Ill posed problems constitute the mathematical model of a large variety of applications. Aim of this paper is to define an iterative algorithm finding the solution of a regularization problem. The method minimizes a function constituted by a least squares term and a generally nonlinear regularization term, weighted by a regularization parameter. The proposed method computes a sequence of iterat...

There are two key issues for the kernel-based regularization method: the kernel structure design and the hyper-parameter estimation. In this contribution, we introduce a new family of kernel structures based on state space models. It has more flexible and more general structure, and includes some of stable spline kernels and diagonal correlated kernels as special cases. We also tested a differe...

A new approach to the construction of finite-difference methods is presented. It is shown how the multi-point differentiators can generate regularizing algorithms with a stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iteratively regularized scheme for solving the numerical differentiation problem in the form of Volterra integral eq...

In this short paper, we formulate parameter estimation for finite mixture models in the context of discrete optimal transportation with convex regularization. The proposed framework unifies hard and soft clustering methods for general mixture models. It also generalizes the celebrated k-means and expectation-maximization algorithms in relation to associated Bregman divergences when applied to e...

— This paper is devoted to the introduction and analysis of regularization in state space for nonlinear illposed inverse problems. Applications to parameter estimation problems are given and numerical experiments are described. Résumé. — Nous introduisons et analysons la régularisation dans l'espace d'état pour les problèmes inverses non linéaires. Nous donnons des applications aux problèmes d'...

We present a Bayesian framework for reconstruction of subsurface hydraulic properties from nonlinear dynamic flow data by imposing sparsity on the distribution of the solution coefficients in a compression transform domain. Sparse representation of the subsurface flow properties in a compression transform basis lends itself to a natural regularization approach, i.e. sparsity regularization, whi...

1. Introduction and historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 2. Spectral estimators, parameter estimators and nonlinear problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 3. Non-Hermitian quantum mechanics: connection to the harmonic inversion problem (HIP) . . . . . . . . . 167 4. HIP can be solve...

Probabilistic graphical models can simulate and classify high dimensional, heterogeneous data and serve as underlying formalism in the data analysis of biology, physics, computer vision, natural language processing and others. Their parameter dimension is a function of the treewidth of the data's conditional independence graph and the data domain. Even if most dependencies are ignored and a pai...

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...