Showing papers on "Asymptotic analysis published in 2021"
TL;DR: In this article, the Riemann-Hilbert method was used for the initial problem of the vector Gerdjikov-Ivanov equation, and the formula for its N-soliton solution was obtained as a ratio of (N + 1) × (N+ 1) determinant and N × N determinant.
Abstract: We consider the Riemann–Hilbert method for initial problem of the vector Gerdjikov–Ivanov equation, and obtain the formula for its N-soliton solution, which is expressed as a ratio of (N + 1) × (N + 1) determinant and N × N determinant. Furthermore, by applying asymptotic analysis, the simple elastic interactions of N-soliton are confirmed, and the shifts of phase and position are also explicitly displayed.
68 citations
TL;DR: This work presents an optimization based method that can accurately compute the phase factors using standard double precision arithmetic operations and demonstrates the performance of this approach with applications to Hamiltonian simulation, eigenvalue filtering, and the quantum linear system problems.
Abstract: Quantum signal processing (QSP) is a powerful quantum algorithm to exactly implement matrix polynomials on quantum computers. Asymptotic analysis of quantum algorithms based on QSP has shown that asymptotically optimal results can in principle be obtained for a range of tasks, such as Hamiltonian simulation and the quantum linear system problem. A further benefit of QSP is that it uses a minimal number of ancilla qubits, which facilitates its implementation on near-to-intermediate term quantum architectures. However, there is so far no classically stable algorithm allowing computation of the phase factors that are needed to build QSP circuits. Existing methods require the use of variable precision arithmetic and can only be applied to polynomials of a relatively low degree. We present here an optimization-based method that can accurately compute the phase factors using standard double precision arithmetic operations. We demonstrate the performance of this approach with applications to Hamiltonian simulation, eigenvalue filtering, and quantum linear system problems. Our numerical results show that the optimization algorithm can find phase factors to accurately approximate polynomials of a degree larger than $10\phantom{\rule{0.16em}{0ex}}000$ with errors below ${10}^{\ensuremath{-}12}$.
59 citations
TL;DR: Based on the Hirota method and symbolic computation, Wang et al. as mentioned in this paper obtained some X-type, Y-type and periodic lump-stripe soliton solutions and observed the fusion phenomenon between the periodic lump and stripe soliton.
42 citations
TL;DR: The asymptotic analysis of the outage performance of reconfigurable intelligent surface (RIS) aided multi-input multi-output (MIMO) communications reveals that the number of reflecting elements at the RIS should not be smaller than the total number of MIMO transmit and receive antennas to get rid of the rank deficiency of the cascaded MIMo channels.
Abstract: We thoroughly investigate the outage performance of reconfigurable intelligent surface (RIS) aided multi-input multi-output (MIMO) communications by exploiting statistical channel state information (CSI). Kornecker channel model is adopted to characterize the impact of spatial correlations among MIMO antennas and reconfigurable reflectors. Mellin transform and random matrix theory are then utilized to derive the outage probability, with which we further conduct the asymptotic outage analysis to obtain insightful findings. In particular, the asymptotic analysis reveals that the number of reflecting elements at the RIS should not be smaller than the total number of MIMO transmit and receive antennas to get rid of the rank deficiency of the cascaded MIMO channels. Moreover, the asymptotic outage probability is a monotonically increasing and convex function with respect to the transmission rate. The numerical outcomes not only corroborate our analytical results, but also demonstrate the negative impact of the spatial correlation and the benefit of increasing the number of reconfigurable reflectors. Finally, we apply the asymptotic results to optimally devise the phase shifts with a low computational complexity.
32 citations
TL;DR: An approach to solving coefficient inverse problems for nonlinear reaction-diffusion-advection equations is proposed and a use of asymptotic analysis methods to select a good initial guess in a gradient method for minimizing a cost functional that occurs when solving the coefficient inverse problem.
31 citations
TL;DR: In this paper, the ergodic secrecy rate (ESR) of a reconfigurable intelligent surface (RIS)-assisted communication system in the presence of multiple eavesdroppers (Eves), and by assuming discrete phase shifts at the RIS was investigated.
Abstract: This letter investigates the ergodic secrecy rate (ESR) of a reconfigurable intelligent surface (RIS)-assisted communication system in the presence multiple eavesdroppers (Eves), and by assuming discrete phase shifts at the RIS. In particular, a closed-form approximation of the ESR is derived for both non-colluding and colluding Eves. The analytical results are shown to be accurate when the number of reflecting elements of the RIS ${N}$ is large. Asymptotic analysis is provided to investigate the impact of ${N}$ on the ESR, and it is proved that the ESR scales with $\log \,_{2} N$ for both non-colluding and colluding Eves. Numerical results are provided to verify the analytical results and the obtained scaling laws.
29 citations
TL;DR: In this article, the authors study how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations, or in other words, to investigate optimal rigidity estimates for the Eigenvalues.
Abstract: The goal of this article is to study how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations—or in other words, to investigate optimal rigidity estimates for the eigenvalues. We do this in the setting of one-cut regular unitary invariant ensembles of random Hermitian matrices, the Gaussian unitary ensemble (GUE) being the prime example of such an ensemble. Our approach to this question combines extreme value theory of log-correlated stochastic processes, and in particular the theory of multiplicative chaos, with asymptotic analysis of large Hankel determinants with Fisher–Hartwig symbols of various types, such as merging jump singularities, size-dependent impurities, and jump singularities approaching the edge of the spectrum. In addition to optimal rigidity estimates, our approach sheds light on the fractal geometry of the eigenvalue counting function.
26 citations
TL;DR: This article examines sparse networks in conjunction with finite dimensional sub-Gaussian mixtures as covariates under moderate separation conditions and proposes a simple optimization framework which improves clustering accuracy when the two sources carry partial information about the cluster memberships, and hence perform poorly on their own.
Abstract: In this article, we investigate community detection in networks in the presence of node covariates. In many instances, covariates and networks individually only give a partial view of the cluster s...
26 citations
TL;DR: In this article, the free energy of the integrable O(4) sigma model in a magnetic field was analyzed using Volin's method to extract high number of perturbative coefficients with very high precision.
Abstract: We analyze the free energy of the integrable two dimensional O(4) sigma model in a magnetic field. We use Volin’s method to extract high number (2000) of perturbative coefficients with very high precision. The factorial growth of these coefficients are regulated by switching to the Borel transform, where we perform several asymptotic analysis. High precision data allowed to identify Stokes constants and alien derivatives with exact expressions. These reveal a nice resurgence structure which enables to formulate the first few terms of the ambiguity free trans-series. We check these results against the direct numerical solution of the exact integral equation and find complete agreement.
25 citations
TL;DR: In this article, a general synthetic iterative scheme (GSIS) is proposed to find steady-state solutions of the gas kinetic equations for molecular gas flows accurately and efficiently, where the kinetic equations are solved together with the macroscopic synthetic equations that expedite solutions towards the steady state.
23 citations
TL;DR: In this article, double Wronskian solitons for the WBK system were constructed by the Darboux transformation together with the Wronkian technique, and the determinant identities were developed in the verification of the solutions.
Abstract: In this paper, by the Darboux transformation together with the Wronskian technique, we construct new double Wronskian solutions for the Whitham-Broer-Kaup (WBK) system. Some new determinant identities are developed in the verification of the solutions. Based on analyzing the asymptotic behavior of new double Wronskian functions as t → ±∞, we make a complete characterization of asymptotic solitons for the non-singular, non-trivial and irreducible soliton solutions. It turns out that the solutions are the linear superposition of two fully-resonant multi-soliton configurations, in each of which the amplitudes, velocities and numbers of asymptotic solitons are in general not equal as t → ±∞. To illustrate, we present the figures for several examples of soliton interactions occurring in the WBK system.
04 Mar 2021
TL;DR: In this article, the interface conditions between the layers of a three-dimensional composite structure in the framework of coupled thermoelasticity are studied by means of an asymptotic analysis.
Abstract: This work proposes new interface conditions between the layers of a three-dimensional composite structure in the framework of coupled thermoelasticity. More precisely, the mechanical behavior of two linear isotropic thermoelastic solids, bonded together by a thin layer, constituted of a linear isotropic thermoelastic material, is studied by means of an asymptotic analysis. After defining a small parameter e, which tends to zero, associated with the thickness and constitutive coefficients of the intermediate layer, two different limit models and their associated limit problems, the so-called soft and hard thermoelastic interface models, are characterized. The asymptotic expansion method is reviewed by taking into account the effect of higher-order terms and defining a generalized thermoelastic interface law which comprises the above aforementioned models, as presented previously. A numerical example is presented to show the efficiency of the proposed methodology, based on a finite element approach developed previously.
TL;DR: This work studies the relationship between capacity and performance for a service firm with spatial operations, in the sense that requests arrive with origin-destination pairs.
Abstract: We study the relationship between capacity and performance for a service firm with spatial operations, in the sense that requests arrive with origin-destination pairs. An example of such a system i...
TL;DR: In this paper, an asymmetric three-layered laminate with prescribed stresses along the faces is considered, where the outer layers are assumed to be much stiffer than the inner one.
Abstract: An asymmetric three-layered laminate with prescribed stresses along the faces is considered. The outer layers are assumed to be much stiffer than the inner one. The focus is on long-wave low-frequency anti-plane shear. Asymptotic analysis of the original dispersion relation reveals a low-frequency harmonic supporting a slow quasi-static (or static at the limit) decay along with near cut-off wave propagation. In spite of asymmetry of the problem, the leading order shortened polynomial dispersion relation factorises into two simpler ones corresponding to the fundamental mode and the aforementioned harmonic. The associated 1D equations of motion derived in the paper are also split into two second-order operators in line with the factorisation of the shortened dispersion relation. Asymptotically justified boundary conditions are established using the Saint-Venant’s principle modified by taking into account the high-contrast properties of the laminate.
TL;DR: In this paper, the motion of a damped rigid body pendulum in which its suspended point travels along the Lissajous trajectory was explored. And the stability criteria were carried out in addition to simulations of the evolution of the non-linear equations of the considered system using the approach of nonlinear stability analysis.
TL;DR: In this paper, the authors provided an asymptotic analysis of the third-order JMGT model with respect to the Westervelt equation and showed that the corresponding solutions converge in a strong topology of the phase space to a limit.
Abstract: The (third order in time) JMGT equation [Jordan (J Acoust Soc Am 124(4):2491–2491, 2008) and Cattaneo (C Sulla conduzione del calore Atti Sem Mat Fis Univ Modena 3:83–101, 1948)] is a nonlinear (quasi-linear) partial differential equation (PDE) model introduced to describe a nonlinear propagation of sound in an acoustic medium. The important feature is that the model avoids the infinite speed of propagation paradox associated with a classical second-order in time equation referred to as Westervelt equation. Replacing Fourier’s law by Maxwell–Cattaneo’s law gives rise to the third-order in time derivative scaled by a small parameter $$\tau >0$$
, the latter represents the thermal relaxation time parameter and is intrinsic to the medium where the dynamics occur. In this paper, we provide an asymptotic analysis of the third-order model when $$\tau \rightarrow 0 $$
. It is shown that the corresponding solutions converge in a strong topology of the phase space to a limit which is the solution of Westervelt equation. In addition, rate of convergence is provided for solutions displaying higher-order regularity. This addresses an open question raised in [20], where a related JMGT equation has been studied and weak star convergence of the solutions when $$\tau \rightarrow 0$$
has been established. Thus, our main contribution is showing strong convergence on infinite time horizon, along with related rates of convergence valid on a finite time horizon. The key to unlocking the difficulty owns to a tight control and propagation of the “smallness” of the initial data in carrying the estimates at three different topological levels. The rate of convergence allows one then to estimate the relaxation time needed for the signal to reach the target. The interest in studying this type of problems is motivated by a large array of applications arising in engineering and medical sciences.
TL;DR: In this paper, the Fokas-Lenells equations are investigated via bilinear approach, and the authors obtain solutions to the classical and non-local versions of the problem, which behave like (multi-)periodic waves with algebraic decay.
Abstract: In this paper, the Fokas-Lenells equations are investigated via bilinear approach. We bilinearize the unreduced Fokas-Lenells system, derive double Wronskian solutions, and then, by means of a reduction technique we obtain variety of solutions of the reduced equations. This enables us to have a full profile of solutions of the classical and nonlocal Fokas-Lenells equations. Some obtained solutions are illustrated based on asymptotic analysis. As a notable new result, we obtain solutions to the Fokas-Lenells equation, which are related to real discrete eigenvalues and not reported before in the analytic approaches. These solutions behave like (multi-)periodic waves or solitary waves with algebraic decay.
TL;DR: In this paper, the authors theoretically provided the analytical bright soliton solution in the two-dimensional Bose-Einstein condensation model and their interesting interaction, and proved that the Hirota bilinear method and asymptotic analysis method are powerful and effective techniques in physical sciences and engineering for analyzing nonlinear mathematical-physical equations and their solutions.
TL;DR: In this article, an asymptotic analysis of target fluxes in the 3D narrow capture problem is presented, which concerns a diffusive search process in which the targets are much smaller than the targets.
Abstract: We develop an asymptotic analysis of target fluxes in the three-dimensional (3D) narrow capture problem. The latter concerns a diffusive search process in which the targets are much smaller than th...
TL;DR: In this article, the authors describe the dynamics of hepatitis B by a fractional-order model in the Caputo sense and present the basic results of the fractional hepatitis B model.
Abstract: This paper describes the dynamics of hepatitis B by a fractional-order model in the Caputo sense. The basic results of the fractional hepatitis B model are presented. Two equilibrium point for the model exists, the disease-free point and the infected point. The local and global stability analysis of the system is given in terms of the basic reproductive number. We execute the global stability analysis using the extended Barbalat’s lemma to the fractional-order system. The results show that the extension of Barbalat’s Lemma is a robust tool for the asymptotic analysis of the fractional dynamic systems. Moreover, numerical simulations by Nonstandard Finite Difference Schemes show that the solutions converge to an equilibrium point as predicted in the stability analysis.
TL;DR: In this paper, the possibilities of using asymptotic analysis for solving the inverse problem of restoring the parameters of the source of nitrogen oxide industrial emissions into the atmosphere were demonstrated.
Abstract: The possibilities of using asymptotic analysis for solving the inverse problem of restoring the parameters of the source of nitrogen oxide industrial emissions into the atmosphere are demonstrated
TL;DR: In this paper, a unified mathematical analysis of a family of non-local diffuse interface models for tumor growth is provided. But the model in consideration couples a nonlocal Cahn-Hilliard equation for the tumor phase variable with a reaction-diffusion equation for nutrient concentration, and takes into account also significant mechanisms such as chemotaxis and active transport.
Abstract: This paper provides a unified mathematical analysis of a family of non-local diffuse interface models for tumor growth. These are non-local variants of the corresponding local model proposed by H. Garcke et al. (2016), and take into account the long-range interactions occurring in biological phenomena. The model in consideration couples a nonlocal Cahn-Hilliard equation for the tumor phase variable with a reaction-diffusion equation for the nutrient concentration, and takes into account also significant mechanisms such as chemotaxis and active transport. The system depends on two relaxation parameters: a viscosity coefficient and parabolic-regularization coefficient on the chemical potential. The first part of the paper is devoted to the analysis of the system with both regularizations. Here, a rich spectrum of results is presented. Weak well-posedness is first addressed, also including singular potentials. Then, under suitable conditions, existence of strong solutions enjoying the separation property is proved. This allows also to obtain a refined stability estimate with respect to the data, including both chemotaxis and active transport. The second part of the paper is devoted to the study of the asymptotic behaviour of the system as the relaxation parameters vanish. The asymptotics are analyzed when the parameters approach zero both separately and jointly, and exact error estimates are obtained. As a by-product, well-posedness of the corresponding limit systems is established.
TL;DR: In this article, the long-time asymptotics of high-order soliton for the Hirota equation were analyzed using the inverse scattering method and Darboux transformation.
TL;DR: In this article, the peristaltic flow phenomenon for non-Newtonian Jeffrey fluid inside an asymmetric channel subject to large magnetic field is addressed. And the boundary layer regions are determined where the magnetic field dominates the viscous force.
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TL;DR: In this article, a multi-agent derivative-free consensus-based optimization (DBOO) method is proposed to solve the problem of global minimization of a possibly non-smooth and non-convex objective function constrained on the unit hypersphere.
Abstract: In this paper we are concerned with the global minimization of a possibly non-smooth and non-convex objective function constrained on the unit hypersphere by means of a multi-agent derivative-free method. The proposed algorithm falls into the class of the recently introduced Consensus-Based Optimization. In fact, agents move on the sphere driven by a drift towards an instantaneous consensus point, which is computed as a convex combination of agent locations, weighted by the cost function according to Laplaces principle, and it represents an approximation to a global minimizer. The dynamics is further perturbed by an anisotropic random vector field to favor exploration. The main results of this paper are about the proof of convergence of the numerical scheme to global minimizers provided conditions of well-preparation of the initial datum. The proof of convergence combines a mean-field limit result with a novel asymptotic analysis, and classical convergence results of numerical methods for SDE. The main innovation with respect to previous work is the introduction of an anisotropic stochastic term, which allows us to ensure the independence of the parameters of the algorithm from the dimension and to scale the method to work in very high dimension. We present several numerical experiments, which show that the algorithm proposed in the present paper is extremely versatile and outperforms previous formulations with isotropic stochastic noise.
TL;DR: In this paper, a general methodology is proposed to derive the hydrodynamic limits of the LBM in the discrete setting, which relies on a Taylor expansion in the limit of low Knudsen numbers.
TL;DR: In this article, the authors studied an elastic arch subject to an end-shortening that evolves linearly with time, i.e. at a constant rate, and showed that the delay in snapthrough is a consequence of delayed bifurcation and occurs even in the perfectly elastic case when viscous (and viscoelastic) effects are negligible.
Abstract: We study elastic snap-through induced by a control parameter that evolves dynamically. In particular, we study an elastic arch subject to an end-shortening that evolves linearly with time, i.e. at a constant rate. For large end-shortening the arch is bistable but, below a critical end-shortening, the arch becomes monostable. We study when and how the arch transitions between states and show that the end-shortening at which the fast ‘snap’ happens depends on the rate at which the end-shortening is reduced. This delay in snap-through is a consequence of delayed bifurcation and occurs even in the perfectly elastic case when viscous (and viscoelastic) effects are negligible. We present the results of numerical simulations to determine the magnitude of this delay (and the associated time lag) as the loading rate and the importance of external viscous damping vary. We also present an asymptotic analysis of the geometrically-nonlinear problem that reduces the salient dynamics to that of an ordinary differential equation; the form of this reduced equation is generic for snap-through instabilities in which the relevant control parameter is ramped linearly in time. Moreover, this asymptotic reduction allows us to derive analytical results for the delay observed in snap-through that are in good agreement with the results of our simulations. Finally, we discuss scaling laws for the delay that should be expected in other examples of delayed bifurcation in elastic instabilities.
TL;DR: In this paper, an asymptotic analysis of a coupled system of kinetic and fluid equations is presented, dealing with the nonlinear Vlasov-fokker-planck equation coupled with the compressible isentropic isentropy.
Abstract: We study an asymptotic analysis of a coupled system of kinetic and fluid equations. More precisely, we deal with the nonlinear Vlasov–Fokker–Planck equation coupled with the compressible isentropic...
TL;DR: In this paper, the structural loss factor (SLF) of a honeycomb reinforced composite laminated structure is determined by applying the homogenous asymptotic method (HAM) and representative unit cell (RUC) approach, considering the frequency and temperature-dependent properties of the viscoelastic damping material (VDM), the constitutive equations of the laminated composite structure is obtained.