parameter dictates the reconstructed optical image quality and is typically chosen empirically or based on prior experience...

In this article we study the regularization of optimization problems by Tikhonov regularization. The optimization problems are subject to pointwise inequality constraints in L2(Ω). We derive a-priori regularization error estimates if the regularization parameter as well as the noise level tend to zero. We rely on an assumption that is a combination of a source condition and of a structural assu...

We address the classical issue of appropriate choice of the regularization and discretization level for the Tikhonov regularization of an inverse problem with imperfectly measured data. We focus on the fact that the proper choice of the discretization level in the domain together with the regularization parameter is a key feature in adequate regularization. We propose a discrepancy-based choice...

Regularization is one of the crucial ingredients of deep learning, yet the term regularization has various definitions, and regularization methods are often studied separately from each other. In our work we present a systematic, unifying taxonomy to categorize existing methods. We distinguish methods that affect data, network architectures, error terms, regularization terms, and optimization p...

In this paper we derive a generalizing concept of G-norms, which we call G-values, which is used to characterize minimizers of nondifferentiable regularization functionals. Moreover, the concept is closely related to the definition of slopes as published in a recent book by Ambrosio, Gigli, Savaré. A paradigm of regularization models fitting in this framework is robust bounded variation regular...

We consider the local regularization problem for integral equations of the first kind, generalizing previous work which applied only to problems of Volterra type. Our approach allows for local control of the regularization process, allowing for resolution of fine/sharp features of solutions without having to resort to nondifferentiable optimization techniques. In addition we present examples il...