Second-order Expansion Research Articles

Superradiant lasing in inhomogeneously broadened ensembles with spatially varying coupling.

Background: Theoretical studies of superradiant lasing on optical clock transitions predict a superb frequency accuracy and precision closely tied to the bare atomic linewidth. Such a superradiant laser is also robust against cavity fluctuations when the spectral width of the lasing mode is much larger than that of the atomic medium. Recent predictions suggest that this unique feature persists even for a hot and thus strongly broadened ensemble, provided the effective atom number is large enough. Methods: Here we use a second-order cumulant expansion approach to study the power, linewidth and lineshifts of such a superradiant laser as a function of the inhomogeneous width of the ensemble including variations of the spatial atom-field coupling within the resonator. Results: We present conditions on the atom numbers, the pump and coupling strengths required to reach the buildup of collective atomic coherence as well as scaling and limitations for the achievable laser linewidth. Conclusions: We show how sufficiently large numbers of atoms subject to strong optical pumping can induce synchronization of the atomic dipoles over a large bandwidth. This generates collective stimulated emission of light into the cavity mode leading to narrow-band laser emission at the average of the atomic frequency distribution. The linewidth is orders of magnitudes smaller than that of the cavity as well as the inhomogeneous gain broadening and exhibits reduced sensitivity to cavity frequency noise.

Digital filter‐based grid synchronization for autonomous microgrids

Having a robust and accurate grid synchronization unit for distributed generators (DGs) has been accentuated due to rapidly growing integration of inverter-based DGs to the autonomous microgrids. Although most of the synchronization methods are based on the phased locked loops (PLL) technique, digital filters could result in more salient dynamic performance in non-ideal sinusoidal situations. This paper presents a novel digital filter as a grid synchronization method that overcomes the flaws of conventional PLLs under harmonic distorted and unbalanced cases as well as large delay of window-based digital filters. For the sake of computational advantages, the proposed method is executed in two distinct offline and online stages. In the offline stage, the second-order Taylor series expansion approximation of the bandpass oscillatory signal calculates the state-transition matrices and state vectors. Then, by applying the concept of Kalman filter, the state vectors are updated in the online calculation section much easier and faster compared to all-online approach. Compared to the existing methods, the proposed method generates a dynamic response with less settling time, more accuracy, and immunity against harmonic distortion, imbalances, and noise.

A High-Order Solver of EM Scattering From Uncertain Geometrical Targets

A high-order solution of electromagnetic (EM) scattering from electrically large targets with uncertain shapes is proposed in this letter. At first, the uncertain geometry can be described with several independent variables by using the nonuniform rational B-spline surfaces. Then, the second-order Taylor series expansion is used to present the geometrical uncertainty in terms of the combined field integral equation. As a result, higher accuracy can be obtained by using this high-order solver. Finally, the mean value and variance of the radar cross section (RCS) are derived and can be compared with the traditional Monte Carlo (MC) method. Numerical results show the proposed method can provide higher accuracy than the traditional one-order solver and higher efficiency than the traditional MC method.

A Multiscale Method for Coupled Steady-State Heat Conduction and Radiative Transfer Equations in Composite Materials

Abstract Predictions of coupled conduction-radiation heat transfer processes in periodic composite materials are important for applications of the materials in high-temperature environments. The homogenization method is widely used for the heat conduction equation, but the coupled radiative transfer equation is seldom studied. In this work, the homogenization method is extended to the coupled conduction-radiation heat transfer in composite materials with periodic microscopic structures, in which both the heat conduction equation and the radiative transfer equation are analyzed. Homogenized equations are obtained for the macroscopic heat transfer. Unit cell problems are also derived, which provide the effective coefficients for the homogenized equations and the local temperature and radiation corrections. A second-order asymptotic expansion of the temperature field and a first-order asymptotic expansion of the radiative intensity field are established. A multiscale numerical algorithm is proposed to simulate the coupled conduction-radiation heat transfer in composite materials. According to the numerical examples in this work, compared with the fully resolved simulations, the relative errors of the multiscale model are less than 13% for the temperature and less than 8% for the radiation. The computational time can be reduced from more than 300 h to less than 30 min. Therefore, the proposed multiscale method maintains the accuracy of the simulation and significantly improves the computational efficiency. It can provide both the average temperature and radiation fields for engineering applications and the local information in microstructures of composite materials.

Development of SD-HACNEM neutron noise simulator based on high order nodal expansion method for rectangular geometry

In this study, the SD-HACNEM (Sharif Dynamic - High order Average Current Nodal Expansion Method) neutron noise simulator in two energy groups using a second-order flux expansion method for two-dimensional rectangular X Y-geometry has been developed. In the first step, the calculations were performed for the steady state and results of ACNEM (Average Current Nodal Expansion Method) and HACNEM (High order Average Current Nodal Expansion Method) were examined and compared. To solve the problem, the power iteration algorithm has been used to calculate the distribution of neutron flux and neutron multiplication factor by considering the coarse-mesh (each fuel assembly one node). To validate the steady state calculations, the results were compared with the western nuclear reactor BIBLIS-2D (reference model). In the next step, dynamic calculations were performed in the frequency domain. For this purpose, after discretizing the neutron noise equations by the high-order average current nodal expansion method (NEM), a neutron noise source of absorber with variable strength was simulated. The results were verified by performing simulation at zero frequency and adjoint calculations. Then, the distribution of neutron noise for Inadvertent Loading and Operation of a Fuel Assembly in an Improper Position (ILOFAIP) is obtained. The numerical results showed that the use of HACNEM compared with ACNEM, without reducing the size of the meshes in neutron noise simulation, presented more accurate.

Superradiant lasing in inhomogeneously broadened ensembles with spatially varying coupling

Background: Theoretical studies of superradiant lasing on optical clock transitions predict a superb frequency accuracy and precision closely tied to the bare atomic linewidth. Such a superradiant laser is also robust against cavity fluctuations when the spectral width of the lasing mode is much larger than that of the atomic medium. Recent predictions suggest that this unique feature persists even for a hot and thus strongly broadened ensemble, provided the effective atom number is large enough. Methods: Here we use a second-order cumulant expansion approach to study the power, linewidth and lineshifts of such a superradiant laser as a function of the inhomogeneous width of the ensemble including variations of the spatial atom-field coupling within the resonator. Results: We present conditions on the atom numbers, the pump and coupling strengths required to reach the buildup of collective atomic coherence as well as scaling and limitations for the achievable laser linewidth. Conclusions: We show how sufficiently large numbers of atoms subject to strong optical pumping can induce synchronization of the atomic dipoles over a large bandwidth. This generates collective stimulated emission of light into the cavity mode leading to narrow-band laser emission at the average of the atomic frequency distribution. The linewidth is orders of magnitudes smaller than that of the cavity as well as the inhomogeneous gain broadening and exhibits reduced sensitivity to cavity frequency noise.

Perturbation expansions and error bounds for the truncated singular value decomposition

Truncated singular value decomposition is a reduced version of the singular value decomposition in which only a few largest singular values are retained. This paper presents a novel perturbation analysis for the truncated singular value decomposition for real matrices. First, we describe perturbation expansions for the singular value truncation of order r. We extend perturbation results for the singular subspace decomposition to derive the first-order perturbation expansion of the truncated operator about a matrix with rank greater than or equal to r. Observing that the first-order expansion can be greatly simplified when the matrix has exact rank r, we further show that the singular value truncation admits a simple second-order perturbation expansion about a rank-r matrix. Second, we introduce the first-known error bound on the linear approximation of the truncated singular value decomposition of a perturbed rank-r matrix. Our bound only depends on the least singular value of the unperturbed matrix and the norm of the perturbation matrix. Intriguingly, while the singular subspaces are known to be extremely sensitive to additive noises, the newly established error bound holds universally for perturbations with arbitrary magnitude. Finally, we demonstrate an application of our results to the analysis of the mean squared error associated with the TSVD-based matrix denoising solution.

Analog Quantum Simulation of Non-Condon Effects in Molecular Spectroscopy

In this work, we present a linear optical implementation for analog quantum simulation of molecular vibronic spectra, incorporating the non-Condon scattering operation with a quadratically small truncation error. Thus far, analog and digital quantum algorithms for achieving quantum speedup have been suggested only in the Condon regime, which refers to a transition dipole moment that is independent of nuclear coordinates. For analog quantum optical simulation beyond the Condon regime (i.e., non-Condon transitions) the resulting non-unitary scattering operations must be handled appropriately in a linear optical network. In this paper, we consider the first and second-order Herzberg-Teller expansions of the transition dipole moment operator for the non-Condon effect, for implementation on linear optical quantum hardware. We believe the method opens a new way to approximate arbitrary non-unitary operations in analog and digital quantum simulations. We report in-silico simulations of the vibronic spectra for naphthalene, phenanthrene, and benzene to support our findings.

Purely Sequential Point Estimation of a Function of the Mean in an Exponential Distribution

A purely sequential minimum risk point estimation methodology with an associated stopping time N is designed to come up with a unified strategy in an exponential distribution (Expo). We work under a weighted squared error loss due to the estimator $$g(\overline{X}_{N})$$ of $$g(\beta ),$$ a function of the unknown mean parameter $$\beta (>0)$$ , plus linear cost of sampling from an Expo $$(\beta )$$ population. A series of important first-order and second-order asymptotic (as c, the cost per unit sample, $$\rightarrow 0$$ ) results are laid out including (1) the first- and second-order efficiency properties, (2) the risk efficiency property, and (3) the second-order regret expansion under suitable conditions on g(.).

Mixed Precision Fermi-Operator Expansion on Tensor Cores from a Machine Learning Perspective.

We present a second-order recursive Fermi-operator expansion scheme using mixed precision floating point operations to perform electronic structure calculations using tensor core units. A performance of over 100 teraFLOPs is achieved for half-precision floating point operations on Nvidia's A100 tensor core units. The second-order recursive Fermi-operator scheme is formulated in terms of a generalized, differentiable deep neural network structure, which solves the quantum mechanical electronic structure problem. We demonstrate how this network can be accelerated by optimizing the weight and bias values to substantially reduce the number of layers required for convergence. We also show how this machine learning approach can be used to optimize the coefficients of the recursive Fermi-operator expansion to accurately represent the fractional occupation numbers of the electronic states at finite temperatures.

PRICING TIMER OPTIONS: SECOND-ORDER MULTISCALE STOCHASTIC VOLATILITY ASYMPTOTICS

AbstractWe study the pricing of timer options in a class of stochastic volatility models, where the volatility is driven by two diffusions—one fast mean-reverting and the other slowly varying. Employing singular and regular perturbation techniques, full second-order asymptotics of the option price are established. In addition, we investigate an implied volatility in terms of effective maturity for the timer options, and derive its second-order expansion based on our pricing asymptotics. A numerical experiment shows that the price approximation formula has a high level of accuracy, and the implied volatility in terms of its effective maturity is illustrated.

Systematic Convergence of the Numerical Taylor Series to the Best Standard and Its Potential Implication for the Development of Composite Methods

The numerical Taylor series is used as an alternative to systematically converge to a high-level calculation showing that it has the potential to be used in the development of composite methods. Five methods are tested, with two of them differing in the truncation of the series expansion and basis sets, namely, Taylor-Dunning and Taylor-Pople, and three of the interpretations of the G4, ccCA-TZ, and CBS-QB3, which are referred to as G4-Taylor, ccCA-TZ-Taylor, and CBS-QB3-Taylor, respectively. The Taylor-Dunning and Taylor-Pople methods restricted to second-order expansion show mean absolute errors of 1.16 and 1.11 kcal mol-1, respectively, for a training set involving enthalpies of formation, ionization energies, and electronic affinities. The G4-Taylor, ccCA-TZ-Taylor, and CBS-QB3-Taylor methods achieve the lowest mean absolute errors of 0.96, 1.54, and 0.58 kcal mol-1, respectively. The CBS-QB3-Taylor method is submitted to a validation step. The method using either forward or backward derivatives achieves a mean absolute error of 0.82 kcal mol-1. These results show that the numerical Taylor series and possibly other series expansions can be applied systematically for the development of accurate alternatives for composite methods.

Bulk viscosity in relativistic fluids: from thermodynamics to hydrodynamics

The approach of extended irreversible thermodynamics consists of promoting the dissipative fluxes to non-equilibrium thermodynamic variables. In a relativistic context, this naturally leads to the formulation of the theory of Israel and Stewart (1979), which is, to date, one of the most successful theories for relativistic dissipation. Although the generality of the principle makes it applicable to any dissipative fluid, a connection of the Israel–Stewart theory with microphysics has been established, through kinetic theory, only for the case of ideal quantum gases. By performing a convenient change of variables, we provide, for the case of bulk viscosity, an equivalent reformulation of the equations at the basis of extended irreversible thermodynamics. This approach maps any thermodynamic process which contributes to the bulk viscosity into a set of chemical reactions, whose reaction coordinates are abstract parameters describing the displacement from local thermodynamic equilibrium of the fluid element. We apply our new formalism to the case of the relativistic fluids, showing that the Israel–Stewart model for bulk viscosity is just the second-order expansion of a minimal model belonging to a larger class of non-perturbative theories for bulk viscosity which include the nuclear-reaction-mediated bulk viscosity of neutron star matter as a particular case. Furthermore, we show with concrete examples that our formalism provides new ways of computing the bulk viscosity directly and defines a simple prescription for constructing the Israel–Stewart model for a generic bulk-viscous fluid.

Stationary models of magnetized viscous tori around a Schwarzschild black hole

We present stationary solutions of magnetized, viscous thick accretion disks around a Schwarzschild black hole. We assume that the tori are not self-gravitating, are endowed with a toroidal magnetic field, and obey a constant angular momentum law. Our study focuses on the role of the black hole curvature in the shear viscosity tensor and in their potential combined effect on the stationary solutions. Those are built in the framework of a causality-preserving, second-order gradient expansion scheme of relativistic hydrodynamics in the Eckart frame description which gives rise to hyperbolic equations of motion. The stationary models are constructed by numerically solving the general relativistic momentum conservation equation using the method of characteristics. We place constraints in the range of validity of the second-order transport coefficients of the theory. Our results reveal that the effects of the shear viscosity and curvature are particularly noticeable only close to the cusp of the disks. The surfaces of constant pressure are affected by viscosity and curvature and the self-intersecting isocontour - the cusp - moves to smaller radii (i.e. towards the black hole horizon) as the effects become more significant. For highly magnetized disks the shift in the cusp location is smaller. Our findings might have implications on the dynamical stability of constant angular momentum tori which, in the inviscid case, are affected by the runaway instability.

A new Poincaré map for investigating the complex walking behavior of the compass-gait biped robot

The planar compass-gait biped robot is recognized by its simple passive morphological structure and by a complex dynamic walking system modeled by an impulsive hybrid nonlinear dynamics. Such complexity inhibits investigating the biped locomotive mechanism by means of the Poincaré map. Nevertheless, it is very difficult (and even impossible) to establish the exact closed form of the Poincaré mapping. This paper is concerned with the development of an improved closed-form analytical expression of the impact-to-impact Poincaré map for analyzing the complex walking behavior of the passive-dynamics biped walker and its stability. Our methodology consists in linearizing and then approximating the hybrid dynamics of the compass-gait biped robot around a predefined hybrid limit cycle. Based on the second-order Taylor series expansion, we design first an enhanced closed-form expression of the Poincaré map as well as an analytical expression for the computation of the step period of the bipedal locomotion. Furthermore, a simplification of these developed expressions is presented by decreasing the dimension. We provide also an expression of the Jacobian matrix for investigating the stability of the period-1 fixed point of the designed simplified Poincaré map. At the end of this work, we illustrate some simulation results in order to evaluate the validity of the new developed expression of the impact-to-impact Poincaré map in analyzing the stability and the complex walking behavior of passive-dynamics compass biped robot.

Decorrelated unbiased converted measurement for bistatic radar tracking

Bistatic radar target tracking is challenging due to the fact that the measurements are nonlinear functions of the Cartesian state. The converted measurement Kalman filter (CMKF) converts the raw measurement into Cartesian coordinates prior to tracking and is superior to the extended Kalman filter for certain problems. The challenges of CMKF are debiasing the converted measurement and approximating the converted measurement error covariance. Due to no closed form of biases, we utilize the second-order Taylor series expansion of the conventional measurement conversion to find the conversion bias in bistatic radar and propose the unbiased converted measurement (UCM). In order to decorrelate the converted measurement error covariance from the measurement noise, we evaluate the covariance using the prediction in Bayesian recursive filtering, designated as the decorrelated unbiased converted measurement (DUCM). Monte Carlo simulations show that the DUCM is unbiased and consistent, and the DUCM filter exhibits an improved performance compared with the conventional CMKF and the UCM filter in bistatic radar tracking.

Partial derivative with respect to the measure and its application to general controlled mean-field systems

Let (E,E) be an arbitrary measurable space. The paper first focuses on studying the partial derivative of a function f:P2,0(Rd×E)→R defined on the space of probability measures μ over (Rd×E,B(Rd)⊗E) whose first marginal μ1≔μ(⋅×E) has a finite second order moment. This partial derivative is taken with respect to q(dx,z), where μ has the disintegration μ(dxdz)=q(dx,z)μ2(dz) with respect to its second marginal μ2(⋅)=μ(Rd×⋅). Simplifying the language, we will speak of the derivative with respect to the law μ conditioned to its second marginal. Our results extend those of the derivative of a function g:P2(Rd)→R over the space of probability measures with finite second order moment by P.L. Lions (see Lions (2013)) but cover also as a particular case recent approaches considering E=Rk and supposing the differentiability of f over P2(Rd×Rk), in order to use the derivative ∂μf to define the partial derivative (∂μf)1. The second part of the paper focuses on investigating a stochastic maximum principle, where the controlled state process is driven by a general mean-field stochastic differential equation with partial information. The control set is just supposed to be a measurable space, and the coefficients of the controlled system, i.e., those of the dynamics as well as of the cost functional, depend on the controlled state process X, the control v, a partial information on X, as well as on the joint law of (X,v). Through considering a new second-order variational equation and the corresponding second-order adjoint equation, and a totally new method to prove the estimate for the solution of the first-order variational equation, the optimal principle is proved through spike variation of an optimal control and with the help of the tailor-made form of second-order expansion. We emphasize that in our assumptions we do not need any regularity of the coefficients neither in the control variable nor with respect to the law of the control process.

Nonlinear Interval Optimization of Asymmetric Damper Parameters for a Racing Car

Uncertainties in parameters can affect racing car performance. In this study, a nonlinear interval suspension damping optimization method is proposed to improve the road holding of a racing car. To evaluate the dynamic responses of racing cars under a random road input and a bump input with interval uncertain parameters, a quarter car model with a two-stage asymmetric damper is established. Then, a quadratic approximation model with second derivative terms is developed by second-order Taylor series expansion and dimension reduction to calculate the nonlinear dynamic response of the vehicle. Interval analysis of the objective function and constraints is carried out using interval arithmetic to eliminate nesting optimization and make the optimization efficient. The results show that the proposed optimization method can improve road holding performance, effectively suppress the fluctuation range of the road holding performance evaluation index, and ensure the robustness of the design scheme.

Target Adaptive Differential Iterative Reconstruction (TADI): A Robust Algorithm for Real-Time Electrical Impedance Tomography

Electrical impedance tomography (EIT) is a noninvasive functional diagnostic technique that has been successfully applied to human lung and brain. EIT reconstruction is an ill-conditioned problem: the regularization parameters establish a trade-off between acceptable data fitting and acceptable solution stability. This trade-off is necessary to select the optimal regularization parameters for each data frame to continuously obtain high-quality images in real-time monitoring. However, using the objective regularization parameter selection method before each image reconstruction may reduce the reconstruction efficiency, and using heuristic selection for all images may cause significant quality degradation, thereby creating challenges in long-term clinical EIT monitoring. This study explores a robust EIT target adaptive differential iterative reconstruction algorithm that does not require the advance selection of the optimal regularization parameter to facilitate the clinical application of EIT long-term bedside monitoring. Specifically, second-order Taylor approximation expansion and Euler–Lagrange theorem were used to derive the conductivity differential iteration relationship. Furthermore, the reconstruction quality and generalization ability of the proposed algorithm were validated based on comparisons with the L-curve and the generalized cross-validation parameter selection methods through simulations, phantom experiments, and human lung recruitment data. Compared with the L-curve and generalized cross-validation methods, the TADI algorithm reduces the position error by 33.07% and 47.17%, the ringing by 26.49% and 32.69%, and the shape deformation by 18.34% and 19.40% on average, respectively. The results revealed that the TADI algorithm could achieve a significant improvement over the existing state-of-the-art methods in terms of stability, consistency, and efficiency.

Hermite Expansion Model and LMMSE Analysis for Low-Resolution Quantized MIMO Detection

In this paper, the Hermite polynomials are employed to study linear approximation models of narrowband multiantenna signal reception (i.e., MIMO) with low-resolution quantizations. This study results in a novel linear approximation using the second-order Hermite expansion (SOHE). The SOHE model is not based on those assumptions often used in existing linear approximations. Instead, the quantization distortion is characterized by the second-order Hermite kernel, and the signal term is characterized by the first-order Hermite kernel. It is shown that the SOHE model can explain almost all phenomena and characteristics observed so far in the low-resolution MIMO signal reception. When the SOHE model is employed to analyze the linear minimum-mean-square-error (LMMSE) channel equalizer, it is revealed that the current LMMSE algorithm can be enhanced by incorporating a symbol-level normalization mechanism. The performance of the enhanced LMMSE algorithm is demonstrated through computer simulations for narrowband MIMO systems in Rayleigh fading channels.