Steepest Descent Method Research Articles

Functions of a Complex Variable with a Large Parameter and Construction of Regions

The study of the asymptotic behavior of solutions of singularly perturbed equations in complex domains comes down to the study of integrals of exponential functions containing a large parameter. Such integrals differ significantly from the integrals to which the saddle point method is applicable. The saddle point method is not applicable to study such integrals. Thus, the problem arises of constructing domains and choosing integration paths for studying such integrals. In this work, specific examples of integrals show the construction of regions in the complex plane and the choice of integration paths. The chosen integration paths ensure that the integrals are bounded over a large parameter as this parameter tends to infinity. When constructing the domain and choosing integration paths, level lines of some harmonic functions that have zeros and singular points were used. The principle of symmetry is also used. In early works, cases were considered when the eigenvalues ​​of the first approximation matrix of a singularly perturbed equation had only zeros or only poles. Cases where the eigenvalues ​​have zeros and poles were not considered.

Sound transmission through an unbaffled plate

Commonly occurring complex structures can often be modelled as simple finite plates mounted without baffles. Thus, the objective here is to study the sound transmission loss of a simply-supported rectangular finite unbaffled plate impinged upon by an acoustic plane wave. The total pressure jump across the plate surface is expressed as the sum of the pressure jump due to pure diffraction, where the plate normal displacement is zero and the pressure jump due to plate radiation. These pressure jumps are computed using the definition of the free-space Green's function in the wavenumber domain and the Euler equation. Then, the coupled equation of motion is solved for the plate displacement. The method of stationary phase is used to derive the closed-form expression for the improper double integral that appears in the far-field acoustic pressure. Finally, the far-field transmitted power is numerically computed and the sound transmission loss of the unbaffled plate is compared with that of the baffled one.

CSEM Optimization Using the Correspondence Principle

Traditionally, 3D modeling of marine controlled-source electromagnetic (CSEM) data (in the frequency domain) involves high-memory demand, requiring solving a large linear system for each frequency. To address this problem, we propose to solve Maxwell’s equations in a fictitious dielectric medium with time-domain finite-difference methods, with the support of the correspondence principle. As an advantage of this approach, we highlight the possibility of its implementation for execution with GPU accelerators, in addition to multi-frequency data modeling with a single simulation. Furthermore, we explore using the correspondence principle to the inversion of CSEM data by calculating the gradient of the least-squares objective function employing the adjoint-state method to establish the relationship between adjoint fields in a conductive medium and their counterparts in the fictitious dielectric medium, similar to the approach used in forward modeling. We validate this method through 2D inversions of three synthetic CSEM datasets, computed for a simple model consisting of two resistors in a conductive medium, a model adapted from a CSEM modeling and inversion package, and the last one based on a reference model of turbidite reservoirs on the Brazilian continental margin. We also evaluate the differences between the results of inversions using the steepest descent method and our proposed momentum method, comparing them with the limited-memory BFGS (Broyden–Fletcher–Goldfarb–Shanno) algorithm (L-BFGS-B). In all experiments, we use smoothing by model reparameterization as a strategy for regularizing and stabilizing the iterations throughout the inversions. The results indicate that, although it requires more iterations, our modified momentum method produces the best models, which are consistent with results from the L-BFGS-B algorithm and require less storage per iteration.

Uncertainty propagation analysis for an aviation accelerometer using an improved saddlepoint approximation method

Uncertainty propagation analysis for an aviation accelerometer using an improved saddlepoint approximation method

Solitons moving on background waves of the focusing nonlinear Schrödinger equation with step-like initial condition

This work concerns the long-time asymptotic behaviors of the focusing nonlinear Schrödinger equation with step-like initial condition in present of discrete spectrum. The exact step initial-value problem with non-vanishing boundary on one side has been solved, while the step-like initial-value problem with solitons emerging remains open. We study this problem and explore the solitons moving on background waves of the focusing nonlinear Schrödinger equation by classifying all possible locations of discrete spectrum associated with the spectral functions. It is shown that there are five kinds of zones for the discrete spectrum in complex plane, which are called dumbing zone, trapping zone, trapping/waking zone, transmitting/waking zone and transmitting zone, respectively. By means of Deift–Zhou nonlinear steepest-descent method for Riemann–Hilbert problems, the long-time asymptotics of the solution along with the locations of the solitons for each case are formulated. Numerical simulations match very well with the theoretical analysis.

High-Resolution Dipole Remote Detection Logging Based on Optimal Nonlinear Frequency Modulation Excitation.

In order to improve the detection range and imaging resolution of dipole remote detection logging, an optimal nonlinear frequency modulation (ONLFM) excitation method is proposed in this paper. In this method, the optimal waveform model of the sound source is designed with objective functions of SNR and resolution to obtain the highest resolution under the condition of the required SNR. This optimal model is a multi-constraint optimization problem, and the simulated annealing method has been used to solve it. Solving the optimal model can obtain the optimal spectrum, and the ONLFM waveform can be designed by using the stationary phase principle. Compared with the electric pulse sound source used in traditional technology, this ONLFM sound source can improve the energy and extend the frequency band range of the reflected wave, which can provide the higher resolution and SNR. The simulation results show that the ONLFM sound source can effectively improve the SNR and resolution of the reflected wave, and the detection range and imaging resolution of dipole shear wave remote detection will be improved.

Wave resonances and the time-dependent capillary gravity wave motion

The interconnection of time and frequency domains for capillary gravity wave motion in the presence of current is discussed in this article. The general time-dependent problem is solved using Green’s function technique, and the asymptotic solution is derived using the method of stationary phase for large time and space. Also, the frequency domain solution is derived as a special case using the Cauchy Residue theorem. Different types of wave resonances like Trapping, Blocking and Bragg resonances are discussed. The existence of the trapped mode below the cutoff frequency is justified theoretically, and numerical results are obtained using the multipole expansion method. The blocking and Bragg resonances are analyzed above the cutoff frequency. It is found that in the presence of current, when the ripple wavenumber of the bottom undulation equals twice the cosine angle of incidence of wave times the wavenumber of the wave, Bragg resonance occurs. It is found that three propagating modes exist in the case of wave blocking, and the trapped modes exist only for the first propagating mode. Furthermore, because of the negative group velocity inside the blocking zone, the Bragg reflection increases while decreasing outside. The effect of current on the wave energy propagation in the form of group velocity is analyzed and the same is verified in the case of time-dependent problem.

Dynamical Fluctuations of Random Walks in Higher-Order Networks

Although higher-order interactions are known to affect the typical state of dynamical processes giving rise to new collective behavior, how they drive the emergence of rare events and fluctuations is still an open problem. We investigate how fluctuations of a dynamical quantity of a random walk exploring a higher-order network arise over time. In the quenched case, where the hypergraph structure is fixed, through large deviation theory we show that the appearance of rare events is hampered in nodes with many higher-order interactions, and promoted elsewhere. Dynamical fluctuations are further boosted in an annealed scenario, where both the diffusion process and higher-order interactions evolve in time. Here, extreme fluctuations generated by optimal higher-order configurations can be predicted in the limit of a saddle-point approximation. Our study lays the groundwork for a wide and general theory of fluctuations and rare events in higher-order networks. Published by the American Physical Society 2024

Dynamical response of a floating ice sheet due to a forced oscillation in a running stream

This paper explores the response of a floating ice sheet to a forced, time-harmonic oscillatory pressure applied to the ice-covered surface of a uniform, finite depth fluid. The floating ice sheet is modeled as a thin elastic plate following the Euler–Bernoulli beam equation, with the additional consideration of in-plane compressive forces acting on the ice. Employing linear wave theory, the problem is presented as an initial boundary value problem and is solved using Laplace and Fourier transforms to obtain a mathematical expression for the ice sheet's deflection in terms of infinite integrals. These integrals are thereafter solved asymptotically for large time and distance using the stationary phase method. The asymptotic analysis indicates a growing response over time when the poles and stationary points of the phase functions coalesce. The deflection of the floating ice sheet is graphically presented, highlighting the effects of various non-dimensional parameters, such as flexural rigidity, compressive force, uniform current speed, and the angular frequency of the oscillatory pressure. Additionally, the group velocity and phase velocity of flexural gravity waves are derived from the dispersion relation and elucidated using diagrams. The results show that compressive force and current speed significantly influence wave amplitude, enhancing the oscillatory nature, while the flexural rigidity of the elastic plate and the angular frequency of the applied pressure have a substantial impact on the plate's deflection.

Borehole radiation wavefield for multipole logging while drilling with an eccentric collar

Abstract This study investigates the radiation characteristics of the borehole-source in acoustic logging while drilling (ALWD) with an eccentric collar. Understanding these characteristics is crucial for the development of acoustic remote detection technology, which has shown great potential in ALWD. Considering the frequent occurrence of tool eccentricity during drilling, it is essential to take into account its impact on the acoustic wavefield outside the borehole. In this paper, we establish an ALWD model with an eccentric collar in a dual-cylindrical coordinate system. By applying the translational Bessel addition theorem and the steepest descent method, we obtain the far-field asymptotic solution of the problem and model the radiation wavefield. Our findings indicate that the asymmetrical radiation pattern resulting from drill collar eccentricity can effectively resolve ambiguities in determining the reflector azimuth.

Time–energy distribution of photoelectron from atomic states with different magnetic quantum numbers in elliptically polarized laser fields

Abstract A Wigner-distribution-like (WDL) function based on the strong-field approximation (SFA) theory is used to investigate the ionization time of the photoelectron emitted from the initial states with different magnetic quantum number m in elliptically polarized electric fields. The saddle-point method is adopted for comparisons. For different m states, a discrepancy exists in the WDL distributions of the photoelectrons emitted in a direction close to the major axis of the laser field ellipse. Based on the saddle-point analysis, this discrepancy can be ascribed to the interference between electrons ionized from two tunneling instants. Our results show that the relationships between the tunneling instants and kinetic energy of photoelectrons are the same for different m initial states when the Coulomb potential is not considered. Our work sheds some light on the ionization-time information of electrons from different magnetic quantum states.

Two-step crystallization in a supersaturated solution with application to protein crystal growth: Theory and analytical solutions

A theory of two-step nucleation and evolution of crystals with allowance for their diffusion in the space of particle radii is developed. The theory is based on a two-step growth law of crystals appearing in dense liquid precursors, initially evolving within them with a diffusion-limited growth law, and then leaving them and evolving with another growth law. This two-step crystal growth law influences the integrodifferential system of kinetic and balance equations for the crystal-size distribution function and liquid supersaturation. The system of integrodifferential equations for the evolution of polydisperse ensemble of crystals is solved using the integral Laplace transform and saddle-point methods for the Weber–Volmer–Frenkel–Zel’dovich and Meirs kinetic mechanisms. A complete analytical solution to this non-linear system has been constructed in a parametric form. Namely, the particle-size distribution function, liquid supersaturation and time have been found as the functions of decision variable — the maximum size of crystals. The theory is in agreement with the experimental data on the growth of beta-lactoglobulin and insulin crystals.

Approximation for probability of coverage in softly inhibitive cellular networks

AbstractThe Strauss point process is a very popular model for describing the random cellular networks, yet several key statistical properties such as intensity, empty space function, and probability generating functional have remained elusive. This article addresses these issues by first leveraging the Poisson saddle point method to approximate the distance‐conditioned intensity for Strauss point processes. Subsequently, the author derives an analytically tractable expression for the distribution of empty space distance based on a conditional thinning mechanism. Additionally, the author establishes an upper bound for the probability generating functional in Strauss point processes, which is crucial for evaluating the Laplace transform of cumulative interference in relevant cellular networks. These findings facilitate the systematic derivation of spatially averaged probability of coverage, and the accuracy of analytic results is validated through simulations.

On the use of the cumulant generating function for inference on time series

Innovative inference procedures for analyzing time series data are introduced. The methodology covers density approximation and composite hypothesis testing based on Whittle's estimator, which is a widely applied M-estimator in the frequency domain. Its core feature involves the cumulant generating function of Whittle's score obtained using an approximated distribution of the periodogram ordinates. A testing algorithm not only significantly expands the applicability of the state-of-the-art saddlepoint test, but also maintains the numerical accuracy of the saddlepoint approximation. Connections are made with three other prevalent frequency domain techniques: the bootstrap, empirical likelihood, and exponential tilting. Numerical examples using both simulated and real data illustrate the advantages and accuracy of the saddlepoint methods.

Co-seismic and post-seismic slip associated with the 2021 Mw5.9 Arkalochori, Central Crete (Greece) earthquake constrained by geodetic data and aftershocks

The co-seismic and post-seismic deformation field associated with the Mw5.9 Arkalochori main shock that occurred in central Crete (Greece) on 27 September 2021 is analyzed using Copernicus Sentinel-1A & 1B images, GNSS measurements and seismological data. The fault geometry is constrained through the joint inversion of multiple datasets and the slip distribution for the co-seismic and post-seismic period is obtained using a homogeneous half-space elastic model and the Steepest Descent Method. The results indicate a blind normal fault striking 215° with a 55° dip to the northwest and the co-seismic slip model suggests a nearly circular main slip patch (8 × 6 km2) with a maximum slip of 0.98 m. Post-seismic displacements started rapidly after the main shock followed by a gradual decay as highlighted by the calculated InSAR time series. The temporal evolution of post-seismic slip is described by a simple logarithmic function, decaying faster at the southwest part of the fault. The cumulative afterslip model suggests that the maximum post-seismic slip of 0.23 m occurred within a similar depth range compared to the co-seismic one, yet with a shift towards the southwest. Post-seismic slip inside the main shock rupture area is sustained, highlighting the slow recovery of locking in the co-seismic slip region. Afterslip (seismic or aseismic) played a dominant role in the early post-seismic period acting complementarily to the main rupture. Indications suggest that the spatiotemporal evolution of the productive aftershock sequence may be driven afterslip, alongside other potential factors.

Soliton resolution for the modified short pulse equation with the weighted Sobolev initial data

In this paper, the Cauchy problem of the modified short pulse (mSP) equation with initial conditions in weighted Sobolev space is studied by using [Formula: see text]-steepest descent method. Based on the spectral analysis of Lax pair and the transformations of field variables and independent variables, a basic Riemann–Hilbert problem (RHP) is established, and the solution of Cauchy problem for the mSP equation is transformed into the corresponding RHP. The asymptotic expansion of the solution of mSP equation is derived in a fixed space-time cone [Formula: see text] According to this, the soliton resolution conjecture of the mSP equation is given, in which the leading term is characterized by an [Formula: see text]-soliton on the discrete spectrum, the second term comes from the soliton–radiation interactions on the continuum spectrum, and the error term [Formula: see text] is generated by the corresponding [Formula: see text]-problem.

Wave equation on general noncompact symmetric spaces

abstract: We establish sharp pointwise kernel estimates and dispersive properties for the wave equation on noncompact symmetric spaces of general rank. This is achieved by combining the stationary phase method and the Hadamard parametrix, and in particular, by introducing a subtle spectral decomposition, which allows us to overcome a well-known difficulty in higher rank analysis, namely the fact that the Plancherel density is not a differential symbol in general. Consequently, we deduce the Strichartz inequality for a large family of admissible pairs and prove global well-posedness results for the corresponding semi-linear equation with low regularity data as on hyperbolic spaces.

Geometric Amplitude Accompanying Local Responses: Spinor Phase Information from the Amplitudes of Spin-Polarized STM Measurements.

Solving the Hamiltonian of a system yields the energy dispersion and eigenstates. The geometric phase of the eigenstates generates many novel effects and potential applications. However, the geometric properties of the energy dispersion go unheeded. Here, we provide geometric insight into energy dispersion and introduce a geometric amplitude, namely, the geometric density of states (GDOS) determined by the Riemann curvature of the constant-energy contour. The geometric amplitude should accompany various local responses, which are generally formulated by the real-space Green's function. Under the stationary phase approximation, the GDOS simplifies the Green's function into its ultimate form. In particular, the amplitude factor embodies the spinor phase information of the eigenstates, favoring the extraction of the spin texture for topological surface states under an in-plane magnetic field through spin-polarized STM measurements. This work opens a new avenue for exploring the geometric properties of electronic structures and excavates the unexplored potential of spin-polarized STM measurements to probe the spinor phase information of eigenstates from their amplitudes.

An Efficient Scattering Center Extraction Method Based on Stationary Phase Method With Quadratic Patch Models

An Efficient Scattering Center Extraction Method Based on Stationary Phase Method With Quadratic Patch Models

Generalized Linear Model (GLM) Applications for the Exponential Dispersion Model Generated by the Landau Distribution

The exponential dispersion model (EDM) generated by the Landau distribution, denoted by EDM-EVF (exponential variance function), belongs to the Tweedie scale with power infinity. Its density function does not have an explicit form and, as of yet, has not been used for statistical aspects. Out of all EDMs belonging to the Tweedie scale, only two EDMs are steep and supported on the whole real line: the normal EDM with constant variance function and the EDM-EVF. All other absolutely continuous steep EDMs in the Tweedie scale are supported on the positive real line. This paper aims to accomplish an overall picture of all generalized linear model (GLM) applications belonging to the Tweedie scale by including the EDM-EVF. This paper introduces all GLM ingredients needed for its analysis, including the respective link function and total and scaled deviance. We study its analysis of deviance, derive the asymptotic properties of the maximum likelihood estimation (MLE) of the covariate parameters, and obtain the asymptotic distribution of deviance, using saddlepoint approximation. We provide numerical studies, which include estimation algorithm, simulation studies, and applications to three real datasets, and demonstrate that GLM using the EDM-EVF performs better than the linear model based on the normal EDM. An R package accompanies all of these.