Interpolation Operator Research Articles

Dissipativity and contractivity of the second-order averaged L1 method for fractional Volterra functional differential equations

<abstract><p>This paper focuses on the dissipativity and contractivity of a second-order numerical method for fractional Volterra functional differential equations (F-VFDEs). Firstly, an averaged L1 method for the initial value problem of F-VFDEs is presented based on the averaged L1 approximation for Caputo fractional derivative together with an appropriate piecewise interpolation operator for the functional term. Then the averaged L1 method is proved to be dissipative with an absorbing set and contractive with an algebraic decay rate. Finally, the numerical experiments further confirm the theoretical results.</p></abstract>

Univariate range-restricted [formula omitted] interpolation algorithms

In this paper, we provide two O(NlogN) (O(N) if the points are sorted) univariate interpolation algorithms, with one preserving nonnegativity and the other one preserving a finite range [λ,Λ]. The functions produced by the algorithms are, respectively, 12th and 17th-degree C2-splines that are second-order approximation of an optimal C2 interpolant.These algorithms rely on the explicit construction of bounded (nonlinear) range-preserving C2 interpolation operators inspired by the previous works of Jiang and Luli (2021) and Fefferman et al. (2021).

Coarray Polarization Smoothing for DOA Estimation with Coprime Vector Sensor Arrays

In this paper, the problem of direction-of-arrival (DOA) estimation of multiple sources using a coprime electromagnetic vector sensor (EMVS) array is addressed. Each EMVS consists of three mutually orthogonal electric dipoles and three mutually orthogonal magnetic loops, all collocated at a single point in space. By exploiting the polarization diversities embedded in the vector sensors, the coarray polarization smoothing (COPS) is presented to increase the degrees of freedom (DOFs) from the polarization coarray domain. In contrast to the widely used spatial smoothing technique, the COPS offers the following insights: (1) it is source polarization-dependent and is proved to offer 20 DOFs at most, (2) it does not reduce the efficient spatial coarray aperture of the coprime array so that it enables the use of all virtual coarray sensors without resorting to the nontrivial sparse recovery or array interpolation operations, and (3) it can be synthesized with the spatial smoothing to get the DOFs multiplied. Finally, the efficacy of the COPS scheme is verified by numerical examples.

Investigation of the Convergence of a Multi-Grid Algorithm for Solving the Task of Pressure in the Thin Lubricating Film of the Non-Newtonian Fluid

The article describes a multi-grid algorithm for integrating the Reynolds equation for hydrodynamic pressures in the lubricating film of a heavy-loaded journal bearing. This equation is the basic one in solving the tasks of designing friction units of piston- and rotary machines. Lubrication sources of various configurations in the form of grooves and holes located on the friction surfaces were taken into account. The version of the multi-grid algorithm developed by the authors is based on Brandt’s work. At each level of grids, not only the convergence of the solution is controlled, but also the rate of convergence. The pressure equation was approximated by finite differences using the control volume method and passed to a system of algebraic equations, which were solved by the Seidel method. Bessel formulas were used as the interpolation operator. The function for taking into account the non-Newtonian properties of the lubricant is based on the power law. Comparison of the developed algorithm with other versions showed high efficiency. The use of multi-grid algorithms makes it possible to perform multi-variant calculations of the dynamics of heavily loaded bearings. As a result of the calculations, the characteristics of the connecting rod bearing of the heat engine, as well as the multilayer bearing of the turbocharger, are presented.

Coarse-proxy reduced basis methods for integral equations

In this paper, we introduce a new reduced basis methodology for accelerating the computation of large parameterized systems of high-fidelity integral equations. Core to our methodology is the use of coarse-proxy models (i.e., lower resolution variants of the underlying high-fidelity equations) to identify important samples in the parameter space from which a high quality reduced basis is then constructed. Unlike the more traditional POD or greedy methods for reduced basis construction, our methodology has the benefit of being both easy to implement and embarrassingly parallel. We apply our methodology to the under-served area of integral equations, where the density of the underlying integral operators has traditionally made reduced basis methods difficult to apply. To handle this difficulty, we introduce an operator interpolation technique, based on random sub-sampling, that is aimed specifically at integral operators. To demonstrate the effectiveness of our techniques, we present two numerical case studies, based on the radiative transport equation and a boundary integral formation of the Laplace equation respectively, where our methodology provides a significant improvement in performance over the underlying high-fidelity models for a wide range of error tolerances. Moreover, we demonstrate that for these problems, as the coarse-proxy selection threshold is made more aggressive, the approximation error of our method decreases at an approximately linear rate. Finally, we provide a public repository of our source code with easy instructions for reproducing all results in this paper.

Ensemble feature extraction-based prediction of fetal arrhythmia using cardiotocographic signals

The classification of Cardiotocography (CTG) signal abnormalities is critical in the identification of fetal anomalies. The non-stationary nature of CTG, as well as the dataset asymmetry, make this classification task complicated. Manual CTG analysis interpretation can easily result in a low diagnosis rate, which is mainly caused by a variety of subjective variables. Hence, in this research a new Ensemble Feature Extraction (EFE) model is implemented to analyze the nonlinear cardiotocographic signals. First, the collected signal artifacts are removed with interpolation operation, then down sampled to eliminate the complexity during classification. Followed by, a new EFE module is applied to extract the features, here learning model SE-ResNet-50 (Squeeze and Excitation -ResNet-50) is applied to separate the features from the regular and Time-Frequency data transformed (via CWT transform) signal. Wavelet Packet Transform is exploited to extract T-F signals. Then the enhanced feature set dimension is lessened by using Linear Discriminant Analysis (LDA) model. Finally, the features are fused and fed to the XG-booster for classification, and the patient is advised to seek therapy based on the outcome. The proposed method achieved 96.34% accuracy, 96.7% precision, 97.3% recalls, and 95.64% Quality Index (QI) on two-class classifications. This ensemble feature extraction increases the classification results and proves very advantageous to cardiotocographic signal classification. The results of the experiments show how effective the suggested EFE is.

On the theory of operator interpolation in spaces of rectangular matrixes

The problem of constructing and studying interpolation operator polynomials of an arbitrary fixed degree, defined in spaces of rectangular matrices, which would be generalisations of the corresponding interpolation formulas in the case of square matrices, is considered. Linear interpolation formulas of various structures are constructed for rectangular matrices. Matrix polynomials, with respect to which the resulting interpolation formulas are invariant, are indicated. As a generalisation of linear formulas, formulas for quadratic interpolation and interpolation by polynomials of arbitrary fixed degree in the space of rectangular matrices are constructed. Particular cases of the obtained formulas are considered: when square matrices are chosen as nodes or when the values of the interpolated function are square matrices, as well as the case when both of these conditions are satisfied. For the last variant, the possibilities of different and identical matrix orders for nodes and function values are explored. The obtained results are based on the application of some well-known provisions of the theory of matrices and the theory of interpolation of scalar functions. The presentation of the material is illustrated by a number of examples.

The supercloseness property of the Stoke projection for the transient Navier–Stokes equations and global superconvergence analysis

In this paper, we derive the supercloseness properties and global superconvergence results for the implicit Euler scheme of the transient Navier–Stokes equations. Using a prior estimate of finite element solutions, the properties of the Stokes projection and Stokes operator, the derivative transforming skill and the \(H^{-1}\)-norm estimate, we deduce the supercloseness properties of the Stokes projection for the velocity in \(L^\infty (H^1)\)-norm and pressure in \(L^\infty (L^2)\)-norm. Then the supercloseness properties of the interpolation operators are obtained for two pairs of rectangular element: the bilinear-constant element and the Bernadi–Raugel element. Finally, by the interpolation postprocessing technique, we obtain the global superconvergent results. Compared with previous results, no time step restriction is required in the analysis, and moreover, the supercloseness analysis is based on the Stokes projection, which makes the proof more concise.

A modelling framework for efficient reduced order simulations of parametrised lithium-ion battery cells

AbstractIn this contribution, we present a modelling and simulation framework for parametrised lithium-ion battery cells. We first derive a continuum model for a rather general intercalation battery cell on the basis of non-equilibrium thermodynamics. In order to efficiently evaluate the resulting parameterised non-linear system of partial differential equations, the reduced basis method is employed. The reduced basis method is a model order reduction technique on the basis of an incremental hierarchical approximate proper orthogonal decomposition approach and empirical operator interpolation. The modelling framework is particularly well suited to investigate and quantify degradation effects of battery cells. Several numerical experiments are given to demonstrate the scope and efficiency of the modelling framework.

A priori and a posteriori error analysis of hp spectral element discretization for optimal control problems with elliptic equations

. A priori and a posteriori error estimates of hp spectral element method for an elliptic optimal control problem with L2-norm control constraint are investigated in this paper. We then present the optimality conditions and set up hp spectral element discretization scheme. Based on some important interpolation operators and suitable immediate variable, a priori error estimates for this problem in hp spectral element discretization are established carefully. Furthermore, using some interpolation operators, a rigorous posteriori error estimates of hp spectral element approximation are also proved for control and the coupled state approximation in L2−H1-norm and L2−L2-norm, respectively. Such estimators can be used to construct reliable adaptive spectral element methods for optimal control problems. Finally, the error analysis results are confirmed by numerical results.

Convergence analysis of a symmetric dual-wind discontinuous Galerkin method for a parabolic variational inequality

This paper investigates a symmetric dual-wind discontinuous Galerkin (DG) method for solving parabolic variational inequalities. By employing a symmetric dual-wind DG discretization in space and a backward Euler discretization in time, we propose a fully discrete scheme to solve a time-dependent obstacle problem. Under reasonable regularity assumptions on the exact solution, we prove the convergence of numerical solutions with rates in the L∞(L2) and L2(H1)-like energy errors by introducing a new interpolation operator which is a combination of the standard interpolation operator and a positive-preserving interpolation operator. Numerical experiments are provided to validate the effectiveness of the proposed method.

An adaptive nonhydrostatic dynamical core using a multimoment finite‐volume method on a cubed sphere

Abstract: An adaptive nonhydrostatic atmospheric dynamical core was developed on a Cartesian grid and extended to spherical geometry with the application of a cubed‐sphere grid. To assure a practical Courant‐Friedrichs‐Lewy number for stability, a horizontally explicit and vertically implicit algorithm was applied in this model through a third‐order implicit–explicit Runge–Kutta scheme. A two‐dimensional adaptive grid was applied in the horizontal directions to generate the computational meshes, with variable resolutions according to the evolution of the predicted variables during the simulations, while the vertical grid is fixed considering the characteristics of atmospheric flows and the computational efficiency in practical applications. The Berger–Oliger block‐structured adaptive algorithm was adopted in this study. Blocks with different resolutions can be constructed straightforwardly on each patch of the cubed sphere and an algorithm to exchange both solution and block information between adjacent patches was designed in order to implement a global model. The nonhydrostatic governing equations are solved by a three‐point multimoment constrained finite‐volume scheme. Using the compact stencil for spatial reconstructions, the interpolation operations between coarse–fine blocks can be implemented efficiently and the local‐based scheme is also helpful to suppress the computational modes around the coarse–fine interfaces due to the abrupt change of grid resolution. Additionally, flux corrections were conducted along the block boundaries and the resulting dynamical core is rigorously conservative. The proposed model was evaluated by calculating several idealized benchmark tests, and the effectiveness of the adaptive model in saving computational costs was verified in this study.

A new type of fixed point theorem via interpolation of operators with application in homotopy theory

The purpose of this paper is to introduce the class of multi-valued operators by the technique of interpolation of operators. Our results extend and generalize several results from the existing literature. Moreover, we also study the data dependence problem of the fixed point set and Ulam–Hyers stability of the fixed point problem for the operators introduced herein. Moreover, as an application, we obtain a homotopy result.

Reconstruction of multidimensional digital signals

We study the reconstruction of multidimensional digital signals in multirate digital signal processing. We show that the composition of sampling operator and interpolation operator forms the frame operator of multivariate discrete time wavelet (MDTW) system of scale matrix [Formula: see text]. By taking a collection of [Formula: see text] filters, we construct dual MDTW tight frames. Also, we discuss the existence and construction of MDTW Parseval frames in multidimensional sequence space. The necessary and sufficient condition for the existence of orthonormal basis of scale matrix [Formula: see text] is discussed. Using dual MDTW tight frame, we characterize biorthogonal pair of Riesz bases. Further, we define multivariate Parseval frame of weighted exponentials, dual multivariate tight frame of weighted exponentials of scale matrix [Formula: see text] in periodic space [Formula: see text]. Finally, we give the applications of MDTW frames in the theory of frames of weighted exponentials of scale matrix [Formula: see text] in periodic space [Formula: see text].

Interpolation graph convolutional network for 3D point cloud analysis

The feature analysis of point clouds, a popular representation of three-dimensional (3D) objects, is rising as a hot research topic nowadays. Point cloud data bear a sparse and unordered nature, making many commonly used feature extraction methods, for example, Convolutional Neural Networks (CNNs) inapplicable, while previous models suitable for the task are usually complex. We aim to reduce model complexity by reducing the number of parameters while achieving better (or at least comparable) performance. We propose an Interpolation Graph Convolutional Network (IGCN) for extracting features of point clouds. IGCN uses the point cloud graph structure and a specially designed Interpolation Convolution Kernel to mimic the operations of CNN for feature extraction. On the basis of weight postfusion and multilevel-resolution aggregation, IGCN not only reduces the cost of calculating the interpolation operation but also improves the model's performance. We validate the performance of IGCN on both point cloud classification and segmentation tasks and explore the contribution of each module of our model through ablation experiments. Furthermore, we embed the IGCN point cloud feature extraction module as a plug-and-play module into other frameworks and perform point cloud registration experiments.

Modeling electrokinetic flows with the discrete ion stochastic continuum overdamped solvent algorithm.

In this article we develop an algorithm for the efficient simulation of electrolytes in the presence of physical boundaries. In previous work the discrete ion stochastic continuum overdamped solvent (DISCOS) algorithm was derived for triply periodic domains, and was validated through ion-ion pair correlation functions and Debye-Hückel-Onsager theory for conductivity, including the Wien effect for strong electric fields. In extending this approach to include an accurate treatment of physical boundaries we must address several important issues. First, the modifications to the spreading and interpolation operators necessary to incorporate interactions of the ions with the boundary are described. Next we discuss the modifications to the electrostatic solver to handle the influence of charges near either a fixed potential or dielectric boundary. An additional short-ranged potential is also introduced to represent interaction of the ions with a solid wall. Finally, the dry diffusion term is modified to account for the reduced mobility of ions near a boundary, which introduces an additional stochastic drift correction. Several validation tests are presented confirming the correct equilibrium distribution of ions in a channel. Additionally, the methodology is demonstrated using electro-osmosis and induced-charge electro-osmosis, with comparison made to theory and other numerical methods. Notably, the DISCOS approach achieves greater accuracy than a continuum electrostatic simulation method. We also examine the effect of under-resolving hydrodynamic effects using a "dry diffusion" approach, and find that considerable computational speedup can be achieved with a negligible impact on accuracy.

Co-training based virtual sample generation for solving the small sample size problem in process industry

With the development of industrialization, the production scale and complexity of process industries are getting larger and larger. But, limited by the small amounts of samples and the uneven sample distribution in the process industry, it is difficult to establish accurate and efficient data-driven soft sensor models to predict some variables. To further develop the application of soft sensor models, generating new virtual samples based on the original sample distribution to extend the sample set is an ideal approach to solve this problem. In this paper, a novel virtual sample generation method based on the co-training of two K-Nearest Neighbor (KNN) models is proposed. First, according to the sparse parameter, sparse regions in each dimension of the feature space are identified. Second, the input features of virtual samples are generated in these sparse regions by performing interpolation operations. Third, the outputs of virtual samples are predicted by double KNN regressors based on co-training. The qualified virtual samples are screened and the model is updated using these virtual samples to improve the prediction accuracy of the double KNN models. To verify the effectiveness and superiority of the proposed virtual sample generation method based on the co-training (CTVSG), case studies are conducted using two standard functions and a Purified Terephthalic Acid (PTA) industrial dataset, where the effectiveness of CTVSG is confirmed.

An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem

AbstractIn this article, an effective finite element method based on dimension reduction scheme is proposed for a fourth-order Steklov eigenvalue problem in a circular domain. By using the Fourier basis function expansion and variable separation technique, the original problem is transformed into a series of radial one-dimensional eigenvalue problems with boundary eigenvalue. Then we introduce essential polar conditions and establish the discrete variational form for each radial one-dimensional eigenvalue problem. Based on the minimax principle and the approximation property of the interpolation operator, we prove the error estimates of approximation eigenvalues. Finally, some numerical experiments are provided, and the numerical results show the efficiency of the proposed algorithm.

A multigrid preconditioner for spatially adaptive high-order meshless method on fluid–solid interaction problems

We present a monolithic geometric multigrid preconditioner for solving fluid–solid interaction problems in Stokes limit. The problems are discretized by a spatially adaptive high-order meshless method, the generalized moving least squares (GMLS) with adaptive h-refinement. For solving fluid–solid interaction problems, we need to deal with a tightly coupled system consisting of the flow field and solid bodies, resulting in a linear system of equations with a block structure. In Stokes limit, solid kinematics can be dominated by the singularities governing the lubrication effects. Resolving those singularities with adaptive h-refinement can lead to an ill-conditioned linear system of equations. The key ingredients of the multigrid preconditioner include the interpolation and restriction operators and the smoothers. For constructing the interpolation and restriction operators, we utilize the geometric information of hierarchical sets of GMLS nodes generated in adaptive h-refinement. We build decoupled smoothers through physics-based splitting and then combine them via a multiplicative overlapping Schwarz approach. Through numerical examples with the inclusion of different numbers and shapes of solid bodies, we demonstrate the performance and assess the scalability of the designed preconditioner. As the total degrees of freedom and the number of solid bodies increase, the proposed monolithic geometric multigrid preconditioner can ensure convergence and good scalability when using the Krylov iterative method for solving the linear systems of equations generated from the spatially adaptive GMLS discretization. More specifically, for a fixed number of solid bodies, as the discretization resolution is incrementally refined, the number of iterations of the linear solver can be maintained at the same level, indicating nearly linear scalability of our preconditioner with respect to the total degrees of freedom. When the number of solid bodies Ns increases, the number of iterations is nearly proportional to Ns, implying the sublinear optimality with respect to the number of solid bodies.

A posteriori error estimation for a C1 virtual element method of Kirchhoff plates

A residual-type a posteriori error estimation is developed for a C 1 -conforming virtual element method (VEM) to solve a Kirchhoff plate bending problem. To derive the reliability and efficiency of the a posteriori error bound, the inverse inequalities and norm equivalence are developed over the underlying C 1 -conforming virtual element space, and a weak interpolation operator together with its error estimates is given as well. As an outcome of the error estimator, an adaptive VEM is introduced by means of the mesh refinement strategy with the one-hanging-node rule. Numerical results on various benchmark tests confirm the robustness of the proposed error estimator and show the efficiency of the resulting adaptive VEM.