Barycentric Interpolation Research Articles

Detection and Estimation of Diffuse Signal Components Using the Periodogram

One basic limitation of using the periodogram as a frequency estimator is that any of its significant peaks may result from a diffuse (or spread) frequency component rather than a pure one. Diffuse components are common in applications such as channel estimation, in which a given periodogram peak reveals the presence of a complex multipath distribution (unresolvable propagation paths or diffuse scattering, for example). We present a method to detect the presence of a diffuse component in a given peak based on analyzing the projection of the data vector onto the span of the signature’s derivatives up to a given order. Fundamentally, a diffuse component is detected if the energy in the derivatives’ subspace is too high at the peak’s frequency, and its spread is estimated as the ratio between this last energy and the peak’s energy. The method is based on exploiting the signature’s Vandermonde structure through the properties of discrete Chebyshev polynomials. We also present an efficient numerical procedure for computing the data component in the derivatives’ span based on barycentric interpolation. The paper contains a numerical assessment of the proposed estimator and detector.

A Terminal Residual Vibration Suppression Method of a Robot Based on Joint Trajectory Optimization

Vibration problems have become one of the most important factors affecting robot performance. To this end, a terminal residual vibration suppression method based on joint trajectory optimization is proposed to improve the accuracy and stability of robot motion. Firstly, based on the characteristics of the friction nonlinearity due to joint coupling and physical feasibility of dynamic parameters, a semidefinite programming method is used to identify dynamic parameters with actual physical meaning, thereby obtaining an accurate dynamic model. Then, based on the result of the residual vibration time domain analysis, a joint trajectory optimization model with the goal of minimizing joint tracking error is established. The Chebyshev collocation method is used to discretize the optimization model. The dynamic model is used as the optimization constraint, and barycentric interpolation is used to obtain the optimized joint motion trajectory. Finally, industrial robot experiments prove that the vibration suppression method proposed in this article can reduce the maximum acceleration amplitude of residual vibration by 62% and the vibration duration by 71%. Compared with the input shaping method, the method proposed in this paper can reduce the terminal residual vibration more effectively and ensure the consistency of running time and trajectory.

Interpolant Method in Analyzing the Laser-Induced Plasma Spectroscopy

A numerical algorithm that effectively solves the Abel integral equation in laser-induced plasma spectroscopy is introduced. The proposed algorithm utilizes a barycentric interpolation and a collocation method with second-kind Chebyshev nodes. This approach reduces the Abel integral equation to a linear system of equations, resulting in efficient approximate solutions. Analytical studies provide an error estimation for the proposed method, which further validates its effectiveness. Through numerical experiments, we demonstrate the ease of implementation and showcase the efficiency of the proposed method. Our results indicate that the proposed algorithm can provide valuable insight for researchers in the field of laser-induced plasma spectroscopy and beyond.

An efficient technique based on barycentric interpolation collocation method for the time fractional Allen-Cahn equation

An efficient technique based on barycentric interpolation collocation method for the time fractional Allen-Cahn equation

Learning Compliant Box-in-Box Insertion through Haptic-Based Robotic Teleoperation.

In modern logistics, the box-in-box insertion task is representative of a wide range of packaging applications, and automating compliant object insertion is difficult due to challenges in modelling the object deformation during insertion. Using Learning from Demonstration (LfD) paradigms, which are frequently used in robotics to facilitate skill transfer from humans to robots, can be one solution for complex tasks that are difficult to mathematically model. In order to automate the box-in-box insertion task for packaging applications, this study makes use of LfD techniques. The proposed framework has three phases. Firstly, a master-slave teleoperated robot system is used in the initial phase to haptically demonstrate the insertion task. Then, the learning phase involves identifying trends in the demonstrated trajectories using probabilistic methods, in this case, Gaussian Mixture Regression. In the third phase, the insertion task is generalised, and the robot adjusts to any object position using barycentric interpolation. This method is novel because it tackles tight insertion by taking advantage of the boxes' natural compliance, making it possible to complete the task even with a position-controlled robot. To determine whether the strategy is generalisable and repeatable, experimental validation was carried out.

Barycentric Interpolation Collocation Method for Solving Fractional Linear Fredholm-Volterra Integro-Differential Equation

In this article, barycentric interpolation collocation method (BICM) is presented to solve the fractional linear Fredholm-Volterra integro-differential equation (FVIDE). Firstly, the fractional order term of equation is transformed into the Riemann integral with Caputo definition, and this integral term is approximated by the Gauss quadrature formula. Secondly, the barycentric interpolation basis function is used to approximate the unknown function, and the matrix equation of BICM is obtained. Finally, several numerical examples are given to solve one-dimensional differential equation.

Lagrangian motion magnification with double sparse optical flow decomposition

Microexpressions are fast and spatially small facial expressions that are difficult to detect. Therefore, motion magnification techniques, which aim at amplifying and hence revealing subtle motion in videos, appear useful for handling such expressions. There are basically two main approaches, namely, via Eulerian or Lagrangian techniques. While the first one magnifies motion implicitly by operating directly on image pixels, the Lagrangian approach uses optical flow (OF) techniques to extract and magnify pixel trajectories. In this study, we propose a novel approach for local Lagrangian motion magnification of facial micro-motions. Our contribution is 3-fold: first, we fine tune the recurrent all-pairs field transforms (RAFT) for OFs deep learning approach for faces by adding ground truth obtained from the variational dense inverse search (DIS) for the OF algorithm applied to the CASME II video set of facial micro expressions. This enables us to produce OFs of facial videos in an efficient and sufficiently accurate way. Second, since facial micro-motions are both local in space and time, we propose to approximate the OF field by sparse components both in space and time leading to a double sparse decomposition. Third, we use this decomposition to magnify micro-motions in specific areas of the face, where we introduce a new forward warping strategy using a triangular splitting of the image grid and barycentric interpolation of the RGB vectors at the corners of the transformed triangles. We demonstrate the feasibility of our approach by various examples.

Structured triangular mesh generation method for free-form gridshells based on conformal mapping and virtual interaction forces

With the rapid development of computer-aided design, manufacturing, and construction techniques, free-form single-layer reticulated shells (SLRSs) are increasingly applied in modern architectures. However, generating a uniform and smooth mesh directly on a three-dimensional free-form surface is challenging. This paper proposes a fast and automatic method to generate structured triangular mesh for free-form SLRSs. The difficulty is reduced by generating the mesh on the two-dimensional (2D) parametric plane obtained by conformal mapping. The virtual interaction forces (VIFs) combined with the area map are used to automatically adjust the distribution of the 2D structured base mesh to reduce distortions. Subsequently, mesh clipping and barycentric interpolation are used to map the planar mesh back to the original surface. Finally, the VIF-based mesh smoothing is conducted to improve the mesh uniformity and smoothness. Case studies show that the proposed method successfully generates high-quality structured triangular meshes on various types of free-form SLRSs. Moreover, this method can precisely control the overall size and direction of the mesh, providing architects and engineers with more choices of mesh patterns.

A flexible mixed model for age-dependent performance: application to golf

Abstract We present a new mixed linear model for the relationship between age and performance. The model allows for random effects at the nodes of a barycentric interpolation, such that performance evolves with age in a non-prescriptive way. We use the model to investigate the effects of age on performance in golf and find that performance peaks in the 30s and then declines after that. We disaggregate performance into its constituent components and find that driving, which tends to require power and speed, deteriorates consistently from the early 20s, whilst putting, which requires touch and finesse, remains strong until the late 40s. Our model can be used in other settings, and requires only that measures of performance exist.

A coordinate transformation based barycentric interpolation collocation method and its application in bending, free vibration and buckling analysis of irregular Kirchhoff plates

AbstractIn this article, a meshless barycentric interpolation collocation method (BICM) with coordinate transformation is proposed to solve the governing partial differential equations (PDEs) for bending, free vibration and buckling problems of irregular plates. First, the coordinate transformation equations for first to fourth derivatives are introduced. Then the PDEs and boundary conditions defined in a physical domain are transformed into a rectangular domain (reference domain) with the derived differential interpolation matrices. The presented BICM can solve the PDEs based on the reference domain and is adaptive to irregular problem domains. Finally, the bending, free vibration and buckling problems of Kirchhoff plates with irregular shapes are studied with the proposed method. To illustrate the effectiveness of the proposed method, the results are compared with those given by theoretical solutions, finite element methods and literature.

Level set function based on conformal geometry theory: An efficient approach for optimization of shell structure with arbitrary Gaussian curvature

Topology optimization based on the level set (LS) theory is of great significance to shell structure design. However, the Gaussian curvature of the shell structure can turn complex in many cases, and the establishing process of the LS function will become extremely cumbersome, which consumes overmuch time to solve the Hamilton-Jacobi(H-J) equation. In order to improve the efficiency of optimization, an LS function based on the conformal geometry theory (LSF-CGT) is proposed in this paper. Firstly, the shell structure with arbitrary Gaussian curvature is conformally mapped into 2D Euclidean space to generate the orthogonal mesh of the preset initial LS function. Then the barycentric interpolation method is used to reveal the transformation relationship between the orthogonal and shell structure mesh. Finally, the LS function of the orthogonal mesh is assigned to the mapped shell structure, and the LS function of the 3D shell structure is subsequently obtained. Three typical shell structures with positive, zero, and negative Gaussian curvature are used to verify the effectiveness of LSF-CGT in topology optimization. Results show that LSF-CGT can effectively reduce the strain energy of shell structures with arbitrary Gaussian curvatures by more than 30%. As a method for optimizing shell structures with arbitrary Gaussian curvatures, LSF-CGT can not only stably and quickly converge to the optimal structure in the 2D domain but also effectively maintain the initial geometric characteristics of the shell structure.

Convex optimization-based structure-preserving filter for multidimensional finite element simulations

In simulation sciences, capturing the real-world problem features as accurately as possible is desirable. Methods popular for scientific simulations such as the finite element method (FEM) and finite volume method (FVM) use piecewise polynomials to approximate various characteristics of a problem, such as the concentration profile and the temperature distribution across the domain. Polynomials are prone to creating artifacts such as Gibbs oscillations while capturing a complex profile. An efficient and accurate approach must be applied to deal with such inconsistencies to obtain accurate simulations. This often entails dealing with negative values for the concentration of chemicals, exceeding a percentage value over 100, and other such problems. We consider these inconsistencies in the context of partial differential equations (PDEs). We propose an innovative filter based on convex optimization to deal with the inconsistencies observed in polynomial-based simulations. In two or three spatial dimensions, additional complexities are involved in solving the problems related to structure preservation. We present the construction and application of a structure-preserving filter with a focus on multidimensional PDEs. Methods used such as the Barycentric interpolation for polynomial evaluation at arbitrary points in the domain and an optimized root-finder to identify points of interest, improve the filter efficiency, usability, and robustness. Lastly, we present numerical experiments in 2D and 3D using discontinuous Galerkin formulation and demonstrate the filter's efficacy to preserve the desired structure. As a real-world application, implementation of the mathematical biology model involving platelet aggregation and blood coagulation has been reviewed and the issues around FEM implementation of the model are resolved by applying the proposed structure-preserving filter.

Wasserstein barycenter regression for estimating the joint dynamics of renewable and fossil fuel energy indices

In order to characterize non-linear system dynamics and to generate term structures of joint distributions, we propose a flexible and multidimensional approach, which exploits Wasserstein barycentric coordinates for histograms. We apply this methodology to study the relationships between the performance in the European market of the renewable energy sector and that of the fossil fuel energy one. Our methodology allows us to estimate the term structure of conditional joint distributions. This optimal barycentric interpolation can be interpreted as a posterior version of the joint distribution with respect to the prior contained in the past histograms history. Once the underlying dynamics mechanism among the set of variables are obtained as optimal Wasserstein barycentric coordinates, the learned dynamic rules can be used to generate term structures of joint distributions.

Planar and Toroidal Morphs Made Easier

We present simpler algorithms for two closely related morphing problems, both based on the barycentric interpolation paradigm introduced by Floater and Gotsman, which is in turn based on Floater’s asymmetric extension of Tutte’s classical spring-embedding theorem. First, we give a very simple algorithm to construct piecewise-linear morphs between planar straight-line graphs. Our algorithm entirely avoids the classical edge-collapsing strategy dating back to Cairns; instead, in each morphing step, we interpolate the pair of weights associated with a single edge. Specifically, given equivalent straight-line drawings $\Gamma_0$ and $\Gamma_1$ of the same 3-connected planar graph $G$, with the same convex outer face, we construct a morph from $\Gamma_0$ to $\Gamma_1$ that consists of $O(n)$ unidirectional morphing steps, in $O(n^{1+\omega/2})$ time, where $n$ is the number of vertices and $\omega$ is the matrix multiplication constant. Second, we describe a natural extension of barycentric interpolation to geodesic graphs on the flat torus. Barycentric interpolation cannot be applied directly in this setting, because the linear systems defining intermediate vertex positions are not necessarily solvable. We describe a simple scaling strategy that circumvents this issue. Computing the appropriate scaling requires $O(n^{\omega/2})$ time, after which we can compute the drawing at any point in the morph in $O(n^{\omega/2})$ time. Our algorithm is considerably simpler than the recent algorithm of Chambers, Erickson, Lin, and Parsa, and produces more natural morphs. Our techniques also yield a simple proof of a conjecture of Connelly, Henderson, Ho, and Starbird for geodesic torus triangulations.

A numerical investigation of nonlinear Schrödinger equation using barycentric interpolation collocation method

<abstract><p>In this paper, we will present a collocation approach based on barycentric interpolation functions and finite difference formulation to study the approximate solution of nonlinear Schrödinger equation. We discretize the time derivative by Crank-Nicolson scheme and bring barycentric interpolation functions into action for spatial discretization. Furthermore, consistency analysis of semi discrete collocation scheme is given. For the nonlinear term, we use Newton iterative method to derive the corresponding linear algebraic equations. Finally, numerical examples show that the numerical scheme has high precision and satisfies the mass and energy conservation.</p></abstract>

The SAV Scheme Based on the Barycentric Interpolation Collocation Method for the Allen-Cahn Equation

The SAV Scheme Based on the Barycentric Interpolation Collocation Method for the Allen-Cahn Equation

Barycentric rational collocation method for semi-infinite domain problems

<abstract><p>The barycentric rational collocation method for solving semi-infinite domain problems is presented. Following the barycentric interpolation method of rational polynomial and Chebyshev polynomial, matrix equation is obtained from discrete semi-infinite domain problem. Truncation method and transformation method are presented to solve linear and nonlinear differential equation defined on the semi-infinite domain problems. At last, three numerical examples are presented to valid our theoretical analysis.</p></abstract>

Linear barycentric rational collocation method for solving a class of generalized Boussinesq equations

<abstract><p>This paper is concerned with solving a class of generalized Boussinesq shallow-water wave (GBSWW) equations by the linear barycentric rational collocation method (LBRCM), which are nonlinear partial differential equations (PDEs). By using the method of direct linearization, those nonlinear PDEs are transformed into linear PDEs which can be easily solved, and the corresponding differentiation matrix equations of their discretization linear GBSWW equations are also given by a Kronecker product. Based on the error estimate of a barycentric interpolation, the rates of convergence for numerical solutions of GBSWW equations are obtained. Finally, three examples are presented to show theoretical results.</p></abstract>

A numerical method based on barycentric interpolation collocation for nonlinear convection-diffusion optimal control problems

<abstract><p>This paper describes a study of the barycentric interpolation collocation method for the optimal control problem governed by a nonlinear convection-diffusion equation. Using Lagrangian multipliers, we obtain the continuous optimality system which is composed of state equations, adjoint equations and optimality conditions. Then, barycentric interpolation collocation methods are applied to discretize the optimality system and the nonlinear term is treated by Newton's iteration. Furthermore, the corresponding consistency analyses of discrete schemes are demonstrated. Finally, the validity of the proposed schemes is demonstrated through several numerical experiments. Compared with the classical finite difference method, collocation schemes can yield the higher-order accurate solutions with fewer nodes.</p></abstract>

Adaptive techniques for solving chaotic system of parabolic-type

Time-dependent partial differential equations of parabolic type are known to have a lot of applications in biology, mechanics, epidemiology and control processes. Despite the usefulness of this class of differential equations, the numerical approach to its solution, especially in high dimensions, is still poorly understood. Since the nature of reaction-diffusion problems permit the use of different methods in space and time, two important approximation schemes which are based on the spectral and barycentric interpolation collocation techniques are adopted in conjunction with the third-order exponential time-differencing Runge-Kutta method to advance in time. The accuracy of the method is tested by considering a number of time-dependent reaction-diffusion problems that are still of current and recurring interests in one and high dimensions.