Maximum Interpolation Error Research Articles

Efficient Table-based Function Approximation on FPGAs Using Interval Splitting and BRAM Instantiation

This article proposes a novel approach for the generation of memory-efficient table-based function approximation circuits for edge devices in general and FPGAs in particular. Given a function f(x) to be approximated in a given interval [ x 0 , x 0+a ) and a maximum approximation error E a , the goal is to determine a function table implementation with a minimized memory footprint, i.e., number of entries that need to be stored. Rather than state-of-the-art work performing an equidistant sampling of the given interval by so-called breakpoints and using linear interpolation between two adjacent breakpoints to determine f(x) at the maximum error bound, we propose and compare three algorithms for splitting the given interval into sub-intervals to reduce the required memory footprint drastically based on the observation that in sub-intervals of low gradient, a coarser sampling grid may be assumed while guaranteeing the maximum interpolation error bound E a . Experiments on elementary mathematical functions show that a large fraction in memory footprint may be saved. Second, a hardware architecture implementing the sub-interval selection, breakpoint lookup, and interpolation at a latency of just 9 clock cycles is introduced. Third, for each generated circuit design, BRAMs are automatically instantiated rather than synthesizing the reduced footprint function table using LUT primitives, providing an additional degree of resource efficiency. The approach presented here for FPGAs can equally be applied to other circuit technologies for fast and, at the same time, memory-optimized function approximation at the edge.

Determination of discrete sampling locations minimizing both the number of samples and the maximum interpolation error: Application to measurements of carbonate chemistry in surface ocean

Over the past three decades, a variety of programs have conducted extensive measurements of ocean properties at fixed stations throughout the water column, as well as in the surface ocean via oceanographic ships and ships of opportunity. Ships of opportunity were particularly used to determine the air-sea CO2 fluxes from automated measurements of sea-surface temperature, salinity, and CO2 fugacity. These underway measurements, often recorded at a frequency of every minute, generate large data files that need to be quality controlled, stored and analyzed. For practical use these data are often binned by 1° latitude x 1° longitude. Unfortunately, by doing so, there is a consequential loss of accuracy for these data sets.Here, using the original 2010 underway data sets of sea-surface temperature, sea-surface salinity, total alkalinity and total inorganic carbon, along the cruise track from Hobart (Tasmania) to Dumont D'Urville (Antarctica), we show what would had been a more appropriate sampling strategy for each of these properties, maintaining their full measurement accuracy, while improving their interpolation accuracy. Furthermore, this analysis illustrates a general methodology for objectively determining, under suitable conditions, the appropriate locations for each property measurement according to a required accuracy. These results should greatly facilitate future cruise preparation and reduce the cost of measurements, while improving their scientific value.

МЕТОД КОДОВОЇ ЛІНІЙНОЇ ІНТЕРПОЛЯЦІЇ ДЛЯ ФОРМУВАННЯ ВІДРІЗКІВ ПРЯМИХ

Graphic images are formed using graphic primitives. These are the smallest, indivisible from the point of view of application programs, graphic elements used as the basis for building more complex images. Among the graphic primitives, the segments of straight lines, for the formation of which provides linear interpolation, have the highest specific weight. The performance of forming a graphic scene depends on the time of vector generation, therefore the question of increasing the performance of linear interpolation is relevant, especially for dynamic images. With the use of matrix screens and matrix executive bodies in registration devices, the possibility of one-stroke reproduction of row or column elements appears, which allows you to significantly increase the speed of these devices. This mode of operation is promising. Its organization requires the development of interpolation methods that allow in one interpolation cycle to receive the increment code in a row or column (code interpolation). A coded linear interpolation method is proposed, the feature of which is determined in each interpolation clock of digital segments, which includes the number of increments of the same type with the same ordinate (abscissa). For this cycle, prepare for interpolation a larger increment of the line segment to a smaller one. In the future, this ratio and the remainder of the division are used to determine digital segments. In the proposed method, the remainder of the division of a larger increment by a smaller one is accumulated, which is equal to the smaller increment. This allows you to eliminate the accumulation of error and ensure that the end point of the straight line segment is reached. The maximum interpolation error in this case does not exceed half of the discretization step, which is due to the symmetry of the error. The code linear interpolation algorithm is proposed. The research carried out in the work can be used to build high-performance computer graphics tools.

Determination of Discrete Sampling Locations Minimizing Both the Number of Samples and the Maximum Interpolation Error: Application to Measurements of Surface Ocean Properties

Determination of Discrete Sampling Locations Minimizing Both the Number of Samples and the Maximum Interpolation Error: Application to Measurements of Surface Ocean Properties

On the generation of enhanced lookup tables for wheel-rail contact models

In railway dynamics, the interpolation of lookup tables (LUTs) is a procedure utilized to reduce the computational effort when computing the wheel-rail contact forces. However, the generation of LUTs with multiple inputs and multiple outputs is a challenging task for which aspects such as their minimal size and uniform accuracy over the LUT domain have not been systematically addressed in the literature. Thus, this work presents a comprehensive methodology for a detailed analysis of general LUTs, and identifies ways to improve them. For that, an analysis of the variation of the input parameters is made and the interpolation error is assessed on the cells and edges of the original table, then, based on this analysis, two enhanced LUTs are proposed. The first one is approximately 5 times smaller than the original but holds similar accuracy. The second table exhibits half of the maximum interpolation error of the original LUT but holds an identical size. The methodology is demonstrated here using the recently published Kalker Book of Tables for Non-Hertzian contact (KBTNH) but it can be used by any other LUT approach in order to improve accuracy and/or to reduce size.

High-Precision Modeling for Aerospace TT&C Channels Using Piecewise Osculating Interpolation

Polynomial interpolation is widely used in channel modeling for aerospace tracking, telemetry, and command channel simulators, to accurately regenerate the satellite motion parameters with a high sampling rate from the available low sampling rate motion data. In this paper, a novel piecewise osculating interpolation approach is proposed to improve modeling precision. It uses the available distance and its higher-order derivatives at each pair of adjacent knots to construct the piecewise interpolation polynomial, thus achieving a better precision and continuity of the resulting motion parameters. Moreover, it is efficiently implemented with an interpolation filter based on a modified Farrow structure, and the designed filter can interpolate the motion data obtained both through uniform or nonuniform sampling. Additionally, a sampling optimization algorithm is proposed. It optimizes the motion data sampling by minimizing the maximum interpolation error given the average sampling interval and interpolation polynomial order, and therefore, improves the precision further without increasing the costs. The validity of the proposed approach is verified by numerical simulation, and its advantages are demonstrated by comparison with existing methods.

An efficient radial basis functions mesh deformation with greedy algorithm based on recurrence Choleskey decomposition and parallel computing

The mesh deformation method based on radial basis functions (RBF) has many advantages and is widely used. RBF based mesh deformation method mainly has two steps: data reduction and displacement interpolation. The data reduction step includes solving interpolation weight coefficients and searching for the node with the maximum interpolation error. The data reduction schemes based on greedy algorithm is used to select an optimum reduced set of surface mesh nodes. In this paper, a parallel mesh deformation method based on parallel data reduction and displacement interpolation is proposed. The proposed recurrence Choleskey decomposition method (RCDM) can decrease the computational cost of solving interpolation weight coefficients from O(Nc4) to O(Nc3), where Nc denotes the number of support nodes. The technology of parallel computing is used to accelerate the searching for the node with the maximum interpolation error and displacement interpolation. The combination of parallel data reduction and parallel interpolation can greatly improve the efficiency of mesh deformation. Two typical deformation problems of the ONERA M6 and DLR-F6 wing-body-Nacelle-Pylon configuration are taken as the test cases to validate the proposed approach and can get up to 19.57 times performance improvement with the proposed approach. Finally, the aeroelastic response of HIRENASD wing-body configuration is used to verify the efficiency and robustness of the proposed method.

On Optimal Bilinear Quadrilateral Meshes

The novelty of this work is in presenting interesting error properties of two types of asymptotically ‘optimal’ quadrilateral meshes for bilinear approximation. The first type of mesh has an error equidistributing property, where the maximum interpolation error is asymptotically the same over all elements. The second type has faster than expected ‘super-convergence’ property for certain saddle-shaped data functions. The ‘super-convergent’ mesh may be an order of magnitude more accurate than the error equidistributing mesh. Both types of mesh are generated by a coordinate transformation of a regular mesh of squares. The coordinate transformation is derived by interpreting the Hessian matrix of a data function as a metric tensor. The insights in this work may have application in mesh design near known corner or point singularities.

A link between Lebesgue constants and Hermite-Fejér interpolation

A review of the development of estimates for Lebesgue constants associated with Lagrange interpolation on the one hand, and estimates for the rate of convergence of Hermite-Fejér interpolation on the other hand, provides a historical perspective for the following surprising, close link between these apparently diverse concepts. Denoting by Λn (T) the Lebesgue constant of order n and by Δn (T) the maximum interpolation error for functions of class Lip 1 by Hexmite-Fejér interpolation polynomials of degree not exceeding 2n − 1, based on the zeros of the Chebyshev polynomial of first kind, we discover that, for even values of n, Λn(T) = n Δn(T).

THE PRINCIPLES OF INTERPOLATION FOR HIGHER ORDER CURVES

So far in numerical-controlled machine tools the principal methods of interpolation arethe straight line method and the circular are method. But with the development of scienceand technology, the requirement for processing complicate-shaped parts arises in some indus-trial departments and production areas. Most of the parts, for example, are of higher orderor cubic curves as in the wings of an airplane, the hull of a ship and, the nonsphericalcurves.So far as we know the higher order curves are treated by dividing them into anumber of small segments and every one of which can be approximated by a straight line ora circular are. However there are some shortcomings in this treatment as stated below: theerror produced in the interpolation may be great, the prceision in processing low. and theprograming complicated; in addition, more programing fields may be required to deal withand the tape may be very long, so it is difficult to carry out.It has been proposed in thispaper that the profile in higher order curve can be interpolated more accurately so long assome simple data are given. The distinguishing characteristics of this method are simple inoperation and easy in speed control. The maximum interpolation error is estimated as lessthan 0.707 pulse equivalent.