Interpolation Approximation Research Articles

Improved hole repairing algorithm for livestock point clouds based on cubic B-spline for region defining

Because of the occlusion of the railings, the collected animal point clouds become fragmentary and have the large holes. The incomplete point clouds have a great impact on body size measurements of pigs. This paper presents an improved hole repairing algorithm based on cubic B-spline curve for region defining. Meanwhile, by applying approximation fitting and interpolation based on cubic B-spline curve, we can repair the holes of the pig point clouds after slicing longitudinally in the mid-axis plane of the pig body point clouds. In the experiment, comparisons were made among polar coordinate transformation algorithm, surface fitting algorithm, triangularization algorithm, and the algorithm proposed by us to verify hole repairing effect and estimation accuracy of abdominal circumference. The results show that we have proposed an algorithm with a better repairing effect and a higher accuracy in the abdominal circumference measurement of pigs.

A Hermite interpolation element-free Galerkin method for piezoelectric materials

Piezoelectric materials have played an important role in industry due to a number of beneficial properties. However, most numerical methods for the piezoelectric materials need mesh, in which the mesh generation and remeshing are prominent difficulties. This paper proposes a Hermite interpolation element-free Galerkin method (HIEFGM) for piezoelectric materials, where the Hermite approximate approach and interpolation element-free Galerkin method (IEFGM) are combined. Based on the constitutive equation, geometric equation, and Galerkin integral weak form, the HIEFGM formulation for piezoelectric materials is established. In the proposed method, the problem domain is discretized by many nodes rather than the meshes, so the pre-processing of numerical computation is simplified. Furthermore, a new approximation technique based on the moving least squares method and Hermite approximate approach is used to derive the approximation function of field quantities. The derived approximation function has the Kronecker delta property and considers the field quantity normal derivatives of boundary nodes, which avoids the problem of imposing the essential boundary conditions and improves the accuracy of meshless approximation. The effects of the scaling factor, node density, and node arrangement on the accuracy of the proposed method are investigated. Numerical examples are given for assessing the proposed method and the results uniformly demonstrate the proposed method has excellent performance in analyzing piezoelectric materials.

Using the Concept of Fuzzy Logic and Controller System in Interpolation and Functions Approximation

The main aim of this contribution is to present an alternative approach for interpolation and function approximation from a given data. Instead of the traditional interpolation methods we consider and propose a numerical procedure for interpolation using the concept of fuzzy logic and membership functions. The method can be used for interpolating data resulting from physical experiments, engineering, medicine, applied sciences etc. Fuzzification will be applied to the given data according to Mamdani technique and membership functions will be chosen to satisfy the interpolation mathematical condition. The defuzzification process will be implemented to get the crisp values of the interpolation. The procedure will be implemented on the mathematical code MATLAB and its Simulink fuzzy logic features. Finally the applicability and efficiency of the procedure is illustrated by numerical examples.

Medical Surgical Video Recognition and Retrieval Based on Novel Unified Approximation

Video retrieval recognition is a linear characterized action constituted by many frame similarity-based videos. This medical video recognition and classification can be a great extent in medical research, such as Endoscopic, radiological, pathological, and applied health informatics. General Video Retrieval Recognition (GVRR) cannot address a problem with recognition alone. GVRR can be solving the Multi-Input-Multi-Output (MIMO) interface mixed video retrieval system. To generalize the conventional video retrieval interface like Multi-user MIMO, WiMAX MIMO, single-user MIMO, several types of research made excused. In fine-tuning existing video retrieval, this research gives the authentic procedure for a frame-based cognitive operation called Secure Approximation and sTability Based Secure Video Retrieval recognition (SAT-SR) recognition proposed. In this research article, the process of recognition has three processes generalized by the video retrieval system. Initially, the virtual dissection and connection weights of input video were established using the mathematical and numerical analysis of interpolation estimation. Secondly, the interpolation approximation and activation function were figured out using the Open Mcrypt Stimulus (oMs) for video security fragments. Similarly, systematic investigations are accomplished for approximation error computation. The result for this widely circulated utilization of three processes on the video retrieval recognition prevents the occurrence of the cybercrime abuse of stored video registers. The proposed technique was used to identify the virtual dissection, interpolation, and activation function for decoding the videos. Using this information, the abusers identified cybercrime rate might be reduced considerably.

Generalizations of the constrained mock-Chebyshev least squares in two variables: Tensor product vs total degree polynomial interpolation

The constrained mock-Chebyshev least squares interpolation is a univariate polynomial interpolation technique exploited to cut-down the Runge phenomenon. It takes advantage of the optimality of the interpolation on the mock-Chebyshev nodes, i.e. the subset of the uniform grid formed by nodes that mimic the behavior of Chebyshev–Lobatto nodes. The other nodes of the grid are not discarded, rather they are used in a simultaneous regression to improve the accuracy of the approximation of the mock-Chebyshev subset interpolant. In this paper we extend the univariate constrained mock-Chebyshev least squares interpolation to the bivariate case in two different ways, relying on the tensor product interpolation and on the interpolation at the mock-Padua nodes. Numerical experiments demonstrate the effectiveness of such extensions.

A Shape Optimisation with the Isogeometric Boundary Element Method and Adjoint Variable Method for the Three-Dimensional Helmholtz Equation

This paper presents a shape optimisation system to design the shape of an acoustically-hard object in the three-dimensional open space. The boundary element method (BEM) is suitable to analyse such an exterior field. However, the conventional BEM, which is based on piecewise polynomial shape and approximate (interpolation) functions, can require many design variables because they are usually chosen as a part of the nodes of the underlying boundary element mesh. In addition, it is not easy for the conventional method to compute the gradient of the sound pressure on the surface, which is necessary to compute the shape derivative of our interest, of a given object. To overcome these issues, we employ the isogeometric boundary element method (IGBEM), which was developed in our previous work. With using the IGBEM, we can design the shape of surfaces through control points of the NURBS surfaces of the target object. We integrate the IGBEM with the nonlinear programming software through the adjoint variable method (AVM), where the resulting adjoint boundary value problem can be also solved by the IGBEM with a slight modification. The numerical verification and demonstration validate our shape optimisation framework.

Two new approximations to Caputo–Fabrizio fractional equation on non-uniform meshes and its applications

We present two numerical approximations with non-uniform meshes to the Caputo–Fabrizio derivative of order α (0 < α < 1). First, the L1 formula is obtained by using the linear interpolation approximation for constructing the second-order approximation. Next, the quadratic interpolation approximation is used for improving the accuracy in the temporal direction. Besides, we discretize the spatial derivative using the compact finite difference scheme. The accuracy of the suggested schemes is not dependent on the fractional α. The coefficients and the truncation errors are carefully investigated for two schemes, separately. Three examples are carried out to support the convergence orders and show the efficiency of the suggested scheme.

Block-centered local refinement methods for the time-fractional equations

In this paper, we present and analyze two block-centered local refinement (BLR) methods for solving the time-fractional equations. The main difference between the two methods is that different approximation methods are used for the value of the pressure at the slave nodes. One adopts a piecewise constant interpolation approximation (PCIA) method, which is called simple block-centered local refinement (S-BLR) method. The other utilizes a piecewise linear interpolation approximation (PLIA) method, which called more accurate block-centered local refinement (MA-BLR) method. The stability analysis is proved carefully. It is estimated that the discrete L2 errors for the velocity and pressure are O(▵t2−α+h3/2) and O(▵t2−α+h3/2) in use of S-BLR and MA-BLR methods, respectively. Where Δt is the time step and h is the maximal mesh size. These error estimate results are all established on locally refinement composite grids. Finally, a numerical experiment is presented to show that the convergence rates are in agreement with the theoretical analysis.

An Improved High-Dimensional Kriging Surrogate Modeling Method through Principal Component Dimension Reduction

The Kriging surrogate model in complex simulation problems uses as few expensive objectives as possible to establish a global or local approximate interpolation. However, due to the inversion of the covariance correlation matrix and the solving of Kriging-related parameters, the Kriging approximation process for high-dimensional problems is time consuming and even impossible to construct. For this reason, a high-dimensional Kriging modeling method through principal component dimension reduction (HDKM-PCDR) is proposed by considering the correlation parameters and the design variables of a Kriging model. It uses PCDR to transform a high-dimensional correlation parameter vector in Kriging into low-dimensional one, which is used to reconstruct a new correlation function. In this way, time consumption of correlation parameter optimization and correlation function matrix construction in the Kriging modeling process is greatly reduced. Compared with the original Kriging method and the high-dimensional Kriging modeling method based on partial least squares, the proposed method can achieve faster modeling efficiency under the premise of meeting certain accuracy requirements.

АНАЛИЗ УСТОЙЧИВОСТИ ЖЕСТКИХ СИСТЕМ ОБЫКНОВЕННЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ

A method for analyzing stability in the sense of Lyapunov for systems of ordinary differentialequations is proposed. The method is based on stability criteria in the form of necessary and sufficientconditions obtained on the basis of vector-matrix transformations of difference numericalintegration schemes. The varieties of criteria in multiplicative, additive and matrix form are presented.The design of the criteria implies the possibility of their programmatic realization. To increasethe reliability of the stability analysis, the approximations of the solution included in theconstruction of the criteria are based on piecewise interpolation approximation by Lagrange polynomialsconverted to a form with numerical coefficients. A programming and numerical experimentis carried out to analyze the stability of the Belousov-Jabotinsky periodic reaction model,which belongs to the class of rigid systems, under given initial conditions. The analysis is carriedout on the basis of the presented criteria and the results of the program clearly determine the natureof the stability in real time. Based on the results of the experiment, it can be argued that replacingthe difference approximations of the solution with piecewise interpolation approximationsincreases the reliability of the stability analysis, reduces the study time, and makes it possible todetermine the asymptotic properties of the solution. In general, the proposed approach is an alternativeto the methods of the qualitative theory of differential equations and makes it possible toreliably determine the stability of rigid systems of ordinary differential equations in real time.

Volumetric Lattice Boltzmann Models in General Curvilinear Coordinates: Theoretical Formulation

A theoretical formulation of lattice Boltzmann models on a general curvilinear coordinate system is presented. It is based on a volumetric representation so that mass and momentum are exactly conserved as in the conventional lattice Boltzmann on a Cartesian lattice. In contrast to some previously existing approaches for arbitrary meshes involving interpolation approximations among multiple neighboring cells, the current formulation preserves the fundamental one-to-one advection feature of a standard lattice Boltzmann method on a uniform Cartesian lattice. The new approach is built on the concept that a particle is moving along a curved path. A discrete space-time inertial force is derived so that the momentum conservation is exactly ensured for the underlying Euclidean space. We theoretically show that the new scheme recovers the Navier-Stokes equation in general curvilinear coordinates in the hydrodynamic limit, along with the correct mass continuity equation.

An Experimental Investigation of Viscoelastic Flow in a Contraction Channel.

In order to assess the predictive capability of the S–MDCPP model, which may describe the viscoelastic behavior of the low-density polyethylene melts, a planar contraction flow benchmark problem is calculated in this investigation. A pressure-stabilized iterative fractional step algorithm based on the finite increment calculus (FIC) method is adopted to overcome oscillations of the pressure field due to the incompressibility of fluids. The discrete elastic viscous stress splitting (DEVSS) technique in combination with the streamline upwind Petrov-Galerkin (SUPG) method are employed to calculate the viscoelastic flow. The equal low-order finite elements interpolation approximations for velocity-pressure-stress variables can be applied to calculate the viscoelastic contraction flows for LDPE melts. The predicted velocities agree well with the experimental results of particle imagine velocity (PIV) method, and the pattern of principal stress difference calculated by the S-MDCPP model has good agreement with the results measured by the flow induced birefringence (FIB) device. Numerical and experimental results show that the S-MDCPP model is capable of accurately capturing the rheological behaviors of branched polymers in complex flow.

An improved higher-order explicit time integration method with momentum corrector for linear and nonlinear dynamics

An improved explicit time integration method is proposed for linear and nonlinear dynamics. Its calculation procedure is obtained with cubic B-spline interpolation approximation and weighted residual method. In the formulation, a momentum corrector is used to improve actual computation accuracy, especially for some special discontinuous loads. Analytical solutions of the local truncation errors, algorithmic damping and period elongation have been deduced to obtain the influence of algorithmic parameters on these basis algorithmic properties. The proposed method possesses at least second-order accuracy and can achieve at most third-order accuracy for no physical damping case. With free algorithmic parameters, the proposed method has controllable stability and numerical dissipation. Some demonstrative numerical examples are tested to confirm high efficiency of the proposed method for a variety of dynamic problems such as, dynamic response analysis of linear systems under various representative applied loads, finite element analysis (FEA) for dynamic response of engineering structures, and nonlinear dynamic analysis for strong nonlinear system.

Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs

By combining a certain approximation property in the spatial domain, and weighted 𝓁2-summability of the Hermite polynomial expansion coefficients in the parametric domain obtained in Bachmayr et al. [ESAIM: M2AN 51 (2017) 341–363] and Bachmayr et al. [SIAM J. Numer. Anal. 55 (2017) 2151–2186], we investigate linear non-adaptive methods of fully discrete polynomial interpolation approximation as well as fully discrete weighted quadrature methods of integration for parametric and stochastic elliptic PDEs with lognormal inputs. We construct such methods and prove convergence rates of the approximations by them. The linear non-adaptive methods of fully discrete polynomial interpolation approximation are sparse-grid collocation methods which are certain sums taken over finite nested Smolyak-type indices sets of mixed tensor products of dyadic scale successive differences of spatial approximations of particular solvers, and of successive differences of their parametric Lagrange interpolating polynomials. The Smolyak-type sparse interpolation grids in the parametric domain are constructed from the roots of Hermite polynomials or their improved modifications. Moreover, they generate in a natural way fully discrete weighted quadrature formulas for integration of the solution to parametric and stochastic elliptic PDEs and its linear functionals, and the error of the corresponding integration can be estimated via the error in the Bochner space L1 (ℝ∞, V, γ) norm of the generating methods where γ is the Gaussian probability measure on ℝ∞ and V is the energy space. We also briefly consider similar problems for parametric and stochastic elliptic PDEs with affine inputs, and problems of non-fully discrete polynomial interpolation approximation and integration. In particular, the convergence rates of non-fully discrete polynomial interpolation approximation and integration obtained in this paper significantly improve the known ones.

Piecewise interpolation solution of ordinary differential equations with application to numerical modeling problems

Piecewise interpolation approximation of functions of one real variable, derivatives and integrals is constructed with the help of Lagrange and Newton interpolation polynomials. The polynomials are transformed to the form of algebraic polynomials with numerical coefficients by means of restoring the polynomial coefficients by its roots. Formulas different from Vieta’s formulas are applied. In the resulting form, the polynomials interpolate the right-hand sides of ordinary differential equations, the expression of the antiderivative ones is used to approximate the solution. Iterative refinement is performed. Error estimates and results of numerical experiments for stiff and non-stiff problems in physical, chemical models and other processes are presented.

Parameter identification in multidimensional hyperbolic partial differential equations using wavelet collocation method

In this article, we have proposed a highly efficient and accurate collocation method based on Haar wavelet for the parameter identification in multidimensional hyperbolic partial differential equations (PDEs). In the proposed method, highest order derivative is represented in terms of Haar wavelet and required term of the PDE is obtained using successive integration. Due to compact support and orthogonality of Haar wavelet, this method leads to significant reduction in the computational cost. Cubic spline interpolation and Taylor series approximation are used for the parameter identification. Error analysis is carried out in order to prove the convergence of the method. The obtained numerical results and central processing unit (CPU) time ensure the efficiency and accuracy of the method.

Generation and Optimization of interpolation approximation algorithm based on Laser Icon Contour Generation

Point-by-point comparison interpolation is a widely used algorithm in laser marking system. This algorithm plays a very important role in controlling the laser trajectory and fitting the image trajectory. Because the point-by-point comparison interpolation method has the characteristics of intuitive operation, uniform pulse output and convenient adjustment, it is a kind of interpolation algorithm which is more suitable for the application of marking machine system. In order to improve the marking efficiency of vector text in laser marking system, by analyzing the outline structure of TrueType font and the properties of Bezier curve, an optimization algorithm for straight line approximation of TrueType font outline is presented, and the theoretical analysis and experimental verification are carried out. Firstly, the Bezier curve in the TrueType font outline is segmented according to the deCasteljau recursive algorithm, and then the curve is replaced by the connection of the head and tail control points. Finally, the approximate line segment is interpolated to generate marking data, and the straight line approximation outline is compared with the standard TrueType font outline. The results show that under the condition of satisfying the accuracy, the algorithm has the advantages of simple calculation process and a small number of nodes, and improves the marking efficiency of vector text. The traditional point-by-point comparison interpolation algorithm has some problems such as low precision and slow speed. In order to improve the performance of the algorithm, the traditional point-by-point comparison interpolation algorithm is improved, and the calculation and simulation are carried out. The results show that the accuracy and speed of the improved interpolation algorithm are obviously improved. The patch is extracted from the point cloud data, and the feature patch is generated by patch growth and fusion, and the feature line is constructed quickly according to the proposed patch two-way index method, and finally the point cloud plane contour is generated. Experiments show that this algorithm can generate point cloud contours quickly, accurately and effectively.

Real-Time Estimation of R0 for COVID-19 Spread

We propose a real-time approximation of R0 in an SIR-type model that applies to the COVID-19 epidemic outbreak. A very useful direct formula expressing R0 is found. Then, various type of models are considered, namely, finite differences, cubic splines, Piecewise Cubic Hermite interpolation and linear least squares approximation. Preserving the monotonicity of the formula under consideration proves to be of crucial importance. This latter property is preferred over accuracy, since it maintains positive R0. Only the Linear Least Squares technique guarantees this, and is finally proposed here. Tests on real COVID-19 data confirm the usefulness of our approach.

Unconditional convergence of iterative approximation methods

In this paper, we propose several iterative formats for quasi-interpolation by using positive definite radial basis functions. Firstly, we present the methodology of approximate interpolation, and then employ the diagonally compensated reduction to preprocess the iterative method. We show that the proposed iterative methods are convergent no matter which shape parameter is used, therefore it can transform a nonconvergent iterative format into a convergent one. The proposed iterative methods can approximate the corresponding exact interpolant with arbitrary precision. Numerical results illustrate the effectiveness of our methods.

A Novel Unified Framework for Solving Reachability and Invariance Problems

The level set method is a widely used tool for solving reachability and invariance problems. However, some shortcomings, such as large amount of storage space consumption and the difficulty of constructing terminal conditions for solving the Hamilton-Jacobi partial differential equation, limit the application of the level set method in some problems, especially those with irregular target sets. This paper proposes a method that can effectively avoid the above tricky issues and thus has better generality. In the proposed method, the reachable or invariant sets with different time horizons are characterized by some non-zero sublevel sets of a value function. This value function is not obtained by solving a viscosity solution of the partial differential equation but by recursion and interpolation approximation. At the end of this paper, some examples are taken to illustrate the accuracy and generality of the proposed method.