Consider the problem of computing an approximate solution of an overdetermined system of linear equations. The usual approach to the problem is least squares, in which the 2--norm of the residual isminimized. This produces the minimum variance unbiased estimator of the solution when the errors in the observations are independent and normally distributed with mean 0 and constant variance. It is well known, however, that the least squares solution is not robust if outliers occur, i.e., if some of the observations are contaminated by large error. In this case, alternate approaches have been proposed which judge the size of the residual in a way that is less sensitive to these components. These include the Huber M-function, the Talwar function, the logistic function, the Fair function, and the $\ell_1$ norm. In this paper, new algorithms are proposed to compute the solution to these problems efficiently, in particular when the matrix $A$ has small displacement rank. Matrices with small displacement rank include matrices that are Toeplitz, block-Toeplitz, block-Toeplitz with Toeplitz blocks, Toeplitz plus Hankel, and a variety of other forms. For exposition, only Toeplitz matrices are considered here, but the ideas apply to all matrices with small displacement rank. Algorithms are also presented to compute the solution efficiently when a regularization term is included to handle the case when the matrix of the coefficients is ill-conditioned or rank-deficient. The techniques are illustrated on a problem of FIR system identification.

In this paper we will present a general framework for solving linear systems of equations. The solver is based on the Levinson-idea for solving Toeplitz systems of equations. We will consider a general class of matrices, defined as the class of simple $ (p_1,p_2)$-Levinson conform matrices. This class incorporates, for instance, semiseparable, band, companion, arrowhead and many other matrices. For this class, we will derive a solver of complexity $O(p_1 p_2 n).$ The system solver is written inductively, and uses in every step k, the solution of a so-called $k$-th order Yule-Walker-like equation. The algorithm obtained first has complexity algorithm $O(p_1 p_2 n^2).$ Based, however on the specific structure of the simple $ (p_1,p_2)$-Levinson conform matrices, we will be able to further reduce the complexity of the presented method, and get an order $O(p_1 p_2 n)$ algoritm. Different examples of matrices are given for this algorithm. Examples are presented for: general dense matrices, upper triangular matrices, higher order generator semiseparable matrices, quasiseparable matrices, Givens-vector representable semiseparable matrices, band matrices, companion matrices, confederate matrices, arrowhead matrices, fellow matrices and many more. Finally, the relation between this method and an upper triangular factorization of the original matrix is given and also details concerning possible look ahead methods are presented.

Magnetic resonance spectroscopic imaging (MRSI) provides information about the spatial metabolic heterogeneity of an organ in the human body. In this way, MRSI can be used to detect tissue regions with abnormal metabolism, e.g. tumor tissue. The main drawback of MRSI in clinical practice is that the analysis of the data requires a lot of expertise from the radiologists. In this article, we present an automatic method that assigns each voxel of a spectroscopic image of the brain to a histopathological class. The method is based on Canonical Correlation Analysis (CCA), which has recently been shown to be a robust technique for tissue typing. In CCA, the spectral as well as the spatial information about the voxel is used to assign it to a class. This has advantages over other methods that only use spectral information since histopathological classes are normally covering neighbouring voxels. In this paper, a new CCA-based method is introduced in which MRSI and MR imaging information is integrated. The performance of tissue typing is compared for CCA applied to the whole MR spectra and to sets of features obtained from the spectra. Tests on simulated and in vivo MRSI data show that the new method is very accurate in terms of classification and segmentation. The results also show the advantage of combining spectroscopic and imaging data.

The physical behavior of a class of mesoscopic models for multiphase flows is analyzed in details near interfaces. In particular, an extended pseudopotential method is developed, which permits to tune the equation of state and surface tension independently of each other. The spurious velocity contributions of this extended model are shown to vanish in the limit of high grid refinement and/or high order isotropy. Higher order schemes to implement self-consistent forcings are rigorously computed for 2d and 3d models. The extended scenario developed in this work clarifies the theoretical foundations of the Shan-Chen methodology for the lattice Boltzmann method and enhances its applicability and flexibility to the simulation of multiphase flows to density ratios up to O(100).

We consider pointwise mean squared errors of several known Bayesian wavelet estimators, namely, posterior mean, posterior median and Bayes Factor, where the prior imposed on wavelet coe±cients is a mixture of an atom of probability zero and a Gaussian density. We show that for the properly chosen hyperparameters of the prior, all the three estimators are (up to a log-factor) asymptotically minimax within any prescribed Besov ball. We discuss the Bayesian paradox and compare the results for the pointwise squared risk with those for the global mean squared error

A priori estimates for solutions to homogeneous Neumann problems for uniformly elliptic equations in open subsets ­ of R^n are established, with data in the limiting space L^{n\2} or, more generally, in the Lorentz spaces L^{n\2, q}(\Omega). These estimates are optimal as far as either constants or norms are concerned.

The strongest possible norms in a priori estimates for solutions to a class of second-order nonlinear elliptic boundary value problems are exhibited.

We study the behaviour of an initially spherical bunch of particles accelerated along trajectories parallel to the symmetry axis of a rotating black hole. We find that, under suitable conditions, curvature and inertial strains compete to generate jet-like structures. This is a purely kinematical effect which does not account by itself for physical processes underlying the formation of jets. In our analysis a crucial role is played by the property of the electric and magnetic part of the Weyl tensor to be Lorentz-invariant along the axis of symmetry in Kerr spacetime.

Objective. Empirical observations show that rowers demonstrate higher efficiency working on the rowing ergometer than working on water. The aim of the study was to determine the difference in efficiency levels and to verify its validity using a model based on fluid dynamics principles. Methods. A part of work done by rowers is lost in boat deformation and in the friction caused by the oarblades moving through the water. Load conditions for training on a rowing ergometer need to be increased by a certain quantity with respect to those set for on-water training. Results. Times were measured during on-water runs and rowing ergometer tests. Discussion and Conclusions. Times measured during on-water runs matched fairly well those calculated on rowing ergometer tests.