In this paper the fundamental concept of repeated linear interpolation and its possible applications in computer-aided geometric design, and start considering basic constructive methods for curves ...and surfaces. We discuss here a repeated linear interpolation method that we commonly find in computer graphics and geometric modelling. Repeated linear Interpolation means to calculate a polynomial by using several points. For a given sequence of points, this means to estimate a curve that passes through every single point. The purpose of this paper is to construct a polynomial of degree less than or equal to n, by using repeated linear Interpolation.
A generalization of the spherical linear interpolation (or slerp) for the finite rotations to the case of more than two control variables on SO(3) is introduced to design an objective FE-formulation ...for the non-linear space Cosserat rod model. The interpolation uses the De Casteljau’s algorithm.
In this way, the same interpolation degree can be used for the placement of the centroid curve and for the finite rotation of the cross-section. A recursive formula is obtained for the interpolation of the rotations. Similar recursive formulas are derived for the spin and the curvature vector, leading to a generalization of the Bézier basis functions on the manifold SO(3).
The rod formulation so obtained is invariant under a rigid rotation (objective), in the sense that the patch-test with respect to the rigid body motion is satisfied. Furthermore, an optimal path-independence is achieved as verified by several numerical investigations.
•Spherical Interpolation for more than two control finite rotations on SO(3).•De Casteljau’s algorithm for the spin and the Darboux vector.•Generalization of the Bézier basis functions on the manifold SO(3).•Objective and path-independent FE-formulation for Cosserat rods.
Summary
In this article, we develop a dynamic version of the variational multiscale (D‐VMS) stabilization for nearly/fully incompressible solid dynamics simulations of viscoelastic materials. The ...constitutive models considered here are based on Prony series expansions, which are rather common in the practice of finite element simulations, especially in industrial/commercial applications. Our method is based on a mixed formulation, in which the momentum equation is complemented by a pressure equation in rate form. The unknown pressure, displacement, and velocity are approximated with piecewise linear, continuous finite element functions. To prevent spurious oscillations, the pressure equation is augmented with a stabilization operator specifically designed for viscoelastic problems, in that it depends on the viscoelastic dissipation. We demonstrate the robustness, stability, and accuracy properties of the proposed method with extensive numerical tests in the case of linear and finite deformations.
The proposed algorithm, named “EAE-LPI” (Exploration by Adjustment Entropy via Linear and Polynomial Interpolation), aims to enhance exploration within the Proximal Policy Optimization (PPO) ...algorithm by addressing two crucial aspects that have been identified as underdeveloped in previous research perspectives. The first aspect involves the introduction of entropy into the algorithm, adjusted using linear interpolation, to promote exploration. This reduces randomness, distinguishing random fluctuations from significant policy improvements. The second aspect involves the incorporation of polynomial interpolation, creating a Lagrange polynomial from existing data points. This allows the utilization of knowledge from neighboring states obtained through interpolation, enabling exploration of previously uncharted areas and reinforcing interactions with the environment. This research introduces the EAE-LPI method, aiming to overcome the limitations of static entropy effects and basic entropy regularization strategies (linear, polynomial, exponential) in exploration control.
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•A flow-cell-based electrochemical analyzer was developed.•It enabled the fast determination of additive concentrations in a Cu plating bath.•Linear interpolation was used to analyze ...the concentrations.•The isotherm-based plotting provided better linearity with higher R2 values.
Automatic Cu plating bath analyzers are required for the long-term maintenance of the bath quality in the semiconductor and printed circuit board industries. This study reports the development of a flow cell-based electrochemical analyzer for the simple and fast determination of additive concentrations in a Cu plating bath. To determine the additive concentrations, cyclic voltammetry was sequentially performed using a flow cell system for the following solutions: blank, blank + sample, and two references. The stripping charges for each solution were obtained. The concentrations of the additives in the target solution were determined by the linear interpolation of the stripping charge between the two references. To improve the determination accuracy, the stripping charges were reprocessed based on adsorption isotherms. The proposed approach enables the fast and highly accurate determination of organic additives in a Cu electroplating solution.
Satellite time series data, widely used for land cover classification, often contain missing values due to cloud contamination, which can negatively affect classification. Numerous strategies have ...been developed to reconstruct the missing values to produce regular time series for machine learning classifiers, among which the compositing followed by the linear interpolation is most widely used. However, the classification improvement of linear interpolation for land cover classification has not been examined. Recently developed deep learning models such as long short term memory (LSTM) and Transformer allow such examination as they can classify time series with missing values. In this study, we compared the time series composites with missing values (without linear interpolation) and the linearly interpolated time series composites (without missing values) for land cover classification. About 18 thousand Harmonized Landsat Sentinel-2 (HLS) images acquired over Amur River Basin of China (890,308 km2) in 2021 were composited to 14 16-day periods. Two time series composites were classified, i.e., (i) the 16-day composites without interpolation that have on average 15.35% 16-day periods with missing values and (ii) the linearly interpolated 16-day composites with no missing values. The classifications showed that (1) between classifications with and without linear interpolation there was < 0.2% overall accuracy differences for the bidirectional LSTM (Bi-LSTM) and < 0.5% for the Transformer both of which were smaller than model training randomness; and (2) the computation time can be saved using composites without linear interpolation. The findings suggested that it is unnecessary to use the time-consuming linear interpolation in Bi-LSTM and Transformer-based land cover classifications. The findings were confirmed by experiments for sensitivity to the number of cloud-free composites and to different classification legends using crop type classifications. It implied the linear interpolation algorithm cannot reconstruct reliable time series for land cover classifications and historical use of such method is more about mitigating the inability of traditional classifiers to handle missing values rather than improving classifications. Linear interpolation is not necessary for LSTM and Transformer with capability to handle missing values. The training datasets and developed codes in this study are made publicly available.
Model quantization is a prevalent method to compress and accelerate neural networks. Most existing quantization methods usually require access to real data to improve the performance of quantized ...models, which is often infeasible in some scenarios with privacy and security concerns. Recently, data-free quantization has been widely studied to solve the challenge of not having access to real data by generating synthetic data, among which generator-based data-free quantization is an important type. Previous generator-based methods focus on improving the performance of quantized models by optimizing the spatial distribution of synthetic data, while ignoring the study of changes in synthetic data from a temporal perspective. In this work, we reveal that generator-based data-free quantization methods usually suffer from the issue that synthetic data show homogeneity in the mid-to-late stages of the generation process due to the stagnation of the generator update, which hinders further improvement of the performance of the quantization model. To solve the above issue, we propose introducing the discrepancy between the full-precision and quantized models as new supervision information to update the generator. Specifically, we propose a simple yet effective adversarial Gaussian-margin loss, which promotes continuous updating of the generator by adding more supervision information to the generator when the discrepancy between the full-precision and quantized models is small, thereby generating heterogeneous synthetic data. Moreover, to mitigate the homogeneity of the synthetic data further, we augment the synthetic data with linear interpolation. Our proposed method can also promote the performance of other generator-based data-free quantization methods. Extensive experimental results show that our proposed method achieves superior performances for various settings on data-free quantization, especially in ultra-low-bit settings, such as 3-bit.
The paper concerns an analysis regarding geometry of a grinding wheel used in grinding of cylindrical worm threads characterized by various geometry. It presents geometrical and physical factors ...having impact on the grinding process and a quality of grinding wheel. In particular, an attention is paid on the accuracy of grinding wheel’s outline if it is approximated by linear interpolation line segments which are obtained in the case of dressing on a CNC machine tool. On the basis of two grinding wheel outlines (Archimedes and circular-arched one), the relevance of the length of the segment line on the geometrical accuracy of wheel’s outline has been presented. In the case of development of a wheel dresser path on the CNC machine tool, the necessity of an customized approach has also been indicated.
Laptop is a desktop personal computer (PC) whose dimensions are reduced to increase flexibility in its use. However, the large number of products will make it difficult for consumers to choose a ...laptop that suits the needs of consumers who want to buy it.The purpose of this research is to help buyers who want to buy laptop products according to their needs by making a Decision Support System (DSS). There are 12 criteria considered in this research, price, processor, RAM capacity, hard disk capacity, SSD capacity, V-RAM capacity, maximum RAM upgrade capacity, laptop weight, screen size, screen type, screen refresh rate, and screen resolution. Choosing a laptop product there is a criterion value of a laptop product and a value of preference criteria from the buyer as a decision maker. Also the criteria values on laptop products have different contributions to the overall value of the laptop product. Thus, the methods used are Analytical Hierarchy Process (AHP), Profile Matching (PM) with linear interpolation, and Simple Addictive Weighting (SAW) to determine the recommended options. Lastly, SPK that has been made will be able to provide recommendations best alternative choices and best suit the needs of buyers for selecting laptop products.
In this paper, we study the online learning of real-valued functions where the hidden function is known to have certain smoothness properties. Specifically, for q≥1, let Fq be the class of absolutely ...continuous functions f:0,1→R such that ‖f′‖q≤1. For q≥1 and d∈Z+, let Fq,d be the class of functions f:0,1d→R such that any function g:0,1→R formed by fixing all but one parameter of f is in Fq. For any class of real-valued functions F and p>0, let optp(F) be the best upper bound on the sum of pth powers of absolute prediction errors that a learner can guarantee in the worst case. In the single-variable setup, we find new bounds for optp(Fq) that are sharp up to a constant factor. We show for all ε∈(0,1) that opt1+ε(F∞)=Θ(ε−12) and opt1+ε(Fq)=Θ(ε−12) for all q≥2. We also show for ε∈(0,1) that opt2(F1+ε)=Θ(ε−1). In addition, we obtain new exact results by proving that optp(Fq)=1 for q∈(1,2) and p≥2+1q−1. In the multi-variable setup, we establish inequalities relating optp(Fq,d) to optp(Fq) and show that optp(F∞,d) is infinite when p<d and finite when p>d. We also obtain sharp bounds on learning F∞,d for p<d when the number of trials is bounded.
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