For dimensions close to D = 4, the Feynman integrals in momentum space derived in Chapter 4 do not converge since their integrands fall off too slowly at large momenta. Divergences arising from this short-wavelength region of the integrals are called ultraviolet (UV)-divergences. For massive fields, these are the only divergences of the integrals. In the zero-mass limit relevant for critical ph...

In this paper we develop a novel criterion for choosing regularization parameters for nonsmooth Tikhonov functionals. The proposed criterion is solely based on the value function, and thus applicable to a broad range of functionals. It is analytically compared with the local minimum criterion, and a posteriori error estimates are derived. An efficient numerical algorithm for computing the minim...

In system identification, the Akaike Information Criterion (AIC) is a well known method to balance the model fit against model complexity. Regularization here acts as a price on model complexity. In statistics and machine learning, regularization has gained popularity due to modeling methods such as Support Vector Machines (SVM), ridge regression and lasso. But also when using a Bayesian approa...

We present an efficient method for the reduction of model equations in the linearized diffuse optical tomography (DOT) problem. We first implement the maximum a posteriori (MAP) estimator and Tikhonov regularization, which are based on applying preconditioners to linear perturbation equations. For model reduction, the precondition is split into two parts: the principal components are consid...

Most seale-spaee eoneepts have been expressed as parabolie or hyperbolie partial differential equations (PDEs). In this paper we extend our work on seale-spaee properties of elliptie PDEs arising from regularization methods: we study linear and nonlinear regularization methods that are applied iteratively and with different regularization parameters. For these so-ealled nonstationary iterative ...