Multiquadric Function Research Articles

Improved multi-strategy adaptive Grey Wolf Optimization for practical engineering applications and high-dimensional problem solving

The Grey Wolf Optimization (GWO) is a highly effective meta-heuristic algorithm leveraging swarm intelligence to tackle real-world optimization problems. However, when confronted with large-scale problems, GWO encounters hurdles in convergence speed and problem-solving capabilities. To address this, we propose an Improved Adaptive Grey Wolf Optimization (IAGWO), which significantly enhances exploration of the search space through refined search mechanisms and adaptive strategy. Primarily, we introduce the incorporation of velocity and the Inverse Multiquadratic Function (IMF) into the search mechanism. This integration not only accelerates convergence speed but also maintains accuracy. Secondly, we implement an adaptive strategy for population updates, enhancing the algorithm's search and optimization capabilities dynamically. The efficacy of our proposed IAGWO is demonstrated through comparative experiments conducted on benchmark test sets, including CEC 2017, CEC 2020, CEC 2022, and CEC 2013 large-scale global optimization suites. At CEC2017, CEC 2020 (10/20 dimensions), CEC 2022 (10/20 dimensions), and CEC 2013, respectively, it outperformed other comparative algorithms by 88.2%, 91.5%, 85.4%, 96.2%, 97.4%, and 97.2%. Results affirm that our algorithm surpasses state-of-the-art approaches in addressing large-scale problems. Moreover, we showcase the broad application potential of the algorithm by successfully solving 19 real-world engineering challenges.

Efficient Scheme for the Economic Heston–Hull–White Problem Using Novel RBF-FD Coefficients Derived from Multiquadric Function Integrals

This study presents an efficient method using the local radial basis function finite difference scheme (RBF-FD). The innovative coefficients are derived from the integrals of the multiquadric (MQ) function. Theoretical convergence rates for the coefficients used in function derivative approximation are provided. The proposed scheme utilizes RBF-FD estimations on three-point non-uniform stencils to construct the final approximation on a tensor grid for the 3D Heston–Hull–White (HHW) PDE, which is relevant in economics and mathematical finance. Numerical evidence and comparative analyses validate the results and the proposed scheme.

Shape preserving fractal multiquadric quasi-interpolation

In this article, we construct a novel self-referential fractal multiquadric function which is symmetric about the origin. The scaling factors are suitably restricted to preserve the differentiability and the convexity of the underlying classical multiquadric function. Based on the translates of a fractal multiquadric function defined on a grid, we propose two fractal quasi-interpolants LCαf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_C^{\\alpha }f$$\\end{document} and LDαf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_D^{\\alpha }f$$\\end{document} to approximate smooth and irregular functions. We study the convergence of LCαf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_C^{\\alpha }f$$\\end{document} and LDαf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_D^{\\alpha }f$$\\end{document} to f using uniform error estimates. We investigate the linear polynomial reproducing property, convexity/concavity and monotonicity features of these quasi-interpolation operators. The advantages of fractal quasi-interpolants over the classical quasi-interpolants are demonstrated by various examples.

An Optimal Multiquadric Variable Shape Parameter for Boundary Value Problems Using Particle Swarm Optimization

The multiquadric radial basis function method has been widely used to solve partial differential equations-based problems regarding its flexibility and meshfree characteristics. The accuracy and stability of this method are derived and based on the use of a free-shape parameter that sensibly controls the comportment of the technique. Significant improvements have already been reported and show that variable shape parameters conduct the method to handle problems with striking results compared to global-based techniques. Nevertheless, choosing a suitable set of shape parameters is still an open topic because of the complexity of the method when the number of collocation points increases. The current work proposes a variant particle swarm optimization based on local displacement with attractors to determine the multi-quadratic function's ``best'' optimal variable shape parameter in solving boundary value problems. Based on an initially random set of variable shape parameters, the proposed algorithm first performs and evaluates the errors between the expected exact solution and the approximate solution thoroughly. In the first stage, the particle swarm algorithm search for an optimal set of shape parameters that minimize the error and the conditioning number of the radial basis system matrix. In the second stage, the obtained optimal set of shape parameters is applied to solve the considered problem. In this way, when the number of collocation points increases, the first stage based on particle swarm optimization stabilizes the strategy. It proposes an ``acceptable'' set of shape parameters for the given problem. The proposed method is applied to a set of well-known boundary value problems in one and two-dimensional spaces and compared to other techniques published in the literature. The results show that the proposed method achieves more accurate solutions than recently proposed in the literature.

High accuracy quasi-interpolation using a new class of generalized multiquadrics

A new generalization of multiquadric functions φ(x)=c2d+||x||2d, where x∈Rn, c∈R, d∈N, is presented to increase the accuracy of quasi-interpolation further. With the restriction to Euclidean spaces of odd dimensionality, the generalization can be used to generate a quasi-Lagrange operator that reproduces all polynomials of degree 2d−1. In contrast to the classical multiquadric, the convergence rate of the quasi-interpolation operator can be significantly improved by a factor h2d−n−1, where h>0 represents the grid spacing. Among other things, we compute the generalized Fourier transform of this new multiquadric function. Finally, an infinite regular grid is employed to analyse the properties of the aforementioned generalization in detail. We also present numerical results to demonstrate the advantages of our new multiquadric functions.

Numerical investigation of high-dimensional option pricing PDEs by utilizing a hybrid radial basis function - finite difference procedure

The target of this research is to resolve high-dimensional partial differential equations (PDEs) for multi-asset options, modeled as parabolic time-dependent PDEs. We present a hybrid radial basis function - finite difference (RBF-FD) solver, which combines the advantages of Gaussian and multiquadric functions. Additionally, we employ the Krylov subspace method on the resulting system of ordinary differential equations, reducing the computation load for finding the numerical solution. Computational tests, up to seven dimensions, and comparisons support the superiority of our hybrid solver.

Integrated intelligence of inverse multiquadric radial base neuro‐evolution for radiative MHD Prandtl–Eyring fluid flow model with convective heating

AbstractIn this communication, a new stochastic numerical paradigm is introduced for thorough scrutinization of the magnetohydrodynamic (MHD) boundary layer flow of Prandtl–Eyring fluidic model along a stretched sheet impact on thermal radiation as well as convective heating scenarios by exploiting the accurate approximation knacks of ANNs (artificial‐neural‐networks) modeling by using the inverse multiquadric (IMQ) function optimized/trained with well‐reputed global search with genetic algorithms (GAs) and local search efficacy of sequential quadratic programming (SQP), that is, ANNs‐IMQ‐GA‐SQP. The mathematical model of the Prandtl–Eyring fluid flow is portrayed in the form of PDEs and then converted in the form of a nonlinear system of ODEs by utilizing suitable similarity transformation to calculate/analyze the solution dynamics. The kinematic and thermodynamic properties of MHD Prandtl–Eyring fluid flow are examined/observed by varying the sundry physical parameters. The obtained intelligent computing designed solver‐based numerical results are consistently found in good agreement with reference results obtained through the Adams numerical technique. First and second‐order cumulant‐based statistical assessments are extensively exploited to endorse trustily the efficacy of ANNs‐IMQ‐GA‐SQP solver based on exhaustive simulations for the Prandtl–Eyring fluidic system.

Spherical interpolation of scattered data using least squares thin-plate spline and inverse multiquadric functions

Spherical interpolation of scattered data using least squares thin-plate spline and inverse multiquadric functions

Solving time fractional partial differential equations with variable coefficients using a spatio-temporal meshless method

<p>This paper presents a novel spatio-temporal meshless method (STMM) for solving the time fractional partial differential equations (TFPDEs) with variable coefficients based on the space-time metric. The main idea of the STMM is to directly approximate the solutions of fractional PDEs by using a multiquadric function with the space-time distance within a space-time scale framework. Compared with the existing methods, the present meshless STMM entirely avoids the difference approximation of fractional temporal derivatives and can be easily applied to complicated irregular geometries. Furthermore, both regular and irregular nodal distribution can be used without loss of accuracy. For these reasons, this new space-time meshless method could be regarded as a competitive alternative to the conventional numerical algorithms based on difference decomposition for solving the TFPDEs with variable coefficients. Numerical experiments confirm the ability and accuracy of the proposed methodology.</p>

Nonsingular Meshless Method in an Acoustic Indoor Problem

An efficiency of the nonsingular meshless method is analyzed in an acoustic indoor problem. The solution is assumed in the form of the series of radial bases functions. The Hardy’s multiquadratic functions, as the bases, are taken into account. The room acoustic field with uniform, impedance walls is considered. The representative, rectangular cross section of the room is chosen. Practical combinations of acoustic boundary conditions, expressed through absorption coefficient values, are considered. The classical formulation of the boundary problem is used. It is established any coefficient in the multiquadratic functions depend on the number of influence points, the frequency and the absorption coefficient. All approximate results are calculated in relation to the exact ones. This way, it is proved that the meshless method based on the multiquadratic functions is simple and efficient method in the description of the complicated acoustic boundary problems for the low and medium ranges of frequency.

An efficient radial basis function generated finite difference meshfree scheme to price multi-dimensional PDEs in financial options

In this paper, the purpose is to focus on proposing a new solver for pricing financial option at the presence of several assets. Toward this goal, first we introduce the weighting coefficients of the radial basis function-finite difference (RBF-FD) procedure for approximating the derivatives of the functions under the inverse multi-quadric function (IMQ). The weights are employed on a nonuniform mesh of points when tackling multi-dimensional partial differential equations (PDEs). A new solver based on RBF-FD methodology is then constructed employing matrix notations for solving option pricing problems. The theoretical discussions will be confirmed via numerical experiments on several test problems consisting of four to five underlying assets.

Statistical Assessment of Some Interpolation Methods for Building Grid Format Digital Bathymetric Models

As far as the knowledge of the seabed is concerned, both for safe navigation and for scientific research, 3D models, particularly digital bathymetric models (DBMs), are nowadays of fundamental importance. This work aimed to evaluate the quality of DBMs according to the interpolation methods applied to obtain grid format 3D surfaces from scattered sample points. Other complementary factors affecting DBM vertical accuracy, such as seabed morphological complexity and surveyed points sampling density, were also analyzed by using a factorial ANOVA experimental design. The experiments were performed on a multibeam dataset provided by the Italian Navy Hydrographic Institute (IIM) with an original resolution of 1 m × 1 m grid spacing, covering a surface of 0.24 km2. Six different sectors comprising different seabed morphologies were investigated. Eight sampling densities were randomly extracted from every sector, each with four repetitions. Finally, four different interpolation methods were tested, including: radial basis multiquadric function (RBMF), ordinary kriging (OK), universal kriging (UK) and Gaussian Markov random fields (GMRF). The results demonstrated that both RBMF and OK produced very accurate DBM in areas characterized by low levels of seabed ruggedness at sampling densities of only 0.0128 points/m2 (equivalent grid spacing of 8.84 m). In contrast, a higher density of 0.1024 points/m2 (3.13 m grid spacing) was required to produce accurate DBM in areas with more complex seabed topography. On the other hand, UK and GMRF were strongly influenced by morphology and sampling density, yielding higher vertical random errors and more prone to slightly overestimate seabed depths. In addition, sampling density and morphology were the factors that most influenced the vertical accuracy of the interpolated DBM. In this sense, the highly statistically significant influence of the interaction between sampling density and morphology on the vertical accuracy of the interpolated DBM confirms the need to perform a preliminary analysis of seabed morphological complexity in order to increase, if necessary, the number of surveyed points in cases of complex morphologies.

Inverse Multiquadric Function to Price Financial Options under the Fractional Black–Scholes Model

The inverse multiquadric radial basis function (RBF), which is one of the most important functions in the theory of RBFs, is employed on an adaptive mesh of points for pricing a fractional Black–Scholes partial differential equation (PDE) based on the modified RL derivative. To solve this problem, discretization along space is carried out on a non-uniform grid in order to focus on the hot area, at which the initial condition of the pricing model, i.e., the payoff, has discontinuity. The L1 scheme having the convergence order 2−α is used along the time fractional variable. Then, our proposed numerical method is built by matrices of differentiations to be as efficient as possible. Computational pieces of evidence are brought forward to uphold the theoretical discussions and show how the presented method is efficient in contrast to the exiting solvers.

An ensemble machine learning approach for enhanced path loss predictions for 4G LTE wireless networks

SummaryAccurate path loss prediction models are indispensable in modern wireless communication systems. In recent times, several path loss prediction models have been proposed to improve network performance. However, most of these models have not addressed the fundamental issues. The problem of deploying a single path loss prediction model that fits well in all wireless propagation environments remains. To address this problem, we present machine learning‐based ensemble methods to path loss predictions. Specifically, ensemble methods have been introduced to improve signal prediction accuracy and performance. Additionally, the radial basis function (RBF) and multilayer perceptron (MLP) neural network models were deployed and improved by adding more network parameters to their respective input layers. Results revealed that the RBF model with an increase in the number of centroids reduces the mean square error (MSE). Also, the Gaussian kernel function gives lower MSE compared to the multiquadric and inverse multiquadric functions. The bagging and blending ensemble path loss prediction models were introduced for optimal performance. An RBF neural network of different clusters was incorporated into the bagging algorithm as base learners. The RBF, MLP, blending, and bagging were examined using standard metrics for the training, testing, and validation dataset for model validation. The developed bagging ensemble path loss prediction model gave the lowest errors (MSE = 0.0011 dB, SSE = 0.6069 dB, MAE = 0.0245 dB, &R = 0.7484 dB) for the datasets. The bagging ensemble method acts as a variance and error reduction mechanism because it predicted path loss closest to measured data and is suitable for near precise path loss predictions.

Analytical Solution of Electromagnetic Force on Nanofluid Flow with Brownian Motion Effects Between Parallel Disks

The innovation of the present paper is the analytical study of Brownian motion effects on nanofluid flow and electromagnetic force between parallel disks with a heat source. Nanoparticle effects on nondimensional temperature field and velocity of fluid flow were analyzed using Akbari-Ganji’s Method and radial basis function approximation based on Hardy multiquadric function. Akbari-Ganji’s Method (AGM) is a strong analytical method that solves any linear and nonlinear differential equation with any degree of variables. Radial basis functions is an approximation method for analyzing functions and equations at high degrees, especially when it is necessary to apply the interpolation problem for scattered data on irregular geometry. The results signified that the maximum difference between AGM and RBF methods, for nondimensional horizontal velocity on CuO nanofluid at and is 0.2251 and the minimum difference for the nondimensional vertical velocity of Al2O3 nanofluid at is equal to 0.0018. Also, the effects of the Hartmann number (Ha) on nondimensional horizontal and vertical velocities field for Al2O3 nanoparticles at have a slight difference from the other Hartmann values using the AGM method. The maximum of nondimensional horizontal velocities at and is equal to 1.9354.

A Meshless Method with Radial Basis Function for Solving Unsaturated Flow in Heterogeneous Porous Media

This paper presents a study for solving unsaturated flow in heterogeneous porous media using the meshless method with the radial basis function (RBF). For modeling the nonlinear hydrological process in unsaturated zone, an exponential model is introduced in the Richards equation such that we may obtain the linearized Richards equation. We adopt the multiquadric function as the RBF in the meshless method for solving the linearized Richards equation. For simulating the unsaturated flow problems in layered heterogeneous soils, the flux and the head must satisfy the continuity condition at the interface. Several examples are carried out for modeling the hydrological process in multi–layered unsaturated soils. The results demonstrate that we only discretize by a set of points without tedious mesh generation and significantly enhance the applicability for solving unsaturated flow problems, especially in heterogeneous multi–layered soils with extreme physical property contrasts.

Localized kernel‐based meshless method for pricing financial options underlying fractal transmission system

The variation in the option pricing of the fractal transmission system is modelled by the time fractional Black–Scholes equation (TFBSE). This paper proposes an efficient local meshless method for the numerical simulation of the TFBSE. At the first step, a difference formula of L1 type is employed to get a semi‐discrete algorithm in the temporal variable with a accuracy of order 2 − α in the case of smooth solutions, where 0 < α ≤ 1 is the fractional‐order derivative. At the second step, a localized radial basis function finite difference is adopted to derive a full‐discrete scheme. Moreover, the unconditional stability and convergence of the proposed method are analyzed based on energy norm. The exact expressions for the weights of the first and second derivatives are used by imposing a multiquadric function generated by finite difference. The proposed technique produces linear systems with tridiagonal and diagonal matrices. Numerical experiments highlight the performance of the method.

Constructing the weighting coefficients for the RBF-Hermite FD scheme under the multiquadric function on irregular meshes

It is well known that the order of finite difference estimates on nonuniform grids reduce dramatically, particularly when higher order derivatives are required. This paper contributes how an optimization of the order of approximation formulas can be investigated and done on such grids for a sufficiently smooth function. To generalize the idea as well as get the optimized orders, the notion of radial basis function—Hermite finite difference (RBF–HFD) approach is used. The new weighting coefficients are worked out and proved to possess higher convergence rates. Several tests are also given to demonstrate the theoretical discussions.

Use of Multiquadric Functions for Multivariable Representation of the Aerodynamic Coefficients of Airfoils

Given an array (or matrix) of values for a function of one or more variables, it is often desired to find a value between two given points. Multivariable interpolation and approximation by radial basis functions are important subjects in approximation theory that have many applications in Science and Engineering fields. During the last decades, radial basis functions (RBFs) have found increasingly widespread use for functional approximation of scattered data. This research work aims at benchmarking two different approaches: an approximation by radial basis functions and a piecewise linear multivariable interpolation in terms of their effectiveness and efficiency in order to conclude about the advantages and disadvantages of each approach in approximating the aerodynamic coefficients of airfoils. The main focus of this article is to study the main factors that affect the accuracy of the multiquadric functions, including the location and quantity of centers and the choice of the form factor. It also benchmarks them against piecewise linear multivariable interpolation regarding their precision throughout the selected domain and the computational cost required to accomplish a given amount of solutions associated with the aerodynamic coefficients of lift, drag and pitching moment. The approximation functions are applied to two different multidimensional cases: two independent variables, where the aerodynamic coefficients depend on the Reynolds number (Re) and the angle-of-attack (α), and four independent variables, where the aerodynamic coefficients depend on Re, α, flap chord ratio (cflap), and flap deflection (δflap).

Spatial variability and risk assessment of groundwater pollution in El-Outaya region, Algeria

Groundwater contaminations’ spatial distribution and risk assessment are among the most important issues for human consumption, water resources managers and decision makers. For this purpose, stochastic and deterministic methods are employed in this paper based on the empirical Bayesian Kriging (EBK), ordinary kriging (OK), and radial basis function (RBF) with multi-quadratic function, which help to predict the spatial distribution and risk assessments. These methods are based on the contamination data from 68 wells in El-Outaya plain (Biskra, southeastern Algeria. Nitrate-nitrogen (NO3–N) is the most important contamination in this region. Among the methods indicator Kriging (IK) is used to elaborate probability maps based on three thresholds, namely, [NO3–N] > 3 mg/l, [NO3–N] > 8.9 mg/l and [NO3–N] > 11.3 mg/l. The probability maps are then transformed to categories using binary maps of two of exceeding thresholds levels as (P ≥ 0.5 and P ≥ 0.9). The determination coefficient (R2), the root mean square error (RMSE) and the average standard error (ASE) of the obtained raster maps are used to compare the different interpolation methods and to select the best one. The results show that all the used interpolation techniques have accurately predicted the spatial distribution of [NO3–N], which provides opportunity to predict the contamination values even at unsampled locations in the region. The raster maps indicate the same overall spatial distribution of groundwater quality categories and their measurement errors indicate the superiority of RBF over EBK and OK. RBF is an accurate interpolation technique to identify groundwater pollution in El-Outaya plain. The transformation of probability maps to categorical maps based on IK is important for decision makers to identify regions with high nitrate-N contamination probability and high risk to human health.