Second-order Expansion Research Articles

Numerical simulation of the geometrical-optics reduction of CE2 and comparisons to quasilinear dynamics

Zonal flows have been observed to appear spontaneously from turbulence in a number of physical settings. A complete theory for their behavior is still lacking. Recently, a number of studies have investigated the dynamics of zonal flows using quasilinear (QL) theories and the statistical framework of a second-order cumulant expansion (CE2). A geometrical-optics (GO) reduction of CE2, derived under an assumption of separation of scales between the fluctuations and the zonal flow, is studied here numerically. The reduced model, CE2-GO, has a similar phase-space mathematical structure to the traditional wave-kinetic equation, but that wave-kinetic equation has been shown to fail to preserve enstrophy conservation and to exhibit an ultraviolet catastrophe. CE2-GO, in contrast, preserves nonlinear conservation of both energy and enstrophy. We show here how to retain these conservation properties in a pseudospectral simulation of CE2-GO. We then present nonlinear simulations of CE2-GO and compare with direct simulations of quasilinear (QL) dynamics. We find that CE2-GO retains some similarities to QL. The partitioning of energy that resides in the zonal flow is in good quantitative agreement between CE2-GO and QL. On the other hand, the length scale of the zonal flow does not follow the same qualitative trend in the two models. Overall, these simulations indicate that CE2-GO provides a simpler and more tractable statistical paradigm than CE2, but CE2-GO is missing important physics.

Simple Modifications of the SCAN Meta-Generalized Gradient Approximation Functional.

We analyzed various possibilities to improve upon the SCAN meta-generalized gradient approximation density functional obeying all known properties of the exact functional that can be satisfied at this level of approximation. We examined the necessity of locally satisfying a strongly tightened lower bound for the exchange energy density in single-orbital regions, the nature of the error cancellation between the exchange and correlation parts in two-electron regions, and the effect of the fourth-order term in the gradient expansion of the correlation energy density. We have concluded that the functional can be modified to separately reproduce the exchange and correlation energies of the helium atom by locally releasing the strongly tightened lower bound for the exchange energy density in single-orbital regions, but this leads to an unbalanced improvement in the single-orbital electron densities. Therefore, we decided to keep the FX ≤ 1.174 exact condition for any single-orbital density, where FX is the exchange enhancement factor. However, we observed a general improvement in the single-orbital electron densities by revising the correlation functional form to follow the second-order gradient expansion in a wider range. Our new revSCAN functional provides more-accurate atomization energies for the systems with multireference character, compared to the SCAN functional. The nonlocal VV10 dispersion-corrected revSCAN functional yields more-accurate noncovalent interaction energies than the VV10-corrected SCAN functional. Furthermore, its global hybrid version with 25% of exact exchange, called revSCAN0, generally performs better than the similar SCAN0 for reaction barrier heights. Here, we also analyzed the possibility of the construction of a local hybrid from the SCAN exchange and a specific locally bounded nonconventional exact exchange energy density. We predict compatibility problems since this nonconventional exact exchange energy density does not really obey the strongly tightened lower bound for the exchange energy density in single-orbital regions.

Generalized Darboux transformation and parameter-dependent rogue wave solutions to a nonlinear Schrödinger system

In this work, we are concerned with a generalized nonlinear Schrodinger equation, which can describe the propagation of nonlinear wave phenomena in plasma, optics and in deep water field. We construct, up to the second-order expansion, rogue wave solutions and give general formula to obtain higher-order ones to this system. We investigate the effects of the higher-order nonlinearity on the rogue waves dynamics. We make use of the generalized Darboux transformation, based on the Darboux matrix method.

Derivation and Application of a New Transmission Loss Formula for Power System Economic Dispatch

The expression and calculation of transmission loss (TL) play key roles for solving the power system economic dispatch (ED) problem. ED including TL must compute the total TL and incremental transmission loss (ITL) by executing power flow equations. However, solving the power flow equations is time-consuming and may result in divergence by the iteration procedure. This approach is unsuitable for real-time ED in practical power systems. To avoid solving nonlinear power flow equations, most power companies continue to adopt the TL formula in ED. Traditional loss formulas are composed of network parameters and in terms of the generator’s real power outputs. These formulas are derived by several assumptions, but these basic assumptions sacrifice accuracy. In this study, a new expression for the loss formula is proposed to improve the shortcomings of traditional loss formulas. The coefficients in the new loss formula can be obtained by recording the power losses according to varying real and reactive power outputs without any assumptions. The simultaneous equations of the second-order expansion of the Taylor series are then established. Finally, the corresponding coefficients can be calculated by solving the simultaneous equations. These new coefficients can be used in optimal real and reactive power dispatch problems. The proposed approach is tested by IEEE 14-bus and 30-bus systems, and the results are compared with those obtained from the traditional B coefficient method and the load flow method. The numerical results show that the proposed new loss formula for ED can hold high accuracy for different loading conditions and is very suitable for real-time applications.

Second-order Asymptotics on Distributions of Maxima of Bivariate Elliptical Arrays

Let $\{ (\xi_{ni}, \eta_{ni}), 1\leq i \leq n, n\geq 1 \}$ be a triangular array of independent bivariate elliptical random vectors with the same distribution function as $(S_{1}, \rho_{n}S_{1}+\sqrt{1-\rho_{n}^2}S_{2})$, $\rho_{n}\in (0,1)$, where $(S_{1},S_{2})$ is a bivariate spherical random vector. For the distribution function of radius $\sqrt{S_{1}^2+S_{2}^2}$ belonging to the max-domain of attraction of the Weibull distribution, Hashorva (2006) derived the limiting distribution of maximum of this triangular array if convergence rate of $\rho_{n}$ to $1$ is given. In this paper, under the refinement of the rate of convergence of $\rho_{n}$ to $1$ and the second-order regular variation of the distributional tail of radius, precise second-order distributional expansions of the normalized maxima of bivariate elliptical triangular arrays are established.

Hole Hopping Rates in Organic Semiconductors: A Second-Order Cumulant Approach.

Second-order cumulant expansion of the time dependent reduced density matrix has been employed to evaluate hole hopping rates in pentacene, tetracene, picene, and rubrene homodimers. The cumulant expansion is a full quantum mechanical approach, which enables the use of the whole set of nuclear coordinates in computations and the inclusion of both the effects of the equilibrium position displacements and of normal mode mixing upon hole transfer. The time dependent populations predicted by cumulant approach are in good agreement with those obtained by numerical solution of time dependent Schrödinger equation, even for ultrafast processes, where the Fermi Golden Rule fails.

Compartmental Modeling of Opportunistic Signals for Energy Efficient Optimal Clustering in WSN

This letter proposes the compartmental model-based cluster size optimization using opportunistic signals in wireless sensor networks. The opportunistic signals, namely, Wi-Fi, acoustic, and/or visible light, can be utilized depending upon good availability of the signals in the given area of interest. The compartmental model, which is an attenuation model, describes the variation of opportunistic signal power with propagation distance. In order to minimize the energy consumption, the optimal number of clusters is computed for different orders of the Taylor series expansion of the compartmental model. We illustrate that the second-order Taylor series expansion approximates well enough the exact performance of the compartmental model. A theoretical analysis of the compartmental model performance has been conducted with parameters derived from experimental measurements.

Intersystem crossing rate dependent dual emission and phosphorescence from cyclometalated platinum complexes: a second order cumulant expansion based approach.

Rates of intersystem crossing (kISC) of two platinum(ii) complexes containing acetylacetonate (acac) and extended cyclometalated ppy (Hppy = 2-phenylpyridine) (1) and thpy (Hthpy = 2-(2' thienyl)pyridine) (2) ligands are calculated using the Condon approximation to the Golden Rule and employing the second-order cumulant expansion method. The emission wavelengths obtained at the RI-CC2 level for the lowest excited singlet (S1) and triplet (T1) states of the two complexes are well in agreement with the experimental results. Our analysis based on kISC evinces that the major pathway involved with the phosphorescence process in complex 1 arises from the S1 → T2 intersystem crossing while the S1 → T1 intersystem crossing is the key step towards the commencement of dual emission in complex 2. Furthermore, it is found that the different pathways are mostly guided by two factors namely, the energy gap and the spin-orbit interaction between the concerned states. Interestingly, the calculated kISC for complex 1 is found to be 107 times larger than that of complex 2, which suggests a rapid depletion of the S1 state population vis-à-vis radiative emission only by phosphorescence from the internally converted lowest excited triplet state while for complex 2, the relatively lower kISC is attributed to the dual emission from this complex.

Discrete-Time Statistical Inference for Multiscale Diffusions

We study statistical inference for small-noise-perturbed multiscale dynamical systems under the assumption that we observe a single time series from the slow process only. We construct estimators for both averaging and homogenization regimes, based on an appropriate misspecified model motivated by a second-order stochastic Taylor expansion of the slow process with respect to a function of the time-scale separation parameter. In the case of a fixed number of observations, we establish consistency, asymptotic normality, and asymptotic statistical efficiency of a minimum contrast estimator (MCE), the limiting variance having been identified explicitly; we furthermore establish consistency and asymptotic normality of a simplified MCE, which is, however, not, in general, efficient. These results are then extended to the case of high-frequency observations under a condition restricting the rate at which the number of observations may grow vis-a-vis the separation of scales. Numerical simulations illustrate the ...

An interval Taylor-based method for transient stability assessment of power systems with uncertainties

Transient stability assessment needs to be further studied when confronted with uncertainties brought in by the integration of renewable energy and measurement errors. This paper outlines a novel method based on interval Taylor expansion incorporated optimal solution technique to solve the uncertainty model for transient stability assessment. Uncertainty assessment framework for power systems transient stability is described as ordinary differential equations with interval variables, which can be equivalently transformed into three sets of ordinary differential equations with only deterministic parameters using the interval Taylor method with second-order expansion. In order to further reduce the overestimation, a quadratic programming model is constructed to estimate upper and lower boundaries of state variables. Application of the proposed method is implemented in SIMB system and IEEE-30 system and the results are demonstrated in details. Comparisons with affine arithmetic (AA)-based method and Monte Carlo method verify the effectiveness and better performance of the proposed method.

An expansion in the model space in the context of utility maximization

In the framework of an incomplete financial market where the stock price dynamics are modeled by a continuous semimartingale (not necessarily Markovian), an explicit second-order expansion formula for the power investor’s value function—seen as a function of the underlying market price of risk process—is provided. This allows us to provide first-order approximations of the optimal primal and dual controls. Two specific calibrated numerical examples illustrating the accuracy of the method are also given.

Reducing Approximation Error in the Fourier Flexible Functional Form

The Fourier Flexible form provides a global approximation to an unknown data generating process. In terms of limiting function specification error, this form is preferable to functional forms based on second-order Taylor series expansions. The Fourier Flexible form is a truncated Fourier series expansion appended to a second-order expansion in logarithms. By replacing the logarithmic expansion with a Box-Cox transformation, we show that the Fourier Flexible form can reduce approximation error by 25% on average in the tails of the data distribution. The new functional form allows for nested testing of a larger set of commonly implemented functional forms.

Numerical study of submicroparticle acoustophoresis using higher-order modes in a rectangular microchannel

Manipulation of submicrometer particles in Lab-on-a-Chip systems using acoustophoresis is challenging due to the effect of acoustic streaming. We numerically study the transition from radiation force dominated to streaming-induced drag force dominated acoustophoresis using the fundamental and higher-order resonances of a water-filled rectangular microchannel. We consider the cases of single mode excitation and simultaneous double mode excitation. The acoustic fields at resonance are calculated using a second-order perturbation expansion of the thermoviscous acoustic problem. We show that the acoustophoretic forces using simultaneous mode excitation can be obtained from a linear combination of the single mode forces. We find that the critical size of suspended particles at the transition scales inversely with the square root of the resonance frequency. Particle tracing shows radiation-dominated concentration of 800 nm diameter polystyrene particles using the fifth-order resonance at 9.8 MHz. For smaller particles we find a streaming-assisted concentration regime where particles are concentrated into the streaming regions close to the walls. In case of double mode excitation, the particle concentrations increase a factor 4 to 18 times for 200 nm to 800 nm particles respectively. We include the numerical model, consisting of a COMSOL implementation and MATLAB control script, as supplemental material.

Second-Order Analytical Uncertainty Analysis in Life Cycle Assessment

Life cycle assessment (LCA) results are inevitably subject to uncertainties. Since the complete elimination of uncertainties is impossible, LCA results should be complemented by an uncertainty analysis. However, the approaches currently used for uncertainty analysis have some shortcomings: statistical uncertainty analysis via Monte Carlo simulations are inherently uncertain due to their statistical nature and can become computationally inefficient for large systems; analytical approaches use a linear approximation to the uncertainty by a first-order Taylor series expansion and thus, they are only precise for small input uncertainties. In this article, we refine the analytical uncertainty analysis by a more precise, second-order Taylor series expansion. The presented approach considers uncertainties from process data, allocation, and characterization factors. We illustrate the refined approach for hydrogen production from methane-cracking. The production system contains a recycling loop leading to nonlinearities. By varying the strength of the loop, we analyze the precision of the first- and second-order analytical uncertainty approaches by comparing analytical variances to variances from statistical Monte Carlo simulations. For the case without loops, the second-order approach is practically exact. In all cases, the second-order Taylor series approach is more precise than the first-order approach, in particular for large uncertainties and for production systems with nonlinearities, for example, from loops. For analytical uncertainty analysis, we recommend using the second-order approach since it is more precise and still computationally cheap.

On fundamentally-based and thermodynamically-consistent description of dilute multicomponent solutions: Formal results, microscopic interpretations, and modeling implications

We propose a thermodynamically consistent truncated composition expansion, around the infinite dilute quaternary reference system, to describe the thermodynamic behavior of dilute mixed solutes dissolved in mixed-solvents. The development involves the thermodynamically necessary second-order composition expansion of the species partial molar quantities, whose expansion coefficients encompass true molecular meaning based on their underlying links with the solvation properties of the corresponding infinite dilute reference system. Thus, we provide a rigorous molecular-based interpretation of the expansion coefficients in terms of precisely defined microscopic quantities that describe the mixed-solvent environment around all species in solution. Consequently, the proposed second-order truncated composition perturbation expansion applies to any type of quaternary system regardless of its aggregation state, in particular, it converges to the rigorous expressions for quaternary mixtures of imperfect gases obeying the truncated virial equation z=1+BP/kT, and it reduces naturally to its second-order truncated ternary and binary counterparts through the identification, and consequent manipulation, of the molecular asymmetries among species giving rise to the system nonidealities. Finally, we highlight the origin and key implications of the thermodynamic inconsistencies encountered in currently used first-order truncated expansions.

Optimal Bandwidth for Multitaper Spectrum Estimation

A systematic method for bandwidth parameter selection is desired for Thomson multitaper spectrum estimation. We give a method for determining the optimal bandwidth based on a mean squared error (MSE) criterion. When the true spectrum has a second-order Taylor series expansion, one can express quadratic local bias as a function of the curvature of the spectrum, that can be estimated by using a simple spline approximation. This is combined with a variance estimate, obtained by jackknifing over individual spectrum estimates, to produce an estimated MSE for the log spectrum estimate for each choice of time-bandwidth product. The bandwidth that minimizes the estimated MSE then gives the desired spectrum estimate. Additionally, the bandwidth obtained by using our method is also optimal for cepstrum estimates. We give an example of a damped oscillatory (Lorentzian) process in which the approximate optimal bandwidth can be written as a function of the damping parameter. The true optimal bandwidth agrees well with that given by minimizing estimated the MSE in these examples.

A new direct second-order reliability analysis method

• A new study of the second-order reliability method (SORM) is presented. • All the information from the fitted quadratic polynomial model of the limit state function is used. • The cumulant generating function of the fitted quadratic polynomial surface in normalized standard variables is derived analytically. • The saddlepoint approximation is utilized for reliability analysis. • The proposed method is more accurate than the conventional SORM. In the second-order reliability method, the limit state function in arbitrarily distributed random variables is approximated by a quadratic polynomial of standard normal variables. The fitted quadratic polynomial is then used to calculate the probability of failure of the limit state. However, a closed-form solution for the probability of failure of a general quadratic polynomial surface is not available. As such, in this paper, a new second-order reliability method for reliability analysis is presented using saddlepoint approximation. The second-order approximation of the limit state function is obtained by the second-order Taylor series expansion at the most probable point. The cumulant generating function of the fitted quadratic polynomial of standard normal variables is derived analytically. The saddlepoint approximation is utilized to generate the probability density function, cumulative distribution function and probability of failure of the limit state. Finally, three numerical examples are used to compare the performance of the proposed second-order reliability with that of the first-order reliability method, the conventional second-order reliability method, and Monte Carlo simulation. The comparisons show that the proposed second-order reliability method gives accurate, convergent, and computationally efficient estimates of the probability of failure.

Simulation-Based Determination of Hydrodynamic Derivatives and 6DOF Motion Analysis for Underwater Vehicle

This paper introduces a simulation-based determination method for hydrodynamic derivatives and 6DOF (degrees-of-freedom) motion analysis for an underwater vehicle. Hydrodynamic derivatives were derived from second-order modulus expansion and composed of the added mass, and linear and nonlinear damping coefficients. The added mass coefficients were analytically obtained using the potential theory. All of the linear and nonlinear damping coefficients were determined using CFD simulation, which were performed for various cases based on the actual operating condition. Then, the linear and nonlinear damping coefficients were determined by fitting the CFD results, which referred to 6DOF forces and moments acting on an underwater vehicle, with the least square method. To demonstrate the applicability of the current study, 6DOF simulations for three different scenarios (L-, U-, and S-turn) were carried out, and the results were validated on the basis of physical plausibility.

Analytical and numerical studies on the second-order asymptotic expansion method for European option pricing under two-factor stochastic volatilities

ABSTRACTThe celebrated Black–Scholes model made the assumption of constant volatility but empirical studies on implied volatility and asset dynamics motivated the use of stochastic volatilities. Christoffersen in 2009 showed that multi-factor stochastic volatilities models capture the asset dynamics more realistically. Fouque in 2012 used it to price European options. In 2013, Chiarella and Ziveyi considered Christoffersen’s ideas and introduced an asset dynamics where the two volatilities of the Heston type act separately and independently on the asset price, and using Fourier transform for the asset price process and double Laplace transform for the two volatilities processes, solved a pricing problem for American options. This paper considers the Chiarella and Ziveyi model and parameterizes it so that the volatilities revert to the long-run-mean with reversion rates that mimic fast (for example daily) and slow (for example seasonal) random effects. Applying asymptotic expansion method presented by Fouque in 2012, we make an extensive and detailed derivation of the approximation prices for European options. We also present numerical studies on the behavior and accuracy of our first- and second-order asymptotic expansion formulas.

Mutual gravitational potential, force, and torque of a homogeneous polyhedron and an extended body: an application to binary asteroids

Binary systems are quite common within the populations of near-Earth asteroids, main-belt asteroids, and Kuiper belt asteroids. The dynamics of binary systems, which can be modeled as the full two-body problem, is a fundamental problem for their evolution and the design of relevant space missions. This paper proposes a new shape-based model for the mutual gravitational potential of binary asteroids, differing from prior approaches such as inertia integrals, spherical harmonics, or symmetric trace-free tensors. One asteroid is modeled as a homogeneous polyhedron, while the other is modeled as an extended rigid body with arbitrary mass distribution. Since the potential of the polyhedron is precisely described in a closed form, the mutual gravitational potential can be formulated as a volume integral over the extended body. By using Taylor expansion, the mutual potential is then derived in terms of inertia integrals of the extended body, derivatives of the polyhedron’s potential, and the relative location and orientation between the two bodies. The gravitational forces and torques acting on the two bodies described in the body-fixed frame of the polyhedron are derived in the form of a second-order expansion. The gravitational model is then used to simulate the evolution of the binary asteroid (66391) 1999 KW4, and compared with previous results in the literature.