Second-order Asymptotic Expansion Research Articles

Quasi-classical ground states. I. Linearly coupled Pauli–Fierz Hamiltonians

We consider a spinless, non-relativistic particle bound by an external potential and linearly coupled to a quantized radiation field. The energy \mathcal{E}(u,f) of product states of the form u\otimes\Psi_f , where u is a normalized state for the particle and \Psi_f is a coherent state in Fock space for the field, gives the energy of a Klein–Gordon–Schrödinger system. We minimize the functional \mathcal{E}(u,f) on its natural energy space. We prove the existence and uniqueness of a ground state under general conditions on the coupling function. In particular, neither an ultraviolet cutoff nor an infrared cutoff is imposed. Our results establish the convergence in the ultraviolet limit of both the ground state and ground state energy of the Klein–Gordon–Schrödinger energy functional, and provide the second-order asymptotic expansion of the ground state energy at small coupling.

Compressibility-induced destabilisation of falling liquid films: an integral approach

We revisit the classical 2D problem of a gravity-driven liquid layer down an inclined plate (Kapitza, 1948), relaxing the usual assumption of homogeneous fluid. We set out to answer three major issues. When the fluid density is allowed to vary, (i) how does this feature structurally affect the formulation of a low-dimensional depth-averaged model? (ii) To what extent and (iii) by virtue of which physical mechanism does compressibility participate in the long-wave interfacial instability? To provide the relevant answers, (i) we first make use of a second-order asymptotic expansion in the shallowness parameter to develop a weakly-compressible boundary-layer system: starting from a two-equation momentum-integrated model, an additional barotropic equation of state is required for closure purposes. In this respect, (ii) a temporal linear stability analysis is performed: it is revealed that compressibility plays a destabilising role whose magnitude is enhanced at intermediately tilted configurations, and the more the Reynolds number approaches the critical threshold in the incompressible limit. (iii) We finally interpret the ensuing dispersion relation under the convenient framework of two-wave hierarchy, initiated by Whitham (1974): the primary instability gets promoted by the flow compressibility as it contributes to deceleration of dynamic waves most significantly in the low-inertia regime. Indeed, compressibility locally acts as a further boost to the inertia-based mechanism of Kapitza instability by amplifying flow-rate variations within the liquid film.

An implicit unified gas-kinetic wave–particle method for radiative transport process

The unified gas-kinetic wave–particle method (UGKWP) has been developed for the multiscale gas, plasma, and multiphase flow transport processes for the past years. In this work, we propose an implicit UGKWP (IUGKWP) method to remove the Courant–Friedrichs–Lewy time step constraint. Based on the local integral solution of the radiative transfer equation (RTE), the particle transport processes are categorized into the long-λ streaming process and the short-λ streaming process compared to a local physical characteristic time tp. In the construction of the IUGKWP method, the long-λ streaming process is tracked by the implicit Monte Carlo method; the short-λ streaming process is evolved by solving the implicit moment equations; and the photon distribution is closed by a local integral solution of RTE. In the IUGKWP method, the multiscale flux of radiation energy and the multiscale closure of photon distribution are constructed based on the local integral solution. The IUGKWP method preserves the second-order asymptotic expansion of RTE in the optically thick regime and adapts its computational complexity to the flow regime. The numerical dissipation is well controlled, and the teleportation error is significantly reduced in the optically thick regime. The computational complexity of the IUGKWP method decreases exponentially as the Knudsen number approaches zero, and the computational efficiency is remarkably improved in the optically thick regime. The IUGKWP is formulated on a generalized unstructured mesh, and multidimensional 2D and 3D algorithms are developed. Numerical tests are presented to validate the capability of IUGKWP in capturing the multiscale photon transport process. The algorithm and code will apply in the engineering applications of inertial confinement fusion.

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.

Imaging of mass distributions from partial domain measurement

Abstract This paper deals with an inverse source problem governed by the Poisson equation. The aim is to reconstruct multiple anomalies from internal observation data within an arbitrary sub-region. The considered model problem is motivated by various applications such as the identification of geological anomalies underneath the Earth’s surface. To overcome the ill-posedness, the inverse problem is formulated as a self-regularized topology optimization one. A least-square functional measuring the misfit between the observed quantities and the values provided by the model problem is introduced. The misfit function involves a regularization term penalizing the relative perimeter of the unknown domain. Finally, it is minimized with respect to a finite number of circular-shaped anomalies. The existence and stability of the solution to the inverse problem are demonstrated. The reconstruction process is based on the topological derivative method. The variation of the shape functional with respect to a small geometric perturbation is studied, and the leading terms of a second-order topological asymptotic expansion are derived. A non-iterative reconstruction algorithm is developed via the minimization of the obtained asymptotic formula with respect to the location and size parameters of the unknown set of anomalies. The efficiency and accuracy of the proposed approach are justified by some numerical experiments.

Second-order fast–slow dynamics of non-ergodic Hamiltonian systems: Thermodynamic interpretation and simulation

A class of fast–slow Hamiltonian systems with potential Uɛ describing the interaction of non-ergodic fast and slow degrees of freedom is studied. The parameter ɛ indicates the typical timescale ratio of the fast and slow degrees of freedom. It is known that the Hamiltonian system converges for ɛ→0 to a homogenised Hamiltonian system. We study the situation where ɛ is small but positive.First, we rigorously derive the second-order corrections to the homogenised (slow) degrees of freedom. They can be decomposed into explicitly given terms that oscillate rapidly around zero and terms that trace the average motion of the corrections, which are given as the solution to an inhomogeneous linear system of differential equations.Then, we analyse the energy of the fast degrees of freedom expanded to second-order from a thermodynamic point of view. In particular, we define and expand to second-order a temperature, an entropy and external forces and show that they satisfy to leading-order, as well as on average to second-order, thermodynamic energy relations akin to the first and second law of thermodynamics.Finally, we analyse for a specific fast–slow Hamiltonian system the second-order asymptotic expansion of the slow degrees of freedom from a numerical point of view. Their approximation quality for short and long time frames and their total computation time are compared with those of the solution to the original fast–slow Hamiltonian system of similar accuracy.

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.

Capturing implied correlation skew from options prices via multiscale stochastic volatility models

The aim of this paper is to derive a second-order asymptotic expansion for the price of European options written on two underlying assets, whose dynamics are described by multiscale stochastic volatility models. In particular, the second-order expansion of option prices can be translated into a corresponding expansion in implied correlation units. The resulting approximation for the implied correlation curve turns out to be quadratic in the log-moneyness, capturing the convexity of the implied correlation skew. Finally, we describe a calibration procedure where the model parameters can be estimated using option prices on individual underlying assets.

On the Second-Order Asymptotics of the Partially Smoothed Conditional Min-Entropy & Application to Quantum Compression

Recently, Anshu et al. introduced "partially" smoothed information measures and used them to derive tighter bounds for several information-processing tasks, including quantum state merging and privacy amplification against quantum adversaries [arXiv:1807.05630 [quant-ph]]. Yet, a tight second-order asymptotic expansion of the partially smoothed conditional min-entropy in the i.i.d. setting remains an open question. Here we establish the second-order term in the expansion for pure states, and find that it differs from that of the original "globally" smoothed conditional min-entropy. Remarkably, this reveals that the second-order term is not uniform across states, since for other classes of states the second-order term for partially and globally smoothed quantities coincides. By relating the task of quantum compression to that of quantum state merging, our derived expansion allows us to determine the second-order asymptotic expansion of the optimal rate of quantum data compression. This closes a gap in the bounds determined by Datta and Leditzky [IEEE Trans. Inf. Theory 61, 582 (2015)], and shows that the straightforward compression protocol of cutting off the eigenspace of least weight is indeed asymptotically optimal at second order.

Second-order asymptotics of the fractional perimeter as <i>s</i> → 1

In this note we provide a second-order asymptotic expansion of the fractional perimeter Ps(E), as $s\to 1^-$, in terms of the local perimeter and of a higher order nonlocal functional.

Corrections to “On the Groundwave Excited by a Vertical Hertzian Dipole Over a Planar Conductor: Second-Order Asymptotic Expansion With Applications to Plasmonics”

The surface electromagnetic fields radiated by a vertical electric dipole over a lossy conducting half-space are investigated. A simple derivation is presented of the second-order asymptotic groundwave expression, using a modified saddle-point method based on integration around vertical branch cuts. The method takes into account the influence of the Sommerfeld pole, which in the plasmonic case contributes a surface plasmon polariton. The proposed groundwave formulation is shown to be superior to the state-of-the-art Norton–Bannister theory for noble metals at optical wavelengths.

A rational framework for dynamic homogenization at finite wavelengths and frequencies.

In this study, we establish an inclusive paradigm for the homogenization of scalar wave motion in periodic media (including the source term) at finite frequencies and wavenumbers spanning the first Brillouin zone. We take the eigenvalue problem for the unit cell of periodicity as a point of departure, and we consider the projection of germane Bloch wave function onto a suitable eigenfunction as descriptor of effective wave motion. For generality the finite wavenumber, finite frequency homogenization is pursued in via second-order asymptotic expansion about the apexes of 'wavenumber quadrants' comprising the first Brillouin zone, at frequencies near given (acoustic or optical) dispersion branch. We also consider the junctures of dispersion branches and 'dense' clusters thereof, where the asymptotic analysis reveals several distinct regimes driven by the parity and symmetries of the germane eigenfunction basis. In the case of junctures, one of these asymptotic regimes is shown to describe the so-called Dirac points that are relevant to the phenomenon of topological insulation. On the other hand, the effective model for nearby solution branches is found to invariably entail a Dirac-like system of equations that describes the interacting dispersion surfaces as 'blunted cones'. For all cases considered, the effective description turns out to admit the same general framework, with differences largely being limited to (i) the eigenfunction basis, (ii) the reference cell of medium periodicity, and (iii) the wavenumber-frequency scaling law underpinning the asymptotic expansion. We illustrate the analytical developments by several examples, including Green's function near the edge of a band gap and clusters of nearby dispersion surfaces.

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.

On the Groundwave Excited by a Vertical Hertzian Dipole Over a Planar Conductor: Second-Order Asymptotic Expansion With Applications to Plasmonics

On the Groundwave Excited by a Vertical Hertzian Dipole Over a Planar Conductor: Second-Order Asymptotic Expansion With Applications to Plasmonics

An optimal execution problem in the volume-dependent Almgren–Chriss model

In this study, we introduce an explicit trading-volume process into the Almgren–Chriss model, which is a standard model for optimal execution. We propose a penalization method for deriving a verification theorem for an adaptive optimization problem. We also discuss the optimality of the volume-weig hted average-price strategy of a risk-neutral trader. Moreover, we derive a second-order asymptotic expansion of the optimal strategy and verify its accuracy numerically.

An Optimal Execution Problem in the Volume-Dependent Almgren-Chriss Model

In this study, we introduce an explicit trading-volume process into the Almgren-Chriss model, which is a standard model for optimal execution. We propose a penalization method of deriving a verification theorem for an adaptive optimization problem. We also discuss the optimality of the volume-weighted average-price strategy of a risk-neutral trader. Moreover, we derive a second-order asymptotic expansion of the optimal strategy and verify its accuracy numerically.

Bayesian Sequential Estimation of the Inverse of the Pareto Shape Parameter

The problem addressed is that of developing a sequential procedure for estimating the inverse of the shape parameter of the Pareto distribution under the squared loss, assuming that the shape parameter is the value of a random variable having a density function with compact support and that the cost per observation is one unit. A stopping time is proposed and a second-order asymptotic expansion is obtained for the Bayes regret incurred by the proposed procedure.

Second-order two-scale finite element algorithm for dynamic thermo–mechanical coupling problem in symmetric structure

The new second-order two-scale (SOTS) finite element algorithm is developed for the dynamic thermo-mechanical coupling problems in axisymmetric and spherical symmetric structures made of composite materials. The axisymmetric structure considered is periodic in both radial and axial directions and homogeneous in circumferential direction. The spherical symmetric structure is only periodic in radial direction. The dynamic thermo-mechanical coupling model is presented and the equivalent compact form is derived. Then, the cell problems, effective material coefficients and the homogenized thermo-mechanical coupling problem are obtained successively by the second-order asymptotic expansion of the temperature increment and displacement. The homogenized material obtained is manifested with the anisotropic property in the circumferential direction. The explicit expressions of the homogenized coefficients in the plane axisymmetric and spherical symmetric cases are given and both the derivation of the analytical solutions of the cell functions and the quasi-static thermoelasticity problems are discussed. Based on the SOTS method, the corresponding finite-element procedure is presented and the unconditionally stable implicit algorithm is established. Some numerical examples are solved and the mutual interaction between the temperature and displacement field is studied under the condition of structural vibration. The computational results demonstrate that the second-order asymptotic analysis finite-element algorithm is feasible and effective in simulating and predicting the dynamic thermo-mechanical behaviors of the composite materials with small periodic configurations in axisymmetric and spherical symmetric structures. This may provide a vital computational tool for analyzing composite material internal temperature distribution and structural deformation induced by the dynamic thermo-mechanical coupling response under strong aerothermodynamic environment. Second-order two-scale (SOTS) expansions for dynamic thermo-mechanical coupling problems in symmetric structure are obtained.SOTS finite element algorithm is proposed and unconditional stable implicit scheme is established.Anisotropic material is obtained by homogenization with enhanced strength and minor thermal expansion in circumferential direction.The mutual interaction and simultaneous vibration of the temperature and displacement field are simulated.A new computational tool is presented for analyzing thermo-mechanical response of composites under strong aerothermodynamic environment.

SEQUENTIAL ESTIMATION OF THE SQUARE OF THE RAYLEIGH PARAMETER

The problem addressed is that of sequentially estim ating the square of the parameter of the Rayleigh distribution, subject to a weighted squared loss pl us cost of sampling. We propose a sequential procedure and provide a second-order asymptotic expansion for the incurred regret. It is seen that the asymptotic regret is negative for a range of values of the parameter.

A method based on non-steady heat diffusion problems for detecting the location of inclusions

This work is concerned with the resolution of inverse problems for the detection of defects inside a homogeneous medium using non-steady heat diffusion problem, under the assumption of small contrast on the value of the conductivity coefficient between the matrix material and that of the defect. This is the so-called small amplitude, small contrast or small aspect ratio assumption. Following the idea developed by Allaire and Gutiérrez for optimal design problems, we develop a second-order asymptotic expansion with respect to the aspect ratio, which allows us to simplify the inverse problem, considering it as an optimization problem. According to this, we can develop a gradient-type algorithm, that reduces, in the time interval being considered, the difference between the boundary values obtained from a problem that is numerically solved with full knowledge of the defect distribution and boundary values obtained from solving another problem based on an assumption on the distribution of the defect. In general, by the use of non-steady problems, we can obtain substantially better information of the defects location compared to using steady problems.