The area of mathematical inverse problems is quite broad and involves the qualitative and quantitative analysis of a wide variety of physical models. Applications include, for example, the problem of inverse heat conduction, image reconstruction, tomography, the inverse scattering problem, and the determination of unknown coefficients or boundary parameters appearing in partial differential equ...

Instead of the Tikhonov regularization method which with a scalar being the regularization parameter, Liu et al. [1] have proposed a novel regularization method with a vector as being the regularization parameter. As a continuation we further propose an optimally scaled vector regularization method (OSVRM) to solve the ill-posed linear problems, which is better than the Tikhonov regularization ...

The effectiveness of regularization to improve SNR in parallel imaging techniques has been reported in previous works [1-2], but how to optimize the regularization parameter remains a problem. The regularization parameter controls the degree of regularization and thereby determines the compromise between SNR and artifacts. Over-regularization causes high level of artifact, while under-regulariz...

We discuss the framework of Transductive Support Vector Machine (TSVM) from the perspective of the regularization strength induced by the unlabeled data. In this framework, SVM and TSVM can be regarded as a learning machine without regularization and one with full regularization from the unlabeled data, respectively. Therefore, to supplement this framework of the regularization strength, it is ...

In learning theory, the convergence issues of the regression problem are investigated with the least square Tikhonov regularization schemes in both the RKHS-norm and the L -norm. We consider the multi-penalized least square regularization scheme under the general source condition with the polynomial decay of the eigenvalues of the integral operator. One of the motivation for this work is to dis...

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 ...