Graph-Directed Coalescence Hidden Variable Fractal Interpolation Functions

Fractal interpolation function (FIF) is a special type of continuous function which interpolates certain data set and the attractor of the Iterated Function System (IFS) corresponding to a data set is the graph of the FIF. Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is both self-affine and non self-affine in nature depending on the free variables and constrained free variables for a generalized IFS. In this article, graph directed iterated function system for a finite number of generalized data sets is considered and it is shown that the projection of the attractors on is the graph of the CHFIFs interpolating the corresponding data sets.

Radial Basis Function Interpolation and Galerkin Projection for Direct Trajectory Optimization and Costate Estimation

This work presents a novel approach combining radial basis function(RBF)interpolation with Galerkin projection to efficiently solve general optimal control problems.The goal is to develop a highly flexible solution to optimal control problems,especially nonsmooth problems involving discontinuities,while accounting for trajectory accuracy and computational efficiency simultaneously.The proposed solution,called the RBF-Galerkin method,offers a highly flexible framework for direct transcription by using any interpolant functions from the broad class of global RBFs and any arbitrary discretization points that do not necessarily need to be on a mesh of points.The RBF-Galerkin costate mapping theorem is developed that describes an exact equivalency between the Karush-Kuhn-Tucker(KKT)conditions of the nonlinear programming problem resulted from the RBF-Galerkin method and the discretized form of the first-order necessary conditions of the optimal control problem,if a set of discrete conditions holds.The efficacy of the proposed method along with the accuracy of the RBF-Galerkin costate mapping theorem is confirmed against an analytical solution for a bang-bang optimal control problem.In addition,the proposed approach is compared against both local and global polynomial methods for a robot motion planning problem to verify its accuracy and computational efficiency.

TRIANGULAR C^m INTERPOLATION BY RATIONAL FUNCTIONS

Two general local Cm triangular inter polation schemes by rational functions from Cm data are proposed for any nonnegative integer m. The sohemes can have either 2m+1 order algebraic precision if the required data are given on vertices and edges, or m+E[m/2]+1 or m+1 order algebraic precision if the data are given only at vertices. The orders of the inter error are estimated. Examples that show the correctness and effectiveness of the scheme are presented.

Spatio-temporal variability of surface chlorophyll a in the Yellow Sea and the East China Sea based on reconstructions of satellite data of 2001-2020

Chlorophyll-a(Chl-a)concentration is a primary indicator for marine environmental monitoring.The spatio-temporal variations of sea surface Chl-a concentration in the Yellow Sea(YS)and the East China Sea(ECS)in 2001-2020 were investigated by reconstructing the MODIS Level 3 products with the data interpolation empirical orthogonal function(DINEOF)method.The reconstructed results by interpolating the combined MODIS daily+8-day datasets were found better than those merely by interpolating daily or 8-day data.Chl-a concentration in the YS and the ECS reached its maximum in spring,with blooms occurring,decreased in summer and autumn,and increased in late autumn and early winter.By performing empirical orthogonal function(EOF)decomposition of the reconstructed data fields and correlation analysis with several potential environmental factors,we found that the sea surface temperature(SST)plays a significant role in the seasonal variation of Chl a,especially during spring and summer.The increase of SST in spring and the upper-layer nutrients mixed up during the last winter might favor the occurrence of spring blooms.The high sea surface temperature(SST)throughout the summer would strengthen the vertical stratification and prevent nutrients supply from deep water,resulting in low surface Chl-a concentrations.The sea surface Chl-a concentration in the YS was found decreased significantly from 2012 to 2020,which was possibly related to the Pacific Decadal Oscillation(PDO).

An efficient large-scale mesh deformation method based on MPI/Open MP hybrid parallel radial basis function interpolation

An efficient MPI/OpenMP hybrid parallel Radial Basis Function (RBF) strategy for both continuous and discontinuous large-scale mesh deformation is proposed to reduce the computational cost and memory consumption.Unlike the conventional parallel methods in which all processors use the same surface displacement and implement the same operation,the present method employs different surface points sets and influence radius for each volume point movement,accompanied with efficient geometry searching strategy.The deformed surface points,also called Control Points (CPs),are stored in each processor.The displacement of spatial points is interpolated by using only 20-50 nearest control points,and the local influence radius is set to 5-20 times the maximum displacement of control points.To shorten the searching time for the nearest control point clouds,an Alternating Digital Tree (ADT) algorithm for 3D complex geometry is designed based on an iterative bisection technique.Besides,an MPI/OpenMP hybrid parallel approach is developed to reduce the memory cost in each High-Performance Computing (HPC) node for large-scale applications.Three 3D cases,including the ONERA-M6 wing and a commercial transport airplane standard model with up to 2.5 billion hybrid elements,are used to test the present mesh deformation method.The robustness and high parallel efficiency are demonstrated by a wing deflection case with a maximum bending angle of 450 and more than 80% parallel efficiency with 1024 MPI processors.In addition,the availability for both continuous and discontinuous surface deformation is verified by interpolating the projecting displacement with opposite directions surface points to the spatial points.

Lacunary Interpolation by Fractal Splines with Variable Scaling Parameters

For a prescribed set of lacunary data{(x_(ν),f_(ν),f^(″)_(ν)):ν=0,1,...,N}with equally spaced knot sequence in the unit interval,we show the existence of a fam-ily of fractal splines S^(α)_(b)∈C 3[0,1]satisfying S^(α)_(b)(x_(ν))=f_(ν),(S^(α)_(b))^(2)(x_(ν))=f^(″)_(ν)forν=0,1,...,N and suitable boundary conditions.To this end,the unique quintic spline introduced by A.Meir and A.Sharma[SIAM J.Numer.Anal.10(3)1973,pp.433-442]is generalized by using fractal functions with variable scaling pa-rameters.The presence of scaling parameters that add extra“degrees of freedom”,self-referentiality of the interpolant,and“fractality”of the third derivative of the in-terpolant are additional features in the fractal version,which may be advantageous in applications.If the lacunary data is generated from a functionΦsatisfying certain smoothness condition,then for suitable choices of scaling factors,the corresponding fractal spline S^(α)_(b)satisfies||Φ^(r)−(S^(α)_(b))(r)||∞→0 for 0≤r≤3,as the number of partition points increases.

The Formulation of Finite Difference RBFWENO Schemes for Hyperbolic Conservation Laws:An Alternative Technique

To solve conservation laws,efficient schemes such as essentially nonoscillatory(ENO)and weighted ENO(WENO)have been introduced to control the Gibbs oscillations.Based on radial basis functions(RBFs)with the classical WENO-JS weights,a new type of WENO schemes has been proposed to solve conservation laws[J.Guo et al.,J.Sci.Comput.,70(2017),pp.551–575].The purpose of this paper is to introduce a new formulation of conservative finite difference RBFWENO schemes to solve conservation laws.Unlike the usual method for reconstructing the flux functions,the flux function is generated directly with the conservative variables.Comparing with Guo and Jung(2017),the main advantage of this framework is that arbitrary monotone fluxes can be employed,while in Guo and Jung(2017)only smooth flux splitting can be used to reconstruct flux functions.Several 1D and 2D benchmark problems are prepared to demonstrate the good performance of the new scheme.

Degradation ofβ‑Actin mRNA and 18S rRNA in Mouse Spleen Cells after Death

We observed degradation ofβ‑actin mRNA and 18S rRNA in mouse spleen cells under constant temperature conditions in the different temperature group during postmortem intervals(PMIs)of 0-72 h.Thirty‑nine mice were sacrificed by cervical dislocation and kept at constant temperatures of 10℃,15℃,20℃,25℃,and 30℃.From 0 to 72 h after death,total RNA in spleen cells was extracted every 6 h.The cycle threshold(Ct)values ofβ‑actin mRNA and 18S rRNA were obtained by real‑time‑quantitative polymerase chain reaction.The results showed that,under the conditions of different and constant temperatures after mouse death at 72 h,the Ct values ofβ‑actin and 18S,Ct ratios ofβ‑actin to 18S,and relative ratios ofβ‑actin to 18S were significantly correlated with PMI.In addition,the relative degradation rates ofβ‑actin and 18S appeared to change from fast to slow with the increase of temperature.By interpolation and fitting analysis of the data,we obtained a ternary quintic equation of the relationship between the change in the relative ratios and PMI,which can be used to infer PMI within a certain temperature range(10℃-30℃).

Estimation of the Postmortem Interval by Measuring Blood Oxidation‑reduction Potential Values

Accurate estimation of the postmortem interval(PMI)is an important task in forensic practice.In the last half-century,the use of postmortem biochemistry has become an important ancillary method in determining the time of death.The present study was carried out to determine the correlation between blood oxidation-reduction potential(ORP)values and PMIs,and to develop a three-dimensional surface equation to estimate the PMI under various temperature conditions.A total of 48 rabbits were placed into six groups and sacrificed by air embolism.Blood was obtained from the right ventricle of each rabbit,and specimens were stored at 10℃,15℃,20℃,25℃,30℃,and 35℃.At different PMIs(once every 4 h),the blood ORP values were measured using a PB-21 electrochemical analyzer.Statistical analysis and curve fitting of the data yielded cubic polynomial regression equations and a surface equation at different temperatures.Result:The results showed that there was a strong positive correlation between the blood ORP values at different temperatures and the PMI.This study provides another example of using a three-dimensional surface equation as a tool to estimate the PMI at various temperature conditions.