Steepest Descent Method Research Articles

Role of topographical disturbances on hydroelastic response of a floating membrane over a slippery bottom

Some disruptions occur at the bottom of the sea due to underwater explosions and natural disasters (e.g. tsunamis, submarine earthquakes, landslides, rock and asteroid falls, drastic changes in weather conditions, etc.). The current paper aims to study how a transitory ground disturbance affects the membrane deflection in a viscous fluid over a porous slippery bottom at a finite water depth in the presence of a floating elastic membrane. Three types of ground disturbance are considered such as H 0 ( x ) = δ ( x ) , H 0 ( x ) = e ( − x 2 / 2 ) and H 0 ( x ) = e − | x | Kundu et al. [Waves generated on running stream due to a disturbance on sea bed. Int J Appl Comput Math. 2020;6(2):1–17.] and Maiti and Mandal [Waves generated by bottom disturbance in an ice-covered ocean. Int J Appl Mech Eng. 2007;12(2):477–486.]. The Stokes stream function and wave potential boundary value problems are used to formulate this problem. The membrane deflection is obtained as infinite integrals using the Fourier, and Laplace transforms. These integrals are then evaluated using the steepest descent method. The effects of the bottom slip parameter, the fluid's viscosity, and the floating membrane's tension and mass on the membrane deflection are analyzed. The study reveals that the amplitude of the membrane deflection decreases, and the oscillatory pattern of the deflection curve is changed by reducing the value of viscosity. Furthermore, the membrane's deflection amplitude decreases with an increase in the values of mass and tension of the floating membrane. Moreover, the slippery bottom plays a vital role in reducing the amplitude of the membrane deflection.

Convergence and complexity guarantees for a wide class of descent algorithms in nonconvex multi-objective optimization

We address conditions for global convergence and worst-case complexity bounds of descent algorithms in nonconvex multi-objective optimization. Specifically, we define the concept of steepest-descent-related directions. We consider iterative algorithms taking steps along such directions, selecting the stepsize according to a standard Armijo-type rule. We prove that methods fitting this framework automatically enjoy global convergence properties. Moreover, we show that a slightly stricter property, satisfied by most known algorithms, guarantees the same complexity bound of O(ϵ−2) as the steepest descent method.

Modified saddle-point method applied to high-order above-threshold ionization and high-order harmonic generation: Slater-type versus asymptotic ground-state wave function

Modified saddle-point method applied to high-order above-threshold ionization and high-order harmonic generation: Slater-type versus asymptotic ground-state wave function

Stochastic path power and the Laplace transform

Transition probabilities for stochastic systems can be expressed in terms of a functional integral over paths taken by the system. Approximately evaluating this integral by the saddle point method in the weak-noise limit leads to a remarkable mapping between dominant stochastic paths through the potential V and conservative, Hamiltonian mechanics with an effective potential −|∇V|2 . The conserved ‘energy’ in this effective system has dimensions of power. We show that this power, H, can be identified with the Laplace parameter of the time-transformed dynamics. As H → 0, corresponding to the long-time limit, the equilibrium Boltzmann density is recovered. However, keeping H finite leads to insights into the non-equilibrium behaviour of the system. Moreover, it facilitates the explicit summation over families of trajectories, which is far harder in the time domain, and turns out to be essential for making contact with the long-time limit in some cases. We illustrate the validity of these results using simple examples that can be explicitly solved by other means.

Molecular dynamics simulation of the effect of temperature on the conformation of ubiquitin protein.

Ubiquitin, a ubiquitous small protein found in all living organisms, is crucial for tagging proteins earmarked for degradation and holds pivotal importance in biomedicine. Protein functionality is intricately linked to its structure. To comprehend the impact of diverse temperatures on ubiquitin protein structure, our study delved into the energy landscape, hydrogen bonding, and overall structural stability of ubiquitin protein at varying temperatures. Through meticulous analysis of root mean square deviation and root mean square fluctuation, we validated the robustness of the simulation conditions employed. Within our simulated system, the bonding energy and electrostatic potential energy exhibited linear augmentation, while the van der Waals energy demonstrated a linear decline. Additionally, our findings highlighted that the α-Helix secondary structure of the ubiquitin protein gradually transitions toward helix destabilization under high-temperature conditions. The secondary structure of ubiquitin protein experiences distinct changes under varying temperatures. The outcomes of our molecular simulations offer a theoretical framework that enhances our comprehension of how temperature impacts the structural stability of ubiquitin protein. These insights contribute not only to a deeper understanding of iniquity's behavior but also hold broader implications in the realm of biomedicine and beyond. All the MD simulations were performed using the GROMACS software with GROMOS96 force field and SPC for water. The ubiquitin protein was put in the center of a cubic box with a length of 8nm, a setting that allowed > 0.8nm in the minimal distance between the protein surface and the box wall. To remove the possible coordinate collision of the configurations, in the beginning, the steepest descent method was used until the maximum force between atoms was under 100kJ/mol·nm with a 0.01nm step size. Minimization was followed by 30ps of position-restrained MD simulation. The protein was restrained to its initial position, and the solvent was freely equilibrated. The product phase was obtained with the whole system simulated for 10ns without any restraint using an integral time step of 1fs with different temperatures. The cutoff for short-range electronic interaction was set to 1.5nm. The long-range interactions were treated with a particle-mesh Ewald (PME) method with a grid width of 1.2nm.

Sub-cycle strong-field tunneling dynamics in solids

Tunneling ionization is a crucial process in the interaction between strong laser fields and matter which initiates numerous nonlinear phenomena including high-order harmonic generation, photoelectron holography, etc. Both adiabatic and nonadiabatic tunneling ionization are well understood in atomic systems. However, the tunneling dynamics in solids, especially nonadiabatic tunneling, has not yet been fully understood. Here, we study the sub-cycle resolved strong-field tunneling dynamics in solids via a complex saddle-point method. We compare the instantaneous momentum at the moment of tunneling and the tunneling distances over a range of Keldysh parameters. Our results demonstrate that for nonadiabatic tunneling, tunneling ionization away from Γ point is possible. When this happens the electron has a nonzero initial velocity when it emerges in the conduction band. Moreover, consistent with atomic tunneling, a reduced tunneling distance as compared to the quasi-static case is found. Our results provide remarkable insight into the basic physics governing the sub-cycle electron tunneling dynamics with significant implications for understanding subsequent strong-field nonlinear phenomena in solids.

P-values and confidence intervals for weighted log-rank tests under truncated binomial design based on clustered medical data

ABSTRACT The randomization design employed to gather the data is the basis for the exact distributions of the permutation tests. One of the designs that is frequently used in clinical trials to force balance and remove experimental bias is the truncated binomial design. The exact distribution of the weighted log-rank class of tests for censored cluster medical data under the truncated binomial design is examined in this paper. For p-values in this class, a double saddlepoint approximation is developed using the truncated binomial design. With the right censored cluster data, the saddlepoint approximation’s speed and accuracy over the normal asymptotic make it easier to invert the weighted log-rank tests and find nominal 95 % confidence intervals for the treatment effect.

Strong-field processes induced by an ultrashort linearly polarized pulse with two carrier frequencies

We perform a detailed semianalytical study of the characteristics of strong-field processes induced by an ultrashort linearly polarized pulse with two carrier frequencies. We show that the photoelectron spectra depend to a great extent on the absolute phases of the field components and on the duration of the pulse. By using the saddle-point method, we show that this dependence can be exploited to control the electron dynamics in the laser field. Besides the photoelectron spectra, we also investigate the spectra of high-order harmonics. Again, we find a strong dependence on the absolute phases and the duration of the pulse. We present the spectra using a false color scale in the absolute phase-harmonic energy plane, and show that particular regions in this plane can be assessed using the simple man's model founded on the classical solution of the Newton equation of motion. Finally, we investigate the symmetry properties of the photoelectron and high-order harmonic spectra with respect to the transformation which includes the change of the absolute phase, and compare them with those valid for a long driving pulse with a flat envelope. Published by the American Physical Society 2024

An analytical approach for unsupervised learning rate estimation using rectified linear units.

Unsupervised learning based on restricted Boltzmann machine or autoencoders has become an important research domain in the area of neural networks. In this paper mathematical expressions to adaptive learning step calculation for RBM with ReLU transfer function are proposed. As a result, we can automatically estimate the step size that minimizes the loss function of the neural network and correspondingly update the learning step in every iteration. We give a theoretical justification for the proposed adaptive learning rate approach, which is based on the steepest descent method. The proposed technique for adaptive learning rate estimation is compared with the existing constant step and Adam methods in terms of generalization ability and loss function. We demonstrate that the proposed approach provides better performance.

Magneto-acousto-electrical tomography using nonlinearly frequency-modulated ultrasound.

Objective. In this study, nonlinearly frequency-modulated (NLFM) ultrasound was applied to magneto-acousto-electrical tomography (MAET) to increase the dynamic range of detection.Approach. Generation of NLFM signals using window function method-based on the principle of stationary phase-and piecewise linear frequency modulation method-based on the genetic algorithm-was discussed. The MAET experiment systems using spike, linearly frequency-modulated (LFM), or NLFM pulse stimulation were constructed, and three groups of MAET experiments on saline agar phantom samples were carried out to verify the performance-respectively the sensitivity, the dynamic range, and the longitudinal resolution of detection-of MAET using NLFM ultrasound in comparison to that using LFM ultrasound. Based on the above experiments, a pork sample was imaged by ultrasound imaging method, spike MAET method, LFM MAET method, and NLFM MAET method, to compare the imaging accuracy.Main results. The experiment results showed that, through sacrificing very little main-lobe width of pulse compression or equivalently the longitudinal resolution, the MAET using NLFM ultrasound achieved higher signal-to-interference ratio (and therefore higher detection sensitivity), lower side-lobe levels of pulse compression (and therefore larger dynamic range of detection), and large anti-interference capability, compared to the MAET using LFM ultrasound.Significance. The applicability of the MAET using NLFM ultrasound was proved in circumferences where sensitivity and dynamic range of detection were mostly important and slightly lower longitudinal resolution of detection was acceptable. The study furthered the scheme of using coded ultrasound excitation toward the clinical application of MAET.

Observables from classical black hole scattering in Scalar-Tensor theory of gravity from worldline quantum field theory

In this paper, we compute the two observables, impulse and waveform, in a black hole scattering event for the Scalar-Tensor theory of gravity with a generic scalar potential using the techniques of Worldline Quantum Field Theory. We mainly investigate the corrections to the above mentioned observables due to the extra scalar degree of freedom. For the computation of impulse, we consider the most general scenario by making the scalar field massive and then show that each computed diagram has a smooth massless limit. We compute the waveform for scalar and graviton up to 2PM, taking the scalar as massless. Furthermore, we discuss if the scalar has mass and how the radiation integrals get more involved than the massless case. We also arrive at some analytical results using stationary phase approximation. Interestingly, we also show that the λ4φ4 interaction vertex does not contribute to the radiation by showing that the integral has no non-zero finite value.

Gravitational waves with dark matter minispikes: Fourier-domain waveforms of eccentric intermediate-mass-ratio-inspirals

Abstract An intermediate mass black hole (IMBH) may have a dark matter (DM) minihalo around it and develop a spiky structure called DM minispike. Gravitational waves (GWs) can be produced if a stellar compact object, such as a black hole (BH) or neutron star, inspirals into the IMBH. This kind of systems are known as itermediate-mass-ratio-inspirals (IMRIs) and may be observed by space-based gravitational wave detectors including LISA, Taiji and Tianqin. In this paper, we lay the foundations for the construction of analytic expressions for Fourier-domain gravitational waves produced by eccentric IMRIs with DM minispikes in a post-circular or small-eccentricity approximation (e < 0.4). We take the effect of dynamical friction from the DM as a perturbation, and decompose the dynamical equations into perturbed part and unperturbed part. The equations are solved in a series expansion form about zero initial eccentricity to eighth order. The time-dependent, “plus” and “cross” polarizations are expanded in Bessel functions, which are then self-consistently reexpanded in a power series about zero initial eccentricity. The stationary-phase approximation is then employed to obtain the explicit DM-modified analytic expressions for the Fourier transform of the post-circular expanded, time-domain signal. We exemplify this framework by considering a typical IMRI with a DM minispike and find the GW detectability strongly depends on the radial profile of the DM distribution. When the density of DM is large enough, the signal to noise ratio (SNR) will be degraded significantly and a detection loss may occur if we use a template without the effect of DM to treat a signal including the DM effect. With the Fourier-domain gravitational waveforms we also estimate the accuracy of the measurement of the DM minispike parameters in our reference model. Our framework hold the promise to construct a “ready-to-use” Fourier-domain waveforms for data analysis of eccentric IMRIs with DM minispikes.

In Silico and Chromatographic Methods for Analysis of Biotransformation of Prospective Neuroprotective Pyrrole-Based Hydrazone in Isolated Rat Hepatocytes.

In the current study, chromatographic and in silico techniques were applied to investigate the biotransformation of ethyl 5-(4-bromophenyl)-1-(2-(2-(2-hydroxybenzylidene) hydrazinyl)-2-oxoethyl)-2-methyl-1H-pyrrole-3-carboxylate (11b) in hepatocytic media. The initial chromatographic procedure was based on the employment of the conventional octadecyl stationary phase method for estimation of the chemical stability. Subsequently, a novel and rapid chromatographic approach based on a phenyl-hexyl column was developed, aiming to separate the possible metabolites. Both methods were performed on a Dionex 3000 ThermoScientific (ACM 2, Sofia, Bulgaria) device equipped with a diode array detector set up at 272 and 279 nm for analytes detection. An acetonitrile: phosphate buffer of pH 3.5: methanol (17:30:53 v/v/v) was eluted isocratically as a mobile phase with a 1 mL/min flow rate. A preliminary purification from the biological media was achieved by protein precipitation with methanol. A validation procedure was carried out, where the method was found to correspond to all ICH (Q2) and M10 set criteria. Additionally, an in silico-based approach with the online server BioTransformer 3.0 was applied in an attempt to predict the possible metabolites of the title compound 11b. It was hypothesized that four CYP450 isoforms (1A2, 2C9, 3A4, and 2C8) were involved in the phase I metabolism, resulting in the formation of 12 metabolites. Moreover, docking studies were conducted to evaluate the formation of stable complexes between 11b and the aforementioned isoforms. The obtained data indicated three metabolites as the most probable products, two of which (M9_11b and M10_11b) were synthesized by a classical approach for verification. Finally, liquid chromatography with a mass detector was implemented for comprehensive and summarized analysis, and the obtained results revealed that the metabolism of the 11b proceeds possibly with the formation of glucuronide and glycine conjugate of M11_11b.

Defect detection in additively manufactured AlSi10Mg and Ti6Al4V samples using laser ultrasonics and phase shift migration

Laser ultrasonics (LU) is a non-contact and non-destructive method with a high data acquisition rate, making it a promising candidate for in-situ monitoring of defects in different additive manufacturing (AM) processes, including laser powder bed fusion (LPBF) and directed energy deposition, as well as final part inspection. In order to see the effect of various artificial defect types on an LU sub-surface reconstruction, AlSi10Mg samples with side through-holes, as well as Ti6Al4V samples with bottom blind holes and trapped powder were printed using LPBF, and then ultrasound B-scans of the samples were obtained using an LU system. The resulting scan data was processed using a custom frequency domain phase shift migration (PSM) algorithm, to reconstruct the defects and their locations. Novel ways of pre-processing the B-scan, used as an input to PSM, and taking advantage of its frequency representation, are demonstrated. Newton’s method was used to find a stationary phase approximation, used to account in the frequency domain for the fixed offset emitter-receiver arrangement within the PSM calculation. The Newton’s method calculation time was reduced by 33%, by using an approximation of the phase function to find an initial guess. The smallest defects that were detected using this method were in the size range between 200 to 300μm for the bottom hole defects, using an 8 ns laser pulse duration. The effect of the laser on the surface of a part being built, and the challenges and further work needed to integrate LU in a LPBF machine for in-situ inspection are discussed.

A Dai-Liao Hybrid PRP and DY Schemes for Unconstrained Optimization

This article presents a new conjugate gradient (CG) method that requires first-order derivatives but overcomes the slow convergence issue associated with the steepest descent method and does not require the computation of second-order derivatives, as needed in the Newton method. The CG update parameter is suggested from the extended conjugacy condition as a convex combination of Polak, Ribiére, and Polyak (PRP) and Dai and Yuan (DY) algorithms by employing the optimal choice of the modulating parameter 't'. Numerical computations show that the algorithm is robust and efficient based on the number of iterations and CPU time. The scheme converges globally under Wolfe line search and adopts an inexact line search to obtain the step-size that generates a descent property, without requiring exalt computation of the step size. Conjugate gradient method, Descent property, Dai-Liao conjugacy condition, Global convergence, Numerical methods

Damped least squares method for nonlinear mixed additive and multiplicative errors model

Measurement data in the field of modern geodesy contains not only additive errors but also multiplicative errors related to signal strength. The existing models for dealing with mixed additive and multiplicative errors are mainly based on the linear form of unknown parameters and observations, and there are few studies on the nonlinear form of the two. In the parameter estimation method of the nonlinear mixed additive and multiplicative errors model, the initial value of the Gauss–Newton parameter estimation method is selected by previous experience. The initial value determined by this method deviates far from the true value due to a lack of experience, which will lead to inaccurate parameter estimation results. In order to solve this problem, based on the least squares principle and the introduction of the damping factor, this paper deduces the damping least squares parameter solution formula for the parameter estimation of the nonlinear mixed additive and multiplicative errors model. The superiority of the damping least squares algorithm is reflected in the adjustment of the damping factor, taking into account the advantages of the Gauss–Newton method and the steepest descent method, and some weighted selection is obtained in the two algorithms. The calculation and comparative analysis of the simulated cases show that the damped least squares method is more suitable for handling geodetic data with this nonlinear mixed additive and multiplicative errors model when the initial value deviates far from the true value.

Effects of topographical disturbances on flexural wave motion in a viscous fluid

Main causes of ground disturbances are geological events such as earthquakes, underwater gravity mass flows, volcanic eruptions, and bottom explosions. This paper investigates the effects of transitory ground disturbance on the generation of flexural waves in the presence of a thin floating elastic plate. The problem is formulated using the Stokes stream function and wave potential boundary value problems in a viscous fluid. The expression of the plate deflection is obtained as multiple infinite integrals using the Fourier and Laplace transforms, which is further solved by the steepest descent method. Three types of ground disturbances such as H0(x)=e(−x2/2), H0(x)=e−|x|, and H0(x)=δ(x) are considered. The deflection of the floating elastic plate is investigated in connection to ground disturbances, fluid viscosity, and structural parameters such as mass per unit length of the plate and flexural rigidity. The study reveals that with higher viscosity and flexural rigidity, the amplitude of the plate deflection is reduced. Moreover, these two parameters play a great role in the reduction of plate deflection.

Geometric approach for determining stationary phase points in radar imaging

This paper proposes a geometric approach to find out the stationary phase points in radar imaging, namely the coordinates of the transmitting and the receiving antennas that have the greatest contribution to a specified point in the wavenumber spectrum of the scattered field. The foundation of the proposed approach is that, the Green’s function can be approximated by a locally plane wave when the distance between the individual antennas and the object is much larger than the wavelength. In this way, the stationary phase points can be easily found through geometric calculations, much more convenient than the prevailing algebraic approach of letting the first derivative of the phase term to be zero. Six examples for different propagation backgrounds and antenna setups are provided in this paper. It is demonstrated that, the proposed method is beneficial in eliminating extraneous roots of the stationary phase points, which could be generated by letting the first derivative of the phase term to be zero in the stationary phase method.

Saddlepoint approximations for the P-values and probability mass functions of some bivariate sign tests

Bivariate analysis is essential in several applied fields, such as medicine, engineering, biology, and econometrics. Many parametric and non-parametric procedures can be used for bivariate analysis. Non-parametric procedures require fewer conditions than their parametric counterparts. Therefore, this article proposes an accurate approximation of the p-values and the probability mass functions of some non-parametric bivariate tests, including bivariate sign tests and medians. The proposed approximation, saddlepoint approximation, is compared to the traditional approximation method, asymptotic normal approximation method, and Edgeworth expansion through three real data examples and a simulation study. The numerical results show that the proposed approximation method is more accurate than the asymptotic method and Edgeworth expansion. Furthermore, it is computationally less demanding than the simulation method, which is permutation-based and time-consuming.

Saddlepoint p-values for a class of location-scale tests under randomized block design

This paper deals with a class of nonparametric two-sample location-scale tests. The purpose of this paper is to approximate the exact p-value of the considered class under a randomized block design. The exact p-value of the considered class is approximated by the saddlepoint approximation method, also by the traditional method which is the normal approximation method. The saddlepoint approximation method is more accurate than the normal approximation method in approximating the exact p-value, and does not take a lot of time like the simulation method. This accuracy is proved by applying the mentioned methods to two real data sets and a simulation study.