Approximation Method Research Articles

Structure Preserving Polytopal Discontinuous Galerkin Methods for the Numerical Modeling of Neurodegenerative Diseases

Many neurodegenerative diseases are connected to the spreading of misfolded prionic proteins. In this paper, we analyse the process of misfolding and spreading of both α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-synuclein and Amyloid-β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}, related to Parkinson’s and Alzheimer’s diseases, respectively. We introduce and analyze a positivity-preserving numerical method for the discretization of the Fisher-Kolmogorov equation, modelling accumulation and spreading of prionic proteins. The proposed approximation method is based on the discontinuous Galerkin method on polygonal and polyhedral grids for space discretization and on ϑ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\vartheta -$$\\end{document}method time integration scheme. We prove the existence of the discrete solution and a convergence result where the Implicit Euler scheme is employed for time integration. We show that the proposed approach is structure-preserving, in the sense that it guarantees that the discrete solution is non-negative, a feature that is of paramount importance in practical application. The numerical verification of our numerical model is performed both using a manufactured solution and considering wavefront propagation in two-dimensional polygonal grids. Next, we present a simulation of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-synuclein spreading in a two-dimensional brain slice in the sagittal plane. The polygonal mesh for this simulation is agglomerated maintaining the distinction of white and grey matter, taking advantage of the flexibility of PolyDG methods in the mesh construction. Finally, we simulate the spreading of Amyloid-β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document} in a patient-specific setting by using a three-dimensional geometry reconstructed from magnetic resonance images and an initial condition reconstructed from positron emission tomography. Our numerical simulations confirm that the proposed method is able to capture the evolution of Parkinson’s and Alzheimer’s diseases.

Bayesian reduced-order deep learning surrogate model for dynamic systems described by partial differential equations

We propose a reduced-order deep-learning surrogate model for dynamic systems described by time-dependent partial differential equations. This method employs space–time Karhunen–Loève expansions (KLEs) of the state variables and space-dependent KLEs of space-varying parameters to identify the reduced (latent) dimensions. Subsequently, a deep neural network (DNN) is used to map the parameter latent space to the state variable latent space.An approximate Bayesian method is developed for uncertainty quantification (UQ) in the proposed KL-DNN surrogate model. The KL-DNN method is tested for the linear advection–diffusion and nonlinear diffusion equations, and the Bayesian approach for UQ is compared with the deep ensembling (DE) approach, commonly used for quantifying uncertainty in DNN models. It was found that the approximate Bayesian method provides a more informative distribution of the PDE solutions in terms of the coverage of the reference PDE solutions (the percentage of nodes where the reference solution is within the confidence interval predicted by the UQ methods) and log predictive probability. The DE method is found to underestimate uncertainty and introduce bias.For the nonlinear diffusion equation, we compare the KL-DNN method with the Fourier Neural Operator (FNO) method and find that KL-DNN is 10% more accurate and needs less training time than the FNO method.

Quantitative stability for stochastic tensor variational inequalities

ABSTRACT The goal of this article is to investigate a class of stochastic tensor variational inequalities (denoted by STVI), which is a generalization of the stochastic linear variational inequalities (denoted by SLVI) and a subclass of stochastic variational inequalities (denoted by SVI). The quantitative stability of the expected residual minimization (ERM) problem for the STVI is analysed. Firstly, the existence of solutions of the ERM problem and its perturbed problem are considered. Then, the quantitative stability of the ERM problem is derived under suitable probability metrics. Finally, under some suitable assumptions, the rates of convergence of the sample average approximation (SAA) method for the optimal solution sets of the perturbed problems are discussed.

Implementation of unknown parameter estimation procedure for hybrid and discrete non‐linear systems

AbstractThe application of the hybrid extended Kalman filter (HEKF), hybrid unscented Kalman filter (HUKF), hybrid particle filter (HPF), and hybrid extended Kalman particle filter (HEKPF) is discussed for hybrid non‐linear filter problems, when prediction equations are continuous‐time and the update equations are discrete‐time, and also the discrete extended Kalman filter (DEKF), discrete unscented Kalman filter (DUKF), discrete particle filter (DPF), and discrete extended Kalman particle filter (DEKPF) for discrete‐time non‐linear filter problems, when prediction equations and update equations are discrete‐time. In order to assess the performance of the filters, the authors consider the non‐linear dynamics for a re‐entry vehicle. The filters are used in two hybrid and discrete states to estimate the position, velocity, and drag parameter associated with the re‐entry vehicle. Theoretical topics concerning estimating the drag parameter of a vehicle in re‐entry phase have been dealt with. Drag parameter estimation is carried out using a combination of hybrid filters and discrete filters as an effective estimator and fixed value, forgetting factor, and Robbins‐Monro stochastic approximation methods as the noise covariance matrix adjuster of the parameter.

Gauss–Bonnet Theorem Related to the Semi-Symmetric Metric Connection in the Heisenberg Group

In this paper, we introduce the notion of the semi-symmetric metric connection in the Heisenberg group. Moreover, by using the method of Riemannian approximations, we define the notions of intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on a surface, and the intrinsic Gaussian curvature of the surface away from characteristic points in the Heisenberg group with the semi-symmetric metric connection. Finally, we derive the expressions of those curvatures and prove the Gauss–Bonnet theorem related to the semi-symmetric metric connection in the Heisenberg group.

Dynamic capacity allotment planning for airlines via travel agencies

In this paper, we study the dynamic capacity allocation problem of an airline using allotment contracts among travel agencies. The capacity allocation problem is formulated as a dynamic programming model where the initial fixed allotment is determined on the basis of a contract between agencies and the airline while the variable allotment is distributed over the planning horizon. We develop approximation methods based on linear programming and Lagrangian relaxation to solve the underlying dynamic programming model. The computational experiments are designed to illustrate the performance of the proposed approaches using real data. The numerical results show that the capacity allotment policy in terms of fixed and variable allotments is significantly affected by price and demand patterns. Even though fixed allotments can provide a cushion against the loss due to potential empty seats, they may not be in the best interest of the airline. Moreover, the booking rules and the airline’s capacity allocation strategy have an impact on the airline’s revenue and the agency’s profit.

Research on the hydraulic parameters and leak rate of natural gas flow in natural gas pipelines based on the methods of characteristic line and successive approximation

Pressure and flow rate are the most important hydraulic parameters in natural gas pipeline flow, and leak rate is the most crucial parameter after a leak accident occurs. The study of these parameters is vital to the safey of natural gas pipelines and risk assessment after accidents. In this research, based on the conservation laws universally applicable to the motion of objects, we establish the fundamental control equation group for natural gas flow. Then, for both normal and leak conditions of pipelines, we use the characteristic line method to derive the corresponding difference equations for the fundamental control equations, thereby providing calculation methods for pressure and flow rate. Finally, we investigate the calculation method of natural gas pipeline leak rate under different leak aperture sizes, and validate the accuracy of this method through simulation examples.

Stochastic Optimization and Learning for Two-Stage Supplier Problems

The main focus of this paper is radius-based (supplier) clustering in the two-stage stochastic setting with recourse, where the inherent stochasticity of the model comes in the form of a budget constraint. In addition to the standard (homogeneous) setting where all clients must be within a distance \(R\) of the nearest facility, we provide results for the more general problem where the radius demands may be inhomogeneous (i.e., different for each client). We also explore a number of variants where additional constraints are imposed on the first-stage decisions, specifically matroid and multi-knapsack constraints, and provide results for these settings. We derive results for the most general distributional setting, where there is only black-box access to the underlying distribution. To accomplish this, we first develop algorithms for the polynomial scenarios setting; we then employ a novel scenario-discarding variant of the standard Sample Average Approximation (SAA) method, which crucially exploits properties of the restricted-case algorithms. We note that the scenario-discarding modification to the SAA method is necessary in order to optimize over the radius.

A class of polynomial approximation methods to second-order delay differential equations

This paper studies special high-order methods to numerically solve special second-order ordinary differential equations and second-order differential equations with constant delay, which are based on vector field expansion along the shifted and scaled Legendre polynomial orthonormal basis. In the framework of method design, we obtain the second-order perturbation results by transforming the second-order differential equations into first-order ones, and the error accuracy estimation of numerical solutions is achieved by using perturbation results for the problem under consideration. In addition, some numerical tests on specific second-order delay Hamiltonian systems are shown, the purpose of which is to verify the effectiveness of the theoretical findings.

Solution of the internal and external problems of electrodynamics for a symmetrical vibrator with a reflector made of parallel straight conductors

Background. The work is aimed at the development and research of rigorous methods for calculating multi-element emitting and re-emitting structures, consisting mainly of the same type of elements, as well as studying the physical processes occurring in them. An iterative approach to solving the internal problem is proposed, which allows minimizing the cost of computer time and computer memory. Aim. In this work, using the proposed approach, we solve the internal and external problems of electrodynamics for a symmetrical vibrator with a reflector made of parallel straight conductors. The convergence of the iterative process is studied, the currents on the elements of the structure, its input resistance and radiation characteristics are calculated. Methods. The research is based on a strict electrodynamic approach, within the framework of which, for the specified structure in the thin-wire approximation, an integral representation of the electromagnetic field is formed, which, when considered on the surface of conductors together with boundary conditions, is reduced to a system of Fredholm integral equations of the second kind, written relative to unknown current distributions on conductors (internal task). The solution of the internal problem within the framework of the method of moments is reduced to solving a SLAE with a block matrix. Results. A mathematical model of a radiating structure is proposed, which is a symmetrical vibrator with a reflector made of parallel straight conductors. The internal and external problems of electrodynamics are formulated and solved for given parameter values. An effective algorithm for calculating the block matrix of SLAEs is proposed. Recommendations are given for the selection of systems of projection functions within the framework of the method of moments. The convergence of the iterative process for solving an internal problem of electrodynamics has been studied. The input resistance of the structure and the basic characteristics of its radiation are determined. Conclusion. A rational choice of systems of projection functions, based on the properties of the structure and its constituent elements, makes it possible to significantly reduce the size of the SLAE matrix and, accordingly, reduce computational costs. Taking into account the properties of the structure and its constituent elements also makes it possible to construct effective algorithms for calculating the SLAE matrix. It is shown that the convergence of the iterative process may be absent near resonant frequencies, therefore, when solving the internal problem, a combined approach should be used, which involves the use of both rigorous and approximate methods for solving SLAEs. At non-resonant frequencies, the iterative approach demonstrates its effectiveness. It is concluded that for such structures it is advisable to distinguish between resonant and non-resonant operating modes. In the non-resonant mode, the current functions have a fairly smooth dependence on frequency; in the resonant mode, these dependences become quite sharp and difficult to predict. In this regard, the task of developing an iterative approach to solving the internal problem of electrodynamics in resonant operating modes seems relevant.

MFRWA: A Multi-Frequency Rayleigh Wave Approximation Method for Concrete Carbonation Depth Evaluation

Carbonation depth is essential to determine the durability and predict the remaining service life of concrete structures. This study proposes a multi-frequency Rayleigh wave approximation method (MFRWA) to evaluate carbonation depth by exploiting the frequency-dependent penetration depths of ultrasonic Rayleigh waves. A series of numerical simulations are conducted to investigate the effective penetration depth of Rayleigh waves and the feasibility of the proposed MFRWA method on carbonation depth evaluation. Subsequently, the accelerated carbonation experiment is conducted to evaluate the carbonation depth using low-frequency and high-frequency Rayleigh waves, and the measured results from the Rayleigh wave method are compared with the ones from the phenolphthalein indicator and thermalgravimetric analysis (TGA) method. The results show that carbonation depth measured by Rayleigh wave method meets well with the one from TGA technique, demonstrating that the proposed method could provide a non-destructive and precise carbonation depth estimation. The proposed MFRWA method contributes a novel scheme for concrete carbonation evaluation and holds substantial potential in both laboratory and field applications.

Reliability-based G1 continuous arc spline approximation

This paper introduces an algorithm for approximating a set of data points with G1 continuous arcs, leveraging covariance data associated with the points. Prior approaches to arc spline approximation typically assumed equal contribution from all data points, resulting in potential algorithmic instability when outliers are present. To address this challenge, we propose a robust method for arc spline approximation, taking into account the 2D covariance of each data point. Beginning with the definition of models and parameters for single-arc approximation, we extend the framework to support multiple-arc approximation for broader applicability. Finally, we validate the proposed algorithm using both synthetic noisy data and real-world data collected through vehicle experiments conducted in Sejong City, South Korea.

Study of optical properties of porous silicon by DFT, comparison to experimental and effective medium approximation methods

Study of optical properties of porous silicon by DFT, comparison to experimental and effective medium approximation methods

Improving the accuracy of halo mass based statistics for fast approximate N-body simulations

ABSTRACT Approximate N-body methods, such as fastpm and cola, have been successful in modelling halo and galaxy clustering statistics, but their low resolution on small scales is a limitation for applications that require high precision. Full N-body simulations can provide better accuracy but are too computationally expensive for a quick exploration of cosmological parameters. This paper presents a method for correcting distinct haloes identified in fast N-body simulations, so that various halo statistics improve to a percent level accuracy. The scheme seeks to find empirical corrections to halo properties such that the virial mass is the same as that of a corresponding halo in a full N-body simulation. The modified outer density contour of the corrected halo is determined on the basis of the fastpm settings and the number of particles inside the halo. This method only changes some parameters of the halo finder, and does not require any extra CPU-cost. We demonstrate that the adjusted halo catalogues of fastpm simulations significantly improve the precision of halo mass-based statistics from redshifts $z=0.0$ to 1.0, and that our calibration can be applied to different cosmologies without needing to be recalibrated.

Sample-Based Continuous Approximate Method for Constructing Interval Neural Network.

In safety-critical engineering applications, such as robust prediction against adversarial noise, it is necessary to quantify neural networks' uncertainty. Interval neural networks (INNs) are effective models for uncertainty quantification, giving an interval of predictions instead of a single value for a corresponding input. This article formulates the problem of training an INN as a chance-constrained optimization problem. The optimal solution of the formulated chance-constrained optimization naturally forms an INN that gives the tightest interval of predictions with a required confidence level. Since the chance-constrained optimization problem is intractable, a sample-based continuous approximate method is used to obtain approximate solutions to the chance-constrained optimization problem. We prove the uniform convergence of the approximation, showing that it gives the optimal INN consistently with the original ones. Additionally, we investigate the reliability of the approximation with finite samples, giving the probability bound for violation with finite samples. Through a numerical example and an application case study of anomaly detection in wind power data, we evaluate the effectiveness of the proposed INN against existing approaches, including Bayesian neural networks, highlighting its capability to significantly improve the performance of applying INNs for regression and unsupervised anomaly detection.

Enhancement of electron–phonon coupling due to increased magnetism and applied hydrostatic pressure in FeSe

Inspired by the parabolic trend of the superconducting transition temperature (Tc ) of bulk FeSe under hydrostatic pressure, we investigated the effect of magnetism and hydrostatic pressure on the electron–phonon coupling (EPC) in FeSe using density-functional perturbation theory. We found that both magnetism and hydrostatic pressure enhanced EPC. The enhancement of the EPC is mainly attributed to phonon softening and deformation potential induced by magnetism, rather than Fermi surface nesting. Furthermore, we investigated the effect of spin fluctuations on superconductivity by applying the random phase approximation method. A possible application of our results to the phase diagram of FeSe under hydrostatic pressure was discussed, and we demonstrated that when EPC and spin fluctuations are both considered, a parabolic superconducting Tc may be obtained, providing a plausible explanation for the phase of FeSe under hydrostatic pressure.

EXPRESS: Constrained Assortment Optimization under the Cross-Nested Logit Model

We study the assortment optimization problem under general linear constraints, where the customer choice behavior is captured by the Cross-Nested Logit model. In this problem, there is a set of products organized into multiple subsets (or nests), where each product can belong to more than one nest. The aim is to find an assortment to offer to customers so that the expected revenue is maximized. We show that, under the Cross-Nested Logit model, the unconstrained assortment problem is NP-hard even when there are only two nests, and the problem is generally NP-hard to approximate to any constant factors. To tackle this challenging problem, we develop a new discretization mechanism to approximate the problem by a linear fractional program with a performance guarantee of [Formula: see text], for any accuracy level ε > 0. We then show that optimal solutions to the approximate problem can be obtained by solving mixed-integer linear programs. We further show that our discretization approach can also be applied to solve a joint assortment optimization and pricing problem, as well as an assortment problem under a mixture of Cross-Nested Logit models to account for multiple classes of customers. Our empirical results on a large number of randomly generated test instances demonstrate that, under a performance guarantee of 90% (i.e., expected revenues are guaranteed to be at least 90% of the optimal revenue), the percentage gaps between the objective values obtained from our approximation methods and the optimal expected revenues are no larger than 1.2%.

Delay Segmented Tristable Stochastic Resonance System Driven by Non-Gaussian Colored Noise and Its Application in Bearing Fault Detection

Abstract This study investigates the Delayed Segmented Tristable Stochastic Resonance (DSTSR) system under the influence of additive non-Gaussian colored noise. The research employs an improved segmented tristable potential function, wherein the equilibrium points and barrier heights can be independently controlled by parameters. Simultaneously, the segmented function on both sides reduces the restrictions of higher-order terms on the walls of the potential wells. The equivalent Langevin equation for the DSTSR system is obtained using the path integral method, the unified colored noise approximation method, and the small-delay approximation. Subsequently, the theoretical expressions for the steady-state probability density, mean first passage time (MFPT), and Signal-to-Noise Ratio (SNR) are derived from the resulting equations, and the impact of variations in system parameters on these performance metrics is discussed. Additionally, Monte Carlo simulations for MFPT are conducted to verify the accuracy of the theoretical derivations. Combining the results from the theoretical section and the impact of parameters on system performance, the article employs an adaptive genetic algorithm to optimize system parameters. This algorithm is then applied to simulation experiments and bearing fault detection. In the simulation experiments, the DSTSR system is compared with other systems. The results indicate that the DSTSR system exhibits the highest SNR improvement. Furthermore, in bearing fault detection under non-Gaussian colored noise, the DSTSR system shows higher spectral amplitude and SNR at the fault frequency compared to the tristable stochastic resonance system and the segmented tristable stochastic resonance system without time delay feedback. This suggests that stochastic resonance can effectively detect weak signals in non-Gaussian non-white noise scenarios, and the introduction of time delay contributes to the occurrence of stochastic resonance to a certain extent.

Иглоподобные листовые органы хвойных. Часть I. Моделирование периметра поперечного сечения иглы

Despite the availability of measuring systems for estimating the surface area of leaf organs of higher plants, the need for simple standard methods for determining this indicator area remains relevant for plant physiologists. The methods for estimating the surface area of needle-like leaf organs of conifers, based on the geometry of an individual needle rest on the general principle of calculating the needle surface area as the product of its length by the perimeter of its cross-section. This makes the cross-section perimeter one of the most important parameters needed to characterize the needle surface area. The strong variability of this parameter depending on the species necessitates the development of individual models of the cross-section of individual needles. The aim of this study has been to create a universal model for estimating the needle cross-section perimeter, irrespective of the tree species. For the practical implementation of the aim, a method was proposed for estimating the perimeter of the needle cross-section, based on the well-known fact that any closed line is transformable into an equivalent circle, while the length of the closed line does not change. The perimeter of the equivalent circle can be related to the parameters of the geometric figure before the transformation. This approach allows us to relate the width and thickness of the needle cross-section to its perimeter. The developed universal model of the needle cross-section has been verified on cross-sections of Siberian fir (Abies sibirica L.) and common juniper (Juniperus communis L.) needles. The samples of needles of these woody plants have been collected from a bilberry-sphagnum spruce forest in the boreal zone of the north-east of the European part of Russia (Knyazhpogostkiy district, the Komi Republic). Statistical analysis has shown the significance and adequacy of the model. It can be used to assess the perimeter of coniferous needles, irrespective of their species. In this case, the accuracy of perimeter estimation is comparable to the accuracy of direct perimeter measurement by the piecewise linear approximation method.

Chaotic vibration of a curved CNT conveying magnetic fluid in the thermo-magnetic field considering the surface effects

The nonlinear chaotic vibration of curved single-walled carbon nanotube (CSWCNT) conveying magnetic fluid is studied based on the nonlocal Euler-Bernoulli beam model. The governing equation of CSWCNT is presented considering the axial thermo-magnetic load and the surface effect. By employing the Galerkin decomposition approximation method along with the admissible beam shape function satisfying the cantilevered beam boundary conditions, the nonlinear partial differential equation of the system is reduced to a nonlinear ordinary differential equation and numeric integration procedures are used to solve it. By considering the non-dimensional damping, cubic and quadratic terms, and amplitude of external force as controlling parameters, the bifurcation diagram and the largest Lyapunov exponent are employed to distinguish the chaotic, periodic, and quasi-periodic critical parameters of the curved CNT dynamics and validate the predicted results. Also, the phase plane and Poincare map for these critical parameters are provided. The results show that decreasing values of damping coefficient and quadratic term leads to quasi-periodic or chaotic motion, while system has periodic behaviour for smaller values of the external force and nonlinear cubic term. Also, magnetic field intensity and length of the CSWCNT help preventing the chaotic motion. Furthermore, adverse effects of higher values of temperature change, flow velocity and its correction factor are seen based on the obtained results.