Spectral Method Research Articles

A Spectral Method for a Fokker-Planck Equation in Neuroscience with Applications in Neuron Networks with Learning Rules

A Spectral Method for a Fokker-Planck Equation in Neuroscience with Applications in Neuron Networks with Learning Rules

A Low-Rank Matrix Approach to Compute Polynomial Approximations of Smooth Two-Dimensional Functions

Polynomial approximation of smooth functions is becoming increasingly important in fields like numerical analysis and scientific computing. These approximations are vital in models that rely on spectral methods. To reduce the memory costs for large dimensional problems, various methods to provide data-sparse representations have been proposed, including methods based on singular value decomposition, adaptive cross approximation, and matrices with hierarchical low-rank structures, to mention a few. This work presents implementation details on the polynomial approximation of univariate smooth functions through the Polynomial1 class, and of bivariate smooth functions by low-rank matrix representation via the Polynomial2 class. These approaches are explained within Tau Toolbox, a mathematical software library for solving integro-differential problems by the spectral Tau method.

Efficient computational hybrid method for the solution of 2D multi-term fractional order advection-diffusion equation

Abstract Multi-term time-fractional advection diffusion equations are vital for simulating a wide range of physical phenomena, including fluid dynamics and environmental transport processes. However, due to their natural complexity, these equations pose challenges for conventional numerical approaches. In this article, we develop a high order accurate method to solve the multi-term time-fractional advection diffusion equations. We combine the Laplace transform (LT) to integrate the considered equations in time, with Chebyshev spectral method (CSM) for spatial terms The proposed method produces highly accurate solutions with remarkably low computational cost as compared to finite difference method. The propose numerical scheme first employs the LT which reduces the considered problem into a finite set of elliptic equations which may be solved in parallel. Then, the CSM is employed for the disctrezation of spatial operators, which makes it possibly to accurately represent the solution chebyshev grid. Finally, numerical inversion of LT is used to convert the obtain solution from the Laplace domain into the real domain. This work utilizes the modified Talbot’s method and Stehfest’s method for numerical inversion of the LT. To measure the performance, efficiency, and accuracy of the suggested approach, numerical approximations of three models are acquired and verified against the exact solution. The outcomes presented in tables and figures demonstrate that the modified Talbot’s method performed better as compared to Stehfest’s method.

Regional evaluation of global geopotential models and three types of digital elevation models with ground-based gravity and GNSS/levelling data using several techniques over Sudan

Abstract To evaluate the performance of the global geopotential models (GGMs) in a more unbiased way, ground-based gravity and GNSS/levelling datasets are highly required. In this study, the eight latest releases of the satellite-only and combined GGMs are evaluated on the regional scale using the available terrestrial gravity and GNSS/Levelling data over Sudan, considering the spectral consistency issue by applying the spectral enhancement method (SEM). The evaluation process consists of three stages: firstly, the eight GGMs are evaluated globally with each other by using different degree variances in terms of geoid heights, gravity anomalies, and signal-to-noise ratio (SNR); secondly, the GGMs are compared against the Earth Gravitational Model 2008 (EGM2008) on a regional scale over Sudan; thirdly, apply the SEM strategy by incorporating high (SEM_WITHOUT_RTM technique) and very-high (SEM technique) frequencies of the gravity field spectrum from the EGM2008 and high-resolution residual terrain model (RTM), respectively. For reliable robustness of the latter evaluation process, three different DEMs are used, namely, SRTM30, ASTER30, and GTOPO30. Our findings on the evaluation process using SEM_WITHOUT_RTM technique show improved gravity anomalies solutions regarding differences of standard deviations (STD) from 19–20.7 mGal to about 14 mGal. When applying the SEM technique, more improvements are achieved, providing STD differences in gravity anomalies and geoid heights of about 12 mGal and 45 cm, respectively. Among the three applied DEMs, it has been found that despite the slight refinements, the ASTER30 and GTOPO30 models show better performance than the SRTM30 model.

Advanced stability analysis of a fractional delay differential system with stochastic phenomena using spectral collocation method.

In recent years, there has been a growing interest in incorporating fractional calculus into stochastic delay systems due to its ability to model complex phenomena with uncertainties and memory effects. The fractional stochastic delay differential equations are conventional in modeling such complex dynamical systems around various applied fields. The present study addresses a novel spectral approach to demonstrate the stability behavior and numerical solution of the systems characterized by stochasticity along with fractional derivatives and time delay. By bridging the gap between fractional calculus, stochastic processes, and spectral analysis, this work contributes to the field of fractional dynamics and enriches the toolbox of analytical tools available for investigating the stability of systems with delays and uncertainties. To illustrate the practical implications and validate the theoretical findings of our approach, some numerical simulations are presented.

A prediction model of rubber content in the dried root of Taraxacum kok-saghyz Rodin based on near-infrared spectroscopy

BackgroundTaraxacum kok-saghyz Rodin (TKS) is a highly potential source of natural rubber (NR) due to its wide range of suitable planting areas, strong adaptability, and suitability for mechanized planting and harvesting. However, current methods for detecting NR content are relatively cumbersome, necessitating the development of a rapid detection model. This study used near-infrared spectroscopy technology to establish a rapid detection model for NR content in TKS root segments and powder samples. The K445 strain at different growth stages within a year and 129 TKS samples hybridized with dandelion were used to obtain their near-infrared spectral data. The rubber content in the root of the samples was detected using the alkaline boiling method. The Monte Carlo sampling method (MCS) was used to filter abnormal data from the root segments of TKS and powder samples, respectively. The SPXY algorithm was used to divide the training set and validation set in a 3:1 ratio. The original spectrum was preprocessed using moving window smoothing (MWS), standard normalized variate (SNV), multiplicative scatter correction (MSC), and first derivative (FD) algorithms. The competitive adaptive reweighted sampling (CARS) algorithm and the corresponding chemical characteristic bands of NR were used to screen the bands. Partial least squares (PLS), random forest (RF), Lightweight gradient augmentation machine (LightGBM), and convolutional neural network (CNN) algorithms were employed to establish a model using the optimal spectral processing method for three different bands: full band, CARS algorithm, and chemical characteristic bands corresponding to NR. The model with the best predictive performance for high rubber content intervals (rubber content > 15%) was identified.ResultThe results indicated that the optimal rubber content prediction models for TKS root segments and powder samples were MWS–FD CASR–RF and MWS–FD chemical characteristic band RF, respectively. Their respective RP2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext{R}}_{{\ ext{P}}}^{2}$$\\end{document}, RMSEP, and RPDP values were 0.951, 0.979, 1.814, 1.133, 4.498, and 6.845. In the high rubber content range, the model based on the LightGBM algorithm had the best prediction performance, with the RMSEP of the root segments and powder samples being 0.752 and 0.918, respectively.ConclusionsThis research indicates that dried TKS root powder samples are more appropriate for constructing a rubber content prediction model than segmented samples, and the predictive capability of root powder samples is superior to that of root segmented samples. Especially in the elevated rubber content range, the model formulated using the LightGBM algorithm has superior predictive performance, which could offer a theoretical basis for the rapid detection technology of TKS content in the future.

A preliminary study on generation of axle box vibration spectra for metro vehicles in China using the developed spectral synthesis method

ABSTRACT In-service metro vehicles in China exhibit diverse vibration behaviours. This study focuses on analysing the vibration characteristics of these vehicles, with a particular focus on the complex non-stationary axle box vibrations measured on-site, as they closely correlate with wheel-rail interaction dynamics. A developed PSD-based synthesis method is first proposed to quantify the axle box vibration. The axle box vibration spectra of the metro vehicles in China are then generated from the measurements for various vehicles. Furthermore, a multi-line segment flat spectrum, designed specifically for the shaker loading, has been produced. The results show that the proposed method can determine the parameters (segment number, quantile) by the energy conservation law, which avoids the empirical value selection in previous methods and can be effectively utilized to generate the vibration spectra of the axle box. In addition, a notable distinction exists between the IEC 61,373 standard spectrum and the axle box vibration spectra of Chinese metro vehicles. While the former represents a single-peak spectrum ranging from 20 ~ 100 Hz, the latter consists of a multi-peak spectrum with peak frequency bands exceeding 100 Hz. The wear condition of the wheel-rail surface has a pronounced contribution to the vibration spectral magnitude. In cases of wheel-rail surface severe wear, the vibration spectrum values in the Z direction surpass the standard spectrum values across most frequencies, with the highest spectral value notably observed within 50 ~ 200 Hz. Such observations should be thoroughly taken into account in the analysis of fatigue damage to mounted equipment.

Optimized Transfer Learning for Chlorophyll Content Estimations across Datasets of Different Species Using Sun-Induced Chlorophyll Fluorescence and Reflectance

Remote sensing-based techniques have been widely used for chlorophyll content (Cab) estimations, while they are challenging when transferred across different species. Sun-induced chlorophyll fluorescence (SIF) provides a new approach to address these issues. This research explores whether SIF has transferability for Cab estimation and to enhance between-species transferability. Here, three rice datasets and a rapeseed dataset were collected. Initially, direct transfer models were constructed using partial least squares regression (PLSR) based on SIF yield (SIFY) and reflectance, respectively. Subsequently, methods were employed within the rice datasets to improve the models’ transferability. Finally, the between-species transferability of two data sources was validated in the rapeseed dataset. Direct transfer models indicated that the reflectance-based model exhibited a higher accuracy in predicting Cab when the training dataset acquired sufficient features, whereas the SIFY-based model showed better performance with fewer features. Spectral preprocessing methods can enhance the transferability, especially for SIFY-based models. In addition, supplementing 10% of out-of-sample data significantly improved the transferability. The proposed methods only require a small amount of new data to extend the original model for predicting Cab in other species. Specifically, the new method reduced the average RMSE based on SIFY and reflectance models by 23.59% and 35.51%, respectively.

Magnitude estimation and site characterization in southwestern British Columbia: application to earthquake early warning

AbstractIn this study, we took a close look at the Ocean Networks Canada’s earthquake early warning system in southwestern British Columbia through analysis of the magnitude estimates by this system and characterization of site conditions for both onshore and offshore stations. Using magnitude values estimated at each station, over hundreds of notifications, we provided station terms to correct the magnitudes for stations that systematically generate high or low magnitude values. Moreover, by compiling a rich ground motion amplitude dataset and applying the horizontal-to-vertical spectral ratio method from Fourier amplitude of acceleration and response spectral acceleration, we investigated site characterization through evaluation of non-linear site response behavior and estimation of the site dominant frequency (fpeak) and its peak amplitude (Apeak) for each station. In general, no strong evidence of non-linearity is observed at any stations considering the magnitude-distance distribution of ground motions in this study. Offshore sites show fpeak and Apeak in the range of approximately 1.7–6 Hz and 0.4–1.2 (in base-10 log unit), respectively, whereas onshore sites show approximately 1–6 Hz for fpeak and 0.3–0.7 (in base-10 log unit) for Apeak.

Evaluation of EEG Signals by Spectral Peak Methods and Statistical Correlation for Mental State Discrimination Induced by Arithmetic Tasks

Evaluation of EEG Signals by Spectral Peak Methods and Statistical Correlation for Mental State Discrimination Induced by Arithmetic Tasks

A worldwide coastal analysis of the climate wave systems

Wind generated waves of a sea state are generally the result of the superposition of wind sea and swells, making the frequency-direction wave energy distribution crucial for comprehending this behavior. Wave spectral partitioning methods provide groups of waves with similar characteristics, thus they have been usually applied to identify wind sea and swell. In addition, several swells can coexist in a sea state. This study develops a method to estimate the wave systems and analyze their characteristics over the coast worldwide using 32year (1989-2020) historical information and more than 10.000 locations. A wave system is considered as the long-term climate conditions prevailing over a frequency-direction wave energy area collecting similar environmental and physical characteristics. The method is applied for the hourly time series of the directional wave spectra. First, the watershed clustering algorithm is used and the partitions found are classified as wind sea or swells based on a wave age criterion. The information obtained from the swell spectral partitions is then used to estimate the probability of their occurrence within specific frequency-direction bins and the clustering algorithm is applied anew to this population in order to identify the number of significant long-term climate wave systems locally and their characteristics. Outcomes reveal that on average swells coexist with wind sea in approximately 70% of the global coast, whereas about 25% is predominantly dominated by pure swells and the wind sea dominates only in the 5%. Only the 2% of the global coast line presents one swell wave system. About 50% of the global coastal locations exhibit three and four, whereas the 15% presents two swell wave systems. The analysis shows that about 30% of the coastal locations present at least five swell wave systems, mostly on Pacific islands and enclosed seas.

Assessing Pressure Wave Components for Aortic Stiffness Monitoring through Spectral Regression Learning

Abstract Aims The aging process notably induces structural changes in the arterial system, primarily manifesting as increased aortic stiffness, a precursor to cardiovascular events. While wave separation analysis is a robust tool for decomposing the components of blood pressure waveform, its relationship with cardiovascular events, such as aortic stiffening, is incompletely understood. Furthermore, its applicability has been limited due to the need for concurrent measurements of pressure and flow. Our aim in this study addresses this gap by introducing a spectral regression learning method for pressure-only wave separation analysis. Methods and results Leveraging data from the Framingham Heart Study (2,640 individuals, 55% women), we evaluate the accuracy of pressure-only estimates, their interchangeability with reference method based on ultrasound-derived flow waves, and their association with Carotid-femoral pulse wave velocity. Method-derived estimates are strongly correlated with the reference ones for forward wave amplitude (R2=0.91), backward wave amplitude (R2=0.88), reflection index (R2=0.87), and moderately correlated with time delay between forward and backward waves (R2=0.38). The proposed pressure-only method shows interchangeability with reference method through covariate analysis. Adjusting for age, sex, body size, mean blood pressure, and heart rate, results suggest that both pressure-only and pressure-flow evaluations of wave separation parameters yield similar model performance for predicting Carotid-femoral pulse wave velocity with forward wave amplitude as the only significant factor (p < 0.001; 95% confidence interval, 0.056-0.097). Conclusions We propose an interchangeable pressure-only wave separation analysis method and demonstrate its clinical applicability in capturing aortic stiffening. The proposed method provides valuable non-invasive tool for assessing cardiovascular health.

Seismic analysis of gravity dam-foundation systems using stochastic spectral finite element method

Seismic analysis of dam-foundation systems is inherently complex due to uncertainties in material properties. This study investigates the effects of material randomness and heterogeneity on the seismic response of gravity dam-foundation systems using the stochastic spectral finite element method (SSFEM) based on the Karhunen Loève expansion and the polynomial chaos expansion. The Pine Plane dam is taken as an analysis example. First, the appropriate Karhunen Loève expansion modes and polynomial chaos order are determined for SSFEM by comparing them with the Monte-Carlo simulations. Subsequently, the seismic response of the dam is analyzed in the stochastic and deterministic models, respectively. Results indicate that material randomness and heterogeneity may reduce the seismic response of gravity dams, and the deterministic model may overestimate dynamic responses. Subsequently, the parametric analysis of the correlation length, which measures the spatial correlation between variables in different positions and indicates the size of heterogeneity material patches, is conducted. It is demonstrated that the correlation length would not significantly affect the mean responses, but a larger correlation length leads to greater dispersions in responses. Finally, the effects of randomness in the foundation are investigated. The heterogeneities in the foundation affect the response of the dam in the high-frequency range. Therefore, the uncertainty in material properties should be fully considered in a structural assessment and risk analysis.

Fractional gaussian noise: Spectral density and estimation methods

The fractional Brownian motion (fBm) process, governed by a fractional parameter , is a continuous‐time Gaussian process with its increment being the fractional Gaussian noise (fGn). This article first provides a computationally feasible expression for the spectral density of fGn. This expression enables us to assess the accuracy of a range of approximation methods, including the truncation method, Paxson's approximation, and the Taylor series expansion at the near‐zero frequency. Next, we conduct an extensive Monte Carlo study comparing the finite sample performance and computational cost of alternative estimation methods for under the fGn specification. These methods include two semi‐parametric methods (based on the Taylor series expansion), two versions of the Whittle method (utilising either the computationally feasible expression or Paxson's approximation of the spectral density), a time‐domain maximum likelihood (ML) method (employing a recursive approach for its likelihood calculation), and a change‐of‐frequency method. Special attention is paid to highly anti‐persistent processes with close to zero, which are of empirical relevance to financial volatility modelling. Considering the trade‐off between statistical and computational efficiency, we recommend using either the Whittle ML method based on Paxson's approximation or the time‐domain ML method. We model the log realized volatility dynamics of 40 financial assets in the US market from 2012 to 2019 with fBm. Although all estimation methods suggest rough volatility, the implied degree of roughness varies substantially with the estimation methods, highlighting the importance of understanding the finite sample performance of various estimation methods.

An improved spectral conjugate gradient projection method for monotone nonlinear equations with application

An improved spectral conjugate gradient projection method for monotone nonlinear equations with application

$\ell$-Proca stars

Abstract Initially applied to the scalar case, we extend the applicability of the multi-field generalization with angular momentum of bosonic stars to the vector case, in order to obtain new configurations that generalize the one-field spherical Proca stars. These new objects, which we call $\ell$-Proca stars, arise as stationary and spherically symmetric bosonic stars solutions of the Einstein-(multi)Proca system, whose matter content is formed by an arbitrary odd number of $2\ell+1$ of complex Proca fields with the same mass, time-frequency, radial profile and angular momentum number $\ell$. We analyze the system of constraint and evolution radial equations for the matter content to show the consistency of our proposal, and obtain numerically the ground states of these new solutions for the first few values of $\ell$ using spectral methods.

Relativistic Quantum Chemical Investigation of Actinide Covalency Measured by Electron Paramagnetic Resonance Spectroscopy.

We investigate actinide covalency effects in two [AnCptt3] (An = Th, U) complexes recently studied with pulsed electron paramagnetic resonance spectroscopy, using the Hyperion package to obtain relativistic hyperfine coupling constants from relativistic multiconfigurational wave functions. 1H and 13C HYSCORE simulations using the computed parameters show excellent agreement with the experimental data, highlighting the accuracy of modern relativistic ab initio methods. The extent of covalency indicated from the calculations on [ThCptt3] is in agreement with the original report based on traditional spectral fitting methods, while the covalency in [UCptt3] is found to be previously overestimated. The latter is due to the paramagnetic spin-orbit effect that arises naturally in a relativistic theory of hyperfine coupling and yet was not accounted for in the original study, thus highlighting the necessity of relativistic approaches for the interpretation of magnetic resonance data pertaining to actinides.

Sensitive Detection of OCS Using Thermal Conversion Combined with Spectral Reconstruction Filtering Differential Optical Absorption Spectroscopy.

Carbonyl sulfide (OCS) is a toxic gas produced during industrial processes that poses risks to both human health and industrial equipment. Therefore, detecting OCS concentrations plays a crucial role in early hazard warning. This paper presents an online system for detecting OCS at the ppb level using thermal conversion and spectral reconstruction filtering differential optical absorption spectroscopy (SRF-DOAS). First, OCS, which is not suitable for DOAS due to its weak absorption characteristics, is completely transformed into SO2 with strong absorption characteristics under high-temperature conditions. Then, the spectral reconstruction filtering method (SRF) is proposed to eliminate the noise and interference. The core idea of the method is to arrange the spectrum according to the spectral intensity from small to large rather than wavelength, reconstructing the spectrum into a new spectrum with linear characteristics. The reconstructed spectrum can remove noise and interference by linear fitting and retain the characteristic of SO2 oscillation absorption. Next, we demonstrate the ability of the reconstructed spectral method to remove noise and interference by comparing the spectra of the inverse-reconstructed gas mixture and SO2. The relative deviation of 0.88% at 100 ppb and detection limit of 7.26 ppb*m for OCS were obtained using the SRF-DOAS method. Finally, the reliability of the system was confirmed by measurements of OCS concentrations in mixture gas of OCS and air, as well as in human exhaled breath.

The posteriori analysis of the spectral element discretization of the wave equation

This work focuses on discretizing a second-order linear wave equation using the implicit Euler scheme for time discretization and the spectral element method for spatial discretization. We prove that optimal adaptivity can be achieved by combining adaptive time steps with adaptive spectral mesh. We introduce two sets of error indicators for time and space, respectively, and derive optimal estimates.

Comparative analysis of C n2 estimation methods for sonic anemometer data

Wind speed and sonic temperature measured with ultrasonic anemometers are often utilized to estimate the refractive index structure parameter C n 2, a vital parameter for optical propagation. In this work, we compare four methods to estimate C n 2 from C T 2, using the same temporal sonic temperature data streams for two separated sonic anemometers on a homogenous path. Values of C n 2 obtained with these four methods using field trial data are compared to those from a commercial scintillometer and from the differential image motion method using a grid of light sources positioned at the end of a common path. In addition to the comparison between the methods, we also consider appropriate error bars for C n 2 based on sonic temperature considering only the errors from having a finite number of turbulent samples. The Bayesian and power spectral methods were found to give adequate estimates for strong turbulence levels but consistently overestimated the C n 2 for weak turbulence. The nearest neighbors and structure function methods performed well under all turbulence strengths tested.