In this paper we study the convergence of the well-known Greedy Rank-One Update Algorithm. It is used to construct the rank-one series solution for full-rank linear systems. The existence of the rank one approximations is also not new, but surprisingly the focus there has been more on the applications side more that in the convergence analysis. Our main contribution is to prove the convergence of the algorithm and also we study the required rank one approximation in each step. We also give some numerical examples and describe its relationship with the Finite Element Method for High-Dimensional Partial Differential Equations based on the tensorial product of one-dimensional bases. We illustrate this situation taking as a model problem the multidimensional Poisson equation with homogeneous Dirichlet boundary condition.

Sparse model identification by means of data is especially cumbersome if the sought dynamics live in a high dimensional space. This usually involves the need for large amount of data, unfeasible in such a high dimensional settings. This well‐known phenomenon, coined as the curse of dimensionality, is here overcome by means of the use of separate representations. We present a technique based on the same principles of the Proper Generalized Decomposition that enables the identification of complex laws in the low‐data limit. We provide examples on the performance of the technique in up to ten dimensions.

Ti-6Al-4V titanium is considered a difficult-to-cut material used in critical applications in the aerospace industry requiring high reliability levels. An appropriate selection of cutting conditions can improve the machinability of this alloy and the surface integrity of the machined surface, including the generation of compressive residual stresses. In this paper, orthogonal cutting tests of Ti-6Al-4V titanium were performed using coated and uncoated tungsten carbide tools. Suitable design of experiments (DOE) was used to investigate the influence of the cutting conditions (cutting speed V_c, uncut chip thickness h, tool rake angle γ_n, and the cutting edge radius r_n) on the forces, chip compression ratio, and residual stresses. Due to the time consumed and the high cost of the residual stress measurements, they were only measured for selected cutting conditions of the DOE. Then, the machine learning method based on mathematical regression analysis was applied to predict the residual stresses for other cutting conditions of the DOE. Finally, the optimal cutting conditions that minimize the machining outcomes were determined. The results showed that when increasing the compressive residual stresses at the machined surface by 40%, the rake angle should be increased from negative (−6°) to positive (5°), the cutting edge radius should be doubled (from 16 µm to 30 µm), and the cutting speed should be reduced by 67% (from 60 to 20 m/min).

The dynamic viscosity and rheological properties of two different non-aqueous graphene nano-plates-based nanofluids are experimentally investigated in this paper, focusing on the effects of solid volume fraction and shear rate. For each nanofluid, four solid volume fractions have been considered ranging from 0.1% to 1%. The rheological characterization of the suspensions was performed at 20 ${}^{\circ }$C, with shear rates ranging from ${10}^{-1}{\mathrm{s}}^{-1}$ to ${10}^3{\mathrm{s}}^{-1}$, using a cone-plate rheometer. The Carreau–Yasuda model has been successfully applied to fit most of the rheological measurements. Although it is very common to observe an increase of the viscosity with the solid volume fraction, we still found here that the addition of nanoparticles produces lubrication effects in some cases. Such a result could be very helpful in the domain of heat extraction applications. The dependence of dynamic viscosity with graphene volume fraction was analyzed using the model of Vallejo et al.