Approximate Function Research Articles

The use of interpolation polynomials in the approximation of functions: comparative study

In this study, we present three methods for approximating functions using interpolation polynomials. In engineering, there are many situations where it is necessary to approximate the value of a function at a certain point, knowing only a finite set of experimental data or when the function itself is presented in a form that is not easily usable. We use specific methods in numerical analysis such as Lagrange interpolation, the method of dividing differences and Neville’s method. Given a function f defined on an interval [a,b], the purpose of this work is to construct a function that approximates the function f at a predetermined value. The expression of the approximation functions will be calculated and we will evaluate the obtained error for each approximation.

Mechanical-electric-magnetic-thermal coupled enriched finite element method for magneto-electro-elastic structures

Abstract Magneto-electro-elastic (MEE) materials possess the ability to convert mechanical, electrical, and magnetic energies, playing a critical role in smart devices. To improve the accuracy and efficiency of solving the mechanical properties of MEE structures in mechanical-electrical-magnetic-thermal (MEMT) environments, an MEMT coupled multiphysics enriched finite element method (MP-EFEM) is proposed. Based on the fundamental equations and boundary conditions of MEE materials, the interpolation coverage function is introduced into the MEMT coupled finite element method (FEM) to construct higher-order approximate interpolation displacement shape functions, electric potential shape functions, and magnetic potential shape functions. Combined with the variational principle, MP-EFEM is proposed, and the governing equations of MP-EFEM are derived. Numerical examples validate the accuracy and high efficiency of MP-EFEM in solving the mechanical properties of MEE structures in MEMT environments. When compared to the MEMT coupled FEM (MEMT-FEM), the results show that this method offers higher accuracy and efficiency. Therefore, MP-EFEM can effectively analyze the mechanical properties of MEE structures under multiphysics coupling, providing a new method for the design and development of smart devices.

HIGH-ORDER ZIG-ZAG THEORIES TO STATIC LAMINATED COMPOSITE BEAM ANALYSIS BY THE RAYLEIGH-RITZ METHOD

Technological advancements across various engineering fields have grown demand for highly efficient materials for specific applications. Laminated composite materials have emerged as a promising solution, offering a combination of favorable mechanical properties by layering different materials. Theories, such as zig-zag theories, have been developed to predict the mechanical behavior of these materials under external stresses. This study investigates the outcomes provided by the Rayleigh-Ritz variational method, employing different approximation functions, particularly focusing on unified higher-order zig-zag kinematics. All proposed shape functions agree to the reference values, but the model with polynomial shape function shows more applications possibilities. Keywords: composite materials, composite laminated beams, zig-zag theory, Rayleigh-Ritz method.

Plain Capping for Improved Accuracy of Approximate One- and Two-Electron Densities at Two-Particle Coalescence Points.

The values of the one-electron and intracule densities at two-particle coalescence points that enter the expressions for relativistic corrections to energies of Coulombic systems cannot be efficiently computed with sufficient accuracy from approximate wave functions expressed in terms of cuspless basis functions such as the explicitly correlated Gaussians. A new approach to alleviation of this problem, called plain capping, is proposed. Unlike those offered by the previously published formalisms, such as the expectation value identities and integral transforms, the accuracy improvements effected by the plain capping are attained with negligible computational effort and minimum programming. In the case of the on-top two-electron densities, whose accurate computation is particularly costly, the plain capping constitutes the only viable means of error reduction available at present.

Multi-focus Image Fusion via Blind Deconvolution and Registration based on Photoacoustic Microscopy

Abstract Optical resolution photoacoustic microscopy (OR-PAM), which can present the microstructure of biological tissues is useful in the disease diagnosis. However, conventional OR-PAM can only obtain images with high resolution in a narrow range of depths around the focal plane. In this study, we proposed a multi-focus photoacoustic image fusion method based on the blind deconvolution. This method is divided into two steps. First, we used the method of blind deconvolution to obtain an approximate point spread function (PSF) for each photoacoustic images obtained from multiple layers, and then got the corresponding deblurred photoacoustic images. Second, we used the method of multi-focus images registration and fusion to accurately align the multi-focus photoacoustic images and effectively fuse the features of different images. Finally, the proposed method was applied to process the photoacoustic images obtained from photoacoustic imaging experiments on ants with irregular morphology using fast controllable confocal focus PAM (FC-PAM). The results show that our method improves the quality of photoacoustic images, and can quickly obtain multi-focus fused photoacoustic images with high resolution. It indicates that the proposed multi-focus photoacoustic image fusion method based on FC-PAM has great potential in fast imaging of living biological tissues at multiple imaging depths.

Better approximation of sigmoid function for privacy-preserving neural networks

Abstract With the development of artificial intelligence technology, privacy protection has become a prominent research direction. Homomorphic encryption is an important and promising technology in the construction of privacy-preserving neural networks. However, computing nonlinear activation functions remains a challenging task in the context of homomorphic encryption. Considering Sigmoid is one of the most commonly used activation functions, this paper proposes a polynomial approximation method for the Sigmoid function and applies it to the homomorphic encryption scheme of CKKS. We design polynomial approximation functions using Improved Taylor Expansion Segmentation (ITES), Least Squares method (LS), and Random Noise Addition method, respectively. These methods are implemented through the CKKS scheme in the homomorphic encryption library of SEAL. The experimental results show that the ITES achieves high fitting accuracy, reducing the average absolute error by 11.09% and the mean squared error by 38.06% compared to previous fitting methods. Therefore, experiments have identified methods with better fitting results in the CKKS scheme, which can meet the requirements of using homomorphic encryption in deep learning technologies. Finally, our analysis points out that different technologies have unique advantages and adapt to different scenarios, e.g., ITES has higher accuracy, LS has higher performance in scenarios with limited computing resources.

A novel method for approximating the distribution of chi-squared-type mixtures

. In order to approximate the distribution of chi-squared-type mixtures, Zhang (2005) proposed to use a chi-squared-type random variable of the form α 1 χ d 1 2 + β 1 , where the unknown parameters α 1, β 1, and d 1 are determined by matching the first three cumulants. In this article, we propose a novel method to approximate the distribution of chi-squared-type mixtures by the distribution of a random variable in the form α χ d 2 + β + σξ , where ξ is a standard normal random variable, and the unknown parameters α, β, σ, and d are determined by matching the first four cumulants. The approximating error bound on the distribution functions of the new method approximation is established. The numerical results show that our proposed method can has fewer error bound than some existed methods in some examples.

Residual growth control for general maps and an approximate inverse function result

The need to control the residual of a potentially nonlinear function F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {F}$$\\end{document} arises in several situations in mathematics. For example, computing the zeros of a given map, or the reduction of some cost function during an optimization process are such situations. In this note, we discuss the existence of a curve t↦x(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\mapsto x(t)$$\\end{document} in the domain of the nonlinear map F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {F}$$\\end{document} leading from some initial value x0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_0$$\\end{document} to a value u such that we are able to control the residual F(x(t))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {F}(x(t))$$\\end{document} based on the value F(x0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {F}(x_0)$$\\end{document}. More precisely, we slightly extend an existing result from J.W. Neuberger by proving the existence of such a curve, assuming that the directional derivative of F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {F}$$\\end{document} can be represented by x↦A(x)F(x0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x \\mapsto \\mathcal {A}(x)\\mathcal {F}(x_0)$$\\end{document}, where A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document} is a suitable defined operator. The presented approach covers, in case of A(x)=-Id\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}(x) = -\ extsf{Id}$$\\end{document}, some well known results from the theory of so-called continuous Newton methods. Moreover, based on the presented results, we discover an approximate inverse function result.

A reliability analysis method based on the intersection area division of hypersphere and paraboloid

In terms of the approximate failure probability of the limit state function (LSF), First Order Reliability Method (FORM) directly solves the failure probability through the most probable point (MPP), but it can only achieve high accuracy in the LSF with low linearity and non-linearity. Second Order Reliability Method (SORM) performs second-order expansion on the basis of MPP to improve reliability accuracy. However, the approximation may be wrong, and even has a large error in high-dimensional nonlinear problems. In order to get more precise approximation of SORM, this paper proposes an area partition method based on the Intersection Area Division Method (IADM) of Hypersphere and Paraboloid. IADM constructs a hypersphere and an approximate parabolic function of the LSF at MPP. The intersection area of the hypersphere and the approximate paraboloid is defined as the failure region from the geometric of the approximate paraboloid function, and the area of the failure region is calculated by the spherical integral. Finally, the approximation of the failure probability is determined by integrating the ratio of the area to the area of the hypersphere. Several examples proved that IADM is more stable compared with SORM which avoids the generation of wrong solutions and improves the accuracy.

An Approximation of the Prime Counting Function and a New Representation of the Riemann Zeta Function

Determining the exact number of primes at large magnitudes is computationally intensive, making approximation methods (e.g., the logarithmic integral, prime number theorem, Riemann zeta function, Chebyshev’s estimates, etc.) particularly valuable. These methods also offer avenues for number-theoretic exploration through analytical manipulation. In this work, we introduce a novel approximation function, ϕ(n), which adds to the existing repertoire of approximation methods and provides a fresh perspective for number-theoretic studies. Deeper analytical investigation of ϕ(n) reveals modified representations of the Chebyshev function, prime number theorem, and Riemann zeta function. Computational studies indicate that the difference between ϕ(n) and the logarithmic integral at magnitudes greater than 10100 is less than 1%.

Hybrid differential evolution algorithm for Falkner-Skan flow with rotation

A hybrid differential evolution algorithm is used in this work to study the rotating transport of Falkner-Skan flow. The problem is modeled as an equivalent optimization problem by using Padé rational approximation functions. The primary model of the governing partial differential equations is imposed to subjugate the error between profiles. A hybrid evolutionary algorithm based on differential evolution and a convergent version of Nelder-Mead direct search algorithm is employed to perform global exploratory search along with an enhanced exploitation to improve the accuracy of the proposed solution scheme. The resulting scheme is named as evolutionary Padé approximation (EPA) scheme. The performance of the proposed EPA scheme on the Falkner-Skan boundary value problem is investigated by considering various values of the rotation parameters. The developed optimizer in EPA scheme was able to minimize the residuals up to10−10. Results are displayed graphically in order to study the effect of various types of parameters. EPA scheme determined that angular velocity increases or decreases accordingly as fluid parameter (β) and the rotation parameter (λ) but shows inverse behavior with respect to power law index(n). Similarly, the response of velocity profile along y−axis was decreasing function of β as well as n but increasing function of λ. The performance of the proposed EPA scheme has been demonstrated by comparing results with a hybrid neural network scheme and found in excellent agreement.

T-stress extraction in arbitrarily cracked orthotropic composites with the numerical manifold method and Stroh formalism

The escalating utilization of composites over the past decades has necessitated the fracture investigation of orthotropic materials. The T-stress, namely, the first non-singular term of the normal stress parallel to the crack in William’s expansion, is of great significance for fracture analysis. In this work, the numerical manifold method (NMM) is advanced to assess the T-stress of arbitrary-shaped cracks (including non-intersecting cracks and multi-branched cracks) in two-dimensional orthotropic composites. Attributing to the bi-cover systems, the NMM can conveniently discretize the physical domain and naturally accommodate the discontinuity across crack surface. Meanwhile, the singularity at crack tip can be well captured by the wise choice of local approximation function. Through the application of interaction integral technology in the NMM postprocessing, the T-stress is extracted with the Stroh-form auxiliary fields. The accuracy of the proposed method is verified by comparing with reference solutions and then applied to arbitrarily branched and intersecting cracks. The results indicate that the present approach has convincing accuracy and also considerable convenience in T-stress evaluation. Additionally, the impacts of material orientations, crack geometries and loading conditions on the T-stress are also investigated.

A novel algorithmic multi-attribute decision-making framework for solar panel selection using modified aggregations of cubic intuitionistic fuzzy hypersoft set

To address the shortcomings of the cubic intuitionistic fuzzy sets (CIFSs) for the entitlement of multi-argument approximate function, the cubic intuitionistic fuzzy hypersoft set (Ω-set) is an emerging study area. This type of setting associates the sub-parametric tuples with the collection of CIFSs. Categorizing the evaluation of parameters into their corresponding sub-parametric values based on non-overlapping sets has significance in decision making and optimization related situations. Some operations of Ω-set are proposed in this study, along with certain practical features. We provide the complement, P-order, and R-order subsets, P-union (∪P), R-union (∪R), P-intersection (∩P) and R-intersection (∩R) of Ω-sets. The internal cubic intuitionistic fuzzy hypersoft set (ΩI-set) and the external cubic intuitionistic fuzzy hypersoft set (ΩE-set) are also proposed in this paper, which will aid researchers in applying this new theory to other areas of study. We show a few examples in this context and look into some more aspects of ∪P, ∪R, ∩P and ∩R of ΩI-sets and ΩE-sets. Arguments for a few significant theorems about ΩI-sets and ΩE-sets are also presented. Lastly, an algorithm is presented that assists decision-makers in evaluating appropriate solar panels to establish solar plants. The proposed algorithm uses the idea of ∪P and ∪R for two Ω-sets constructed based on expert opinions of decision makers.

Deep Learning-Meshless Method for Inverse Potential Problems

This paper combines the deep learning method with the meshless method to propose a new numerical method, which is the deep learning-improved element-free Galerkin (DL-IEFG) method, for solving inverse potential problems. In this method, the unknown term in the governing equation of the inverse potential problem is represented by the feedforward neural network (FNN). By employing the improved element-free Galerkin (IEFG) method to solve the inverse potential problem, the solution equations are established to obtain numerical solutions. The training set is constructed on the valid values obtained from discretized observation spatial points. The predicted values at the training sample points are calculated by combining the numerical solutions with the approximation function built by the improved moving least-squares (MLS) approximation. Then, the FNN representing the unknown term is iterated using the loss function. The effectiveness of the DL-IEFG method for solving potential inverse problems is validated through numerical examples. In addition, the factors impacting the calculation accuracy and efficiency of the DL-IEFG method are investigated.

Survival probability surfaces of hysteretic fractional order structures exposed to non-stationary code-compliant stochastic seismic excitation

A novel first-passage probability stochastic incremental dynamics analysis (SIDA) methodology tailored for hysteretic fractional order structural systems under a fully non-stationary seismic excitation vector consistently designated with contemporary aseismic codes provisions (e.g., Eurocode 8) is developed. Specifically, the vector of the imposed seismic excitations is characterised by evolutionary power spectra that stochastically align with aseismic codes elastic response acceleration spectra, defined for specified modal damping ratios and scaled ground accelerations. Leveraging the concepts of stochastic averaging and statistical linearization, the approximative non-stationary response displacement joint probability density function (PDF) is derived, retaining the particularly coveted attribute of computational efficacy. Subsequently, the coupling with the survival probability model allows for the efficient determination of the response first-passage time probability density surfaces and the survival probability surfaces across various limit-state rules and scalable intensity measures. The first-passage time probability serves as a robust engineering demand parameter, effectively monitoring structural behaviour by considering both intensity and timing information, while inherently aligned with pertinent limit-state requirements. Notably, the associated low computational cost and the ability to handle a wide range of complex nonlinear/hysteretic structural behaviours, coupled with its compliance with modern aseismic codes, underscore its potential for applications in the fields of structural and earthquake engineering. A nonlinear system endowed with fractional derivative elements is used to exemplify the method’s reliability. The accuracy of the proposed method is validated in a Monte Carlo-based context, conducting nonlinear response time–history analyses with an extensive ensemble of accelerograms compatible with Eurocode 8 response acceleration spectra.

Numerical study of the multi-dimensional Galilei invariant fractional advection–diffusion equation using direct mesh-less local Petrov–Galerkin method

This article presents a local mesh-less procedure for simulating the Galilei invariant fractional advection–diffusion (GI-FAD) equations in one, two, and three-dimensional spaces. The proposed method combines a second-order Crank–Nicolson scheme for time discretization and the second-order weighted and shifted Grünwald difference (WSGD) formula. This time discretization scheme ensures unconditional stability and convergence with an order of O(τ2). In the spatial domain, a mesh-less weak form is employed based on the direct mesh-less local Petrov–Galerkin (DMLPG) method. The DMLPG method employs the generalized moving least-square (GMLS) approximation in conjunction with the local weak form of the equation. By utilizing simple polynomials as shape functions in the GMLS approximation, the necessity for complex shape function construction in the MLS approximation is eliminated. To validate and demonstrate the effectiveness of the proposed algorithm, a variety of problems in one, two, and three dimensions are investigated on both regular and irregular computational domains. The numerical results obtained from these investigations confirm the accuracy and reliability of the developed approach in simulating GI-FAD equations.

Neural operators for robust output regulation of hyperbolic PDEs

The recently introduced neural operator (NO) has been employed as a gain approximator in the backstepping stabilization control of first-order hyperbolic and parabolic partial differential equation (PDE) systems. Due to the global approximation ability of the DeepONet, the NO provides approximate spatial gain function with arbitrary accuracy. The closed-loop system stability can be ensured by the backstepping controller involving the approximate gain with sufficiently small error. In this paper, the NO theory is leveraged to solve the robust output regulation problem for a class of uncertain hyperbolic PDE systems under the design framework of backstepping-based regulator. The NO is trained offline on a dataset containing a sufficient number of system parameters and corresponding prior solutions of the kernel equation, so as to generate feedback gain for the robust regulator. Once the NO is trained, the kernel equation does not need to be solved ever again, for any new system parameters that do not exceed the range of the training set. Based on the internal model principle, the regulator is inherently robust to a degree of parameter uncertainty and error in approximate gain. Therefore, the tracking error can still converge to 0 if the extended regulator equations are solvable and the parameter uncertainty leads to an asymptotically stable origin. We provide a series of theory proofs and a numerical test under the approximate control and observation gains to demonstrate the robust regulation problem. It is shown that the NO is almost three orders of magnitude faster than PDE solver in generating kernel function, and the loss remains on the order of 10−4 in the test. This provides an opportunity to use the NO methodology for accelerated gain scheduling regulation for PDEs with time-varying system parameters.

Trajectory optimization for aerodynamically controlled missiles by chance-constrained sequential convex programming

The flight environment of aerodynamically controlled missiles is full of complexity and uncertainty. To cope with the uncertainty more effectively and enhance the convergence performance in trajectory optimization problems for aerodynamically controlled missiles simultaneously, the chance-constrained sequential convex programming (CC-SCP) algorithm is proposed in this paper. The uncertainty is regarded as the chance constraint, and a smooth and differential approximation function is designed to transform this chance constraint into the constraint that the convex optimization method can handle. Subsequently, the originally non-convex trajectory optimization problem is reformulated into a series of convex optimization subproblems, in which an initial reference trajectory guess generation strategy is proposed, and a theoretical proof of the exact convex relaxation is given to enhance the algorithm's convergence performance and theoretical value, respectively. Numerical simulations are provided to verify the convergence and effectiveness of the CC-SCP algorithm, and the advantages of using the CC-SCP algorithm to cope with the uncertainty are illustrated. Furthermore, comparative simulation examples show that the proposed algorithm possesses a low conservatism, which means the proposed algorithm can obtain a bigger convergence region and a better solution than other current methods when handling the same chance constraints. Finally, the robustness of the algorithm is discussed.

Approximate moment functions for logistic stochastic differentialequations

Approximate moment functions for logistic stochastic differentialequations

Calculating Molecular Polarizabilities Using Exact Frozen Density Embedding with External Orthogonality.

Frozen density embedding (FDE) with freeze-thaw cycles is a formally exact embedding scheme. In practice, this method is limited to systems with small density overlaps when approximate nonadditive kinetic energy functionals are used. It has been shown that the use of approximate nonadditive kinetic energy functionals can be avoided when external orthogonality (EO) is enforced, and FDE can then generate exact results even for strongly overlapping subsystems. In this work, we present an implementation of exact FDEc-EO (coupled FDE TDDFT with EO) for the calculation of polarizabilities in the Amsterdam density functional program package. EO is enforced using the level-shift projection operator method, which ensures that orbitals between fragments are orthogonal. For pure functionals, we show that only the symmetric EO contributions to the induced density matrix are needed. This leads to a simplified implementation for the calculation of polarizability that can exactly reproduce the supermolecular TDDFT results. We further discuss the limitation of exact FDEc-EO in interpreting subsystem polarizabilities due to the nonunique partitioning of the total density. We show that this limitation is due to the fact that subsystem polarizability partitioning is dependent on how the subsystems are initially polarized. As supermolecular virtual orbitals are exactly reproduced, this dependence is attributed to the description of the occupied orbitals. In contrast, for excitations of subsystems that are localized within one subsystem, we show that the excitation energies are stable with respect to the order of polarization. This observation shows that impacts from the nonunique nature of exact FDE on subsystem properties can be minimized by better fragmentation of the supermolecular systems if the property is localized. For global properties like polarizability, this is not the case, and nonuniqueness remains independent of the fragmentation used.