Interpolation Operator Research Articles

Superconvergence error estimate of Galerkin method for Sobolev equation with Burgers' type nonlinearity

In this paper, based on the implicit Euler scheme in the temporal direction, the superconvergence property is investigated by using the special property of the bilinear element on the rectangular mesh for the Sobolev equation with Burgers' nonlinearity. The existence and uniqueness of the fully-discrete solution is proved. Further, the superconvergence error estimate in L∞(H1)-norm is established in terms of a novel approach, i.e., the technique of the combination of the interpolation operator and projection operator. Finally, a numerical experiment is carried out to confirm the theoretical analysis.

XCycles Backprojection Acoustic Super-Resolution.

The computer vision community has paid much attention to the development of visible image super-resolution (SR) using deep neural networks (DNNs) and has achieved impressive results. The advancement of non-visible light sensors, such as acoustic imaging sensors, has attracted much attention, as they allow people to visualize the intensity of sound waves beyond the visible spectrum. However, because of the limitations imposed on acquiring acoustic data, new methods for improving the resolution of the acoustic images are necessary. At this time, there is no acoustic imaging dataset designed for the SR problem. This work proposed a novel backprojection model architecture for the acoustic image super-resolution problem, together with Acoustic Map Imaging VUB-ULB Dataset (AMIVU). The dataset provides large simulated and real captured images at different resolutions. The proposed XCycles BackProjection model (XCBP), in contrast to the feedforward model approach, fully uses the iterative correction procedure in each cycle to reconstruct the residual error correction for the encoded features in both low- and high-resolution space. The proposed approach was evaluated on the dataset and showed high outperformance compared to the classical interpolation operators and to the recent feedforward state-of-the-art models. It also contributed to a drastically reduced sub-sampling error produced during the data acquisition.

The smooth forcing extension method: A high-order technique for solving elliptic equations on complex domains

High-order numerical methods for solving elliptic equations over arbitrary domains typically require specialized machinery, such as high-quality conforming grids for finite elements method, and quadrature rules for boundary integral methods. These tools make it difficult to apply these techniques to higher dimensions. In contrast, fixed Cartesian grid methods, such as the immersed boundary (IB) method, are easy to apply and generalize, but typically are low-order accurate. In this study, we introduce the Smooth Forcing Extension (SFE) method, a fixed Cartesian grid technique that builds on the insights of the IB method, and allows one to obtain arbitrary orders of accuracy. Our approach relies on a novel Fourier continuation method to compute extensions of the inhomogeneous terms to any desired regularity. This is combined with the highly accurate Non-Uniform Fast Fourier Transform for interpolation operations to yield a fast and robust method. Numerical tests confirm that the technique performs precisely as expected on one-dimensional test problems. In higher dimensions, the performance is even better, in some cases yielding sub-geometric convergence. We also demonstrate how this technique can be applied to solving parabolic problems and for computing the eigenvalues of elliptic operators on general domains, in the process illustrating its stability and amenability to generalization.

Multi-GPU acceleration of large-scale density-based topology optimization

This work presents a parallel implementation of density-based topology optimization using distributed GPU computing systems. The use of multiple GPU devices allows us accelerating the computing process and increasing the device memory available for GPU computing. This increment of device memory enables us to address large models that commonly do not fit into one GPU device. The most modern scientific computers incorporate these devices to design energy-efficient, low-cost, and high-computing power systems. However, we should adopt the proper techniques to take advantage of the computational resources of such high-performance many-core computing systems. It is well-known that the bottleneck of density-based topology optimization is the solving of the linear elasticity problem using Finite Element Analysis (FEA) during the topology optimization iterations. We solve the linear system of equations obtained from FEA using a distributed conjugate gradient solver preconditioned by a smooth aggregation-based algebraic multigrid (AMG) using GPU computing with multiple devices. The use of aggregation-based AMG reduces memory requirements and improves the efficiency of the interpolation operation. This fact is rewarding for GPU computing. We evaluate the performance and scalability of the distributed GPU system using structured and unstructured meshes. We also test the performance using different 3D finite elements and relaxing operators. Besides, we evaluate the use of numerical approaches to increase the topology optimization performance. Finally, we present a comparison between the many-core computing instance and one efficient multi-core implementation to highlight the advantages of using GPU computing in large-scale density-based topology optimization problems.

On the Ra\u015fa Inequality for Higher Order Convex Functions

We study the following (q-1)th convex ordering relation for qth convolution power of the difference of probability distributions mu and nu (ν-μ)∗q≥(q-1)cx0,q≥2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} (\\nu -\\mu )^{*q}\\ge _{(q-1)cx} 0 , \\quad q\\ge 2, \\end{aligned}$$\\end{document}and we obtain the theorem providing a useful sufficient condition for its verification. We apply this theorem for various families of probability distributions and we obtain several inequalities related to the classical interpolation operators. In particular, taking binomial distributions, we obtain a new, very short proof of the inequality given recently by Abel and Leviatan (2020).

Convergence analysis of the scaled boundary finite element method for the Laplace equation

The scaled boundary finite element method (SBFEM) is a relatively recent boundary element method that allows the approximation of solutions to partial differential equations (PDEs) without the need of a fundamental solution. A theoretical framework for the convergence analysis of SBFEM is proposed here. This is achieved by defining a space of semi-discrete functions and constructing an interpolation operator onto this space. We prove error estimates for this interpolation operator and show that optimal convergence to the solution can be obtained in SBFEM. These theoretical results are backed by two numerical examples.

Nielson interpolation operators on an arbitrary triangle with one curved side

We extend some Nielson type interpolation operators to the cases of standard and arbitrary triangles with one curved side. The correspondence between the operators defined on standard triangles and arbitrary triangles is made using barycentric coordinates. We study the interpolation properties of the obtained operators and the interpolation errors. For illustration, we give some numerical examples.

Fast Linear Interpolation

We present fast implementations of linear interpolation operators for piecewise linear functions and multi-dimensional look-up tables. These operators are common for efficient transformations in image processing and are the core operations needed for lattice models like deep lattice networks, a popular machine learning function class for interpretable, shape-constrained machine learning. We present new strategies for an efficient compiler-based solution using MLIR to accelerate linear interpolation. For real-world machine-learned multi-layer lattice models that use multidimensional linear interpolation, we show these strategies run 5-10× faster on a standard CPU compared to an optimized C++ interpreter implementation.

Operator Interpolation and Systems of Linear Equations and Inequalities in Euclidean Spaces

Operator Interpolation and Systems of Linear Equations and Inequalities in Euclidean Spaces

Optimization of the parallel semi‐Lagrangian scheme based on overlapping communication with computation in the YHGSM

AbstractThe Yi‐he global spectral model (YHGSM) is a numerical weather prediction (NWP) model developed by the National University of Defense Technology, China. The YHGSM has two optimization techniques that have recently been adopted for the on‐demand communication version of the semi‐Lagrangian scheme: one‐sided communication and an adaptive strategy for the maximum wind speed. However, the limitation of the original model is that the computation and communication cannot be performed simultaneously, resulting in a waste of resources. This article addresses this limitation by dividing the array to be interpolated into three independent subarrays. The communication and interpolation operations for each subarray are rearranged into three groups. This allows the overlap of the communication with computation between different groups. In addition, by adjusting the synchronization operations, the degree of overlap is further increased. In the optimized scheme, the per‐step overhead of the interpolation phase is greatly reduced, and its speedup decreases first and increases afterwards with the increase of the number of cores. Therefore, when the number of cores is large, the benefits from overlapping the communication with computation will be more substantial.

The Application of Intelligent Algorithms in the Animation Design of 3D Graphics Engines

With the rapid improvement of computer hardware capabilities and the reduction of cost, the quality of game pictures has made a qualitative breakthrough, which has reached or exceeded the picture effect of many dedicated virtual reality engines. On the basis of the design and implementation of the virtual reality 3D engine, the rendering queue management method is proposed to improve the frame rate. Based on the object-oriented design method, emitter regulator particle rendering mode, and traditional bone skin animation technology, the key structure technology in skeletal animation is analyzed, and the animation controller used to control animation playback and key structure interpolation operation is designed, which achieves the ideal animation effect. Finally, a prototype system based on engine is implemented.

Generalization of the Lieb–Thirring–Araki Inequality and Its Applications

The matrix eigenvalue is very important in matrix analysis, and it has been applied to matrix trace inequalities, such as the Lieb–Thirring–Araki theorem and Thompson–Golden theorem. In this manuscript, we obtain a matrix eigenvalue inequality by using the Stein–Hirschman operator interpolation inequality; then, according to the properties of exterior algebra and the Schur-convex function, we provide a new proof for the generalization of the Lieb–Thirring–Araki theorem and Furuta theorem.

CP-PSIS: CRT and polynomial-based progressive secret image sharing

In (k, n) Progressive Secret Image Sharing (PSIS) schemes, n shares are obtained by sharing a secret image. With k to n shares, the secret image can be recovered progressively. No information of the secret image can be constructed by k-1 or fewer shares. Many Secret Image Sharing (SIS) schemes have been developed to achieve the progressive reconstruction property. However, such property is usually based on the pre-processing of the secret image which is inconvenient and would increase the cost of the algorithm. Moreover, in existing PSIS schemes, only if all of n shares are involved, the secret image can be completely reconstructed. That makes the schemes not fault-tolerant. Thus, a solution for the flexible and extendable PSIS scheme is required. In this paper, we propose a Chinese Remainder Theorem (CRT) and Polynomial-based Progressive Secret Image Sharing (CP-PSIS) scheme and a modified version of CP-PSIS scheme. The proposed schemes are the first to utilize the homomorphism of CRT in the Lagrange interpolation operation. The progressive property of the schemes is achieved by CRT. In the (k, n) threshold SIS, any k participants can obtain different versions of the secret image by choosing different parameters. Our proposed schemes achieve a high security based on CRT and polynomial. Theoretical analyses and experimental results have demonstrated that the proposed schemes are general, flexible and extendable for different PSIS applications.

MODELING A DYNAMIC THREE-DIMENSIONAL BODY USING INTERFLATATION AND BLENDING APPROXIMATION

The paper deals with the problem of reconstruction of the internal structure (density, absorption or attenuation coefficient) of a three-dimensional body by the information about it in the form of tomograms, given on a certain system of planes intersecting the object of study. This problem arises in practice in cases where there is no plane among the planes that are included in the experimental data, which consists of one or another set of points that are of interest to the researcher. For example, such a problem may arise after a patient has undergone examinations on a medical tomograph. After analyzing the obtained tomograms, it becomes necessary to find with their help one or more tomograms in the planes intersecting the body, but not coinciding with any of the given planes. It is noted that the operators of interflatation of functions is a natural generalization of the interpolation operators for the functions of three variables. Therefore, as in the case of interpolation, errors in the experimental data (in this case, in tomograms) are also introduced into the interflatation operators. In mathematics, there is an alternative to interpolation operators, namely approximation operators. These are operators constructed by smoothing experimental data using polynomials, rational functions, trigonometric polynomials, wavelets, and the like. An operator of mixed approximation of a function of three variables is constructed using Bernstein polynomials; the general form of the approximation error by the constructed operator and the estimate of this error are given. In the paper a four-dimensional mathematical model of a three-dimensional body that changes over time is also built and studied. A computational experiment is presented to restore the internal structure of the human heart from tomograms lying on a system of mutually perpendicular planes, which come from an actually operating computer tomograph. The article presents some possibilities of working with tomograms when restoring the internal structure of a three-dimensional body from the known projections of this body (tomograms).

Efficient radial basis function mesh deformation methods for aircraft icing

This paper presents an approach to update the moving ice boundary resulting from aircraft icing simulations using radial basis function mesh deformation techniques. State-of-the-art surface and volume point reduction schemes are used to reduce the computational cost of the mesh deformation. The data reduction schemes which are utilised include multi-level greedy surface point selection and volume point reduction. The multi-level greedy surface point selection reduces the number of control points to increase the efficiency of the interpolation operation. While the volume point reduction improves the computational cost of the volume mesh update operation which is important for large data sets. The paper assesses the capabilities of radial basis function mesh deformation for both two and three-dimensional problems. Furthermore, the effectiveness of the deformation technique is assessed for both local, non-smooth deformations and global, smooth deformations. The convergence history of the multi-level greedy point selection is assessed in terms of number of control points and computational cost. The location of the selected control points near the ice accretion illustrates the efficacy of the method for localised deformations. The results show that the radial basis function mesh deformation performs well for both the two and three-dimensional problems. The data-reduction schemes utilised in this work represent a significant improvement to standard radial basis function mesh deformation for aircraft icing problems comprising of large data-sets typical of three-dimensional problems.

Stable Dynamical Adaptive Mesh Refinement

We consider accurate and stable interpolation procedures for numerical simulations utilizing time dependent adaptive meshes. The interpolation of numerical solution values between meshes is considered as a transmission problem with respect to the underlying semi-discretized equations, and a theoretical framework using inner product preserving operators is developed, which allows for both explicit and implicit implementations. The theory is supplemented with numerical experiments demonstrating practical benefits of the new stable framework. For this purpose, new interpolation operators have been designed to be used with multi-block finite difference schemes involving non-collocated, moving interfaces.

A representation of the interpolation polynomial

We provide a decomposition formula for the classical polynomial interpolation operator and obtain the generalized Hermite interpolant through a limiting process. As a consequence of our results, we obtain/reobtain known and new identities related to interpolation theory.

Subpixel sampling moiré method for in-plane displacement measurement considering the symmetric errors induced by interpolation.

Sampling moiré method (SMM) is a highly accurate vision-based deformation measurement method, whose measurement error is minimized when an integer that is closest to the grating pitch is selected as the sampling pitch. We propose a subpixel SMM by applying subpixel interpolation operation before the downsampling in traditional SMM; the sampling pitch was selected as a noninteger nearest to the grating pitch in the subpixel resolution image. Meanwhile, the average filter method was used to eliminate the symmetric error cause by interpolation. As a result, the period of moiré fringe was enlarged greatly, and the measurement accuracy was also increased. To investigate the efficiency of the subpixel SMM, a computer simulation was applied to analyze the accuracy of the subpixel SMM. Then a simple tensile experiment was conducted to validate the efficiency of this method, and the result of the subpixel SMM accorded well with the fiber Bragg grating. In summary, the proposed novel subpixel SMM is capable of being used for high accuracy in-plane deformation measuring.

Nonconforming finite element method for coupled Poisson–Nernst–Planck equations

AbstractA nonconforming finite element method (FEM) is developed and investigated for the coupled Poisson–Nernst–Planck (PNP) equations with low order element. Then, by use of the special properties of this element, that is, the interpolation operator is equivalent to its projection operator, and the consistency error estimate can reach order of O(h2) which is one order higher than that of its interpolation error estimates when the exact solution belongs to H3(Ω), the superclose estimates of order O(h2) and O(h2 + τ) in the broken H1‐norm are derived with new techniques for the semidiscrete scheme and backward Euler fully discrete scheme, respectively. Further, through employing interpolation postprocessing approach, the corresponding global superconvergence results are obtained. Finally, some numerical results are provided to confirm the theoretical analysis. It seems that our results have never been found in the existing literature. Here h and τ denote the mesh size and time step, respectively.

Simple and efficient continuous data assimilation of evolution equations via algebraic nudging

AbstractWe introduce, analyze, and test an interpolation operator designed for use with continuous data assimilation (DA) of evolution equations that are discretized spatially with the finite element method. The interpolant is constructed as an approximation of the L2 projection operator onto piecewise constant functions on a coarse mesh, but which allows nudging to be done completely at the linear algebraic level, independent of the rest of the discretization, with a diagonal matrix that is simple to construct; it can even completely remove the need for explicit construction of a coarse mesh. We prove the interpolation operator has sufficient stability and accuracy properties, and we apply it to algorithms for both fluid transport DA and incompressible Navier–Stokes DA. For both applications we prove the DA solutions with arbitrary initial conditions converge to the true solution (up to discretization error) exponentially fast in time, and are thus long‐time accurate. Results of several numerical tests are given, which both illustrate the theory and demonstrate its usefulness on practical problems.