Barycentric Interpolation Research Articles

Barycentric interpolation collocation method based on Crank-Nicolson scheme for the Allen-Cahn equation

<abstract> This paper proposes a numerical scheme for the Allen-Cahn equation that represents a phenomenological model for anti-phase domain coarsening in a binary mixture. In order to obtain a high order discretization in space, we adopt the barycentric interpolation collocation method. The semi-discretized scheme in space is shown to be consistent. The second-order Crank-Nicolson scheme is used for temporal discretization and the simple iteration method is adopted for nonlinear term. Corresponding algebraic system is derived. Numerical examples are demonstrated to validate the efficiency of the proposed method. </abstract>

Tikhonov regularization for polynomial approximation problems in Gauss quadrature points

This paper is concerned with the introduction of Tikhonov regularization into least squares approximation scheme on [−1, 1] by orthonormal polynomials, in order to handle noisy data. This scheme includes interpolation and hyperinterpolation as special cases. With Gauss quadrature points employed as nodes, coefficients of the approximation polynomial with respect to given basis are derived in an entry-wise closed form. Under interpolatory conditions, the solution to the regularized approximation problem is rewritten in forms of two kinds of barycentric interpolation formulae, by introducing only a multiplicative correction factor into both classical barycentric formulae. An L 2 error bound and a uniform error bound are derived, providing similar information that Tikhonov regularization is able to reduce the operator norm (Lebesgue constant) and the error term related to the level of noise, both by multiplying a correction factor which is less than one. Numerical examples show the benefits of Tikhonov regularization when data is noisy or data size is relatively small.

Application of a collocation method based on linear barycentric interpolation for solving 2D and 3D Klein-Gordon-Schrödinger (KGS) equations numerically

Purpose The purpose of this paper is to obtain accurate numerical solutions of two-dimensional (2-D) and 3-dimensional (3-D) Klein–Gordon–Schrödinger (KGS) equations. Design/methodology/approach The use of linear barycentric interpolation differentiation matrices facilitates the computation of numerical solutions both in 2-D and 3-D space within reasonable central processing unit times. Findings Numerical simulations corroborate the efficiency and accuracy of the proposed method. Originality/value Linear barycentric interpolation method is applied to 2-D and 3-D KGS equations for the first time, and good results are obtained.

An interpolation method for the optimal control problem governed by the elliptic convection–diffusion equation

AbstractWe know that due to the Weierstrass approximation theorem any continuous function over a closed interval can be approximated by a polynomial of sufficiently high degree. Therefore, they are frequently used to approximate functions. We proposed a Lagrange polynomial‐based approach for solving the optimal control problem governed by the elliptic convection–diffusion partial differential equation. This paper solves it by the barycentric interpolation method as a class of strong‐form numerical methods. Thanks to Lagrangian multipliers, optimality system is derived and then the barycentric collocation method is employed to discretize the state variable and adjoint variable. Barycentric interpolation method is a Lagrange polynomial interpolation that is fast and deserves to be known as a method of polynomial interpolation. The convergence of the proposed method is also proved. At the end, numerical experiments are employed to illustrate the theoretical findings.

Numerical solution of special 2D Fredholm-Volterra integral equations using barycentric Gegenbauer interpolation collocation method

In this paper, an efficient collocation method based on two dimensional barycentric Gegenbauer interpolation is used to solve a kind of special two dimensional Fredholm-Volterra integral equations (2D-FVIEs). The explicit barycentric weights for the Gegenbauer-Gauss nodes not only reduce the complicated calculation but also preserve the numerical stability. The combination of the barycentric interpolation and the Legendre-Gauss quadrature formula transforms the 2D-FVIEs into a system of discrete algebraic equations whose solution is a set of the nodal function values. The numerical experiments provide the maximum absolute error and the convergent order to assess the accuracy and the convergence of the method.

Study on the Manifold Cover Lagrangian Integral Point Method Based on Barycentric Interpolation

To achieve numerical simulation of large deformation evolution processes in underground engineering, the barycentric interpolation test function is established in this paper based on the manifold cover idea. A large-deformation numerical simulation method is proposed by the double discrete method with the fixed Euler background mesh and moving material points, with discontinuous damage processes implemented by continuous simulation. The material particles are also the integration points. This method is called the manifold cover Lagrangian integral point method based on barycentric interpolation. The method uses the Euler mesh as the background integral mesh and describes the deformation behavior of macroscopic objects through the motion of particles between meshes. Therefore, this method can avoid the problem of computation termination caused by the distortion of the mesh in the calculation process. In addition, this method can keep material particles moving without limits in the set region, which makes it suitable for simulating large deformation and collapse problems in geotechnical engineering. Taking a typical slope as an example, the results of a slope slip surface obtained using the manifold cover Lagrangian integral point method based on barycentric interpolation proposed in this paper were basically consistent with the theoretical analytical method. Hence, the correctness of the method was verified. The method was then applied for simulating the collapse process of the side slope, thereby confirming the feasibility of the method for computing large deformations.

Enabling parametric design space exploration by non-designers

AbstractIn mass customization, software configurators enable novice end-users to design customized products and services according to their needs and preferences. However, traditional configurators hardly provide an engaging experience while avoiding the burden of choice. We propose a Design Participation Model to facilitate navigating the design space, based on two modules. Modeler enables designers to create customizable designs as parametric models, and Navigator subsequently permits novice end-users to explore these designs. While most parametric designs support direct manipulation of low-level features, we propose interpolation features to give customers more flexibility. In this paper, we focus on the implementation of such interpolation features into Navigator and its user interface. To assess our approach, we designed and performed user experiments to test and compare Modeler and Navigator, thus providing insights for further developments of our approach. Our results suggest that barycentric interpolation between qualitative parameters provides a more easily understandable interface that empowers novice customers to explore the design space expeditiously.

Linear barycentric rational collocation method for solving second-order Volterra integro-differential equation

Second-order Volterra integro-differential equation is solved by the linear barycentric rational collocation method. Following the barycentric interpolation method of Lagrange polynomial and Chebyshev polynomial, the matrix form of the collocation method is obtained from the discrete Volterra integro-differential equation. With the help of the convergence rate of the linear barycentric rational interpolation, the convergence rate of linear barycentric rational collocation method for solving Volterra integro-differential equation is proved. At last, several numerical examples are provided to validate the theoretical analysis.

Mesh adaptation based on transfinite mean value interpolation

In this paper, the efficiency in mesh updating (r-adaptivity) of the Transfinite Mean value Interpolation (TMI) and its generalization (k-TMI) are compared on three standardized problems to the well-known Inverse Distance Weighted interpolation (IDW) and Radial Basis Function interpolation (RBF) for unstructured data points and the new k-Transfinite Barycentric Interpolation (k-TBI) for structured data points such as, for instance, curves or surfaces in 3D. This is achieved by introducing a dynamical version of these interpolations via an ordinary differential equation that can be solved by standard ODE methods that are more economical than, for instance, solving vector partial differential equations as in the pseudo-solid method.A review of the very recent mathematical foundations of the k-TMI and k-TBI constructed from the function alone (standard) or from the function and its derivatives (enhanced) is provided in the first part of the paper.

Numerical solution of a coupled reaction-diffusion model using barycentric interpolation collocation method

In thermal science, chemical and mechanics, the non-linear reaction-diffusion model is very important, and an approximate solution with high precision is always needed. In this article, the barycentric interpolation collocation method is proposed for this purpose. Numerical experiments show that the proposed approach is highly reliable.

Implementation and performance study of lpCMFD acceleration method for multi-energy group k-eigenvalue neutron transport problem in hexagonal geometry

The recently developed lpCMFD acceleration scheme has been shown to be unconditionally stable and effective for SN neutron transport problems in Cartesian rectangular geometry. However, studies of lpCMFD performance for multi-energy group k-eigenvalue neutron transport problem in hexagonal geometries have not been previously investigated. In this paper, linear interpolation method on general convex polygon coarse mesh geometries is developed for lpCMFD in multi-energy group k-eigenvalue neutron transport problems. This method is tested on two hexagonal type reactor benchmark problems discretized with an equilateral triangular fine mesh for SN and different sizes of equilateral triangular and a mixed hexagonal-triangle coarse mesh grid for low order diffusion equation. It was found that lpCMFD was stable and effective for the benchmark problems investigated here with different coarse mesh grids, while CMFD became unstable when coarse mesh size increases. pCMFD was found to be stable in all configurations but was generally slower or as effective as lpCMFD and CMFD. The barycentric interpolation technique developed in this paper for lpCMFD in 2D hexagonal geometry is demonstrated to be effective and can be applied in realistic practical problems.

A New Solution for Optimal Control of Fractional Convection–Reaction–Diffusion Equation Using Rational Barycentric Interpolation

This paper solves an optimal control problem governed by a fractional convection–reaction–diffusion partial differential equation. Using Lagrangian multipliers, necessary conditions are obtained, and then, Barycentric collocation method are applied for discretizing classical derivatives and Grunwald–Letnikov formula for fractional derivative. Barycentric interpolation is a class of Lagrange polynomial interpolation that is fast and deserves to be known as a method of polynomial interpolation and Grunwald–Letnikov formula is a basic extension of the derivative in fractional calculus. Numerical examples are presented to show the effectiveness of the method.

Minimax strategy in approximate model predictive control

It is known that a model predictive control law for a linear dynamical system with a linear or quadratic cost function can be explicitly computed as a piece-wise affine function. However, the number of regions required grows rapidly with the horizon length, the number of states and constraints limiting the deployment of explicit solutions to relatively small MPC problems, and motivating approximate solutions requiring less storage for online implementation. Unfortunately, the offline computation required to generate the approximate solution can be very high using many existing algorithms. In this paper, we propose a new procedure to generate the approximate solution based on barycentric interpolation whilst retaining a (less conservative) certification of the controller. This novel certification procedure requires solving a number of small-scale multiparametric linear programs together with convex optimization problems. During the online implementation of the approximate MPC, only a small linear program has to be solved to evaluate the control law. The efficacy of the proposed approach is demonstrated through both simulation and also in application to a canonical cart–pole stabilization problem.

Barycentric interpolation on Riemannian and semi-Riemannian spaces

ABSTRACT Interpolation of data represented in curvilinear coordinates and possibly having some non-trivial, typically Riemannian or semi-Riemannian geometry is a ubiquitous task in all of physics. In this work, we present a covariant generalization of the barycentric coordinates and the barycentric interpolation method for Riemannian and semi-Riemannian spaces of arbitrary dimension. We show that our new method preserves the linear accuracy property of barycentric interpolation in a coordinate-invariant sense. In addition, we show how the method can be used to interpolate constrained quantities so that the given constraint is automatically respected. We showcase the method with two astrophysics related examples situated in the curved Kerr space–time. The first problem is interpolating a locally constant vector field, in which case curvature effects are expected to be maximally important. The second example is a general relativistic magnetohydrodynamics simulation of a turbulent accretion flow around a black hole, wherein high intrinsic variability is expected to be at least as important as curvature effects.

Characteristic Scheme for the Solution of the Transport Equation on an Unstructured Grid with Barycentric Interpolation

An interpolative-characteristic method is constructed with the approximation not lower than the second order for solving the transport equation on an unstructured grid composed of tetrahedra. The problem of finding a numerical solution by this method, further referred to as the method of short characteristics, is subdivided into two subproblems. The first concerns the resolution of a single simplicial cell. A set of grid values must be specified, which when set on the illuminated faces is mathematically sufficient for obtaining all the remaining grid values in the cell. Depending on the location of the cell and the propagation direction of the radiation there are three different types of illumination. It is proposed to use interpolation of a cell in barycentric coordinates with 14 free coefficients, which enables taking into account the values of the radiation intensity at the nodes and the average integral values of the intensity over the edges and faces without adding new stencil points. This interpolation ensures at least the second order of approximation with additional allowance for the terms with the third-order approximation. Moreover, the method takes into account a conservative redistribution of the outbound flow over the edges of the cell. The second subproblem is associated with choosing the order of tracing the cell and can be solved using methods of graph theory. The numerical calculations confirmed approximately the second order of convergence.

A Highly Accurate Collocation Method for Linear and Nonlinear Vibration Problems of Multi-Degree-Of-Freedom Systems Based on Barycentric Interpolation

Abstract A highly accurate collocation method based on barycentric interpolation (BICM) is proposed for solving linear and nonlinear vibration problems for multi-degree-of-freedom systems in this article. The mathematical model of the linear and nonlinear vibrations of multi-degree-of freedom systems is the initial value problem of the linear and nonlinear differential equations. The numerical solution of the linear differential equations can be directly solved by BICM. The numerical solution of nonlinear differential equations can be solved as following: Firstly, the nonlinear governing equation is converted to linear differential equation by assuming the initial function. Secondly, the linear differential equations are discretized into algebraic equations by using barycentric interpolation differential matrices. Thirdly, the numerical solution can be calculated by iteration method under given control precision. Finally, the numerical solution of calculation examples by using barycentric Lagrange interpolation iteration collocation method (BLIICM) and barycentric rational interpolation iteration collocation method (BRIICM) are compared with the analytical solution. Numerical results illustrate the advantages of proposed methodology are efficient, fast, simple formulations, and high precision. Comparing with BLIICM, BRIICM can still maintain its computational stability when dealing with a large number of nodes, especially the equidistant nodes.

Numerical Simulation of the Lorenz-Type Chaotic System Using Barycentric Lagrange Interpolation Collocation Method

Although some numerical methods of the Lorenz system have been announced, simple and efficient methods have always been the direction that scholars strive to pursue. Based on this problem, this paper introduces a novel numerical method to solve the Lorenz-type chaotic system which is based on barycentric Lagrange interpolation collocation method (BLICM). The system (1) is adopted as an example to elucidate the solution process. Numerical simulations are used to verify the effectiveness of the present method.

The barycentric interpolation collocation method for a class of nonlinear vibration systems

A nonlinear vibration system arises in physics. Besides its mathematical model, it is of great importance to have an accurate and reliable solution to the system. Though there are many analytical methods, such as the variational iteration method and the homotopy perturbation method, numerical approaches are rare. This paper suggests the barycentric interpolation collocation method to solve nonlinear oscillators. The Duffing equation is adopted as an example to elucidate the solution process. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.

Numerical Simulation of a Class of Three‐Dimensional Kolmogorov Model with Chaotic Dynamic Behavior by Using Barycentric Interpolation Collocation Method

This paper numerically simulates three‐dimensional Kolmogorov model with chaotic dynamic behavior by barycentric Lagrange interpolation collocation method. Some numerical examples are studied for finding some new chaotic behaviors and demonstrating some existing chaotic dynamic behaviors of the Kolmogorov model. Results obtained by the present method indicate that the method has merits of small operations and good numerical stability.

Numerical solution of a class of advection-reaction-diffusion system

In this article, the barycentric interpolation collocation methods is proposed for solving a class of non-linear advection-reaction-diffusion system. Compared with other methods, the numerical experiment shows the barycentric interpolation collocation method is a high precision method to solve the advection- reaction-diffusion system.