Second-order Expansion Research Articles

Asymptotic expansion of transmission eigenvalues for anisotropic thin layers

We study the asymptotic expansion of transmission eigenvalues for anisotropic thin layers. We establish a rigorous second-order expansion for simple transmission eigenvalues with respect to the thickness of the layer. The convergence analysis is based on a generalization of Osborn's Theorem to non-linear eigenvalue problems by Moskow [Nonlinear eigenvalue approximation for compact operators. J Math Phys. 2015;56(11):113512]. We also provide formal derivation in more general cases validating the obtained theoretical result.

Predicting reaction barriers in solid-state systems under stress via second-order energy expansion

Solid-state transformations are important in many areas of science and technology. Herein, a model for predicting the relative energies of stationary points along the reaction pathways for stress-induced solid-state transformations is assessed and applied. The model is based on a second-order expansion of the energy of the system with respect to changes in the unit cell, and requires a small number of parameters that can be obtained through quantum chemical calculations. Comparison of the model with the results of quantum chemical calculations indicates that the model accurately reproduces changes in energy occurring during stress-induced transformations over a reasonable range of stresses. A procedure for applying the model to identify stress tensors that are most likely to promote a desired reaction is illustrated. The results also indicate that this procedure provides insight into the connection between the form of a stress tensor and the changes in energy occurring during a stress-induced solid-state transformation.

The bacco simulation project: bacco hybrid Lagrangian bias expansion model in redshift space

ABSTRACT We present an emulator that accurately predicts the power spectrum of galaxies in redshift space as a function of cosmological parameters. Our emulator is based on a second-order Lagrangian bias expansion that is displaced to Eulerian space using cosmological N-body simulations. Redshift space distortions are then imprinted using the non-linear velocity field of simulated particles and haloes. We build the emulator using a forward neural network trained with the simulations of the BACCO project, which covers an eight-dimensional parameter space including massive neutrinos and dynamical dark energy. We show that our emulator provides unbiased cosmological constraints from the monopole, quadrupole, and hexadecapole of a mock galaxy catalogue that mimics the BOSS-CMASS sample down to non-linear scales ($k\sim 0.6{h\, {\rm Mpc}^{-1}}$). This work opens up the possibility of robustly extracting cosmological information from small scales using observations of the large-scale structure of the universe.

Toward more accurate adiabatic connection approach for multireference wavefunctions.

A multiconfigurational adiabatic connection (AC) formalism is an attractive approach to compute the dynamic correlation within the complete active space self-consistent field and density matrix renormalization group (DMRG) models. Practical realizations of AC have been based on two approximations: (i) fixing one- and two-electron reduced density matrices (1- and 2-RDMs) at the zero-coupling constant limit and (ii) extended random phase approximation (ERPA). This work investigates the effect of removing the "fixed-RDM" approximation in AC. The analysis is carried out for two electronic Hamiltonian partitionings: the group product function- and the Dyall Hamiltonians. Exact reference AC integrands are generated from the DMRG full configuration interaction solver. Two AC models are investigated, employing either exact 1- and 2-RDMs or their second-order expansions in the coupling constant in the ERPA equations. Calculations for model molecules indicate that lifting the fixed-RDM approximation is a viable way toward improving the accuracy of existing AC approximations.

The halo bias for number counts on the light cone from relativistic N-body simulations

We present the halo number counts and its two-point statistics, the observable angular power spectrum, extracted for the first time from relativistic N-body simulations. The halo catalogues used in this work are built from the relativistic N-body code gevolution, and the observed redshift and angular positions of the sources are computed using a non-perturbative ray-tracing method, which includes all relativistic scalar contributions to the number counts. We investigate the validity and limitations of the linear bias prescription to describe our simulated power spectra. In particular, we assess the consistency of different bias measurements on large scales, and we estimate up to which scales a linear bias is accurate in modelling the data, within the statistical errors. We then test a second-order perturbative bias expansion for the angular statistics, on a range of redshifts and scales previously unexplored in this context, that is 0.4 ≤ z̅ ≤ 2 up to scales ℓ max ∼ 1000. We find that the angular power spectra at equal redshift can be modelled with high accuracy with a minimal extension of the number of bias parameters, that is using a two-parameter model comprising linear bias and tidal bias. We show that this model performs significantly better than a model without tidal bias but with quadratic bias as extra degree of freedom, and that the latter is inaccurate at z̅ ≥ 0.7. Finally, we extract from our simulations the cross-correlation of halo number counts and lensing convergence. We show that the estimate of the linear bias from this cross-correlation is consistent with the measurements based on the clustering statistics alone, and that it is crucial to take into account the effect of magnification in the halo number counts to avoid systematic shifts in the computed bias.

Heat transfer analysis for tissue with surface heat flux based on the non-linearized form of the three-phase-lag model

The three-phase-lag model of heat conduction has been proposed for considering thermoelastic effect in medium. The bioheat transfer equations based on Taylor's series approximation of the three-phase-lag model were derived in conjunction with a modified energy conservation equation. For exploring the effect of non-linear expansion in the phase lag times, the Taylor's series of second-order expansion was applied. The resulting equation involves mixed derivative terms and higher-order derivatives of temperature with respect to time. The hybrid application of the Laplace transform method and a modified discretization technique was extended to solve the equations and explore the effect of thermoelasticity on the thermal behavior in living tissue with surface heat flux. The influence of thermoelastic parameters and phase lags on heat transfer in tissue has been investigated. The present results illustrate the thermal response oscillation is excited in medium for the thermoelastic effect, the phase lag times significantly affect the amplitude and frequency of the oscillation, and the expansion order of TPL model evidently affects the predicted temperature.

Three Solution Approaches to Stochastic Multi-Period AC Optimal Power Flow in Active Distribution Systems

This paper proposes three novel solution approaches (A1, A2, A3) to solve stochastic multi-period AC optimal power flow (S-MP-OPF) for day-ahead flexibility procurement from distributed energy resources (DER) in active distribution systems (ADSs). The S-MP-OPF is a mixed-integer nonlinear programming (MINLP) problem due to the binary variables modeling the operation of storage devices and flexible loads. The proposed three approaches have a shared common first step, which resorts to a new mixed-integer linear programming (MILP) model approximation of the S-MP-OPF problem. The MILP model employs second-order Taylor series expansion of trigonometric terms and formulates the linear approximations relying on variables such as square of voltage magnitude and voltage angle difference. This first step serves also the purpose of fixing the binary variables to the values computed by the MILP problem. Then, the approach A1 only checks the AC feasibility of MILP solution while approaches A2 and A3 further optimize continuous variables. Specifically, the sophisticated heuristic approach A2 employs sequential linear programming and AC power flow while the approach A3 models and solves directly the remaining nonlinear programming (NLP) problem. The performances of these approaches, a benchmark MINLP solver, and a state-of-the-art method are thoroughly compared in three radial or weakly meshed ADSs of 34, 31, and 191 nodes, respectively. The numerical results indicate that, albeit the approach A2 performs best overall, these approaches present distinct accuracy vs speed trade-offs, which make them suitable to problems of different sizes and diverse accuracy or speed requirements.

Complete Positivity Violation in Higher-order Quantum Adiabatic Elimination

When a composite Lindblad system consists of weakly coupled sub-systems with fast and slow timescales, the description of slow dynamics can be simplified by discarding fast degrees of freedom. This model reduction technique is called adiabatic elimination. While second-order perturbative expansion with respect to the timescale separation has revealed that the evolution of a reduced state is completely positive, this paper presents an example exhibiting complete positivity violation in the fourth-order expansion. Despite the non-uniqueness of slow dynamics parametrization, we prove that complete positivity cannot be ensured in any parametrization. The violation stems from correlation in the initial state.

New second-order optimality conditions in sub-riemannian geometry

We study the geometry of the second-order expansion of the extended end-point map for the sub-Riemannian geodesic problem. Translating the geometric reality into equations we derive new second-order necessary optimality conditions in sub-Riemannian Geometry. In particular, we find an ODE for velocity of an abnormal sub-Riemannian geodesics. It allows to divide abnormal minimizers into two classes, which we propose to call 2-normal and 2-abnormal extremals. In the 2-normal case the above ODE completely determines the velocity of a curve, while in the 2-abnormal case the velocity is undetermined at some, or at all points. With some enhancement of the presented results it should be possible to prove the regularity of all 2-normal extremals (the 2-abnormal case seems to require study of higher-order conditions) thus making a step towards solving the problem of smoothness of sub-Riemannian abnormal geodesics. As a by-product we present a new derivation of Goh conditions. We also prove that the assumptions weaker than these used in [Boarotto, Monti, Palmurella, 2020] to derive third-order Goh conditions, imply piece-wise-C2 regularity of an abnormal extremal. Marty me i Zuzi

High-order asymptotic solutions for gas transport in heterogeneous media with multiple spatial scales

Subsurface structures generally exhibit strong heterogeneities at multiple spatial scales. In this study, two- and three-scale high-order models are developed to effectively predict nonlinear gas transport in heterogeneous porous media with multiscale configurations. The considered heterogeneous porous media are composed of the matrix and inclusions at the mesoscopic and microscopic scales, respectively. First, using the formal two-scale asymptotic analysis, the homogenized solutions, the two-scale first-order and higher-order solutions are derived, with the first-order and second-order cell functions defined at the mesoscopic cell. Second, by further expanding all the mesoscopic cell functions to the microscopic levels, the second-order expansions of the mesoscopic cell functions are established and the upscaled relationships for the permeability tensor from the microscale to the macroscale are developed accordingly. Finally, the three-scale low-order and high-order solutions are constructed by combining the multiscale expansions of the mesoscopic cell functions and the macro–meso two-scale solutions. Several representative cases are simulated to demonstrate the accuracy and reliability of the proposed multiscale solutions. The results show that the high-order solutions can perfectly capture the locally steep pressure fluctuations and non-equilibrium effects caused by the heterogeneities and large permeability contrast in porous media with two- or three-scale configurations. The strategies to obtain the multiscale high-order solutions follow the reverse thought process of the reiteration homogenization method, and can be easily extended to heterogeneous porous media with arbitrary multiple scales.

A novel model order reduction-based two-stage damage detection paradigm for trusses using time–history acceleration

This paper presents a novel model order reduction (MOR)-based two-stage damage detection method for trusses employing time-series acceleration measured by limited sensors. In the first step, an acceleration-based strain energy indicator (ASEI) is newly proposed to evaluate the most doubtfully damaged candidates. Accordingly, the number of design variables defined in an inverse optimization problem of the second phase is reduced significantly. The location and severity of damaged members are then determined by minimizing an objective function with a newly added dynamic penalty parameter to achieve a faster convergence speed and better optimal solutions. Since acceleration signals are incompletely measured at limited sensors, the second-order Neumann series expansion (SNSE) is firstly employed to condense the proportionally damped trusses for inferring the unmeasured time–history information. An adaptive hybrid evolutionary firefly algorithm (AHEFA) is utilized to resolve the above inverse optimization problem. Four numerical examples of 2D and 3D trusses with various damage scenarios including noises are examined to demonstrate the reliability of the proposed methodology. Attained outcomes indicate that the suggested paradigm can reliably diagnose both multidamage sites and extents of trusses with only relatively short time histories and a few measurement sensors.

Second-order asymptotic expansion and thermodynamic interpretation of a fast–slow Hamiltonian system

This article includes a short survey of selected averaging and dimension reduction techniques for deterministic fast–slow systems. This survey includes, among others, classical techniques, such as the WKB approximation or the averaging method, as well as modern techniques, such as the GENERIC formalism. The main part of this article combines ideas of some of these techniques and addresses the problem of deriving a reduced system for the slow degrees of freedom (DOF) of a fast–slow Hamiltonian system. In the first part, we derive an asymptotic expansion of the averaged evolution of the fast–slow system up to second order, using weak convergence techniques and two-scale convergence. In the second part, we determine quantities which can be interpreted as temperature and entropy of the system and expand these quantities up to second order, using results from the first part. The results give new insights into the thermodynamic interpretation of the fast–slow system at different scales.

A privacy-preserving decentralized credit scoring method based on multi-party information

With society's wide-scale adoption of information technology, significant information about borrowers is distributed across various parties, information that can be jointly used to improve credit scoring. However, use of such information faces many challenges, such as the problems of preserving privacy and information redundancy. To address these challenges in leveraging multi-source information for credit scoring, we propose a decentralized multi-party method based on logistic regression. Specifically, we formulate a logistic regression model using the vertical federated learning paradigm. To preserve data privacy during multi-party collaborative model training, we use additively homomorphic encryption based on the second-order Taylor series expansion of the loss function and its gradient. To address information redundancy and to improve the performance of the credit scoring model, we use the proposed HE-DPGD algorithm to estimate the distributed coefficients in a privacy-preserving setting. Empirical evaluation shows that the proposed method can leverage information from multiple parties securely and effectively.

A study on Bayesian method for reliability evaluation of small sample size aviation support systems

In order to realize the reliability evaluation of small sample size aviation support system with known prior information, the Bayes methods of system reliability evaluation are given by using the second-order moment equivalent method, relative least squares method (RLS method), maximum entropy method (ME method) and Chebyshev polynomial method respectively, and the accuracies of the above Bayes synthesis methods are compared and analysed through an example. The results show that the ME method and RLS method have higher evaluation accuracy than the second-order moment equivalent method and Chebyshev expansion method. Finally, ME method and RLS method are applied to the reliability evaluation of aviation support system, and good evaluation results are obtained. At the same time, it provides a valuable reference for the reliability evaluation of complex system with small sample size in engineering practice.

Generalised NURBS interpolator with nonlinear feedrate scheduling and interpolation error compensation

Current nonuniform rational B-spline interpolating methods encounter problems pertaining to the interpolating error, feedrate profile, and machining efficiency; the optimal feedrate profile may not be obtained owing to the complex kinematics of multi-axis machine tools with hybrid structures. To address these problems and obtain an optimal feedrate profile with minimum machining time, this paper establishes a generalised representation for the nonlinear feedrate scheduling model, with static and dynamic feedrate constraints, through a numerical convex transformation. Based on the generalised model representation, a time-optimal feedrate profile can then be solved globally for both series- and parallel-structured mechanisms without problems caused by profile linear simplification or imprecise feedrate evolution. Considering the feedrate fluctuation problem caused by the difference between the theoretical arc length of the curve interpolation and the actual tool path, this paper proposes a novel co-lateral triangle deviation (CTD)-based parameter compensation interpolation algorithm. With the CTD calculation, which is related to the actual feed motion in real time, the proposed interpolation algorithm can determine the interpolating compensation distance for each interpolating segment and obtain the interpolation points using a second-order Taylor series expansion with a minimal interpolating error. Carried on a hybrid mechanism with complex kinematic constraints, the experimental and comparison results validated the effectiveness of the proposed method in feedrate scheduling and feedrate fluctuation reduction.

Asymptotics and Summation of the Effective Properties of Suspensions, Simple Liquids and Composites

We review the problem of summation for a very short truncation of a power series by means of special resummation techniques inspired by the field-theoretical renormalization group. Effective viscosity (EV) of active and passive suspensions is studied by means of a special algebraic renormalization approach applied to the first and second-order expansions in volume fractions of particles. EV of the 2D and 3D passive suspensions is analysed by means of various self-similar approximants such as iterated roots, exponential approximants, super-exponential approximants and root approximants. General formulae for all concentrations are derived. A brief introduction to the rheology of micro-swimmers is given. Microscopic expressions for the intrinsic viscosity of the active system of puller-like microswimmers are obtained. Special attention is given to the problem of the calculation of the critical indices and amplitudes of the EV and to the sedimentation rate in the vicinity of known critical points. Critical indices are calculated from the short truncation by means of minimal difference and minimal derivative conditions on the fixed points imposed directly on the critical properties. Accurate expressions are presented for the non-local diffusion coefficient of a simple liquid in the vicinity of a critical point. Extensions and corrections to the celebrated Kawasaki formula are discussed. We also discuss the effective conductivity for the classical analog of graphene and calculate the effective critical index for superconductivity dependent on the concentration of vacancies. Finally, we discuss the effective conductivity of a random 3D composite and calculate the superconductivity critical index of a random 3D composite.

Variability response functions for statically indeterminate structures

The response variability of statically indeterminate beam structures caused by uncertain material properties modeled by random fields is studied in this paper. By solving the governing equations of the statically indeterminate structure, the response bending moment along the length of the beam, M(x), is expressed as a function of its (random) zero-moment location. This approach, combined with a second-order Taylor series expansion of the random zero-moment location, leads to novel Variability Response Function-based integral expressions for the variance of the response bending moment, Var[M(x)]. Variability Response Functions have numerous desirable attributes including the capability to perform a full sensitivity analysis of the response variability with respect to the spectral characteristics of the random field modeling the uncertain material/system properties and establishing realizable upper bounds of the response variability. The proposed expressions are fundamentally different from all currently existing expressions that are based inherently on first-order Taylor series expansions, as they involve: (i) two distinct integrals in the expression for Var[M(x)] and, (ii) an expression for the mean value ɛ[M(x)] that depends on the stochastic field modeling the uncertain material properties. In general, the proposed second-order Taylor series-based formulation is expected to provide more accurate results. Extensive numerical examples are provided where the accuracy of the results obtained using the proposed formulation is validated using Monte Carlo simulations involving stochastic fields that follow truncated Gaussian and shifted lognormal probability distribution functions. These Monte Carlo simulation results indicate that the proposed Variability Response Functions are probability-distribution-independent.

Nonlinear dynamics of coupled oscillators in 1:2 internal resonance: effects of the non-resonant quadratic terms and recovery of the saturation effect

This article considers the nonlinear dynamics of coupled oscillators featuring strong coupling in 1:2 internal resonance. In forced oscillations, this particular interaction is the source of energy exchange, leading to a particular shape of the response curves, as well as quasi-periodic responses and a saturation phenomenon. These main features are embedded in the simplest system which considers only the two resonant quadratic monomials conveying the 1:2 internal resonance, since they are the proeminent source allowing one to explain these phenomena. However, it has been shown recently that those features can be substantially modified by the presence of non-resonant quadratic terms. The aim of the present study is thus to explain the effect of the non-resonant quadratic terms on the dynamics. To that purpose, the normal form up to the third order is used, since the effect of the non-resonant quadratic terms will be transferred into the resonant cubic terms. Analytical solutions are detailed using a second-order mutliple scale expansion. A thorough investigation of the backbone curves, their stability and bifurcation, and the link to the forced–damped solutions, is detailed, showing in particular interesting features that had not been addressed in earlier studies. Finally, the saturation effect is investigated, and it is shown how to correct the detuning effect of the cubic terms thanks to a specific tuning of non-resonant quadratic terms and resonant cubic terms. This choice, derived analytically, is shown to extend the validity of the saturation effect to larger amplitudes, which can thus be used in all applications where this effect is needed e.g. for control.

Efficient configuration calibration using ground auxiliary receivers at inaccurate locations

The target localization accuracy is very sensitive to the knowledge of radar positions in an airborne distributed system. However, perfect knowledge of radar positions is generally unavailable, leading to the degradation in the target localization accuracy and furthermore the deterioration in signal coherence at the destination. Thus, this work deals with the configuration calibration problem in the airborne distributed system with the assistance of several ground auxiliary receivers (GARs) at inaccurate positions, such that absolute radar positions can be obtainable through jointly using the inter-radar direct path time measurements (DPTMs) and external DPTMs from radars to GARs. The analysis on Bayesian Cramer-Rao lower bound (BCRLB) affirms the required minimum of GARs is two for precise calibration, provided that the position of a radar and orientation from the projection of the radar on the ground to a GAR are known. It proceeds to introduce the idea of iteration and second-order Taylor series expansion to enable the formulation of the pseudolinear estimator relating radar and GAR positions. The previous work is also extended to a more practical scenario, where exact positions of GARs are inaccessible. Simulations confirm well with the theoretical developments.

An all-sapphire fiber temperature sensor for high-temperature measurement

An all-sapphire high-temperature optical fiber sensor with an extrinsic Fabry–Perot interferometer (EFPI) structure is proposed and experimentally demonstrated. The EFPI structure of the sensor is composed of a sapphire ferrule with a sapphire fiber and a polished solid sapphire crystal rod. The first reflection occurs on the left end of the sapphire ferrule, and the second reflection occurs on the right end of the sapphire rod, forming double-beam interference. The interference signal is picked up by the sapphire fiber and transmitted to the white light interferometric demodulator by the multimode fiber. The picked-up signal is demodulated by the Fourier transform and interference algorithms. Experimental results show that the temperature response of the sensor is quadratic, and that the sensitivity changes linearly with a temperature range from room temperature to 1500 °C. The first-order thermal expansion coefficient and the second-order thermal expansion coefficient are 5.4575 × 10−6 °C−1 and 7.3755 × 10−9 °C−2, respectively. The thermal expansion coefficient is linearly related to temperature. Due to its all sapphire structures, this sensor solves the problem of thermal expansion coefficient mismatch, which was observed in a previous sensor made of sapphire and ceramic materials. In addition, it improves the high-temperature resistance, so it can be fully qualified for long-term high-temperature measurements above 1000 °C–1500 °C.