### A Computational Procedure for the Approximation of Random Functions.

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A numerical method of fitting a multiparameter function, non-linear in the parameters which are to be estimated, to the experimental data in the ${L}_{1}$ norm (i.e., by minimizing the sum of absolute values of errors of the experimental data) has been developed. This method starts with the least squares solution for the function and then minimizes the expression ${\sum}_{i}{({x}_{i}^{2}+{a}^{2})}^{1/2}$, where ${x}_{i}$ is the error of the $i$-th experimental datum, starting with an $a$ comparable with the root-mean-square error of the least squares solution...

A two-level algebraic algorithm is introduced and its convergence is proved. The restriction as well as prolongation operators are defined with the help of aggregation classes. Moreover, a particular smoothing operator is defined in an analogical way to accelarate the convergence of the algorithm. A model example is presented in conclusion.

Le Bail fitting method is procedure used in the applied crystallography mainly during the crystal structure determination. As in many other applications, there is a need for a great performance and short execution time. In this paper, we describe utilization of parallel computing for mathematical operations used in Le Bail fitting. We present an algorithm implementing this method with highlighted possible approaches to its aforementioned parallelization. Then, we propose a sample parallel version...

We present an algorithm to generate a smooth curve interpolating a set of data on an $n$-dimensional ellipsoid, which is given in closed form. This is inspired by an algorithm based on a rolling and wrapping technique, described in [11] for data on a general manifold embedded in Euclidean space. Since the ellipsoid can be embedded in an Euclidean space, this algorithm can be implemented, at least theoretically. However, one of the basic steps of that algorithm consists in rolling the ellipsoid, over...

An approximate necessary condition for the optimal bandwidth choice is derived. This condition is used to construct an iterative bandwidth selector. The algorithm is based on resampling and step-wise fitting the bandwidth to the density estimator from the previous iteration. Examples show fast convergence of the algorithm to the bandwidth value which is surprisingly close to the optimal one no matter what is the initial knowledge on the unknown density.

This paper studies the gradient flow of a regularized Mumford-Shah functional proposed by Ambrosio and Tortorelli (1990, 1992) for image segmentation, and adopted by Esedoglu and Shen (2002) for image inpainting. It is shown that the gradient flow with ${L}^{2}\times {L}^{\infty}$ initial data possesses a global weak solution, and it has a unique global in time strong solution, which has at most finite number of point singularities in the space-time, when the initial data are in ${H}^{1}\times {H}^{1}\cap {L}^{\infty}$. A family of fully discrete approximation...

This paper studies the gradient flow of a regularized Mumford-Shah functional proposed by Ambrosio and Tortorelli (1990, 1992) for image segmentation, and adopted by Esedoglu and Shen (2002) for image inpainting. It is shown that the gradient flow with L2 x L∞ initial data possesses a global weak solution, and it has a unique global in time strong solution, which has at most finite number of point singularities in the space-time, when the initial data are in H1 x H1 ∩ L∞. A family of fully...