One-dimensional Interpolation Research Articles

An objective and accurate G1-conforming mixed Bézier FE-formulation for Kirchhoff–Love rods

We present a new invariant numerical formulation suitable for the analysis of Kirchhoff–Love rod model based on the smallest rotation map for which the two kinematic descriptors are the placement of the centroid curve and the rotation angle that describes the orientation of the cross-section. In order to guarantee objectivity with respect to a rigid body motion, the spherical linear interpolation (slerp) for the rotations is introduced. A new hierarchical interpolation for the rotation angle of the cross-section is proposed, composed by the drilling rotation (with respect to the unit tangent of the centroid curve) contained in the geodetic interpolation of the ends’ rotations plus an additional polynomial correction term. This correction term is needed in order to enhance the accuracy of the proposed finite element (FE) formulation. The FE model implicitly guarantees the G1 continuity conditions, thanks to a change in the parametrization of the centroid curve that leads to a generalization of the one-dimensional (1D) Hermitian interpolation. A mixed formulation is used in order to avoid locking and improve the computational efficiency of the method. A symmetric tangent stiffness operator is obtained performing the second covariant derivative of the mixed functional for which the Levi-Civita connection of the configurations manifold of the rod is needed. The objectivity, the robustness, and accuracy of the G1-conforming formulation are confirmed by means of several numerical examples.

A time-accurate inflow coupling for zonal LES

Generating turbulent inflow data is a challenging task in zonal large eddy simulation (zLES) and often relies on predefined DNS data to generate synthetic turbulence with the correct statistics. The more accurate, but more involved alternative is to use instantaneous data from a precursor simulation. Using instantaneous data as an inflow condition allows to conduct high fidelity simulations of subdomains of, e.g. an aircraft including all non-stationary or rare events. In this paper, we introduce a toolchain that is capable of interchanging highly resolved spatial and temporal data between flow solvers with different discretization schemes. To accomplish this, we use interpolation algorithms suitable for scattered data in order to interpolate spatially. In time, we use one-dimensional interpolation schemes for each degree of freedom. The results show that we can get stable simulations that map all flow features from the source data into a new target domain. Thus, the coupling is capable of mapping arbitrary data distributions and formats into a new domain while also recovering and conserving turbulent structures and scales. The necessary time and space resolution requirements can be defined knowing the resolution requirements of the used numerical scheme in the target domain.

High-efficiency simulation of nonstationary wind velocity using diagonal POD of decomposed time-frequency interpolation node spectrum matrices

The classical spectral representation method (SRM) used extensively to simulate nonstationary wind fields suffers from lower efficiency. Although some tries can already accelerate SRM, the efficiency improvement is restricted when the number of simulation points is large. For this reason, this paper proposed a high-efficiency interpolation scheme to boost the efficiency of SRM. Firstly, the decomposed time-frequency interpolation node spectrum matrices can be approximately decouple into the products of time and frequency functions using diagonal proper orthogonal decomposition (POD). Specifically, the POD operation is just for the diagonal elements, which are low-dimensional matrices. Since Cholesky decomposition is merely needed in the time-frequency interpolation point, the time consumed by spectral decomposition will be significantly reduced. Then, one-dimensional (1D) interpolation is implemented for these decoupling terms in the time and frequency directions, respectively. The decoupling results only contains a few time terms, so the number of 1D interpolation operations can be considerably decreased. Moreover, by exchanging the summation order, very few fast Fourier transform operations are required to accelerate harmonic superposition. Therefore, the proposed method may further enhance the simulation efficiency. Finally, through numerical examples, the accuracy and superiority of the proposed method are demonstrated.

Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach

The problems of polynomial interpolation with several variables present more difficulties than those of one-dimensional interpolation. The first problem is to study the regularity of the interpolation schemes. In fact, it is well-known that, in contrast to the univariate case, there is no universal space of polynomials which admits unique Lagrange interpolation for all point sets of a given cardinality, and so the interpolation space will depend on the set Z of interpolation points. Techniques of univariate Newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms. The Newton basis format, with divided-difference algorithm for coefficients, generalizes in a straightforward way when interpolating at nodes on a grid within certain schemes. In this work, we propose a random algorithm for computing several interpolating multivariate Lagrange polynomials, called RLMVPIA (Random Lagrange Multivariate Polynomial Interpolation Algorithm), for any finite interpolation set. We will use a Newton-type polynomials basis, and we will introduce a new concept called Z , z -partition. All the given algorithms are tested on examples. RLMVPIA is easy to implement and requires no storage.

Fast simulation of fully non-stationary wind fields using a new matrix factorization assisted interpolation method

The classical spectral representation method and its improved versions have been widely used to generate non-stationary wind fields. However, they are not efficient for the simulation of non-stationary wind fields with time-varying coherence (i.e., fully non-stationary wind fields) due to the extensive computational demand regarding spectral matrix decomposition or harmonic superposition. To this end, this study develops a new matrix factorization assisted interpolation method, which is composed of an improved interpolation scheme and the monotone projected Barzilai-Borwein based non-negative matrix factorization (MPBB-NMF). Specifically, an improved interpolation scheme is firstly developed to quantify the desired distribution of time-frequency interpolation nodes, and only the decomposed spectra at these nodes are calculated by Cholesky decomposition. Then, MPBB-NMF with a fast global convergence is used to decouple these spectra into products of non-negative time- and frequency-dependent functions. Finally, one-dimensional interpolation is performed for these decoupled functions, which makes it possible to accelerate the harmonic superposition using a small amount of fast Fourier transforms (FFTs). Obviously, the proposed method realizes the decomposition of spectral matrix in the decoupling form. Besides reducing of the number of Cholesky decomposition and invoking FFT, its decoupling only aims at the low dimensional matrix and the interpolation merely involves a few decoupled items. Numerical simulation on the fully non-stationary wind field of a long-span bridge demonstrates the effectiveness and superiority of the proposed method.

A Low-Cost Hardware Design of Learning-Based One-Dimensional Interpolation for Real-Time Video Applications at the Edge

Gradual advances have occurred in the design of micro-artificial intelligence for resource-limited hardware. A high-resolution (HR) image reconstruction module is indispensable for video analytics chips or devices at the edge. This paper proposes a low-cost and learning-based interpolation method for HR image reconstruction. The proposed method generates reconstructed pixels by processing reference pixels with optimal weights, which are pre-trained by solving the minimum mean square error problem for real images. To reduce the number of computation units and the usage of learned weights, a cross-directional interpolation architecture, which includes a vertical kernel and a horizontal kernel, is adopted. Moreover, a one-dimensional feature discriminator is proposed to improve the quality of up-scaled images efficiently. The main benefit of the proposed method is that it requires a small number of computation units but can still produce high-quality images. The hardware architecture of the proposed method was implemented on a field-programmable gate array (FPGA) by using Xilinx UltraScale+ ZCU102 and an application-specific integrated circuit (ASIC) by using TSMC’s 0.13-μm technology. On the ASIC, the proposed hardware required only approximately 60K gate counts and 50 KBytes of memory. The experimental results indicate that the average peak signal-to-noise ratio of the up-scaled images reached 35.92 dB for the Set-5 dataset. The throughput of the proposed hardware was at least 1000 Mpixels/s on the FPGA and 1200 Mpixels/s on the ASIC, which indicates that the proposed hardware can handle a target resolution higher than 4K in real time.

A bivariate Chebyshev polynomials method for nonlinear dynamic systems with interval uncertainties

A bivariate Chebyshev polynomials approach is proposed to estimate the dynamic response bounds of nonlinear systems with interval uncertainties. The existing collocation method directly searches the maximum and minimum values of the surrogate model in the entire interval space by the scanning method (SM). The presence of too many uncertain parameters will lead to expansive computational cost. To overcome this shortcoming, the dynamic response is decomposed by a bivariate function decomposition (BFD), established based on high-order Taylor expansion, into the sum of multiple univariate and bivariate response functions. The above univariate and bivariate functions are fitted using Chebyshev polynomials, and polynomial coefficients are obtained through one-dimensional (1D) and two-dimensional (2D) interpolation points. Thus, the solution of the nonlinear dynamic systems with uncertain parameters can be transformed into that of univariate and bivariate Chebyshev interval functions. The extremum values of the low-dimensional Chebyshev interval functions can be found by SM, and then the bounds of dynamic response are acquired by interval arithmetic. Since SM searches for extreme values only in 1D and 2D uncertain domains, the amount of calculation is reduced compared to searching the whole uncertain space. The efficiency, practicability and effectiveness of the proposed interval uncertainty analysis method are proved by three dynamic examples.

An Improved Hybrid Cartesian/Immersed Boundary Method for Flow Simulation with Moving Boundaries

To simulate flow problems with moving fluid-solid interfaces, a boundary treatment technique based on a hybrid Cartesian/immersed boundary method is presented. In the vicinity of the immersed boundary, a forcing term is added into the governing flow equations to modify the discretization scheme for the viscous term during its spatial discretization. Addition of such a term enforces desired boundary condition on the immersed boundary. A one-dimensional linear interpolation scheme is used to obtained required information for the modified discretization scheme when dealing with a moving boundary. The present method is essentially second-order accurate and can obtain results that are in good agreement with benchmark references when simulating the flow induced by the in-line oscillation of a circular cylinder.

Determination of load distribution in a given medium according to the values of the loads at certain points

Taking into account force, temperature and other loads, the stress and strain state calculations methods of spatial structures involve determining the distribution of the loads in the three-dimensional body of the structure [1, 2].
 In many cases the output data for this distribution can be the values of loadings in separate points of the structure. The problem of load distribution in the body of the structure can be solved by three-dimensional discrete interpolation in four-dimensional space based on the method of finite differences, which has been widely used in solving various engineering problems in different fields. A discrete conception of the load distribution at points in the body or in the environment is also required for solving problems by the finite elements method [3-7].
 From a geometrical point of view, the result of three-dimensional interpolation is a multivariate of the four-dimensional space [8], where the three dimensions are the coordinates of a three-dimensional body point, and the fourth is the loading at this point. Such interpolation provides for setting of the three coordinates of the point and determining the load at that point. The simplest three-dimensional grid in the three-dimensional space is the grid based on a single sided hypercube. The coordinates of the nodes of such a grid correspond to the numbering of nodes along the coordinate axes.
 Discrete interpolation of points by the finite difference method is directly related to the numerical solutions of differential equations with given boundary conditions and also requires the setting of boundary conditions.
 If we consider a three-dimensional grid included into a parallelepiped, the boundary conditions are divided into three types: 1) zero-dimensional (loads at points), where the three edges of the grid converge; 2) one-dimensional (loads at points of lines), where the four edges of the grid converge; 3) two-dimensional (loads at the points of faces), where the five edges of the grid converge. The zero-dimensional conditions are boundary conditions for one-dimensional interpolation of the one-dimensional conditions, which, in turn, are boundary conditions for two-dimensional conditions, and the two-dimensional conditions are boundary conditions for determining the load on the inner points of the grid.
 If a load is specified only at certain points of boundary conditions, then the interpolation problem is divided into three stages: one-dimensional load interpolation onto the line nodes, two-dimensional load interpolation onto the surface nodes and three-dimensional load interpolation onto internal grid nodes.
 The proposed method of discrete three-dimensional interpolation allows, according to the specified values of force, temperature or other loads at individual points of the three-dimensional body, to interpolate such loads on all nodes of a given regular three-dimensional grid with cubic cells.
 As a result of interpolation, a discrete point framework of the multivariate is obtained, which is a geometric model of the distribution of physical characteristics in a given medium according to the values of these characteristics at individual points.

Hybrid Trefftz finite element method for axisymmetric elasticity problems under torsion

An eight-node axisymmetric element AX88T is proposed for predicting the torsional response of axisymmetric elasticity based on the hybrid fundamental solution based finite element method (HFS-FEM). In this approach, two displacement fields are independently assumed in the domain and on the boundary of the element, respectively. The intra-element displacement field is expressed in terms of fundamental solutions satisfying the governing equation of the problem while a conforming frame displacement field is related to the nodal degrees of freedom by standard one-dimensional interpolation functions. By applying the Gauss divergence theorem to the hybrid functional, the resultant finite element formulation completely avoids the need for domain integration, which renders the AX88T element insensitive to distortion. The accuracy and efficiency of HFS-FEM are demonstrated through three numerical examples which includes thick-walled cylinder, flange-shaped cylinder and stepped cylindrical shaft.

Accuracy of compact-stencil interpolation algorithms for unstructured mesh finite volume solver

This study considers the accuracy of cell-to-face centre interpolation of convected quantities in unstructured finite volume meshes with cell-centred storage of variables. The accuracy of the interpolation algorithms were tested in isolation using ideal data to determine their numerical accuracy on both standard and artificially distorted meshes. It was found that the formally second- and third-order accurate interpolations based on one-dimensional interpolation along the line connecting the cells to the right and left of the face under consideration only have first-order accuracy in standard unstructured mesh, and less than first-order accuracy in distorted unstructured mesh. L1 interpolation errors in the distorted unstructured mesh are greater than in standard unstructured mesh. The order of accuracy and L1 errors can be improved by applying spatial corrections. The formally second-order accurate multi-dimensional interpolations tested in this study that are not based on one-dimensional interpolation along lines connecting the neighbour cells have first-order accuracy in both standard and distorted unstructured mesh. Linear interpolation between end vertices produces greatest L1 error in standard mesh; polynomial interpolation, linear interpolation between cell centres and standard QUICK produce the greatest L1 error in distorted mesh. Spatially correct QUICK, spatially correct linear interpolation between cell centres, Laplacian interpolation to face centres, and Taylor series expansion about an upstream cell produce the smallest L1 error in both standard and distorted mesh. Based on accuracy and the simplicity of implementation, Taylor series expansion about an upstream cell is the best choice for use in unstructured mesh.

SYSTEMATIZATION OF INTERPOLATION METHODS IN GEOMETRIC MODELING

This article proposes to show the connections between the practical problems of geometric modeling, for the solution of which there are no adequate interpolation methods. Interpolation methods in mathematics first emerged as part of mathematical modeling, and eventually became a separate area of mathematics. The classical definition of interpolation is the constructive reproduction of a function of a certain class according to its known values or the values of its derivatives at given points.Interpolation problems arise in spaces of different dimensions when solving engineering problems. Elements of such spaces can be not only points but also other geometric shapes. The dimension of space is always equal to the parametric number of the element of space. In applied geometry, interpolation problems have been studied since the 1960s. Since there are so many publications related to interpolation problems, systematization is proposed for their analysis. The largest number of existing publications is devoted to the problems of one-dimensional interpolation in two-dimensional space, both continuous and discrete. A much smaller number of publications are devoted to two-dimensional continuous and discrete interpolation. The variety of interpolation methods is due, first, to different sets of initial geometric conditions. Only in some publications the choice of initial conditions depending on the purpose of interpolation problems in specific sectors of the economy. In many publications, the choice of interpolation method is related to the features of the basic mathematical apparatus. Among the considered publications there is none where the influence of parameters of the set points, namely distances from them to the current point of an interpolant, on its parameter is directly considered. Such interpolation can be the basis for modeling various energy fields, where given points are point sources of energy.

Geostatistical electrical resistivity tomography using random mixing

With modern data acquisition technology, Electrical Resistivity Tomography (ERT) surveys can be deployed easily and run cost-efficiently over large areas and over long periods. Inversion of the electric potential data to image the subsurface resistivity distribution is a key step in the workflow. In contrast to deterministic inversion commonly used in ERT, geostatistical inversion methods construct a variety of possible realizations of resistivity distributions and allow for a better interpretation of results and a quantification of the uncertainty. In this paper we present a geostatistical approach for the inversion of ERT survey data based on Random Mixing. The method of Random Mixing had been developed and tested for inverse groundwater modelling and is applied here to ERT as an example of geophysical inversion for the first time. Random Mixing is based on the idea of constructing realizations of the unknown electric conductivity field by combining random fields representing the spatial correlation of conductivity such that direct measurements of conductivity if available and indirect dipole measurement of electric potentials at electrodes are matched. Implementation is based on the three-dimensional finite element method to solve the forward model where properties in a core region are updated through the Random Mixing iteration. To reduce computational costs for the evaluation of the misfit cost function one-dimensional interpolation is applied. The results of Random Mixing for a synthetic example and a field survey inversion are discussed and compared with deterministic inversion results.

Complex exponential functions: A high-precision hardware realization

Hardware realization of complex exponential functions (CEF) is not practical for over 16-bit operands. This is due to high cost and delay of the required look-up table (LUT) and arithmetic units. Therefore, we decompose larger (up to 64 bits) operands to three parts (e.g., 11+24+29=64). They are handled by minimax approximation, one-dimensional degree-3 interpolation, and pure LUT approaches, respectively. However, the second one is not used for smaller operands. Taking advantage of rectangular multipliers, we have realized high precision CEF, on FPGA platform. More than 30% speedup and 75% lower cost are achieved in comparison with the previous relevant works.

Arbitrage-free interpolation of call option prices

Abstract In this paper, we introduce a new interpolation method for call option prices and implied volatilities with respect to the strike, which first generates, for fixed maturity, an implied volatility curve that is smooth and free of static arbitrage. Our interpolation method is based on a distortion of the call price function of an arbitrage-free financial “reference” model of one’s choice. It reproduces the call prices of the reference model if the market data is compatible with the model. Given a set of call prices for different strikes and maturities, we can construct a call price surface by using this one-dimensional interpolation method on every input maturity and interpolating the generated curves in the maturity dimension. We obtain the algorithm of N. Kahalé [An arbitrage-free interpolation of volatilities, Risk 17 2004, 5, 102–106] as a special case, when applying the Black–Scholes model as reference model.

Radiomics for Distinguishing Myocardial Infarction from Myocarditis at Late Gadolinium Enhancement at MRI: Comparison with Subjective Visual Analysis.

To evaluate whether radiomics features of late gadolinium enhancement (LGE) regions at cardiac MRI enable distinction between myocardial infarction (MI) and myocarditis and to compare radiomics with subjective visual analyses by readers with different experience levels. In this retrospective, institutional review board-approved study, consecutive MRI examinations of 111 patients with MI and 62 patients with myocarditis showing LGE were included. By using open-source software, classification performances attained from two-dimensional (2D) and three-dimensional (3D) texture analysis, shape, and first-order descriptors were compared, applying five different machine learning algorithms. A nested, stratified 10-fold cross-validation was performed. Classification performances were compared through Wilcoxon signed-rank tests. Supervised and unsupervised feature selection techniques were tested; the effect of resampling MR images was analyzed. Subjective image analysis was performed on 2D and 3D image sets by two independent, blinded readers with different experience levels. When trained with recursive feature elimination (RFE), a support vector machine achieved the best results (accuracy: 88%) for 2D features, whereas linear discriminant analysis (LDA) showed the highest accuracy (85%) for 3D features (P <.05). When trained with principal component analysis (PCA), LDA attained the highest accuracy with both 2D (86%) and 3D (89%; P =.4) features. Results found for classifiers trained with spline resampling were less accurate than those achieved with one-dimensional (1D) nearest-neighbor interpolation (P <.05), whereas results for classifiers trained with 1D nearest-neighbor interpolation and without resampling were similar (P =.1). As compared with the radiomics approach, subjective visual analysis performance was lower for the less experienced and higher for the experienced reader for both 2D and 3D data. Radiomics features of LGE permit the distinction between MI and myocarditis with high accuracy by using either 2D features and RFE or 3D features and PCA.© RSNA, 2019Supplemental material is available for this article.

Fundamental-solution-based hybrid finite element with singularity control for two-dimensional mixed-mode crack problems

In this work, a fundamental-solution-based hybrid finite element method is presented for modeling mixed-mode cracks in two-dimensional (2D) isotropic elastic media. In the method, a double-variable hybrid functional for each element is formulated to derive element stiffness equation in terms of nodal displacements, which includes line integrals along the element boundary only. The element interior displacement field is approximated using a linear combination of displacement fundamental solutions at different source locations, while the independent element frame displacement field is approximated by one-dimensional shape function interpolation. To correctly model the behavior of crack-tip displacement, the discontinuous quarter-point crack-tip singular hybrid element formulation is developed, so that crack tip stress intensity factors can be easily evaluated by the near-tip displacement method. Three numerical examples of internal cracks in 2D elastic domains are presented to show the efficiency of the proposed method.

Fast Matrix-Free Evaluation of Discontinuous Galerkin Finite Element Operators

We present an algorithmic framework for matrix-free evaluation of discontinuous Galerkin finite element operators. It relies on fast quadrature with sum factorization on quadrilateral and hexahedral meshes, targeting general weak forms of linear and nonlinear partial differential equations. Different algorithms and data structures are compared in an in-depth performance analysis. The implementations of the local integrals are optimized by vectorization over several cells and faces and an even-odd decomposition of the one-dimensional interpolations. Up to 60% of the arithmetic peak on Intel Haswell, Broadwell, and Knights Landing processors is reached when running from caches and up to 40% of peak when also considering the access to vectors from main memory. On 2×14 Broadwell cores, the throughput is up to 2.2 billion unknowns per second for the 3D Laplacian and up to 4 billion unknowns per second for the 3D advection on affine geometries, close to a simple copy operation at 4.7 billion unknowns per second. Our experiments show that MPI ghost exchange has a considerable impact on performance and we present strategies to mitigate this effect. Finally, various options for evaluating geometry terms and their performance are discussed. Our implementations are publicly available through the deal.II finite element library.

Fast Green Function Evaluation for Method of Moment

In this letter, an approach to accelerate the matrix filling in method of moment (MOM) is presented. Based on the fact that the Green function is dependent on the Euclidean distance between the source and the observation points, we constructed an efficient adaptive one-dimensional interpolation approach to fast calculate the $Exp$ type function values. In the proposed method, several adaptive interpolation tables are constructed based on the maximum and minimum distance between any two integration points with local refinement near zero function values to minimize the relative error. An efficient approach to obtain the sampling points used in the interpolation phase is carefully designed. Then, any function values can be efficiently calculated through a linear interpolation method for Exp and a Lagrange polynomial interpolation method for the Green function. In addition, the error bound of the proposed method is rigorously investigated. The proposed method can be quite easily integrated into the available MOM codes for different integration equation (IE) formulations with few efforts. Comprehensive numerical experiments validate its accuracy and efficiency through several IE formulations. Results show that over 20% efficiency improvement can be achieved without sacrificing the accuracy.

Semi-supervised learning using autodidactic interpolation on sparse representation-based multiple one-dimensional embedding

In this paper, we present a novel semi-supervised classification method based on sparse representation (SR) and multiple one-dimensional embedding-based adaptive interpolation (M1DEI). The main idea of M1DEI is to embed the data into multiple one-dimensional (1D) manifolds satisfying that the connected samples have shortest distance. In this way, the problem of high-dimensional data classification is transformed into a 1D classification problem. By alternating interpolation and averaging on the multiple 1D manifolds, the labeled sample set of the data can enlarge gradually. Obviously, proper metric facilitates more accurate embedding and further helps improve the classification performance. We develop a SR-based metric, which measures the affinity between samples more accurately than the common Euclidean distance. The experimental results on several databases show the effectiveness of the improvement.