Steepest Descent Method Research Articles

PERTURBATIONS FAR FIELDS OF THE INTERFACE SURFACE OF THE DEEP OCEAN AND THE ICE COVER FROM LOCALIZED SOURCES

The problem of far wave fields arising at the interface between ice and an infinitely deep homogeneous liquid in the flow around a localized source of perturbations is solved. An integral representation of the solution is obtained and, using the stationary phase method, an asymptotic representation of the solution is constructed for various modes of wave generation. Numerical calculations show that with a change in the flow velocities and ice thickness, a noticeable qualitative rearrangement of the phase patterns of the excited far wave fields at the interface between ice and liquid occurs.

Two-dimensional study of Rossby waves generated by an initial disturbance

Rossby waves are the fundamental time dependent large-scale motion of the ocean and atmosphere. In this article, we are interested to find analytical solution of Rossby waves produced by an initial disturbance. These waves, as we know, are archetypical low-frequency waves, which play a central role in defining meteorological and oceanographic conditions. Our purpose here is to analyze the properties of such waves in two dimension as a guide to the construction of the dynamical framework appropriate for low-frequency motion. As a precursor to our aim, we revisited the problem of Rossby wave generation in an ocean bounded by two imaginary latitudes for a particular type of external forcing. The characteristics of this motion are explored with the help of group velocity and phase velocity. Subsequently, the two-dimensional case of generation of Rossby wave by a general form of disturbance in an unbounded ocean is solved where the concept of stationary phase method is extensively used. The rules of governing the propagation of wave packet are discussed, and we found for a wave-like disturbance, Rossby wave is not a single wave train but rather several, which moves as a wave front with a definite direction-deviation. It is natural to question what role, if any, the source plays in explicit determination of these radiated waves and their asymptotic behavior. To address these and other concerns, we have developed an analytical model of generation of Rossby waves produced by an initial disturbance in two-dimensional open ocean with a considerable generality.

Estimating linear mixed effect models with non-normal random effects through saddlepoint approximation and its application in retail pricing analytics

ABSTRACT Linear Mixed Effects (LME) models are powerful statistical tools that have been employed in many different real-world applications such as retail data analytics, marketing measurement, and medical research. Statistical inference is often conducted via maximum likelihood estimation with Normality assumptions on the random effects. Nevertheless, for many applications in the retail industry, it is often necessary to consider non-Normal distributions on the random effects when considering the unknown parameters' business interpretations. Motivated by this need, a linear mixed effects model with possibly non-Normal distribution is studied in this research. We propose a general estimating framework based on a saddlepoint approximation (SA) of the probability density function of the dependent variable, which leads to constrained nonlinear optimization problems. The classical LME model with Normality assumption can then be viewed as a special case under the proposed general SA framework. Compared with the existing approach, the proposed method enhances the real-world interpretability of the estimates with satisfactory model fits.

Gravitational-Wave Phasing of Quasicircular Compact Binary Systems to the Fourth-and-a-Half Post-Newtonian Order.

The inspiral phase of gravitational waves emitted by spinless compact binary systems is derived through the fourth-and-a-half post-Newtonian (4.5PN) order beyond quadrupole radiation, and the leading amplitude mode (ℓ,m)=(2,2) is obtained at 4PN order. We also provide the radiated flux, as well as the phase in the stationary phase approximation. Rough numerical estimates for the contribution of each PN order are provided for typical systems observed by current and future gravitational wave detectors.

Generalization of the Ramanujan's integrals for the Volterra μ-functions via complex contours: representations and approximations

In this paper, we consider the inverse Laplace transform of the Volterra μ-function (the Bromwich integral) and evaluate it with respect to the Hankel, Schläfli and Bourguet complex contours. In this sense, we establish the generalized Ramanujan's integral representations of the Volterra μ-function for general variations of the parameters. We also discuss the asymptotic analysis of this function with large parameters using the steepest descent method. Further, we show that the solution of Volterra integral equation with differentiated-order fractional integral operator is the Volterra μ-function.

Nuclear level density in the statistical semiclassical micro-macroscopic approach

Level density ρ is derived for a finite system with strongly interacting nucleons at a given energy E, neutron N, and proton Z particle numbers, projection of the angular momentum M, and other integrals of motion, within the semiclassical periodic-orbit theory (POT) beyond the standard Fermi-gas saddle-point method. For large particle numbers, one obtains an analytical expression for the level density which is extended to low excitation energies U in the statistical micro-macroscopic approach (MMA). The interparticle interaction averaged over particle numbers is taken into account in terms of the extended Thomas - Fermi component of the POT. The shell structure of spherical and deformed nuclei is taken into account in the level density by the Strutinsky shell correction method through the mean-field approach used near the Fermi energy surface. The MMA expressions for the level density ρ reaches the well-known macroscopic Fermi-gas asymptote for large excitation energies U and the finite combinatoric power-expansion limit for low energies U. We compare our MMA results for the averaged level density with the experimental data obtained from the known excitation energy spectra by using the sample method under statistical and plateau conditions. Fitting the MMA ρ to these experimental data on the averaged level density by using only one free physical parameter - inverse level density parameter K - for several nuclei and their long isotope chain at low excitation energies U one obtains the results for K. These values of K might be much larger than those deduced from neutron resonances. The shell, isotopic asymmetry, and pairing effects are significant for low excitation energies.

On the long-time asymptotic for a coupled generalized nonlinear Schrödinger equations with weighted Sobolev initial data

From the matrix ∂̄ problem, we propose a novel integrable coupled hierarchy with the application of recursive operator Λn which includes a coupled generalized nonlinear Schrödinger (cgNLS) equations with its Lax pair as n=0. We successfully derive infinite conservation laws and Hamiltonian function of the cgNLS equations for the first time. Furthermore, we employ the ∂̄ steepest descent method in order to study the Cauchy problem of the cgNLS equations with initial conditions in weighted Sobolev space H1,1(R)=H1(R)∩L2,1(R). In a fixed space–time cone S(x1,x2,v1,v2)={(x,t)∈R2:x=x0+vt,x0∈[x1,x2],v∈[v1,v2]}, the long-time asymptotic behavior of the solutions u(x,t) and v(x,t) is derived. Based on the results of asymptotic behavior, we prove the soliton resolution conjecture of the cgNLS equations which consist of three terms, the soliton term confirmed by |Z(I)|-soliton on discrete spectrum, the t−12 order term on continuous spectrum and the residual error up to O(t−34).

A reliability-based design and optimization strategy using a novel MPP searching method for maritime engineering structures

PurposeIn order to solve the problems faced by First Order Reliability Method (FORM) and First Order Saddlepoint Approximation (FOSA) in structural reliability optimization, this paper aims to propose a new Reliability-based Design Optimization (RBDO) strategy for offshore engineering structures based on Original Probabilistic Model (OPM) decoupling strategy. The application of this innovative technique to other maritime structures has the potential to substantially improve their design process by optimizing cost and enhancing structural reliability.Design/methodology/approachIn the strategy proposed by this paper, sequential optimization and reliability assessment method and surrogate model are used to improve the efficiency for solving RBDO. The strategy is applied to the analysis of two marine engineering structure cases of ship cargo hold structure and frame ring of underwater skirt pile gripper. The effectiveness of the method is proved by comparing the original design and the optimized results.FindingsIn this paper, the proposed new RBDO strategy is used to optimize the design of the ship cargo hold structure and the frame ring of the underwater skirt pile gripper. According to the results obtained, compared with the original design, the structure of optimization design has better reliability and stability, and reduces the risk of failure. This optimization can also better balance the relationship between performance and cost. Therefore, it is recommended for related RBDO problems in the field of marine engineering.Originality/valueIn view of the limitations of FORM and FOSA that may produce multiple MPPs for a single performance function, the new RBDO strategy proposed in this study provides valuable insights and robust methods for the optimization design of offshore engineering structures. It emphasizes the importance of combining advanced MPP search technology and integrating SORA and surrogate models to achieve more economical and reliable design.

Linear rank tests for left-truncated data using randomized block design: saddlepoint p-values and confidence intervals

Survival analysis is most typically characterized by left truncation; left-truncated data are composed of two recorded times, the time of occurrence of an event before the event of interest, and the time of the interested event. In this study, p-values of the permutation distribution of the linear rank tests are computed using saddlepoint approximation method for left-truncated data using a randomized block design. Simulation studies demonstrate the saddlepoint approximation method’s accuracy in approximating p-value of the linear rank tests. Due to the efficiency of the proposed method over the normal approximation, 95% confidence intervals for the treatment impact are performed.

MIMO-SA-Based 3-D Image Reconstruction of Targets Under Illumination of Terahertz Gaussian Beam—Theory and Experiment

For the 3-D terahertz (THz) holographic imaging, this work proposes an extended phase-shift migration (PSM) reconstruction algorithm based on multiple-input-multiple-output (MIMO) array with an orthogonal synthetic aperture (SA). For this MIMO-SA-based imaging mechanism, the accurate expression of the signal spatial spectrum can also be obtained by utilizing the method of stationary phase (MSP). Meanwhile, the analytical expression of the reconstructed 3-D point-spread function (PSF) for target under the illumination of MIMO-SA-based THz Gaussian beam is derived to provide the theoretical spatial resolutions quantitatively. A serial of simulation results with fairly good agreement were given to verify the accuracy of the proposed MIMO-SA-based PSM reconstruction algorithm. Meanwhile, due to the PSM’s characteristic of phase shift compensation along the range direction, the image reconstruction process can also be implemented and accelerated by a GPU solution. Finally, two types of MIMO-SA-based prototype imagers are applied for the proof-of-principle experiments. The first prototype imager is configured with one transmitter and two receivers in 0.3-THz band. The MIMO linear array including arbitrary number of transmitters/receivers can be realized by controlling their positions in a low-cost way with the help of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x$</tex-math> </inline-formula> -axis of a 2-D scanner. The second prototype imager is configured with eight transmitters and eight receivers in 0.2-THz band. The experimental results from the two prototype imagers can validate the effectiveness on image reconstruction quality.

Geometric triangulations and the Teichmüller TQFT volume conjecture for twist knots

We construct a new infinite family of ideal triangulations and H–triangulations for the complements of twist knots, using a method originating from Thurston. These triangulations provide a new upper bound for the Matveev complexity of twist knot complements. We then prove that these ideal triangulations are geometric. The proof uses techniques of Futer and the second author, which consist in studying the volume functional on the polyhedron of angle structures. Finally, we use these triangulations to compute explicitly the partition function of the Teichmüller TQFT and to prove the associated volume conjecture for all twist knots, using the saddle point method.

Quantum evolution with random phase scattering

We consider the quantum evolution of a fermion–hole pair in a d-dimensional gas of non-interacting fermions in the presence of random phase scattering. This system is mapped onto an effective Ising model, which enables us to show rigorously that the probability of recombining the fermion and the hole decays exponentially with the distance of their initial spatial separation. In contrast, without random phase scattering the recombination probability decays like a power law, which is reflected by an infinite mean square displacement. The effective Ising model is studied within a saddle point approximation and yields a finite mean square displacement that depends on the evolution time and on the spectral properties of the deterministic part of the evolution operator.

Differential dynamic programming for finite‐horizon zero‐sum differential games of nonlinear systems

AbstractIn this article, we present an iterative algorithm based on differential dynamic programming (DDP) for finite‐horizon two‐person zero‐sum differential games. The technique of DDP is used to expand the Hamilton–Jacobi–Isaacs (HJI) partial differential equation into higher‐order differential equations. Using value function and saddle point approximations, the DDP expansion is transformed into algebraic matrix equation in integral form. Based on the algebraic matrix equation, a DDP iterative algorithm is developed to learn the solution to the differential games. Strict proof is proposed to guarantee the iterative convergences of the value function and saddle point. The new algorithm is fundamentally different from existing results, in the sense that it overcome the technical obstacle to address the time‐varying behavior of HJI partial differential equation. Simulation examples are given to demonstrate the effectiveness of the proposed method.

Lifting Re-entry Trajectory Optimization

Trajectory optimization of a lypical RLV technology demonstrator is formulated as an optimal control problem. Both planar and out of plane trajectory optimization is carried out during re-entry phase using steepest descent method. Dynamic pressure and lateral acceleration are used as inequality constraints. Terminal equality conslraints on altitude and velocity are also specifed. Constraints qre included in the performance index as exterior penatty functions. Detailed formulations of the optimal control problem along with results are presented in this paperfor some test cases.

Sound source identification using the boundary element method in a sparse framework

Traditional near-field acoustic holography (NAH) based on the boundary element method (BEM) exhibits good performance for sound source identification. However, good reconstruction results can only be guaranteed if a sufficient number of sampling points are used. In this paper, the unitary matrix is used as the sparse basis matrix of source surface velocity and is obtained from the transfer matrix between the source surface sound pressure and vibration velocity by singular value decomposition (SVD). Based on this sparse basis, a sparse Bayesian learning (SBL) algorithm is introduced into BEM-based NAH to address the case of few sampling points. The use of a standard SBL algorithm in BEM-based NAH leads to hyperparameters that are overly sparse due to the large number of iterations, which results in large reconstruction errors. To solve this problem, a modified SBL algorithm is proposed. Good reconstruction results are maintained by selecting appropriate hyperparameters and revising the results using the steepest descent method. A simulated cuboid shell is used to explore the influence of the (i) number of sampling points and (ii) signal-to-noise ratio (SNR) on the reconstruction. Numerical results show that the proposed method can achieve effective reconstruction when the number of sampling points is approximately one-tenth of the number of discrete points on the source surface. The validity of the proposed method is verified by a comparison with experimental results.

Bootstrapping through discrete convolutional methods

AbstractBootstrapping was designed to randomly resample data from a fixed sample using Monte Carlo techniques. However, the original sample itself defines a discrete distribution. Convolutional methods are well suited for discrete distributions, and we show the advantages of utilizing these techniques for bootstrapping. The discrete convolutional approach can provide exact numerical solutions for bootstrap quantities, or at least mathematical error bounds. In contrast, Monte Carlo bootstrap methods can only provide confidence intervals which converge slowly. Additionally, for some problems the computation time of the convolutional approach can be dramatically less than that of Monte Carlo resampling. This article provides several examples of bootstrapping using the proposed convolutional technique and compares the results to those of the Monte Carlo bootstrap, and to those of the competing saddlepoint method.

The birth of the strong components

AbstractIt is known that random directed graphs undergo a phase transition around the point . Earlier, Łuczak and Seierstad have established that as when , the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases from 1 to 0 as goes from to . By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of and provide more statistical insights into the structure of a random digraph around, below and above its transition point. We obtain the limiting probability that a random digraph is acyclic and the probability that it has one strongly connected complex component with a given difference between the number of edges and vertices (called excess). Our result can be extended to the case of several complex components with given excesses as well in the whole range of sparse digraphs. Our study is based on a general symbolic method which can deal with a great variety of possible digraph families, and a version of the saddle point method which can be systematically applied to the complex contour integrals appearing from the symbolic method. While the technically easiest model is the model of random multidigraphs, in which multiple edges are allowed, and where edge multiplicities are sampled independently according to a Poisson distribution with a fixed parameter , we also show how to systematically approach the family of simple digraphs, where multiple edges are forbidden, and where 2‐cycles are either allowed or not. Our theoretical predictions are supported by numerical simulations when the number of vertices is finite, and we provide tables of numerical values for the integrals of Airy functions that appear in this study.

Black hole entropy contributions from Euclidean cores

The entropy of a Schwarzschild black hole, as computed via the semiclassical Euclidean path integral in a stationary phase approximation, is determined not by the on-shell value of the action (which vanishes), but by the Gibbons–Hawking–York boundary term evaluated on a suitable hypersurface, which can be chosen arbitrarily far away from the horizon. For this reason, the black hole singularity seemingly has no influence on the Bekenstein–Hawking area law. In this paper, we estimate how a regular black hole core, deep inside a Euclidean black hole of mass [Formula: see text] and generated via a UV regulator length scale [Formula: see text], affects the black hole entropy. The contributions are suppressed by factors of [Formula: see text]; demanding exact agreement with the area law as well as a self-consistent first law of black hole thermodynamics at all orders, however, demands that these contributions vanish identically via uniformly bounded curvature. This links the limiting curvature hypothesis to black hole thermodynamics.

Exchange effects in strong-field ionization of the excited lithium ion induced by one- and two-component laser fields

Spin-dependent effects in strong-laser-field-induced above-threshold ionization of excited Li+ ions, caused by the requirement that the electron wave function is antisymmetric, are analyzed using the strong-field approximation and saddle-point method. For an excited Li+ ion exposed to a linearly polarized laser field, the minima in the photoelectron momentum distribution in the polarization plane appear if the excited Li+ ion state is the spin singlet state, while for the spin triplet state these minima are absent. The difference between the spectra obtained with these spin states is quantified by the corresponding normalized difference of the differential ionization probabilities. Employing the saddle-point method, we show that, for the spin singlet state, all relevant contributions to the differential ionization probability exhibit minima for approximately the same values of the photoelectron energy and emission angle, thus leading to the minima in the total spectra. Similar conclusions hold for a bicircular driving field. In this case, the range of values of the photoelectron energy and emission angle for which different saddle-point contributions exhibit minima is almost the same for all saddle-point solutions. This is particularly true for the high-energy part of the spectrum, and the minima are more pronounced than for the linearly polarized driving field case. In order to check whether these minima can be detected in an experiment, we perform focal averaging, which takes into account the intensity distribution in the laser focus. For both linearly polarized and bicircular driving fields, the minima survive the focal-averaging procedure. They are slightly blurred in the former case, while in the latter case the focal averaging almost does not affect the minima at all. Finally, we confirm that similar conclusions hold for the photoelectron velocity maps in the plane that contains the laser-field propagation direction.

Modified saddle-point method applied to direct ionization of noble gas atoms

When exposed to strong laser fields, atoms or molecules can absorb more photons from the laser field than is necessary for ionization. This process is called above-threshold ionization (ATI). In analyzing this process, the strong-field approximation (SFA) turns out to be a very useful theoretical tool. In the SFA the differential ionization rate, which is an observable quantity, can be expressed as an integral over the ionization time and can be calculated by numerical integration (NI) or using the saddle-point method (SPM). When we use the Slater orbitals to describe the ground-state wave function of the valence electron, the results obtained using the SPM and NI do not agree. We find the reasons for this disagreement and introduce a modified SPM that leads to excellent agreement between the SPM and NI results for various strong laser fields.