Polynomial Approximation Research Articles

Polynomial and rational convergence rates for Laplace problems on planar domains

Laplace problems on planar domains can be solved by means of least-squares expansions associated with polynomial or rational approximations. Here it is shown that, even in the context of an analytic domain with analytic boundary data, the difference in convergence rates may be huge when the domain is non-convex. Our proofs combine the theory of the Schwarz function for analytic continuation, potential theory for polynomial and rational approximation rates and the theory of crowding of conformal maps.

Brownian particle diffusion in generalized polynomial shear flows.

This study presents a mathematical framework for calculating the diffusion of Brownian particles in generalized shear flows. By solving the Langevin equationsusing stochastic instead of classical calculus, we propose a mathematical formulation that resolves the particle mean-square displacement (MSD) at all timescales for any two-dimensional parallel shear flow described by a polynomial velocity profile. We show that at long timescales, the polynomial order of time of the particle MSD is n+2, where n is the polynomial order of the transverse coordinate of the velocity profile. We generalize the method to resolve particle diffusion in any polynomial shear flow at all timescales, including the order of particle relaxation timescale, which is unresolved in current theories. Particle diffusion at all timescales is then studied for the cases of Couette and plane Poiseuille flows and a polynomial approximation of a hyperbolic tangent flow while neglecting the boundary effects. We observe three main stages of particle diffusion along the timeline for Couette and plane Poiseuille flows and four main stages for hyperbolic tangent flow. The particle MSD is distinctly different across these stages due to different dominant physical mechanisms. Thus, higher temporal and spatial resolution for diffusion processes in shear flows may be realized, suggesting a more accurate analytical approach for the diffusion of Brownian particles.

The Convex Sets in Banach Spaces and Polynomial Approximation

The Convex Sets in Banach Spaces and Polynomial Approximation

Minimising the makespan on parallel identical machines with log-linear position-dependent processing times

This article studies the minimum makespan scheduling problem on parallel identical machines with the log-linear learning/ageing effect, also known as polynomial positional learning/ageing effect. To develop a Fully Polynomial Time Approximation Scheme to the problem, we start with an intermediate artificial variant that rounds the values to integers and restricts the solutions to instances sorted in the shortest/longest processing time order. To this end, we propose a dynamic programming algorithm and show that the difference between its returned value and the minimum makespan of the original problem is independent of the processing times. This then leads to an algorithm with provable guaranteed ϵ-additive approximation and pseudo-polynomial running time algorithm, resulting in the desired fully polynomial time approximation solution to the original, not restricted problem.

Advancing Microplastic Detection Technology through Digital Image Processing, Fractal Analysis, and Polynomial Approximation Methods

Advancing Microplastic Detection Technology through Digital Image Processing, Fractal Analysis, and Polynomial Approximation Methods

On uniform polynomial approximation

On uniform polynomial approximation

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.