Steepest Descent Method Research Articles

A Collocation Numerical Method for Highly Oscillatory Algebraic Singular Volterra Integral Equations

The highly oscillatory algebraic singular Volterra integral equations cannot be solved directly. A collocation numerical method is proposed to overcome the difficulty created by the highly oscillatory algebraic singular kernel. This paper is composed primarily of two methods—the piecewise constant collocation method and the piecewise linear collocation method—in which uniformly distributed nodes serve as collocation points. For the efficient computation of highly oscillatory and algebraic singular integrals, the steepest descent method as well as the Gauss–Laguerre and generalized Gauss–Laguerre quadrature rules are employed. Consequently, the resulting linear system is solved for the unknown function approximated by the Lagrange interpolation polynomial. Detailed theoretical analysis is carried out and numerical experiments showing high accuracy are also presented to confirm our analysis.

Time-dependent fatigue reliability estimation and optimal design of scraper chains under corrosive wear effect

Fatigue damage is one of the main failures of the scraper chain. Moreover, corrosive wear has a detrimental effect on fatigue crack propagation. It is so essential to study the fatigue reliability of the scraper chain considering the interaction of corrosive wear and fatigue. This paper is intended to propose a reliability-based estimation and optimal design of the scraper chain considering the performance degradation of the scraper chain under the interaction of corrosive wear and fatigue. First, the dynamic model of the scraper conveyor is established based on the finite element method. Second, Paris Low is used to empirically describe the occurrence of cracking with corrosion effects. Furthermore, the time-dependent reliability of scraper chains under the interaction of the corrosive wear and fatigue crack propagation is addressed by the fourth-moment saddlepoint approximation (FMSA) method, and the reliability-based design optimization (RBDO) of the scraper chain is then carried out. Finally, the applicability of the proposed method is verified by the SGZ1000/1400 scraper conveyor.

Inexact Exponential Penalty Function with the Augmented Lagrangian for Multiobjective Optimization Algorithms

This paper uses an augmented Lagrangian method based on an inexact exponential penalty function to solve constrained multiobjective optimization problems. Two algorithms have been proposed in this study. The first algorithm uses a projected gradient, while the second uses the steepest descent method. By these algorithms, we have been able to generate a set of nondominated points that approximate the Pareto optimal solutions of the initial problem. Some proofs of theoretical convergence are also proposed for two different criteria for the set of generated stationary Pareto points. In addition, we compared our method with the NSGA-II and augmented the Lagrangian cone method on some test problems from the literature. A numerical analysis of the obtained solutions indicates that our method is competitive with regard to the test problems used for the comparison.

Stress Evolution of the Main Himalayan Thrust Fault Induced by the Mw 7.8 Gorkha Earthquake

ABSTRACT The Mw 7.8 Gorkha earthquake that occurred in 2015 is the largest thrust event in the middle of the main Himalayan thrust fault zone (MHT) in the past 81 yr. Its impact on the regional tectonic stress state and future seismic risks is a significant scientific issue worthy of in-depth analyses. In this study, we inverted the planar fault-slip model (the planar model), the flat-ramp fault-slip model (the flat-ramp model) and the double flat-ramp fault-slip model (the double flat-ramp model) to analyze the effect of the fault geometry, based on the steepest descent method (SDM) and the layered earth model. Compared with the flat-ramp model, the planar model exhibits a wider slip distribution in the down-dip direction of the main rupture zone, whereas the double flat-ramp model shows a larger coseismic slip on the middle ramp at the depth of 7.5–11.5 km. Those slip differences produce larger stress shadow areas in the above models, but for a blind thrust event induced by a low-dip thrust fault, this does not significantly change the distribution mode of coseismic Coulomb stress change (ΔCFS) in the three models. Namely, there is an obvious stress release in the main rupture but an obvious stress loading in the up-dip and down-dip directions of the main rupture zone. Based on the Burgers rheological model, we calculated the postseismic viscoelastic Coulomb stress change (V−ΔCFS) and the cumulative Coulomb stress change (C−ΔCFS) induced by the Gorkha earthquake in the flat-ramp model and comparatively analyzed the evolution pattern of coseismic and postseismic stresses. Our results indicate that the variation trend of postseismic stress in the lithosphere is opposite to that of the coseismic ΔCFS. The postseismic viscoelastic relaxation promotes the slip of the flat-ramp structure at the depth of 10–25 km, and the stress unloads in the shallow and deep part simultaneously. As a blind thrust event, the coseismic ΔCFS still plays a dominant role in the shallow part after 50 yr, whereas the loading C−ΔCFS in the deep part of the MHT is greatly weakened by the postseismic V−ΔCFS. Seismic risks still exist in the unruptured area on the west side of the mainshock and the shallow Main Frontal thrust.

Soliton resolution for the complex short-pulse positive flow with weighted Sobolev initial data in the space-time soliton regions

The Riemann Hilbert approach and ∂‾ steepest descent method are used to study the Cauchy problem of a new complex short-pulse positive flow, from which the long-time asymptotics and soliton resolution conjecture for the complex short-pulse positive flow with initial conditions is obtained in weighted Sobolev spaces. Resorting to the spectral analysis of Lax pair, the introduced transformations of field variables and transformations of independent variables, we establish the basic Riemann-Hilbert problem, and convert the solution of the Cauchy problem of the complex short-pulse positive flow into the corresponding solution of the Riemann Hilbert problem. The jump matrix then expands and deforms continuously over the initial contour. We finally obtain the long-time asymptotics and soliton resolution for the complex short-pulse positive flow in the soliton region by using the ∂‾ steepest descent method. The results also show that N-soliton solutions of the complex short-pulse positive flow are asymptotically stable.

Asymptotic distribution of the zeros of a certain family of generalized hypergeometric polynomials

The primary aim of this paper is to investigate the asymptotic distribution of the zeros of certain classes of hypergeometric q + 1 F q polynomials. We employ classical analytical techniques, including Watson's lemma and the method of steepest descent, to understand the asymptotic behavior of these polynomials: q + 1 F q ( − n , kn + α , … , kn + α + q − 1 q ; kn + β , … , kn + β + q − 1 q ; z ) ( n → ∞ ) , where n is a nonnegative integer, q is a positive integer and the constant parameters α and β are constrained by α < β . By applying the general results established in this paper, we generate numerical evidence and graphical illustrations using Mathematica to show the clustering of zeros on certain curves.

Soliton resolution and asymptotic stability of N-solutions for the defocusing Kundu–Eckhaus equation with nonzero boundary conditions

In this paper, we address interesting soliton resolution, asymptotic stability of N-soliton solutions and the Painlevé asymptotics for the Kundu-Eckhaus (KE) equation with nonzero boundary conditions iqt+qxx−2(∣q∣2−1)q+4β2(∣q∣4−1)q+4iβ∣q∣2xq=0,q(x,0)=q0(x)∼±1,x→±∞. The key to proving these results is to establish the formulation of a Riemann-Hilbert (RH) problem associated with the above Cauchy problem and find its connection with the RH problem of the defocusing NLS equation. With the ∂¯ -steepest descent method and the results of the defocusing NLS equation, we find complete leading order approximation formulas for the defocusing KE equation on the whole (x,t) half-plane including soliton resolution and asymptotic stability of N-soliton solutions in a solitonic region, Zakharov-Shabat asymptotics in a solitonless region and the Painlevé asymptotics in two transition regions.

Saddlepoint approximation of the p-values for the multivariate one-sample sign and signed-rank tests

&lt;p&gt;A multivariate data analysis (MVDA) is a powerful statistical approach to simultaneously analyze datasets with multiple variables. Unlike univariate or bivariate analyses, which simultaneously focus on one or two variables, respectively, MVDA considers the interactions and relationships among multiple variables within a dataset. Several nonparametric tests can be used in the context of one-sample multivariate location problems. The exact distributions of such tests cannot be analytically computed and are usually approximated using an asymptotic approximation. This article proposes the saddlepoint approximation method to approximate the tail probability for multivariate sign and signed-rank tests. It is suggested as a more accurate alternative to the traditional asymptotic approximation method and an alternative to the simulation method. It requires a lot of time as it depends on all possible permutations. Real data examples were provided to illustrate the calculation of p-values, and a simulation study was conducted to compare the accuracy of the saddlepoint approximation method with the simulation method (permutation-based, so time-consuming) and an asymptotic normal approximation method. The study results show that the saddlepoint approximation provides highly accurate approximations to the p-values of the considered statistics, and it often outperforms the normal approximation. Additionally, the results show that the proposed method's computation time is much less than that of the time-consuming simulation method.&lt;/p&gt;

On the method of stationary phase in calculating the propagator of the Coulomb-Kepler problem (II.)

On the method of stationary phase in calculating the propagator of the Coulomb-Kepler problem (II.)

A Discretize-then-Optimize Approach to PDE-Constrained Shape Optimization

We consider discretized two-dimensional PDE-constrained shape optimization problems, in which shapes are represented by triangular meshes. Given the connectivity, the space of admissible vertex positions was recently identified to be a smooth manifold, termed the manifold of planar triangular meshes. The latter can be endowed with a complete Riemannian metric, which allows large mesh deformations without jeopardizing mesh quality; see R. Herzog and E. Loayza-Romero, Math. Comput. 92 (2022) 1-50. Nonetheless, the discrete shape optimization problem of finding optimal vertex positions does not, in general, possess a globally optimal solution. To overcome this ill-possedness, we propose to add a mesh quality penalization term to the objective function. This allows us to simultaneously render the shape optimization problem solvable, and keep track of the mesh quality. We prove the existence of a globally optimal solution for the penalized problem and establish first-order necessary optimality conditions independently of the chosen Riemannian metric. Because of the independence of the existence results of the choice of the Riemannian metric, we can numerically study the impact of different Riemannian metrics on the steepest descent method. We compare the Euclidean, elasticity, and a novel complete metric, combined with Euclidean and geodesic retractions to perform the mesh deformation.

Analysis of the application of neural network technologies in predicting equipment failure in production

The relevance. The article considers the solution to the problem of analyzing and predicting changes in the dynamics of measuring instrument readings and identifying patterns describing the course and possible periods of failures of technological process equipment by constructing a predictive time series model. A critically important infrastructure facility of a thermal power plant was selected as the studied one – a water pump for pumping process water into neutralizer tanks (the first stage of chemical water purification) located in the chemical workshop of the Krasnoyarsk CHP-3. The general methodology of model construction, requirements for input data arrays, preprocessing algorithms for the formation of samples used for training and testing models are described. To build a predictive model, the work used elements such as recurrent layers of long-term short- term memory LSTM, MinMaxScaler regularization, Dropout layer to reduce the effect of retraining, the ‘Adam’ optimizer using error back propagation methods and optimization using the steepest descent method using conjugate gradients, the learning error function is the average root of the error (Root Mean Squared Error, RMSE), as well as plot_predictions visualization. The algorithms are implemented in Python, in the Anaconda Jupyter Nootebook core. Based on the results of testing the training, the model showed sufficient performance of this method, since the time spent on creating the model and obtaining a forecast plays an important role when launching the model in real production conditions. In conclusion, recommendations are given to improve the accuracy of the forecast, the developed interface of the production system is presented and the directions of future research in this area are outlined.

Nonlinear Gradient Field Mapping Using a Spherical Grid Phantom for 3 and 7 Tesla MR Imaging Systems Equipped with High-performance Gradient Coils.

Recent high-performance gradient coils are fabricated mainly at the expense of spatial linearity. In this study, we measured the spatial nonlinearity of the magnetic field generated by the gradient coils of two MRI systems with high-performance gradient coils. The nonlinearity of the gradient fields was measured using 3D gradient echo sequences and a spherical phantom with a built-in lattice structure. The spatial variation of the gradient field was approximated to the 3rd order polynomials. The coefficients of the polynomials were calculated using the steepest descent method. The geometric distortion of the acquired 3D MR images was corrected using the polynomials and compared with the 3D images corrected using the harmonic functions provided by the MRI venders. As a result, it was found that the nonlinearity correction formulae provided by the vendors were insufficient and needed to be verified or corrected using a geometric phantom such as used in this study.

Optimal Selective Maintenance for Complex Systems Under Stochastic Maintenance and Break Durations

The selective maintenance problem (SMP) arises in many settings where systems perform sequences of missions separated by maintenance breaks. The SMP aims to identify the optimal subset of maintenance actions that will maximize the system reliability for the next mission. This paper addresses the SMP when the durations of the break and maintenance actions are stochastic. An exact solution method is proposed which uses the saddlepoint approximation in the computation of the system reliability. Several experiments are used to demonstrate the effectiveness of the proposed solution method for large and complex systems.

Saddlepoint Approximations for Credit Portfolio Distributions with Applications in Equity Risk Management

Saddlepoint Approximations for Credit Portfolio Distributions with Applications in Equity Risk Management

Saddlepoint Approximation Method in Reliability Analysis: A Review

Saddlepoint Approximation Method in Reliability Analysis: A Review

Dimension reduction for constructing high-dimensional response distributions by accounting for unimportant and important variables

Probability distributions of responses have been widely used in structural analysis and design because of their complete statistical information. In practice, the dimensionality of input variables could easily reach hundreds or thousands, making it computationally expensive to obtain accurate distributions. In this paper, a generalized most probable point (MPP) method is developed to effectively build the response distributions of high-dimensional problems. First, a global index based on one-iteration MPPs is presented for dimension reduction, which is to divide the input variables into important and unimportant variables. After fixing the unimportant variables at their one-iteration MPP components, the MPP components of the important variables are obtained by performing the inverse first-order reliability method (FORM) in the reduced space. Predictive models of the all MPP components are then established to quickly estimate the MPPs of other cumulative distribution function (CDF) values. To accurately calculate CDF points of limit state functions with different shapes, a comprehensive uncertainty analysis method that accommodates the contributions of the important and unimportant variables is proposed by multiple combinations of FORM, second-order reliability method, and second-order saddlepoint approximation. Finally, the response distributions are generated based on Gaussian mixture distribution and all CDF points. The effectiveness of the proposed method is verified by a mathematical example and two engineering cases.

Path integral factorization and the gravitational effective action

We discuss the factorization and continuity properties of fields in the Euclidean gravitational path integral with higher dimension operators constructed from powers of the Riemann tensor. We construct the boundary terms corresponding to the microcanonical ensemble and show that the saddle point approximation to the path integral with a quasilocal energy constraint generally yields a saddle point with discontinuous temperature. This extends a previous result for the Euclidean Schwarzschild–de Sitter geometry in Einstein gravity and shows that it is robust against at least some types of quantum corrections from heavy fields. As an application, we compute the entropy of SdS in D = 4 using the BTZ method. Our result matches the entropy calculated using Wald’s formula.

Inside the Jaynes–Cummings sum

It is shown that the atomic inversion in the Jaynes–Cummings model has an exact representation as an integral over the Hankel contour. For a field in a coherent state, the integral is evaluated using the saddle point method. The trajectories of saddle points as a function of time are on the branches of the multi-valued Lambert function. All of them start at the initial moment of time, but make the maximum contribution to the inversion at different times. If the collapse and the first revival are clearly distinguished, then subsequent revivals are determined by the comparable contributions of several trajectories.

Minimization of the weight of the blade of the aerial installation by an adaptive hybrid optimization method

In various fields of engineering practice, such as turbine building, power engineering, elements of thin-walled structures are widely used, which operate under increased loads when interacting with air or water. These include blades of impellers of radial-axial and rotary-blade hydraulic turbines, blades of wind power plants. Designing highly efficient machines and structures with the required level of reliability requires determining their optimal characteristics. The article proposes an adaptive method for finding the minimum of an arbitrary smooth multivariable function. The method was used to solve the benchmark optimization problem of a function in the form of a valley. The essence of the proposed algorithm lies in the sequential approach to the bottom of the valley and the subsequent movement in the direction of decreasing the objective function. Comparison of the results of calculating the minimum point of the function is performed using both non-gradient and gradient methods, namely: Powell, Hook-Jeeves, the steepest descent method and the developed method. It was found that the effectiveness of the proposed method is greater than the usual search algorithms, but it is not without its drawbacks. The method is proposed that represents a number of hybrid methods, which form a hybrid coalition Proposed hybrid algorithm does not provide a satisfactory result in the "single" search. The search algorithm reaches a point where all the values of the function at the surrounding points are greater than the values at the obtained point, and the algorithm cannot overcome the barrier. To solve the problem, it is necessary to take the obtained point as a new starting point and repeat the algorithm for finding the minimum of the function, that is, use the multistart method. The proposed method was used to solve the problem of optimizing the blade of an air installation, which was reduced to the problem of unconditional optimization using the method of penalty functions, but the goal function had a significantly valley structure. The optimal values of section thicknesses were obtained, which made it possible to build a blade with improved characteristics.

Bayesian eikonal tomography using Gaussian processes

&#x0D; Eikonal tomography has become a popular methodology for deriving phase velocity maps from surface wave phase delay measurements. Its high efficiency makes it popular for handling datasets deriving from large-N arrays, in particular in the ambient-noise tomography setting. However, the results of eikonal tomography are crucially dependent on the way in which phase delay measurements are predicted from data, a point which has not been thoroughly investigated. In this work, I provide a rigorous formulation for eikonal tomography using Gaussian processes (GPs) to smooth observed phase delay measurements, including uncertainties. GPs allow the posterior phase delay gradient to be analytically derived. From the phase delay gradient, an excellent approximate solution for phase velocities can be obtained using the saddlepoint method. The result is a fully Bayesian result for phase velocities of surface waves, incorporating the nonlinear wavefront bending inherent in eikonal tomography, with no sampling required. The results of this analysis imply that the uncertainties reported for eikonal tomography are often underestimated.&#x0D;