Steepest Descent Method Research Articles

On the Distribution of the Random Sum and Linear Combination of Independent Exponentiated Exponential Random Variables

The exponentiated exponential distribution has received great attention from many statisticians due to its popularity, many applications, and the fact that it is an efficient alternative to many famous distributions such as Weibull and gamma distributions. Many statisticians have studied the mathematical properties of this distribution and estimated its parameters under different censoring schemes. However, it seems that the distribution of the random sum, the distribution of the linear combination, and the value of the reliability index, R=P(X2<X1), in the case of unequal scale parameters, were not known for this distribution. Therefore, in this article, we present the saddlepoint approximation to the distribution of the random sum, the distribution of linear combination, and the value of the reliability index R=P(X2<X1) for exponentiated exponential variates. These saddlepoint approximations are computationally appealing, and numerical studies confirm their accuracy. In addition to the accuracy provided by the saddlepoint approximation method, it saves time compared to the simulation method, which requires a lot of time. Therefore, the saddlepoint approximation method provides an outstanding balance between precision and computational efficiency.

Contributions of different quantum pathways to nonsequential double ionization by monochromatic and bichromatic laser fields

Abstract When nonsequential double ionization is treated using the strong-field approximation and the saddle-point method, the transition amplitude can be expressed as a coherent sum of the partial amplitudes corresponding to different saddle-point (SP) solutions. For the case of the recollision excitation with subsequent ionization (RESI) mechanism of the nonsequential double ionization, we examine the partial contributions of the SP solutions which correspond to the electron responsible for the excitation. For a monochromatic linearly polarized laser field, we find that, in addition to the pair of the SP solutions with the shortest travel time, other SP solutions may also make a significant contribution to the photoelectron yield. Moreover, the SP solutions appear in pairs and exhibit notable modifications in comparison to those observed in high-order above-threshold ionization. Furthermore, for a bichromatic linearly polarized driving field, we investigate the intensity range obtained using the simpleman’s model for which the RESI mechanism is dominant. We find that this range must be modified if the photoelectron yield corresponding to the SP solution for which the photoelectron has the highest energy upon return to the parent ion is small. This is particularly the case for the excitation channels involving loosely bound excited states.

A Novel Approximation Method for Solving Ordinary Differential Equations Using the Representation of Ball Curves

Numerical methods are frequently developed to investigate concepts for approximately solving ordinary differential equations (ODEs). To achieve minimal error and higher accuracy in approximate solutions, researchers have focused on developing algorithms using various numerical techniques. This study proposes the application of Ball curves, specifically the Said–Ball curve, for estimating solutions to higher-order ODEs. To obtain the best control points of the Said–Ball curve, the least squares method is used. These control points are calculated by minimizing the residual error through the sum of the squares of the residual functions. To demonstrate the proposed method, several boundary value problems are presented, and their performance is compared with existing methods in terms of error accuracy. The numerical results indicate that the proposed method improves error accuracy compared to existing studies, including those employing Bézier curves and the steepest descent method.

Statistical methods for extremely unbalanced data in genome-wide association study (2)

Extremely unbalanced data refers to datasets with independent or dependent variables showing severe imbalances in proportions, which might lead to deviation of classical test statistics from theoretical distribution and difficulties in controlling type Ⅰ error. The increased availability of genome-wide resources from large population cohorts has highlighted the growing demand for efficient and accurate statistical methods for the process of extremely unbalanced data to improve the development of genetic statistical methods. This paper introduces two widely used correction methods in current genome-wide association study for extremely unbalanced data, i.e. Firth correction and saddle point approximation, describes their effectiveness in controlling type Ⅰ errors confirmed by simulation experiments, finally, and summarizes the commonly used software for extremely unbalanced genomic data to provide theoretical reference and suggestion for its application for the statistical analysis on extremely unbalanced data in future.

Saddle-point approximation to the false vacuum decay at finite temperature in one-dimensional quantum mechanics

Abstract We calculate the false-vacuum decay rate in one-dimensional quantum mechanics on the basis of the saddle-point approximation in the Euclidean path integral at finite temperature. The saddle points are the finite-T and shifted bounce solutions, which are finite-period analogs of the (zero-temperature) bounce solution, and the shot solutions. We re-examined the zero-temperature result by Callan and Coleman and compare with the zero-temperature limit of our results. We also perform some numerical calculations to illustrate the temperature dependence of the decay rate and compare it with the result by Affleck.

A research framework for seismic slip distribution inversion considering atmospheric effects and deformation field downsampling

ABSTRACT Traditional Interferometric Synthetic Aperture Radar (InSAR) methods have difficulties in capturing deformation information of small and medium-sized earthquakes due to small surface deformation and atmospheric interference. This study proposes a research framework for seismic slip distribution inversion considering atmospheric effects and deformation field downsampling. The research framework begins by generating multiple interferometric pairs, using the coherence thresholding method to select stable points as reference during phase unwrapping. This step includes averaging the unwrapped pairs post-phase superposition to mitigate atmospheric turbulence effects in the interferograms and effectively extract the faint coseismic deformation field. Additionally, the framework employs a saliency-based quadtree downsampling algorithm to differentiate between major and minor deformation zones, thereby streamlining the data associated with the deformation field. The Bayesian method is then used to determine the geometric parameters of the earthquake fault, with the steepest descent method (SDM) applied to invert the fault slip distribution. The methodology was tested using the March 20, 2020, Ms5.9 earthquake in Dingri County, Tibet, China as a case study. Results demonstrate that this approach reduces atmospheric influences in the deformation field and enhances the accuracy of fault slip inversion, showing high consistency with existing studies and validating the reliability of the proposed method.

Defect Conformal Field Theory from Sachdev-Ye-Kitaev Interactions.

The coupling between defects and extended critical degrees of freedom gives rise to the intriguing theory known as defect conformal field theory (CFT). In this work, we introduce a novel family of boundary and interface CFTs by coupling N Majorana chains with SYK_{q} interactions at the defect. Our analysis reveals that the interaction with q=2 constitutes a new marginal defect. Employing a versatile saddle-point method, we compute unique entanglement characterizations, including the g function and effective central charge, of the defect CFT. Furthermore, we analytically evaluate the transmission coefficient using CFT techniques. Surprisingly, the transmission coefficient deviates from the universal relation with the effective central charge across the defect at the large N limit, suggesting that our defect CFT extends beyond all known examples of Gaussian defect CFT.

Stabilizing complex-Langevin field-theoretic simulations for block copolymer melts.

Complex-Langevin field-theoretic simulations (CL-FTSs) provide an approximation-free method of calculating fluctuation corrections to the self-consistent field theory (SCFT) of block copolymer melts. However, the complex fields are prone to the formation of hot spots, which causes the method to fail. This problem has been attributed to an invariance under complex translations, which allows the system to drift away from the real-valued saddle-point of SCFT. Here, we apply dynamical stabilization to CL-FTSs of diblock copolymer melts, whereby the drift is suppressed by a small imaginary force on the composition field. The force needs to be sufficient to hold the system near the real saddle-point but also small enough not to significantly bias the statistics. Although larger forces are required as the fluctuations become more intense, we are able to lower the invariant polymerization indices of the CL-FTSs by several orders of magnitude before this becomes a problem. The new CL-FTS results are then used to test conventional Langevin simulations (L-FTSs), in which the instability is removed by a partial saddle-point approximation to the pressure field. As found previously, the L-FTSs agree accurately with the CL-FTSs, provided that the comparison is performed using a Morse calibration.

Partial absence of cosine problem in 3D Lorentzian spin foams

Abstract We study the semi-classical limit of the recently proposed coherent spin foam model for (2+1) Lorentzian quantum gravity. Specifically, we analyze the gluing equations derived from the stationary phase approximation of the vertex amplitude. Typically these exhibit two solutions yielding a cosine of the Regge action. However, by inspection of the algebraic equations as well as their geometrical realization, we show in this note that the behavior is more nuanced: when all triangles are either spacelike or timelike, two solutions exist. In any other case, only a single solution is obtained, thus yielding a single Regge exponential.

Local Limit Theorems for Hook Lengths in Partitions

AbstractPartition hook lengths have wide-ranging applications in combinatorics, number theory, physics, and representation theory. We study two infinite families of random variables associated with t-hooks. For fixed $$t\ge 1,$$ t ≥ 1 , if $$Y_{t;\,n}$$ Y t ; n counts the number of hooks of length t in a random integer partition of n, we prove a uniform local limit theorem for $$Y_{t;\,n}$$ Y t ; n on any bounded set of $${\mathbb {R}}.$$ R . To achieve this, we establish an asymptotic formula with a power-saving error term for the number of partitions of n with m many t-hooks. In contrast, we define $${\widehat{Y}}_{t;\,n}$$ Y ^ t ; n as the count of hooks divisible by t in a randomly chosen partition of n. While $${\widehat{Y}}_{t;\,n}$$ Y ^ t ; n converges in distribution, we show that it fails to satisfy the local limit theorem for any $$t \ge 2$$ t ≥ 2 . The proofs employ the multivariable saddle-point method, asymptotic formulas for the number of t-core partitions from Anderson and Lulov–Pittel, and estimates of certain exponential sums. Notably, for $$t=4,$$ t = 4 , the analysis involves the asymptotic behavior of class numbers of imaginary quadratic fields.

Time-dependent wave motion in a running stream due to initial disturbances in Magnetohydrodynamics

Abstract The generation of two-dimensional time-dependent magneto-hydrodynamic surface waves in a flowing stream is investigated in this paper. The waves here arise from some initial disturbances at the free surface of an electrically conducting field-free fluid, which is of finite depth. It is also assumed that a constant magnetic field exists outside the air-water interface. With the help of linearised theory, the present problem is modeled as an initial boundary value problem, and the Laplace-Fourier transform techniques are primarily used to obtain the form of free surface elevation through infinite integrals. These integrals are then interpreted asymptotically for large time t and distance x from the origin, using the method of stationary phase. Dispersion relation, phase, and group velocities are also studied. Numerical results of the asymptotic form of free surface elevation are obtained in a number of figures for different values of current speed, Alfven velocity, and various types of initial disturbances. Graphical representations demonstrate that the presence of a magnetic field has a significant effect on wave motion.

Saddle point analysis on selective enhancement of high-order harmonic generation

We theoretically demonstrate control of selective enhancement of the near-cutoff odd order harmonic generation by dual-color chirped laser field. The sharp enhancements of a single harmonic order ranging from H37 to H59 (λ h =21.6nm-13.5nm) are achieved by adjusting the intensity of the fundamental frequency field. The saddle point method is applied to the analysis of high-order harmonic generation using the distorted-waveform laser field. A series of orbits of principal contribution to enhancement are extracted using the saddle point method, and the enhancement of the key orbits’ interference is reconstructed by quantum orbital theory. Through the analysis of the interference mechanism of enhancement, we give a full explanation of the dependence of enhanced harmonic order on laser intensity.

Kelvin’s method of stationary phase?

Kelvin’s method of stationary phase?

Simultaneous Impacts of Correlated Photovoltaic Systems and Fast Electric Vehicle Charging Stations on the Operation of Active Distribution Grids

This paper presents two novel probabilistic models developed to account for the uncertainties of aggregated fast electric vehicle charging stations (FEVCSs) demand and correlated photovoltaic (PV) injections in active distribution network (ADN) analysis. Both models are more precise than the available ones. A probabilistic model based on the Beta distribution is used for solar radiance, while the shared random variables technique is proposed considering correlated solar radiation random variables. Furthermore, a probabilistic negative exponential load model is extended for modeling the FEVCSs based on the Weibull probability density function. Moreover, the proposed probabilistic load flow (PLF) model is solved using the combined cumulants and saddle-point approximation method. Numerical tests are provided and discussed by applying the IEEE 69-bus distribution system for different PV correlation coefficients and FEVCS load models. The results demonstrate how the uncertainty of PLF outputs is increased by integrating FEVCSs and correlated PV resources into the distribution network. In addition, simulation results validate that the cumulants-based methodology provides satisfactory accuracy with a low computational cost.

Regularity and asymptotics of densities of inverse subordinators

AbstractIn this article, densities (and their derivatives) of subordinators and inverse subordinators are considered. Under minor restrictions, generally milder than the existing in the literature, employing a useful modification of the saddle point method, we obtain the large asymptotic behaviour of these densities (and their derivatives) for a specific region of space and time and quantify how the ratio between time and space affects the explicit speed of convergence. The asymptotics is governed by an exponential term depending on the Laplace exponent of the subordinator and the region represents the behaviour of the subordinator when it is atypically small (the inverse one is larger than usual). As a result, a route to the derivation of novel general or particular fine estimates for densities with explicit constants in the speed of convergence in the region of the lower envelope/the law of iterated logarithm is available. Furthermore, under mild conditions, we provide a power series representation for densities (and their derivatives) of subordinators and inverse subordinators. This representation is explicit and based on the derivatives of the convolution of the tails of the corresponding Lévy measure, whose smoothness is also investigated. In this context, the methods adopted are based on Laplace inversion and strongly rely on the theory of Bernstein functions extended to the cut complex plane. As a result, smoothness properties of densities (and their derivatives) and their behaviour near zero immediately follow.

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

In this work, the Cauchy problem for the generalized complex short pulse equation with initial conditions in the weighted Sobolev space H(R) is studied by using the Riemann-Hilbert method and the ∂‾-steepest descent method. Based on the spectral analysis of the Lax pair, the solution of the Cauchy problem can be expressed as solution of a Riemann-Hilbert problem, which is transformed into a solvable model after a series of deformations. Finally, the long-time asymptotics and soliton resolution of the generalized complex short pulse equation in the soliton region are obtained by resorting to the ∂‾-steepest descent method. The results also indicate that the N-soliton solutions of the generalized complex short pulse equation are asymptotically stable.

A saddlepoint approximation for the smoothed periodogram

Poor variance properties of the periodogram often limit its practical applicability to a wide range of modern spectral estimation and detection applications. The smoothed periodogram, a refined periodogram-based method, is one such nonparametric approach to reducing variance. Neighboring spectral samples are averaged across a spectral window, as opposed to the more common temporal or lag window. Tapered spectral windows and other modifications needed to address the time-bandwidth product and resolution-variance trade-offs complicate the statistical analysis, making it difficult to quantify statistical performance. In addition, approximate distributions for the smoothed periodogram require a priori normalization along with simplifying assumptions to yield computationally tractable results. Here, under mild asymptotic conditions, the distribution derived prior to normalization is shown to be computationally intractable in most cases. First-order statistical approximations are computationally stable but result in sizeable inaccuracies, particularly in the tails. We use a saddlepoint approximation, a second-order asymptotic method, that allows for accurate statistical characterization but is also numerically stable. Monte Carlo simulations are used to validate the results and to illustrate the robustness of the approach. Finally, its utility is demonstrated on a real-world dataset relevant to the side-channel and hardware cybersecurity communities.

Steepest descent method for uncertain multiobjective optimization problems under finite uncertainty set

In this paper, a steepest descent method is developed for the robust counterpart of uncertain multiobjective optimization problems under finite uncertainty. The robust counterpart under consideration is the minimum of objective wise worst case, which is a nonsmooth deterministic multiobjective optimization problem. To solve this robust counterpart with the help of the steepest descent method, a subproblem is constructed and solved to find a descent direction. An Armijo-type inexact line search technique is employed to find a suitable step length. With the help of the descent direction and step length, a sequence is generated. The convergence of the proposed method is justified under some common assumptions. Finally, the algorithm is verified and compared with one existing method with the help of some numerical problems.

Conditional Saddle-point Approximations for Bivariate Compound Distributions

The majority of existing research that is related to our study aims to explain phenomena in various fields of application that rely on bivariate random variables [1,2]. Although these distributions have attracted some attention in the literature, little research exists on the bivariate compound distribution due to computational difficulties in implementing it. This study introduces the conditional saddle-point approximation method, which is more powerful than other approximation methods, to the bivariate compound distribution in continuous and discrete settings. We discuss conditional approximations for cumulative distribution functions of bivariate compound distributions. Furthermore, examples of continuous and discrete distributions from the bivariate compound truncated Poisson compound class are presented and comparisons between saddle-point approximations and exact calculations show the great accuracy of the saddle-point methods.

Electric-magnetic duality of dyonic Kerr-Newman-NUT-AdS spacetimes

We study the (anti-)self-duality conditions under which the electric and magnetic parts of the conserved charges of the dyonic Kerr-Newman-NUT-anti-de Sitter solution become equivalent. Within a holographic framework, the stress tensor and the boundary Cotton tensor are computed from the electric/magnetic content of the Weyl tensor. The holographic stress tensor/Cotton tensor duality is recovered along the (anti-)self-dual curve in parameter space. We show that the latter not only implies a duality relation for the mass but also for the angular momentum. The partition function is computed to first order in the saddle-point approximation and a Bogomol’nyi-Prasad-Sommerfield bound is obtained. The ground state of the theory is enlarged to all the (anti-)self-dual configurations when the SO(4) and U(1) Pontryagin densities are introduced. We demonstrate this at the level of the action and variations thereof. Published by the American Physical Society 2024