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This paper deals with the problem of determining the sea surface topography from geostrophic flow of ocean currents on local domains of the spherical Earth. In mathematical context the problem amounts to the solution of a spherical differential equation relating the surface curl gradient of a scalar field (sea surface topography) to a surface divergence-free vector field (geostrophic ocean flow). At first, a continuous solution theory is presented in the framework of an integral formula involving Green's function of the spherical Beltrami operator. Different criteria derived from spherical vector analysis are given to investigate uniqueness. Second, for practical applications Green's function is replaced by a regularized counterpart. The solution is obtained by a convolution of the flow field with a scaled version of the regularized Green function. Calculating locally without boundary correction would lead to errors near the boundary. To avoid these Gibbs phenomenona we additionally consider the boundary integral of the corresponding region on the sphere which occurs in the integral formula of the solution. For reasons of simplicity we discuss a spherical cap first, that means we consider a continuously differentiable (regular) boundary curve. In a second step we concentrate on a more complicated domain with a non continuously differentiable boundary curve, namely a rectangular region. It will turn out that the boundary integral provides a major part for stabilizing and reconstructing the approximation of the solution in our multiscale procedure.

We discuss the operator transforming the argument of a function in the L 2 -setting. Here this operator is unbounded and closed. For the approximate solution of ill-posed equations with closed operators, we present a new view on the Tikhonov regularization.

Currently, industrial winding processes are often optimized by trial and error. A digital twin of winding processes could be helpful in order to assist industry to optimize the winding processes. Formulating the kinematic equations that form the basis of such a simulation of the winding process is straightforward in principle. However, a major challenge is to model the increase of the package diameter as a function of time or length of wound up yarn, respectively. In this paper, a kinematic model for the winding process is first outlined. The focus of the paper is the description of a workflow in order to find a model for the package diameter increase dependent on the wound yarn length. For that purpose, a new image analysis method is presented to derive the general class of the model function for the diameter increase. Then, the measurement results of a series of experiments are analyzed to find a parameterization of the model function. Here, the input process parameters winding tension, cradle pressure, winding speed, and traverse ratio are varied at two levels. Finally, the linear regression model for the package diameter increase is presented.

An elementary algorithm is used to simulate the industrial production of a fiber of a 3-dimensional nonwoven fabric. The algorithm simulates the fiber as a polyline where the direction of each segment is stochastically drawn based on a given probability density function (PDF) on the unit sphere. This PDF is obtained from data of directions of fiber fragments which originate from computer tomography scans of a real nonwoven fabric. However, the simulation algorithm requires numerous evaluations of the PDF. Since the established technique of a kernel density estimator leads to very high computational costs, a novel greedy algorithm for estimating a sparse representation of the PDF is introduced. Numerical tests for a synthetic and a real example are presented. In a realistic scenario, the introduced sparsity ansatz leads to a reduction of the computation time for 100 fibers from around 80 days to 2.5 hours