Polynomial Approximation Research Articles

The Extended Diamond Difference - Constant Nodal Method with Decoupled Cell Iteration Scheme in Two-Dimensional Discrete Ordinate Transport Problems

Analytical solutions for the Boltzmann transport equation are typically limited to specific scenarios, prompting the research transport community to extensively explore numerical approaches with special attention to the discrete ordinates (SN) formulation. The spectral nodal methods are widely recognized for their capability to preserve the analytical solution of the SN mathematical model. However, these formulations exhibit notable complexity with high-order polynomial approximations for the transverse leakage terms, motivating researchers to pursue simpler hybrid approaches by partially preserving the analytical solution. Both spectral nodal and hybrid spectral nodal methods introduce nonstandard auxiliary equations not amenable to standard iterative schemes due to a highly coupling in angular directions. Following these ideas, we present a finite element – spectral nodal hybrid method referred to as Extended Diamond Difference-Constant Nodal (EDD-CN) method for two-dimensional fixed-source discrete ordinate transport problems with constant nodal approximation for the leakage terms. The proposed method seeks to preserve eigenfunctions associated with the dominant pair of eigenvalues in the transverse integrated SN equations. In addition, we propose a novel iterative scheme called Decoupled Cell Iteration (DCI) scheme which decouples the angular directions in the nonstandard auxiliary equations and follows conventional iteration transport sweeps for solving SN problems.

Assessment of Digital Image Correlation Effectiveness and Quality in Determination of Surface Strains of Hybrid Steel/Composite Structures.

The application of the digital image correlation (DIC) contactless method has extended the possibilities of reliable assessment of structure strain fields and deformations throughout the last years. However, certain weak points in the analyses using the DIC method still exist. The fluctuations of the results caused by different factors as well as certain deficiencies in the evaluation of DIC accuracy in applications for hybrid steel/composite structures with adhesive joints are one of them. In the proposed paper, the assessment of DIC accuracy based on the range of strain fluctuation is proposed. This relies on the use of a polynomial approximation imposed on the results obtained from the DIC method. Such a proposal has been used for a certain correction of the DIC solution and has been verified by the introduction of different error measures. The evaluation of DIC possibilities and accuracy are presented on the examples of the static tensile tests of adhesively bonded steel/composite joints with three different adhesives applied. The obtained results clearly show that in a non-disturbed area, very good agreement between approximated DIC and FEM results is achieved. The relative average errors in an area, determined by comparison of DIC and FEM strains, are below 15%. It is also observed that the use of approximated strains by polynomial function leads to a more accurate solution with respect to FEM results. It is concluded that DIC can be successfully applied for the analyses of hybrid steel/adhesive/composite samples, such as determination of strain fields, non-contact visual detection of faults of manufacturing and their development and influence on the whole structure behavior during the strength tests, including the elastic response of materials.

Approximation of the Hilbert transform on the half–line

The paper concerns the weighted Hilbert transform of locally continuous functions on the semiaxis. By using a filtered de la Vallée Poussin type approximation polynomial recently introduced by the authors, it is proposed a new “truncated” product quadrature rule (VP- rule). Several error estimates are given for different smoothness degrees of the integrand ensuring the uniform convergence in Zygmund and Sobolev spaces. Moreover, new estimates are proved for the weighted Hilbert transform and for its approximation (L-rule) by means of the truncated Lagrange interpolation at the same Laguerre zeros. The theoretical results are validated by the numerical experiments that show a better performance of the VP-rule versus the L-rule.

Nonautonomous Volterra Series Expansion of the Variable Phase Approximation and its Application to the Nucleon-Nucleon Inverse Scattering Problem

Abstract In this paper, the nonlinear Volterra series expansion is extended and used to describe certain types of nonautonomous differential equations related to the inverse scattering problem in nuclear physics. The nonautonomous Volterra series expansion lets us determine a dynamic, polynomial approximation of the variable phase approximation (VPA), which is used to determine the phase shifts from nuclear potentials through first-order nonlinear differential equations. By using the first-order Volterra expansion, a robust approximation is formulated to the inverse scattering problem for weak potentials and/or high energies. The method is then extended with the help of radial basis function neural networks by applying a nonlinear transformation on the measured phase shifts to be able to model the scattering system with a linear approximation given by the first-order Volterra expansion. The method is applied to describe the ${}^1S_0$ NN potentials in neutron+proton scattering below 200 MeV laboratory kinetic energies, giving physically sensible potentials and below $1\%$ averaged relative error between the recalculated and the measured phase shifts.

Perturbation-based nonperturbative method

This paper presents a nonperturbative method for solving eigenproblems. This method applies to almost all potentials and provides nonperturbative approximations for any energy level. The method converts an eigenproblem into a perturbation problem, obtains perturbation solutions through standard perturbation theory, and then analytically continues the perturbative solution into a nonperturbative solution. Concretely, we follow three main steps: (1) Introduce an auxiliary potential that can be solved exactly and treat the potential to be solved as a perturbation on this auxiliary system. (2) Use perturbation theory to obtain an approximate polynomial of the eigenproblem. (3) Use a rational approximation to analytically continue this approximate polynomial into the nonperturbative region.

Weighted holomorphic polynomial approximation

Weighted holomorphic polynomial approximation

Incorporating slow NMDA-type receptors with nonlinear voltage-dependent magnesium block in a next generation neural mass model: derivation and dynamics

We derive a next generation neural mass model of a population of quadratic-integrate-and-fire neurons, with slow adaptation, and conductance-based AMPAR, GABAR and nonlinear NMDAR synapses. We show that the Lorentzian ansatz assumption can be satisfied by introducing a piece-wise polynomial approximation of the nonlinear voltage-dependent magnesium block of NMDAR current. We study the dynamics of the resulting system for two example cases of excitatory cortical neurons and inhibitory striatal neurons. Bifurcation diagrams are presented comparing the different dynamical regimes as compared to the case of linear NMDAR currents, along with sample comparison simulation time series demonstrating different possible oscillatory solutions. The omission of the nonlinearity of NMDAR currents results in a shift in the range (and possible disappearance) of the constant high firing rate regime, along with a modulation in the amplitude and frequency power spectrum of oscillations. Moreover, nonlinear NMDAR action is seen to be state-dependent and can have opposite effects depending on the type of neurons involved and the level of input firing rate received. The presented model can serve as a computationally efficient building block in whole brain network models for investigating the differential modulation of different types of synapses under neuromodulatory influence or receptor specific malfunction.

Cubic Spline Chebyshev Polynomial Approximation for Solving Boundary Value Problems

Cubic Spline Chebyshev Polynomial Approximation for Solving Boundary Value Problems

Increasing measurement accuracy by nonparametric data reconciliation

This paper discusses two approaches to the reconciliation of joint measurements with non-Gaussian errors’ distributions based on considering the functional relationships between measurands: semi-nonparametric based on polynomial approximation or Gram-Charlier series expansions and completely nonparametric based on kernel density estimator. The article presents mathematical expressions for all proposed methods in detail and the results of diverse statistical modelling for determining the limits of their possible applicability. We compared the effectiveness of the mentioned methods in a series of Monte Carlo experiments and showed that the nonparametric approach based on the kernel density estimation is less demanding on sample size than Gram-Charlier approximation for samples of size n > 30. However, in the case of asymmetric densities (skewness greater than 2.0), the reconciling algorithm with Gram-Charlier approximation is preferable for samples of medium size. The paper shows that the recommended measurement amount should be n > 150 for each measured quantity if we use not more than four terms in Gram-Charlier expansion.

Bifurcation analysis and control study of improved full-speed differential model in connected vehicle environment

Abstract In recent years, the traffic congestion problem has become more and more serious, and the research on traffic system control has become a new hot spot. Studying the bifurcation characteristics of traffic flow systems and designing control schemes for unstable pivots can alleviate the traffic congestion problem from a new perspective. In this work, the full-speed differential model considering the vehicle network environment is improved in order to adjust the traffic flow from the perspective of bifurcation control, the existence conditions of Hopf bifurcation and saddle-node bifurcation in the model are proved theoretically, and the stability mutation point for the stability of the transportation system is found. For the unstable bifurcation point, a nonlinear system feedback controller is designed by using Chebyshev polynomial approximation and stochastic feedback control method. The advancement, postponement, and elimination of Hopf bifurcation are achieved without changing the system equilibrium point, and the mutation behavior of the transportation system is controlled so as to alleviate the traffic congestion. The changes in the stability of complex traffic systems are explained through the bifurcation analysis, which can better capture the characteristics of the traffic flow. By adjusting the control parameters in the feedback controllers, the influence of the boundary conditions on the stability of the traffic system is adequately described, and the effects of the unstable focuses and saddle points on the system are suppressed to slow down the traffic flow. In addition, the unstable bifurcation points can be eliminated and the Hopf bifurcation can be controlled to advance, delay, and disappear, so as to realize the control of the stability behavior of the traffic system, which can help to alleviate the traffic congestion and describe the actual traffic phenomena as well.

Compact: Approximating Complex Activation Functions for Secure Computation

Secure multi-party computation (MPC) techniques can be used to provide data privacy when users query deep neural network (DNN) models hosted on a public cloud. State-of-the-art MPC techniques can be directly leveraged for DNN models that use simple activation functions (AFs) such as ReLU. However, these techniques are ineffective and/or inefficient for the complex and highly non-linear AFs used in cutting-edge DNN models. We present Compact, which produces piece-wise polynomial approximations of complex AFs to enable their efficient use with state-of-the-art MPC techniques. Compact neither requires nor imposes any restriction on model training and results in near-identical model accuracy. To achieve this, we design Compact with input density awareness, and use an application specific simulated annealing type optimization to generate computationally more efficient approximations of complex AFs. We extensively evaluate Compact on four different machine-learning tasks with DNN architectures that use popular complex AFs silu, gelu, and mish. Our experimental results show that Compact incurs negligible accuracy loss while being 2x-5x computationally more efficient than state-of-the-art approaches for DNN models with large number of hidden layers. Our work accelerates easy adoption of MPC techniques to provide user data privacy even when the queried DNN models consist of a number of hidden layers, and trained over complex AFs.

Closed‐loop stability analysis of deep reinforcement learning controlled systems with experimental validation

AbstractTrained deep reinforcement learning (DRL) based controllers can effectively control dynamic systems where classical controllers can be ineffective and difficult to tune. However, the lack of closed‐loop stability guarantees of systems controlled by trained DRL agents hinders their adoption in practical applications. This research study investigates the closed‐loop stability of dynamic systems controlled by trained DRL agents using Lyapunov analysis based on a linear‐quadratic polynomial approximation of the trained agent. In addition, this work develops an understanding of the system's stability margin to determine operational boundaries and critical thresholds of the system's physical parameters for effective operation. The proposed analysis is verified on a DRL‐controlled system for several simulated and experimental scenarios. The DRL agent is trained using a detailed dynamic model of a non‐linear system and then tested on the corresponding real‐world hardware platform without any fine‐tuning. Experiments are conducted on a wide range of system states and physical parameters and the results have confirmed the validity of the proposed stability analysis (https://youtu.be/QlpeD5sTlPU).

On simultaneous approximation of certain classes of functions in the Bergman space <i>B</i><sub>2</sub>

The problem of finding the supremums of the best simultaneous polynomial approximations of some classes of functions analytic in the unit disk and belonging to the Bergman space B2 is considered. The indicated function classes are defined by the averaged values of the mth order moduli of continuity of the highest derivative bounded from above by some majorant Φ.

HIGH ORDER ANTI EVEN LEAST SQUARE FOR APPROXIMATING PRIME NUMBERS BELOW 1000

The study of prime numbers, pivotal in mathematics for centuries, holds significant importance in number theory and diverse applications like cryptography and computer science. This article introduces a novel approach, "High Order Anti Even Least Square," for approximating prime numbers below 1000. Integrating Least Square with specialized techniques tailored to complex prime number distributions, this method aims to enhance accuracy compared to traditional approaches. The research methodology involves polynomial approximation using both traditional and anti-even least squares methods, error calculation, visualization, and analysis. Results indicate that while traditional least squares generally performs better, the anti-even least squares method shows promise, particularly at higher polynomial degrees. This study contributes to advancing number theory and its practical applications by presenting a novel method for prime number approximation.

Optimal Piecewise Polynomial Approximation for Minimum Computing Cost by Using Constrained Least Squares.

In this paper, the optimal approximation algorithm is proposed to simplify non-linear functions and/or discrete data as piecewise polynomials by using the constrained least squares. In time-sensitive applications or in embedded systems with limited resources, the runtime of the approximate function is as crucial as its accuracy. The proposed algorithm searches for the optimal piecewise polynomial (OPP) with the minimum computational cost while ensuring that the error is below a specified threshold. This was accomplished by using smooth piecewise polynomials with optimal order and numbers of intervals. The computational cost only depended on polynomial complexity, i.e., the order and the number of intervals at runtime function call. In previous studies, the user had to decide one or all of the orders and the number of intervals. In contrast, the OPP approximation algorithm determines both of them. For the optimal approximation, computational costs for all the possible combinations of piecewise polynomials were calculated and tabulated in ascending order for the specific target CPU off-line. Each combination was optimized through constrained least squares and the random selection method for the given sample points. Afterward, whether the approximation error was below the predetermined value was examined. When the error was permissible, the combination was selected as the optimal approximation, or the next combination was examined. To verify the performance, several representative functions were examined and analyzed.

Discontinuous polynomial approximation in electrical impedance tomography with total variational regularization

In this paper, we present an alternative discrete total variation type functional for image reconstruction in electrical impedance tomography. The modified functional deals with unknown inclusions and values of conductivity. The convergence of the proposed finite element method with uniform refinement is also established: the sequence of discrete solutions contains a subsequence that converges to a solution of the continuous functional. Additionally, we propose an adaptive reconstruction algorithm in which the estimators are based on the discrete state variable, the discrete adjoint variable and the gradient of the discrete functional. A series of numerical examples are provided to illustrate the convergence behavior in diverse scenarios.

Parametric Control via the Algebraic Expression of Lotus-Type Pore Shapes in Metals

Lotus-type porous metals, characterized by low densities, large surface areas, and directional properties, are contemporarily utilized as lightweight, catalytic, and energy-damping materials; heat sinks; etc. In this study, the effects of dimensionless working parameters on the morphology of lotus-type pores in metals during unidirectional solidification were extensively investigated via general algebraic expressions. The independent dimensionless parameters include metallurgical, transport, and geometrical parameters such as Sieverts’ law constant, a partition coefficient, the solidification rate, a mass transfer coefficient, the imposed mole fraction of a solute gas, the total pressure at the top free surface, hydrostatic pressure, a solute transport parameter, inter-pore spacing, and initial contact angle. This model accounts for transient gas pressure in the pore, affected by the solute transfer, gas, capillary, and hydrostatic pressures, and Sieverts’ laws at the bubble cap and top free surface. Solute transport across the cap accounts for solute convection at the cap and the amount of solute rejected by the solidification front into the pore. The shape of lotus-type pores can be described using a proposed fifth-degree polynomial approximation, which captures the major portions between the initial contact angle and the maximum radius at a contact angle of 90 degrees, obtained by conserving the total solute content in the system. The proposed polynomial approximation, along with its working parameters, offers profound insights into the formation and shape of lotus-type pores in metals. It systematically provides deep insights into mechanisms that may not be easily revealed with experimental studies. The prediction of a lotus-type pore shape is thus algebraically achieved in good agreement with the available experimental data and previous analytical results.

An efficient collocation technique based on operational matrix of fractional-order Lagrange polynomials for solving the space-time fractional-order partial differential equations

In this research, we propose a novel and fast computational technique for solving a class of space-time fractional-order linear and non-linear partial differential equations. Caputo-type fractional derivatives are considered. The proposed method is based on the operational and pseudo-operational matrices for the fractional-order Lagrange polynomials. To carry out the method, first, we find the integer and fractional-order operational and pseudo-operational matrix of integration. The collocation technique and obtained operational and pseudo-operational matrices are then used to generate a system of algebraic equations by reducing the given space-time fractional differential problem. The resultant algebraic system is then easily solved by Newton's iterative methods. The upper bound of the fractional-order operational matrix of integration is also provided, which confirms the convergence of fractional-order Lagrange polynomial's approximation. Finally, some numerical experiments are conducted to demonstrate the applicability and usefulness of the suggested numerical scheme.

Dynamic Cube Attacks against Grain-128AEAD

In this paper, we revisit the division property based dynamic cube attack on the full Grain-128 presented by Hao et al. at FSE 2020 and demonstrate that their attack on the full Grain-128 is invalid, that is, no key information could be successfully recovered. The theoretical framework for the dynamic cube attack provided by Hao et al. is correct, but the technique for building the MILP model in the dynamic cube attack has flaws. Besides, strong evidence indicates that their bias estimation method is not applicable to Grain-128AEAD and Grain-128. Accordingly, we introduce the three-subset division property without unknown subset (3SDP/u) into dynamic cube attacks and present a correct MILP modeling technique. In addition, we propose a heuristic technique called Polynomial Approximation with regard to Bias (PAB) to evaluate the bias in superpolies in the dynamic cube attack, which can provide a more accurate bias evaluation for high-dimension cubes. As a result, we implemented the dynamic cube attack based on 3SDP/u on 190-round Grain-128AEAD, and we could recover 3 key bits with a complexity 2103.44 and the success probability was evaluated to be 99.68%. For Grain-128, some zero-sum distinguishers of cube size 80 are given for the first time.

Bifurcation analysis and control of an improved continuous traffic model considering weather effect

PurposeHighway traffic systems are complex and variable, and studying the bifurcation characteristics of traffic flow systems and designing control schemes for unstable bifurcation points can alleviate traffic congestion from a new perspective. Bifurcation analysis is used to explain the changes in system stability, identify the unstable bifurcation points of the system, and design feedback controllers to realize the control of the unstable bifurcation points of the traffic system. It helps to control the sudden changes in the stable behavior of the traffic system and helps to alleviate traffic congestion, which is of great practical significance.Design/methodology/approachIn this paper, we improve the macroscopic traffic flow model by integrating severe weather factors such as rainfall, snowfall, and dust. We use traveling wave transform to convert it into a traffic flow stability model suitable for branching analysis, thus converting the traffic flow problem into a system stability analysis problem. First, this paper derives the existence conditions of the model Hopf bifurcation and saddle-node bifurcation for the improved macroscopic model, and finds the stability mutation point of the system. Secondly, the connection between the stability mutation points and bifurcation points of the traffic system is analyzed. Finally, for the unstable bifurcation point, a nonlinear system feedback controller is designed using Chebyshev polynomial approximation and stochastic feedback control method.FindingsThe Hopf bifurcation is delayed and completely eliminated without changing the equilibrium point of the system, thus controlling the abrupt behavior of the traffic system.Originality/valueCurrently there are fewer studies to explain the changes in the stability of the transportation system through bifurcation analysis, in this paper; we design a feedback controller for the unstable bifurcation point of the system to realize the control of the transportation system. It is a new research method that helps to control the sudden change of the stable behavior of the traffic system and helps to alleviate traffic congestion, which is of great practical significance.