A note on optimal Hermite interpolation in Sobolev spaces

This paper investigates the optimal Hermite interpolation of Sobolev spaces $W_{\infty }^{n}[a,b]$ W ∞ n [ a , b ] , $n\in \mathbb{N}$ n ∈ N in space $L_{\infty }[a,b]$ L ∞ [ a , b ] and weighted spaces $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p< \infty $ 1 ≤ p < ∞ with ω a continuous-integrable weight function in $(a,b)$ ( a , b ) when the amount of Hermite data is n. We proved that the Lagrange interpolation algorithms based on the zeros of polynomial of degree n with the leading coefficient 1 of the least deviation from zero in $L_{\infty }$ L ∞ (or $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p<\infty $ 1 ≤ p < ∞ ) are optimal for $W_{\infty }^{n}[a,b]$ W ∞ n [ a , b ] in $L_{\infty }[a,b]$ L ∞ [ a , b ] (or $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p<\infty $ 1 ≤ p < ∞ ). We also give the optimal Hermite interpolation algorithms when we assume the endpoints are included in the interpolation systems.

The Impact of Preprocessing on the PET-CT Radiomics Features in Non-small Cell Lung Cancer

Purpose: This study aimed to investigate the impact of image preprocessing steps, including Gray Level Discretization (GLD) and different Interpolation Algorithms (IA) on 18F-Fluorodeoxyglucose (18F-FDG) radiomics features in Non-Small Cell Lung Cancer (NSCLC). Materials and Methods: One hundred and seventy-two radiomics features from the first-, second-, and higher-order statistic features were calculated from a set of Positron Emission Tomography/Computed Tomography (PET/CT) images of 20 non-small cell lung cancer delineated tumors with volumes ranging from 10 to 418 cm3 regarding five intensity discretization schemes with the number of gray levels of 16, 32, 64, 128, and 256, and four Interpolation algorithms, including nearest neighbor, tricubic convolution and tricubic spline interpolation, and trilinear were used. Segmentation was based on 3D region growing-based. The Intraclass Correlation Coefficient (ICC), Overall Concordance Correlation Coefficient (OCCC), and Coefficient Of Variations (COV) were calculated to demonstrate the features' variability and select robust features. ICC and OCCC < 0.5 presented weak reliability, ICC and OCCC between 0.5 and 0.75 illustrated appropriate reliability, values within 0.75 and 0.9 showed satisfying reliability, and values higher than 0.90 indicate exceptional reliability. Besides, features with less than 10% COV have been selected as robust features. Results: All morphology family (except four features), statistic, and Intensity volume histogram families were not affected by GLD and IA. And the rest of them, 10 and 61 features showed COV ≤ 5% against GLD and IA, respectively. Ten and 80 features showed excellent reliability (ICC values greater than 0.90) against GLD and IA. Eight and 60 features showed OCCC≥0.90 against GLD and IA, respectively. Based on our results Inverse difference normalized and Inverse difference moment normalized from Grey Level Co-occurrence Matrix (GLCM) were the most robust features against GLD and Skewness from intensity histogram family and Inverse difference normalized and Inverse difference moment normalized from GLCM were the most robust features against IA. Conclusion: Preprocessing can substantially impact the 18F-FDG PET image radiomic features in NSCLC. The impact of gray level discretization on radiomics features is significant and more than Interpolation algorithms.

Lighthill's analogy applied to a automotive turbocharger compressor

Computing transient CFD simulations of turbocharger compressors is computationally very demanding. It is of fundamental importance to resolve turbulent structures at the location of their generation and to establish a fine enough grid to allow propagation of the resolved structures. This results in high-resolution grids, existing of more than 20 million cells. Applying Lighthill's analogy, it is possible to only resolve turbulent structures at their location of generation and compute the pressure propagation by using an additional, not that demanding, acoustic grid. This allows using coarser CFD grids in the inlet and outlet section. For transferring Lighthill's source terms from the CFD to the acoustic grid, advanced interpolation algorithms are used. The simulation results are validated by measurements of a cold gas test rig are considered.

LoopStructural 1.0: time-aware geological modelling

In this contribution we introduce LoopStructural, a new open-source 3D geological modelling Python package (http://www.github.com/Loop3d/LoopStructural, last access: 15 June 2021). LoopStructural provides a generic API for 3D geological modelling applications harnessing the core Python scientific libraries pandas, numpy and scipy. Six different interpolation algorithms, including three discrete interpolators and 3 polynomial trend interpolators, can be used from the same model design. This means that different interpolation algorithms can be mixed and matched within a geological model allowing for different geological objects, e.g. different conformable foliations, fault surfaces and unconformities to be modelled using different algorithms. Geological features are incorporated into the model using a time-aware approach, where the most recent features are modelled first and used to constrain the geometries of the older features. For example, we use a fault frame for characterising the geometry of the fault surface and apply each fault sequentially to the faulted surfaces. In this contribution we use LoopStructural to produce synthetic proof of concepts models and a 86 km × 52 km model of the Flinders Ranges in South Australia using map2loop.