Inverse Multiquadrics Research Articles

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.

Polyharmonic splines interpolation on scattered data in 2D and 3D with applications

Data interpolation is a fundamental problem in many applied mathematics and scientific computing fields. This paper introduces a modified implicit local radial basis function interpolation method for scattered data using polyharmonic splines (PS) with a low degree of polynomial basis. This is an improvement to the original method proposed in 2015 by Yao et al.. In the original approach, only radial basis functions (RBFs) with shape parameters, such as multiquadrics (MQ), inverse multiquadrics (IMQ), Gaussian, and Matern RBF are used. The authors claimed that the conditionally positive definite RBFs such as polyharmonic splines r2nlnr and r2n+1 “failed to produce acceptable results”.In this paper, we verified that when polyharmonic splines together with a polynomial basis is used on the interpolation scheme, high-order accuracy and excellent conditioning of the global sparse systems are gained without selecting a shape parameter. The scheme predicts functions’ values at a set of discrete evaluation points, through a global sparse linear system. Compared to standard implementation, computational efficiency is achieved by using parallel computing. Applications of the proposed algorithms to 2D and 3D benchmark functions on uniformly distributed random points, the Halton quasi-points on regular or Stanford bunny shape domains, and an image interpolation problem confirmed the effectiveness of the method. We also compared the algorithms with other methods available in the literature to show the superiority of using PS augmented with a polynomial basis. High accuracy can be easily achieved by increasing the order of polyharmonic splines or the number of points in local domains, when small order of polynomials are used in the basis. MATLAB code for the 3D bunny example is shared on MATLAB Central File Exchange (Yao, 2023).

The RBF-FD and RBF-FDTD Methods for Solving Time-Domain Electrical Transient Problems in Power Systems

In this paper, the development and application of the radial basis function-finite difference (RBF-FD) method and the RBF-finite difference time domain (RBF-FDTD) method for solving electrical transient problems in power systems that are defined by the time-dependent ordinary differential equations (ODEs) and the time-dependent partial differential equations (PDEs), respectively, are presented. RBFs such as Gaussian (GA), Multiquadric (MQ), Inverse Quadric (IQ), and Inverse Multiquadric (IMQ) are used in these numerical methods to formulate the central finite difference approximations of the first- and second-order derivatives of a function. The algorithm of selecting “optimal” shape parameters for our basis functions is also applied, specifically to increase the accuracy of the suggested methods with regard to high accuracy needs. Finally, the accuracy, effectiveness, and applicability of our new approaches are evaluated through simulations of the switching transient voltages on a typical electrical circuit and a 220 kV single-phase transmission line, lightning-induced voltages on a 110 kV single-phase overhead distribution line, and transient voltages along two horizontal grounding electrodes excited by lightning impulse sources. The obtained numerical results demonstrate that our proposed RBF-based numerical approaches compare favorably to the traditional numerical methods.

Design of inverse multiquadric radial basis neural networks for the dynamical analysis of MHD casson nanofluid flow along a nonlinear stretchable porous surface with multiple slip conditions

The present research, a numerical approach to examine magnetohydrodynamics (MHD) Casson nanofluid flow in a porous medium along a stretchable surface with different slips using artificial neural networks (ANNs) by taking inverse multiquadric (IMQ) radial basis function (RBF) as an activation function. i.e. ANNs-IMQ-RBF. The hybridization of genetic algorithms (GAs) and sequential quadratic programming (SQP) is used for learning in ANNs-IMQ-RBF. The PDEs representing the fluid flow are converted into a nonlinear system of dimensionless form of ODEs through an appropriate transformation while effects of variation in the values of Casson parameter (β), Brownian motion parameter (Nb), Prandtl number (Pr), stretching parameter (n), porosity parameter (P), Lewis number (Le) along with temperature slip parameter (λ2) on velocity, temperature and nanofluid concentration are depicted through graphs. The effectiveness, convergence and accuracy of the proposed solver are validated evidently through boxplot analysis, histograms and cumulative distribution function (CDF) plots.

GLOBAL POSITIONING SYSTEM PRECIPITABLE WATER VAPOR INTERPOLATION USING INVERSE MULTIQUADRIC, ARTIFICIAL NEURAL NETWORK AND INVERSE DISTANCE WEIGHTED

Abstract. Precipitable water vapor (PWV) is one of the most critical data in many meteorological departments. This component has great spatial and temporal changes, so the global positioning system (GPS) always seeks to increase the accuracy of estimating the water vapor component in the troposphere. The waves sent by the satellites of this system are delayed due to passing through atmospheric layers such as the troposphere. In this paper, interpolation methods are used to estimate precipitable water vapor. Inverse multiquadric (IMQ) interpolation which is based on radial basis functions, artificial neural network (ANN) method, and inverse distance weighted (IDW) which are the most common method of interpolation in meteorology. A region in North America with 23 GPS stations was randomly selected. Then, the interpolation of precipitable water vapor on a summer day is done using GPS data. The root mean square error value (RMSE) for the IMQ method was the lowest compared to other methods and was equal to 2.11 mm. Finally, using the IMQ interpolation method, a dense map of Precipitable water vapor changes in the troposphere layer is developed for the study area.

A Mesh-Free Collocation Method Based on RBFs for the Numerical Solution of Hunter–Saxton and Gardner Equations

In this paper, the radial basis function (RBF) collocation method is applied to solve nonlinear partial differential equations (PDEs). First, the given equation is reduced to time-discrete form using Ө-weighted scheme. Then, with the help of RBFs, the given PDEs are transformed into a system of algebraic equations that is easy to solve. The proposed technique is verified by solving Hunter–Saxton and Gardner equations. For the solution of these problems, three types of RBFs, including multiquadric (MQ), inverse multiquadric (IMQ), and Gaussian (GA) have been used. The obtained solution is validated with the help of absolute error, L2, and L∞ error norms. The involved shape parameter is selected by the automatic algorithm that lies in the stable region of the method. The stability of the scheme is discussed by the spectral matrix method and validated computationally.

A Simplified Radial Basis Function Method with Exterior Fictitious Sources for Elliptic Boundary Value Problems

In this article, we propose a simplified radial basis function (RBF) method with exterior fictitious sources for solving elliptic boundary value problems (BVPs). Three simplified RBFs, including Gaussian, multiquadric (MQ), and inverse multiquadric (IMQ) without the shape parameter, are adopted in this study. With the consideration of many exterior fictitious sources outside the domain, the radial distance of the RBF is always greater than zero, such that we can remove the shape parameter from RBFs. Additionally, simplified Gaussian, MQ, and IMQ RBFs and their derivatives in the governing equation are always smooth and nonsingular. Comparative analysis is conducted for three different collocation types, including conventional uniform centers, randomly fictitious centers, and exterior fictitious sources. Numerical examples of elliptic BVPs in two and three dimensions are carried out. The results demonstrate that the proposed simplified RBFs with exterior fictitious sources can significantly improve the accuracy, especially for the Laplace equation. Furthermore, the proposed simplified RBFs exhibit the simplicity of solving elliptic BVPs without finding the optimum shape parameter.

Localized radial basis functions for no-arbitrage pricing of options under stochastic alpha–beta–rho dynamics

Closed-form explicit formulas for implied Black–Scholes volatilities provide a rapid evaluation method for European options under the popular stochastic alpha–beta–rho (SABR) model. However, it is well known that computed prices using the implied volatilities are only accurate for short-term maturities, but, for longer maturities, a more accurate method is required. This work addresses this accuracy problem for long-term maturities by numerically solving the no-arbitrage partial differential equation with an absorbing boundary condition at zero. Localized radial basis functions in a finite-difference mode are employed for the development of a computational method for solving the resulting two-dimensional pricing equation. The proposed method can use either multiquadrics or inverse multiquadrics, which are shown to have comparable performances. Numerical results illustrate the accuracy of the proposed method and, more importantly, that the computed risk-neutral probability densities are nonnegative. These two key properties indicate that the method of solution using localized meshless methods is a viable and efficient means for price computations under SABR dynamics. doi:10.1017/S1446181121000237

A high-order accurate meshless method for solution of incompressible fluid flow problems

Meshless solution to differential equations using radial basis functions (RBF) is an alternative to grid based methods commonly used. Since the meshless method does not need an underlying connectivity in the form of control volumes or elements, issues such as grid skewness that adversely impact accuracy are eliminated. Gaussian, Multiquadrics and inverse Multiquadrics are some of the most popular RBFs used for the solutions of fluid flow and heat transfer problems. But they have additional shape parameters that have to be fine tuned for accuracy and stability. Moreover, they also face stagnation error when the point density is increased for accuracy. Recently, Polyharmonic splines (PHS) with appended polynomials have been shown to solve the above issues and give rapid convergence of discretization errors with the degree of appended polynomials. In this research, we extend the PHS-RBF method for the solution of incompressible Navier-Stokes equations. A fractional step method with explicit convection and explicit diffusion terms is combined with a pressure Poisson equation to satisfy the momentum and continuity equations. Systematic convergence tests have been performed for five model problems with two of them having analytical solutions. We demonstrate fast convergence both with refinement of number of points and degree of appended polynomials. The method is further applied to solve problems such as lid-driven cavity and vortex shedding over circular cylinder. We have also analyzed the performance of this approach for solution of Euler equations. The proposed method shows promise to solve fluid flow and heat transfer problems in complex domains with high accuracy.

Modeling the kinetics of essential oil content and main constituents of mint ( Mentha aquatica L.) leaves during thin‐layer drying process using response surface methodology

The purpose of this study was to investigate changes in essential oil content and the main components of mint (Mentha aquatica L.) during thin-layer drying process. Response surface methodology (RSM) based on radial basis functions was used to model the relationships between drying temperature and different drying time (independent variables) and essential oil content and main essential oil compounds of mint (dependent variables). Results of gas chromatography with flame ionization detectior (GC-FID) and gas chromatography-mass spectrometry (GC-MS) analyses showed linalyl acetate, 1,8-cineol, and linalool are the most abundant constituents of essential oil. Inverse multiquadrics model was the best model to predict the percentage of linalyl acetate, 1,8-cineol, and essential oil content, whereas Gaussian model was the best to predict linalool. The efficiency of the selected models was acquired with R2 of 0.9658, 0.9514, 0.9568, and 0.9828 for essential oil content, linalyl acetate, linalool, and 1,8-cineol, respectively. Practical applications Results suggest that essential oil content M. aquatica and its main constituents can be scaled-up by optimizing drying time and temperature. The superior ability of RSM for modeling and monitoring of essential oil content and main components of M. aquatica during drying process as a rapid, accurate, nondestructive, and online method was also observed.

LOCALIZED RADIAL BASIS FUNCTIONS FOR NO-ARBITRAGE PRICING OF OPTIONS UNDER STOCHASTIC ALPHA–BETA–RHO DYNAMICS

AbstractClosed-form explicit formulas for implied Black–Scholes volatilities provide a rapid evaluation method for European options under the popular stochastic alpha–beta–rho (SABR) model. However, it is well known that computed prices using the implied volatilities are only accurate for short-term maturities, but, for longer maturities, a more accurate method is required. This work addresses this accuracy problem for long-term maturities by numerically solving the no-arbitrage partial differential equation with an absorbing boundary condition at zero. Localized radial basis functions in a finite-difference mode are employed for the development of a computational method for solving the resulting two-dimensional pricing equation. The proposed method can use either multiquadrics or inverse multiquadrics, which are shown to have comparable performances. Numerical results illustrate the accuracy of the proposed method and, more importantly, that the computed risk-neutral probability densities are nonnegative. These two key properties indicate that the method of solution using localized meshless methods is a viable and efficient means for price computations under SABR dynamics.

RBF collocation and hybrid-LHI methods for Stokes systems and its application to controllability problems

The purpose of this article is to introduce radial basis function (RBF) methods for solving both direct Stokes equations and controllability problems for the Stokes system with few internal scalar controls. In both cases, Dirichlet or Navier-slip boundary conditions are considered. We introduce two radial basis function solvers, one global and the other local, to solve Stokes equations. These methods are used to discretize the primal and adjoint systems related to the controllability problems. Both techniques are based on divergence-free global RBFs. A global colocation technique based on Div-free inverse multi-quadrics is formulated and analyzed. A generalization of scalar hybrid kernels to a vector divergence-free hybrid RBFs setting is defined. Based on these kernels, the local Hermite interpolation (LHI) method in vector form is introduced. Due to the properties of the hybrid kernel, we show that due to the properties of the hybrid kernel this local method, can reduce up to double precision, the value of the condition number of the local Gram matrices. Simultaneously, it is proved that the real components of the eigenvalues corresponding to the global LHI matrix are negative and that consequently backward difference formulas are stable for time integration. The conjugate gradient algorithm is adapted to the radial basis function setting to solve the controllability problems. Several benchmarks problems in two dimensions with a non-convex domain (a star shape) are numerically solved by these RBFs methods to display and compare their feasibility. The solutions to these problems are also implemented by finite element techniques to study their relative performance.

A RADIAL BASIS FUNCTION APPROACH TO ESTIMATE PRECIPITATIONS IN BRASOV COUNTY, ROMANIA

This research tackles the problems of spatial distribution of precipitation using five deterministic sub methods related to the radial basis function (RBF) group: thin-plate spline (TPS), spline with tension (ST), multiquadric (MQ), inverse multiquadric (IMQ) and completely regularized spline (CRS). The study is used for retrieving annual precipitation over Brasov County, Romania for a period of two decades (2000-2019) using data from the fifth-generation reanalysis dataset (ERA5) provided by the European Centre for Medium-range Weather Forecasts (ECMWF). Each method was tested for 10 and 60 neighbours, and the results were evaluated through cross-validation, Taylor diagram and six statistical indicators: root mean square error (RMSE), mean error (ME), correlation coefficient (R), determination coefficient (R2), average absolute percent relative error (AAPRE) and average percent relative error (APRE). The result of the study shows a similar pattern between all methods, where the predicted precipitation increases from south to north. The southern part of Brasov County recorded in two decades 1547.7 mm precipitation, and the northern part 2149.5 mm. From all the methods analysed, the most accurate method of predicting precipitation in Brasov County is ST60. The study reveals that the number of neighbours influence the accuracy of prediction. As a result, the best prediction of precipitation was generated by ST60. Not for all methods the increase number of neighbours lead to better results. From all outputs, 66.6% have better results when is used 60 neighbours and 34.4% with 10 neighbours.

A Comprehensive Price Prediction System Based on Inverse Multiquadrics Radial Basis Function for Portfolio Selection

Price prediction plays a crucial role in portfolio selection (PS). However, most price prediction strategies only make a single prediction and do not have efficient mechanisms to make a comprehensive price prediction. Here, we propose a comprehensive price prediction (CPP) system based on inverse multiquadrics (IMQ) radial basis function. First, the novel radial basis function (RBF) system based on IMQ function rather than traditional Gaussian (GA) function is proposed and centers on multiple price prediction strategies, aiming at improving the efficiency and robustness of price prediction. Under the novel RBF system, we then create a portfolio update strategy based on kernel and trace operator. To assess the system performance, extensive experiments are performed based on 4 data sets from different real-world financial markets. Interestingly, the experimental results reveal that the novel RBF system effectively realizes the integration of different strategies and CPP system outperforms other systems in investing performance and risk control, even considering a certain degree of transaction costs. Besides, CPP can calculate quickly, making it applicable for large-scale and time-limited financial market.

Local geoid determination using a generalized regression neural network and interpolation methods: A case study in Kars, Turkey

This study aimed to determine the most suitable local geoid model based on 641 GNSS/leveling points within the borders of Kars Province in eastern Turkey using the generalized regression neural network (GRNN), weighted average (WA), multiquadric (MQ), inverse multiquadric (IMQ) function, and local polynomial (LP) method. Among these methods used in local geoid determination, the studies conducted with the GRNN method are very limited in the literature. To test the performance of the model, 169 GNSS/leveling points were selected as test data. When selecting the reference and test points, care was taken to ensure that the distribution of the points was homogeneous. The criteria of root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R2) were used to assess the accuracy and error rates of the results achieved using the different methods. The analysis showed that the GRNN yielded better results than other interpolation methods (RMSE = 1.215 cm, MAE = 0.467 cm, R2 = 0.99980).

Multiquadrics without the Shape Parameter for Solving Partial Differential Equations

In this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional collocation method that assigns the RBF at each center point coinciding with an interior point, we separated the center points from the interior points, in which the center points were regarded as the fictitious sources collocated outside the domain. The interior, boundary, and source points were therefore collocated within, on, and outside the domain, respectively. Since the radial distance between the interior point and the source point was always greater than zero, the MQ and IMQ RBFs and their derivatives in the governing equation were smooth and globally infinitely differentiable. Accordingly, the shape parameter was no longer required in the MQ and IMQ RBFs. Numerical examples with the domain in symmetry and asymmetry are presented to verify the accuracy and robustness of the proposed method. The results demonstrated that the proposed method using MQ RBFs without the shape parameter acquires more accurate results than the conventional RBF collocation method with the optimum shape parameter. Additionally, it was found that the locations of the fictitious sources were not sensitive to the accuracy.

A direct prediction of the shape parameter—A purely scattered data approach

In this paper we present an approach which predicts directly without search the optimal choice of the shape parameter c contained in the multiquadrics (−1)⌈β2⌉(c2+∥x∥2)β2,β>0, and the inverse multiquadrics (c2+∥x∥2)β2,β<0. Unlike the simplex scheme where the data points are required to be evenly spaced, as in a recent paper of the author, here we allow them to be arbitrarily scattered in the simplex, making it much more useful. Besides this, we aim at helping non-mathematicians use our approach and hence remove some complicated requirements for the domain. The drawback is that its theoretical ground is not so strong as in the evenly spaced data setting. However, experiments show that it works well. The experimentally optimal value of c coincides with the theoretically predicted one. Since the fill distance, which reflects the amount of data points needed, involved is always of reasonable size, this approach is supposed to be practically useful.

Piecewise reproducing kernel-based symmetric collocation approach for linear stationary singularly perturbed problems

The aim of this paper is to develop an accurate symmetric collocation scheme for a class of linear stationary singular perturbation problems with two boundary layers. To adapt to the character of solutions, piecewise reproducing kernels is constructed. In the boundary layers intervals, inverse multiquadrics kernel function is employed. In the regular interval, exponential kernel function is used. On the basis of the piecewise reproducing kernels, a new symmetric collocation technique is presented for the considered linear stationary singular perturbation problems. Results of numerical tests illustrate that our method is easy to implement and is uniformly effective for any small e.

Combination of finite difference method and meshless method based on radial basis functions to solve fractional stochastic advection–diffusion equations

The present article develops a semi-discrete numerical scheme to solve the time-fractional stochastic advection–diffusion equations. This method, which is based on finite difference scheme and radial basis functions (RBFs) interpolation, is applied to convert the solution of time-fractional stochastic advection–diffusion equations to the solution of a linear system of algebraic equations. The mechanism of this method is such that time-fractional stochastic advection–diffusion equation is first transformed into elliptic stochastic differential equations by using finite difference scheme. Then meshfree method based on RBFs has been used to approximate the resulting equation. In other words, the approximate solution of time-fractional stochastic advection–diffusion equation is achieved with discrete the domain in the t-direction by finite difference method and approximating the unknown function in the x-direction by generalized inverse multiquadrics RBFs. In this method, the noise terms are directly simulated at the collocation points in each time step and it is the most important advantage of the suggested approach. Stability and convergence of the scheme are established. Finally, some test problems are included to confirm the accuracy and efficiency of the new approach.

Multi-objective optimization of semi-submersible platforms using particle swam optimization algorithm based on surrogate model

An Innovative Semi-submersible platform Optimization Program (ISOP) has been developed to solve the multi-objective optimization problem for semi-submersible platforms (SEMI). Three types of SEMIs, including semi-submersible floating production unit (SEMI FPU), heave and vortex induced motion (VIM) suppressed semi-submersible (HVS) and semi-submersible floating drilling unit (SEMI FDU) are selected for case studies. The hydrodynamic performances of three types of semi-submersible platforms are analyzed by using panel method and Morison's equation. In order to improve the computing efficiency, the hydrodynamic performances for different hull forms during optimization process are estimated by the surrogate models, which are built by artificial neural network prediction method and Inverse Multi-Quadric (IMQ) radial basis function (RBF). The accuracy of surrogate models is ensured by performing leave-one-out cross validation (LOOCV). The most probable maximum (MPM) heave motion and total weight, representing the safety and economy, respectively, are chosen as the two objectives for optimization. The transverse metacentric height, the MPM surge motion, and the most probable minimum (MPMin) airgap are selected as constraints. Based on surrogate models, multi-objective particle swarm optimization (MOPSO) is employed to search for the Pareto-optimal solutions. A Computational Fluid Dynamics (CFD) tool is adopted to validate the proposed model for the prediction of the motion responses. By comparing the obtained Pareto-optimal solutions with the initial design using simple panel method plus Morison's equation, it is confirmed that the MPM heave motions for SEMI FPU, HVS and SEMI FDU can be suppressed by up to 12.68%, 11.92%, and 14.96%, respectively, and the total weights can be reduced by up to 12.16%, 13.00%, and 24.91%, respectively. Through the detailed analyses of optimization results, the most efficient design strategies for semi-submersible platforms are discussed and proposed.