Approximation Method Research Articles

A fast analysis algorithm for structural vibration modal sensitivity

Vibration modal sensitivity analysis has a wide range of applications in structural optimization design, model correction, fault detection, and reliability analysis. The existing methods for calculating modal sensitivity mainly include modal superposition method, Nelson’s method, and some improved algorithms derived from them. However, the existing algorithms have the drawback of low computational efficiency when applied to modal sensitivity analysis of the large-scale structures. In view of this, the objective of this work is to develop a fast modal sensitivity analysis method that can more efficiently calculate the modal sensitivity of the large structures with thousands of degrees of freedom. This new sensitivity analysis technique integrates Sherman-Morrison-Woodbury formula, subspace iteration and forward-difference to achieve the goal of fast calculation. The innovations of the proposed method mainly include the following aspects. Firstly, the Sherman-Morrison-Woodbury formula is used to quickly calculate the perturbed flexibility matrix due to the minor modification in the structure. Then, the perturbed flexibility matrix is introduced into the subspace iteration method to calculate the perturbed modal parameters. Finally, the forward-difference algorithm is used to calculate the modal sensitivity. Two numerical examples are used to verify the superiority of the proposed modal sensitivity algorithm. Taking the 1952-bar structure as an example, the calculation time of the proposed algorithm is only 8.3 % and 18.2 % of that of the existing approximate modal superposition method and the improved Nelson’s method, respectively. In terms of calculation accuracy, the results obtained by the proposed method are exactly the same as the exact solution when the perturbation scale is set to 10−5. It is found that the proposed method significantly reduces computation time compared to the existing methods on the premise that the calculation accuracy is basically unchanged.

Global Existence and Blow-up of Solutions for a Class of Singular Parabolic Equations with Viscoelastic Term

In this paper, we consider the initial boundary value problem for a class of singular parabolic equations with viscoelastic term and logarithmic term. By using the technique of cut-off and the method of Faedo-Galerkin approximation, the local existence of the weak solution is established. Based on the potential well method, the global existence of the weak solution is derived. Furthermore, we prove that the weak solution blows up in finite time by taking the concavity analysis method.

A Hybrid Finite Element Method – Kirchhoff Approximation Method for Modeling Acoustic Scattering from an Underwater Vehicle Model with Alberich Coatings with Periodic Internal Cavities

Anechoic tiles can significantly reduce the echo intensity of underwater vehicles, thereby increasing the difficulty of detecting such vehicles. However, the computational efficiency of conventional methods such as the finite element method (FEM) and the boundary element method (BEM) has its limitations. A fast hybrid method for modeling acoustic scattering from underwater vehicles with anechoic tiles with periodic internal cavities, is developed by combining the Kirchhoff approximation (KA) and FEM. The accuracy and rapidity of the KA method were validated by FEM. According to the actual situation, the reflection coefficients of rubber materials with two different structures under rigid backing are simulated by FEM. Using the KA method, the acoustic scattering characteristics of the underwater vehicle with anechoic tiles are obtained by inputting the reflection coefficients and the target’s geometric grid. Experiments on the monostatic target strength (TS) in the frequency range of 1 to 20 kHz and time domain echo characteristics of acoustic scattering on a benchmark scale model with anechoic tiles are conducted. The research results indicate that the TS values and echo characteristic curves of the KA solutions closely approximate the experimental results, which verifies the accuracy of the KA method in calculating the TS and echo characteristics of underwater vehicles with anechoic tiles.

Resolving nonuniform flow in gas turbines: challenges, progress, and moving forward

Abstract The flow in gas turbines exhibits a highly unsteady, complex and nonuniform manner, which presents two main challenges. Firstly, it introduces instrumentation errors, contributing to uncertainties when calculating one-dimensional performance metrics during rig or engine tests using fixed-location rakes. Secondly, it raises mechanical concerns, including high-cycle fatigue due to blade row interactions in turbomachines and thermal fatigue caused by hot-streaks at the combustor exit. Experimental characterisation of the flow nonuniformity in gas turbines is highly challenging due to the confined space and harsh environment for instrumentation. This paper presents recent efforts to address this issue by resolving the nonuniform flow in gas turbines using spatially under-sampled measurements. The proposed approach utilises discrete probe data and leverages a ‘Fourier-based approximation’ method developed by the author. The technique has undergone preliminary experimental validation involving reconstruction of the total pressure distribution in a multi-stage axial compressor and the total temperature field at the exit of the combustor and high-pressure turbine. Results show that, in the multi-stage axial compressor environment, reconstruction of inter-stage total pressure is achieved using a reduced dataset covering less than 20% of the annulus with reasonably good accuracy. The reconstructed total pressure yields almost identical mean total pressure values at all spanwise locations, with a maximum deviation of less than 0.02%. Additionally, reconstruction of the total temperature distribution at an engine-representative full annulus combustor is achieved using measurements at ten carefully selected circumferential locations. Results show that the reconstructed temperature field successfully captures the primary features associated with combustor exit temperature flow. The reconstructed temperature field yields excellent agreement in the magnitude of radial temperature distribution factor (RTDF) and overall temperature distribution factor (OTDF) predictions to the experiment, with a deviation of less than 0.5% for RTDF and less than 2.5% for OTDF. Lastly, reconstruction of the total temperature distribution at the exit of the GE E3 high-pressure turbine (HPT) is achieved using measurements at eight carefully selected circumferential locations. Results demonstrate remarkable robustness in resolving the temperature profile at the HPT exit with high fidelity, irrespective of the HPT inlet conditions. The initial validation results are promising, demonstrating that the new probe layout scheme and the Fourier-based approximation method enable effective characterisation of flow nonuniformity in gas turbines, thereby providing valuable insights into the complex flow of gas turbine engines.

Comparative study of some rigorous and approximate methods for calculating the energy loss straggling

Some exact and approximate methods commonly used to calculate corrections to the Bethe stopping formula are modified and adapted by the authors to the calculation of the energy loss straggling (ELS). An intercomparison is carried out for results of approaches developed in the present work. An excellent agreement obtained between the results for ELS, calculated with an exact method previously proposed by one of the authors and the results of the Lindhard–Sørensen method for moderate relativistic energies. It is shown that some modified by authors approximate methods for the ELS calculating have the higher accuracy than the conventional approximate method of Lijian−Qing−Zhengming (LQZ). So, in all the cases considered, the accuracy of the twice modified LQZ method in the ELS calculations is higher than the accuracy of the usual LQZ method. The velocity averages of the relative error for Z = 92–118 of LQZ varies from 0.08% to 0.33%, and of LQZm2 – from 0.03% to 0.09%. A thrice modified LQZ method was also suggested in this work. The average error of LQZm3 becomes lower than the average error of LQZ, starting from Z = 99, and lower than the error of LQZm2, starting from Z = 109. A number of shortened LQZ methods for the ELS calculating have been proposed and developed too.

Bekenstein–Hawking entropy products for NUT class of black holes in AdS space

In this paper, we derive the entropy product rule for Taub–Newman–Unti–Tamburino (Taub–NUT)–de Sitter black hole (BH) and Taub–NUT–anti-de Sitter BH. We show that the entropy products in terms of both the physical horizons are mass-independent. Both perturbative approximation and direct method have been considered. By introducing the cosmological horizon, we show that for Taub–NUT–de Sitter BH, there exists a mass-independent entropy functional relation in terms of three horizons namely event horizon (EH), Cauchy horizon (CH) and cosmological horizon (CHH) which depends on cosmological parameter ([Formula: see text]) and the NUT parameter (N). For Taub–NUT–anti-de Sitter BHs, we determine the mass-independent entropy functional relations in terms of two physical horizons (namely EH and CH) which depends on only NUT parameter. Sometimes some complicated functions of EH entropy and CH entropy are also strictly mass-independent. This is plausible only due to the new formalism developed in [S. Wu and D. Wu, Phys. Rev. D 100, 101501(R) (2019)] for NUT class of BHs. The formalism states that a generic four dimensional Taub–NUT spacetime should be described completely in terms of three or four different types of thermodynamic hairs. They could be defined as the Komar mass ([Formula: see text]), the angular momentum ([Formula: see text]), the gravitomagnetic charge ([Formula: see text]), the dual (magnetic) mass [Formula: see text]. Finally, we could say that this universality is mainly due to the presence of new conserved charges [Formula: see text] which is closely analogue to the Kerr-like angular momentum [Formula: see text].

Pressure-induced evolution of structures and phase transition of technetium diboride

Using the particle swarm optimization algorithm, we conducted an extensive search for the high-pressure stable structure of technetium diboride (TcB2) within the pressure range of 0–400 GPa. At zero pressure, the P63/mmc (hP6-TcB2) structure is considered the ground state configuration. As the pressure increases, a structural transition from hP6-TcB2 to P6/mmm (hP3-TcB2) occurs at approximately 174.9 GPa. We discuss the bonding between the two distinct phases and analyze the contribution of different atomic bonds to maintaining their structural stability. Meanwhile, the temperature–pressure phase diagram of TcB2 was successfully determined for the first time through the quasi-harmonic approximation method. It is predicted that the transition pressure from hP6-TcB2 to hP3-TcB2 can be reduced to about 164 GPa at a room temperature of 300 K. These results provide valuable insights into the behavior of TcB2 under different temperature and pressure conditions and open up new possibilities for exploring its potential applications in a variety of environments.

A Modified Perturbation Method for Global Dynamic analysis of Generalized Mixed Rayleigh–Liénard Oscillator with Cubic and Quintic Nonlinearities

Abstract Deriving analytical relationship between the system parameters and amplitude of the limit cycle is a meaningful and challenging task. Currently, numerous existing analytical approximate methods struggle to achieve this goal when expressions of restoring force or nonlinear damping is complicated. To overcome this shortcoming, this study proposes a modified generalized harmonic function perturbation method. Using the proposed method, a generalized mixed Rayleigh–Liénard oscillator with cubic and quintic nonlinearities was investigated. The analytical relationships between the system parameters and amplitude of the limit cycle, as well as the expression of its characteristic quantity, were derived. By employing these analytical relationships, the existence, stability, number, position, and amplitude of each limit cycle are quantitatively analysed. The homoclinic and heteroclinic bifurcations were also predicted using the above analytical relationships. Additionally, analytical approximate solutions for this oscillator were calculated using the proposed method. All results obtained in this study were subsequently confirmed numerically to demonstrate their feasibility and validity. Consequently, the proposed method can be considered an effective supplement to perturbation-based methods. This also implies that the work presented in this paper has a certain theoretical significance and application value in the research area of quantitative analysis methods for strongly nonlinear oscillators.

Real-time generation of circular patterns in electron beam lithography

Electron beam lithography (EBL) involves the transfer of a pattern onto the surface of a substrate by first scanning a thin layer of organic film (called resist) on the surface by a tightly focused and precisely controlled electron beam (exposure) and then selectively removing the exposed or nonexposed regions of the resist in a solvent (developing). It is widely used for fabrication of integrated circuits, mask manufacturing, photoelectric device processing, and other fields. The key to drawing circular patterns by EBL is the graphics production and control. In an EBL system, an embedded processor calculates and generates the trajectory coordinates for movement of the electron beam, and outputs the corresponding voltage signal through a digital-to-analog converter (DAC) to control a deflector that changes the position of the electron beam. Through this procedure, it is possible to guarantee the accuracy and real-time control of electron beam scanning deflection. Existing EBL systems mostly use the method of polygonal approximation to expose circles. A circle is divided into several polygons, and the smaller the segmentation, the higher is the precision of the splicing circle. However, owing to the need to generate and scan each polygon separately, an increase in the number of segments will lead to a decrease in the overall lithography speed. In this paper, based on Bresenham’s circle algorithm and exploiting the capabilities of a field-programmable gate array and DAC, an improved real-time circle-producing algorithm is designed for EBL. The algorithm can directly generate circular graphics coordinates such as those for a single circle, solid circle, solid ring, or concentric ring, and is able to effectively realizes deflection and scanning of the electron beam for circular graphics lithography. Compared with the polygonal approximation method, the improved algorithm exhibits improved precision and speed. At the same time, the point generation strategy is optimized to solve the blank pixel and pseudo-pixel problems that arise with Bresenham’s circle algorithm. A complete electron beam deflection system is established to carry out lithography experiments, the results of which show that the error between the exposure results and the preset patterns is at the nanometer level, indicating that the improved algorithm meets the requirements for real-time control and high precision of EBL.

On the geometry dependence of the nuclear magnetic resonance chemical shift of mercury in thiolate complexes: A relativistic density functional theory study.

Thiolate containing mercury(II) complexes of the general formula [Hg(SR) ] have been of great interest since the toxicity of mercury was recognized. 199Hg nuclear magnetic resonance spectroscopy (NMR) is a powerful tool for characterization of mercury complexes. In this work, the Hg shielding constants in a series of [Hg(SR) ] complexes are therefore investigated computationally with particular emphasis on their geometry dependence. Geometry optimizations and NMR chemical shift calculations are performed at the density functional theory (DFT) level with both the zeroth-order regular approximation (ZORA) and four-component relativistic methods. The four exchange-correlation (XC) functionals PBE0, PBE, B3LYP, and BLYP are used in combination with either Dyall's Gaussian-type (GTO) or Slater-type orbitals (STOs) basis sets. Comparing ZORA and four-component calculations, one observes that the calculated shielding constants for a given molecular geometry have a constant difference of 1070 ppm. This confirms that ZORA is an acceptable relativistic method to compute NMR chemical shifts. The combinations of four-component/PBE0/v3z and ZORA/PBE0/QZ4P are applied to explore the geometry dependence of the isotropic shielding. For a given coordination number, the distance between mercury and sulfur is the key factor affecting the shielding constant, while changes in bond and dihedral angles and even different side groups have relatively little impact.

The First Global Insight of Cirrus Clouds Characterized by Hollow Ice Crystals From Space‐Borne Lidar

AbstractCirrus clouds often contain numerous hollow ice crystals, which are distinct in scattering properties from solid ice crystals, and will be challenging to microphysical retrieval and radiative forcing assessment. Currently, hollow ice crystals have not been observed by remote sensing methods, and the estimation of their hollowness is a complex task. To address this issue, the Mixed Modal Hollow Columns (MMHC) model for hollow ice crystals is introduced, and its backscattering properties are computed using the physical optics approximation method. Through comparison with spaceborne lidar observations, we identify a specific type of cirrus associated with the MMHC model for the first time. The visible optical depth of this cirrus is less than or equal to 0.1, and the temperature is between −60 and −40°C. The MMHC characteristic cirrus clouds are prevalent in middle and high latitudes but less comm+on in low latitudes. They exhibit distinct patterns in terms of sea and land distribution as well as seasonal variation.

Optimisation of energy efficiency of ambient backscatter communication and reconfigurable intelligent surfaces in non‐orthogonal multiple access downlink

AbstractThe authors study the energy efficiency (EE) of ambient backscatter communication (AmBC) device‐assisted and reconfigurable intelligent surfaces (RIS)‐assisted non‐orthogonal multiple access (NOMA) downlinks. The authors establish two optimisation problems based on the two collaborative devices (AmBC devices, RIS) with the objective of maximising the EE of the system, taking into account the requirements of power limitation and rate limitation, etc. and also obtain the solutions of two problems by optimising the relevant performance metrics based on the alternating optimisation algorithm. For the backscatter device (BD)‐aided downlink NOMA network, the problem is first decoupled into three subproblems, where the power allocation optimisation subproblem is solved by using the quadratic transformation method and the subgradient algorithm. The maximum EE is obtained by iterating according to the Dinkelbach's algorithm. For the RIS‐aided downlink NOMA network, the power allocation problem is solved by the same method as above and the phase optimisation problem is solved by the successive convex approximation method. Numerical results show that the proposed algorithm can achieve convergence after several iterations, and the EE of systems with BD‐assisted and RIS‐assisted have different levels of sensitivity to different influencing factors.

On the issue of contactless alternators on movable objects

BACKGROUND: This article discusses automotive inductor generators of domestic and foreign production with different power and design. A comparative analysis of inductor generators according to the main electromagnetic parameters and design designs is carried out. Based on the results of the comparative analysis, conclusions are drawn about the competitiveness of domestic inductor generators and their superiority over foreign analogues. AIMS: research and comparative analysis of tractor inductor generators in order to determine the optimal design, as well as the main characteristics and parameters. METHODS: The calculation of the magnetic circuit and the idling characteristic is based on the method of successive approximations using iterations. A well-known technique using the Blondel diagram is used to calculate the current-speed characteristic. RESULTS: A review and comparative analysis of automotive inductor generators of domestic and foreign production, with various magnetic systems and constructive equipment, is carried out. The advantages and disadvantages of the magnetic circuit of inductor generators of both domestic and foreign production are determined, which are reflected in comparative tables and current-speed characteristics. To carry out the calculation, the optimal design of the inductor generator with the most effective technical and economic indicators was chosen. Conclusions: The calculation of a three-phase single-pole, single-pack inductor generator 11.3701 with a classical toothed zone and a two-half-period rectifier is carried out. The calculation of the magnetic circuit and the idling characteristics was carried out by the method of successive approximations using iterations. The calculation of the current-velocity characteristic was performed using a well-known technique using the Blondel diagram.

Opinion formation on evolving network: the DPA method applied to a nonlocal cross-diffusion PDE-ODE system

Abstract We study a system of nonlocal aggregation cross-diffusion PDEs that describe the evolution of opinion densities on a network. The PDEs are coupled with a system of ODEs that describe the time evolution of the agents on the network. Firstly, we apply the Deterministic Particle Approximation (DPA) method to the aforementioned system in order to prove the existence of solutions under suitable assumptions on the interactions between agents. Later on, we present an explicit model for opinion formation on an evolving network. The opinions evolve based on both the distance between the agents on the network and the ’attitude areas’, which depend on the distance between the agents’ opinions. The position of the agents on the network evolves based on the distance between the agents’ opinions. The goal is to study radicalisation, polarisation and fragmentation of the population while changing its open-mindedness and the radius of interaction.

Mixed quantum/classical theory (MQCT) approach to the dynamics of molecule-molecule collisions in complex systems.

We developed a general theoretical approach and a user-ready computer code that permit study of the dynamics of collisional energy transfer and ro-vibrational energy exchange in complex molecule-molecule collisions. The method is a mixture of classical and quantum mechanics. The internal ro-vibrational motion of collision partners is treated quantum mechanically using a time-dependent Schrödinger equation that captures many quantum phenomena including state quantization and zero-point energy, propensity and selection rules for state-to-state transitions, quantum symmetry and interference phenomena. A significant numerical speed up is obtained by describing the translational motion of collision partners classically, using the Ehrenfest mean-field trajectory approach. Within this framework a family of approximate methods for collision dynamics is developed. Several benchmark studies for diatomic and triatomic molecules, such as H2O and ND3 collided with He, H2 and D2, show that the results of MQCT are in good agreement with full-quantum calculations in a broad range of energies, especially at high collision energies where they become nearly identical to the full quantum results. Numerical efficiency of the method and massive parallelism of the MQCT code permit us to embrace some of the most complicated collisional systems ever studied, such as C6H6 + He, CH3COOH + He and H2O + H2O. Application of MQCT to the collisions of chiral molecules such as CH3CHCH2O + He, and to molecule-surface collisions is also possible and will be pursued in the future.

On tiling spherical triangles into quadratic subpatches

Various interpolation and approximation methods arising in several practical applications in geometric modeling deal, at a particular step, with the problem of computing suitable rational patches (of low degree) on the unit sphere. Therefore, we are concerned with the construction of a system of spherical triangular patches with prescribed vertices that globally meet along common boundaries. In particular, we investigate various possibilities for tiling a given spherical triangular patch into quadratically parametrizable subpatches. We revisit the condition that the existence of a quadratic parameterization of a spherical triangle is equivalent to the sum of the interior angles of the triangle being π, and then circumvent this limitation by studying alternative scenarios and present constructions of spherical macro-elements of the lowest possible degree. Applications of our method include algorithms relying on the construction of (interpolation) surfaces from prescribed rational normal vector fields.

First-principles study of hydrogen sulfide decomposition on Sc-Ti3C2O2 single-atom catalyst.

Hydrogen sulfide gas poses significant risks to both human health and the environment, with the potential to induce respiratory and neurological effects, and a heightened fatality risk at elevated concentrations. This article investigates the catalytic decomposition of H2S on a Sc-Ti3C2O2 single-atom catalyst(SAC) using the density functional theory-based first-principles calculation approach. Initially, the adsorption behavior of H2S on Ti3C2O2-MXene was examined, revealing weak physical adsorption between them. Subsequently, the transition metal atom Sc was introduced to the Ti3C2O2 surface, and its stability was studied, demonstrating high stability. Further exploration of H2S adsorption on Sc-Ti3C2O2 revealed direct dissociation of H2S gas molecules into HS* and H*, with HS* binding to Sc and H* binding to O on the Ti3C2O2 surface, resulting in OH groups. Using the transition state search method, the dissociation of H2S molecules on the SAC's surface was investigated, revealing a potential barrier of 2.45eV for HS* dissociation. This indicates that the H2S molecule can be dissociated into H2 and S with the action of the Sc-Ti3C2O2 SAC. Moreover, the S atom left on the catalyst surface can aggregate to produce elemental S8, desorbing on the catalyst surface, completing the catalytic cycle. Consequently, the Sc-Ti3C2O2 SAC is poised to be an efficient catalyst for the catalytic decomposition of H2S. The Dmol3 module in Materials Studio software based on density functional theory is used in this study. The generalized gradient approximation method GGA-PBE is used for the exchange-correlation function. The complete LST/QST and the NEB methods in the Dmol3 module were used to study the minimum energy path of the dissociation of hydrogen sulfide molecules on the catalyst surface.

Pressure effects on the structural, electronic, elastic, optical, and vibrational properties of YMg intermetallic compounds: a first-principles study

The properties of YMg in B2 structure have been comprehensively analysed using the first-principles plane-wave pseudopotential method. Specifically, the structural, electronic, elastic, vibrational, and optical properties were investigated using the generalized gradient approximation (GGA) method in the context of density functional theory. The Vienna ab initio simulation package (VASP) was utilized for these calculations. The computed lattice parameter (3.803 Å) and bulk modulus (41.33 GPa) are consistent with the earlier data on ambient pressure. The electronic band structure and energy-dependent density of states reveal the metallic nature of the titled compounds. The Born stability requirements confirmed the mechanical stability. The analysis of Pugh’s and Poisson’s ratios and Cauchy’s pressure reveals that YMg is ductile under the pressures in consideration. According to several anisotropy indices, the compound is noticeably anisotropic both in ambient and under pressure. Our investigation includes an analysis of several fundamental mechanical parameters of the material, including the bulk modulus, the pressure derivative of the Zener anisotropy factor, Poisson’s ratio, isotropic shear modulus and Young’s modulus with a particular focus on their dependence on pressure. We have determined that the elastic constants obtained remain mechanically stable, satisfying the Born Stability conditions even at high pressures of up to 60 GPa. To explore the dynamic stability of YMg, we analysed the material’s phonon dispersion curves. The examined compound displays stability under dynamic conditions from 0 GPa to 30 GPa, as evidenced by its positive vibration frequencies. However, this stability is not sustained under higher pressure, as the compound becomes unstable after 30 GPa to 70 GPa. The electronic band structure and density of states diagrams demonstrate YMg’s metallic properties. At atmospheric pressure (0 GPa), the total density of states (TDOS) near the Fermi level is approximately 1.63 states/eV, with pressure application reducing DOS. The dielectric function, refractive index, and energy loss spectra are examined within the 0–20 eV energy range. YMg exhibits its highest absorption between 4 and 11 eV. The peak optical conductivity is observed around 0.78 eV (equivalent to 1589.5409 nm), while the most significant energy loss occurs at 11.90 eV, roughly corresponding to 2.8 Hz in the ultraviolet spectrum. Moreover, we extensively analyzed the material’s phonon thermodynamic and optical properties, providing insights into its behavior under various conditions. The outcomes acquired at zero pressure are generally coherent with the current theoretical values.

Behaviour Analysis of Modeling and Model Evaluating Methods in System Identification for a Multiprocess Station

Systems are designed to perform specific task by giving certain input which produces the required output in an orderly manner known as process. The input, output, and the state variables should be known that will help in interacting with the system. The relation between these variables can be brought out by building a model that resembles or expresses the original performance of the system. The parameters of the model are estimated using the least squares approximation, maximum likelihood, maximum log-likelihood, and Bayesian parameter estimation methods by utilizing the experimental data from the multiprocess station. The selected parameters are converted to nine different transfer function models that represent the given dynamic system. The models framed are analyzed by the criterion curve technique using seven criterion functions evaluating the fitness of the model. Order of the model is found from Hankel matrix representation methods such as singular value decomposition and determinant method. Response of the models is compared with the original response to choose the best fit model by calculating ISE standard. All the above methods are used to model the system without physical and theoretical laws which is known as system identification.

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.