Steepest Descent Method Research Articles

Asymptotic behavior of localized disturbance in a viscous fluid flow down an incline

We analytically solve the problem of the evolution of small-amplitude waves in a uniform flow of a viscous fluid down an inclined plane. The flow is described in a hydraulic approximation. The flow is supposed to be convectively unstable, and the waves arise as a result of an instantaneous external point disturbance. The solution is presented as a Fourier integral to which the steepest descent method is applied twice. The asymptotics of the growing waves is found analytically as a function of two spatial coordinates and time. We show that the region of growing perturbations is a segment of a circle, that its linear dimensions grow linearly with time, and that it is defined by the characteristics of a system of Saint-Venant differential equations.

SEARCH FOR AN EXTREMUM USING THE STEEPEST DESCENT METHOD UNDER THE CONDITIONS OF EXPERIMENTAL ERRORS

One of the spread first level methods of optimum search is learned by the steepest descent method in conditions when there are mistakes in the experiment. The steepest descent method is investigated and is successfully applied in situations, when, there are no mistakes of experiment. However, in real situations the used means of measurement always have determined errors owing to what the appropriate meanings of the response receives with mistakes. The model of the steepest descent algorithm in created, when the length of the step does not depend on the meaning of the purpose functioning. Stepping process realization algorithm and program provision in MathCAD, computer mathematic, system is designed. The realization outcome mistakes for different meaning are presented, the step movement of the optimum dot direction is shown according to function meaning and argument meaning as well. The amount needed for the tactics necessary to approach the minimum is established, the quake amplitude in the surrounding of different level experiment mistakes at the optimum search efficiency in different step conditions.

A Distorted Wave T-Matrix Approach to Ionization of Atoms by Slow Heavy Particle Impact

We propose a theory for the description of slow inelastic collisions between charged particles and simple atomic systems based on a T-matrix approach. Using an eikonal distortion to describe the heavy particle motion, the T-matrix is obtained by the stationary phase approximation. In result, the ionization cross section is expressed by the product between the width of decay of the "quasimolecular" system formed during the collision and a surviving probability of the heavy particle in elastic channel.

Level density within a micro-macroscopic approach

Statistical level density ρ(E,A) is derived for nucleonic system with a given energy E, particle number A and other integrals of motion in the micro-macroscopic approximation beyond the standard saddle-point method of the Fermi gas model. This level density reaches the two limits; the well-known Fermi gas grand-canonical ensemble limit for a large entropy S related to large excitation energies, and the finite micro-canonical limit for a small combinatorical entropy S at low excitation energies. The inverse level density parameter K as function of the particle number A in the semiclassical periodic orbit theory, taking into account the extended Thomas-Fermi and Strutinsky shell corrections, is calculated and compared with experimental data.

Analytical computation of quantum corrections to a nontopological soliton within the saddle-point approximation

Schr\"{o}dinger field theory with an attractive self-interaction possess non-topological extended solutions with a finite energy in both finite and infinite-volume cases, namely, bright solitons. The analytical form of the solution itself is well-known, though analytical investigation of the quantum fluctuations in this background still requires more thorough investigation, for instance, analytical computation of quantum corrections to this background within the saddle-point approximation. In the present work this gap is filled. Both 2-point Green's function and quantum corrections to the background are analytically computed and properly renormalized by means of momentum cut-off procedure. It is deduced that quantum corrections are indeed small provided that particle number is large. Also, we see that perturbation modes of continuum spectrum at bright soliton background generate a gap in the energy spectrum. Moreover, it turns out that the whole spectrum is continuous modulo zero-modes, which is similar to Sine-Gordon solitons.

Simulation and Analysis of Electromagnetic Scattering from Anisotropic Plasma-Coated Electrically Large and Complex Targets

An efficient physical optics (PO) calculation method is proposed for the electromagnetic (EM) scattering of electrically large targets coated with magnetized plasma characterized by asymmetric tensor dielectric parameters. The outer surface of the arbitrarily shaped target is discretized into triangular elements. According to the principle of tangent plane approximation and by using the plane wave spectrum expansion method, the scattered field from one triangular element is derived as a double integral in the spectral domain. To obtain the solution in the spatial domain, the saddle point method is used to asymptotically calculate the integral. Then, the equivalent surface currents (ESCs) are constructed by calculating the surface field at the outer surface of the planar model, from which the PO solution is derived by using the Stratton–Chu integral. Moreover, to interpret the field propagation process in the plasma layer quantitatively, the total scattered field of the coated planar model is decomposed into the superposition of different mode field components. It is observed that the scattered fields demonstrate an inherent cross-polarization phenomenon due to the nonreciprocal constitutive relation of the plasma, which is a distinct feature and is different from the general anisotropic medium whose dielectric parameters can be diagonalized. The effectiveness of the proposed method is verified by numerical results. Furthermore, the proposed algorithm consumes less calculation time and memory as compared to commercial full solvers.

Gradient-based optimization for spectral-based multiple-leak identification

This study tailors an efficient optimization procedure for multiple-leak identification in the frequency domain. Firstly, the maximum likelihood estimation is adopted to build a multi-dimensional optimization problem whose objective function is only in terms of the leak locations. Then, closed-form formulations to compute the gradient of the proposed objective function are presented. Finally, to demonstrate the effectiveness of the developed analytical formulations, the steepest-descent method with multiple initializations is implemented to solve the optimization problem. An initialization algorithm based on a set of points with maximum leak reflections is proposed. The use of the most classical gradient-based technique to find the optimum allowed for a deeper understanding of the multi-dimensional optimization required to pinpoint multiple leaks in transient-based methods in terms of complexity and solvability of the problem. The cases of three leaks and various numbers of resonant frequencies with a clear manifestation of the performance of gradients and their applications in the steepest descent method are investigated and discussed. The procedure is also validated against a laboratory case study having two leaks. The main advantages of the proposed technique, which distinguishes it from previous studies, are (i) the versatility of the process to find leaks with even a single spatial measurement vector and (ii) the efficiency and accuracy of the underlying optimization, as closed-form formulations are employed in the scheme. This work may provide a benchmark for investigating and assessing other methods dealing with multiple leaks due to its comprehensive presentation of the local extrema and the entire identification process.

$L^p-L^q$ estimates for wave equations with strong time-dependent oscillations

In this paper we consider the following Cauchy problem for the linear wave equation with time-dependent propagation speed: \begin{align} \label{Abstract.Cauchy} \tag{$\star$} \begin{cases} u_{tt}-\lambda^2(t)\omega^2(t)\Delta u=0, & (t,x)\in[0,\infty)\times \mathbb{R}^n, \\ u(0,x)=u_0(x), ~~ u_t(0,x)=u_1(x), & x\in\mathbb{R}^n, \end{cases} \end{align} where $\lambda=\lambda(t)$ is an increasing shape function and $\omega=\omega(t)$ is a bounded oscillating function which has very fast oscillations. The goal is to prove $L^p-L^q$ estimates on the conjugate line for Sobolev solutions of the Cauchy problem \eqref{Abstract.Cauchy} in the case that $\omega=\omega(t)$ has very fast oscillations. Basically, we apply the WKB analysis for the construction of a fundamental system of solutions for ordinary differential equations depending on a parameter. Then, the method of stationary phase yields the asymptotical behaviour of Fourier multipliers with nonstandard phase functions depending on a parameter. This research continues the research of the paper [17].

Nonlinear steepest descent approach to orthogonality on elliptic curves

We consider the recently introduced notion of denominators of Padé-like approximation problems on a Riemann surface. These denominators are related as in the classical case to the notion of orthogonality over a contour. We investigate a specific setup where the Riemann surface is a real elliptic curve with two components and the measure of orthogonality is supported on one of the two real ovals. Using a characterization in terms of a Riemann–Hilbert problem, we determine the strong asymptotic behaviour of the corresponding orthogonal functions for large degree. The theory of vector bundles and the non-abelian Cauchy kernel play a prominent role even in this simplified setting, indicating the new challenges that the steepest descent method on a Riemann surface has to overcome.

Effects of bottom permeability on wave generation by a moving oscillatory disturbance in magneto-hydrodynamics

Generation of magneto-hydrodynamic surface wave by a moving oscillatory disturbance in a finite depth ocean bounded below by a horizontal porous bottom is presented. The governing initial-boundary value problem is solved using Laplace-Fourier transform and an integral form of the surface elevation is obtained. The asymptotic solutions for the free surface elevation are obtained using stationary phase method. The results show that both the steady-state and transient components exist where the latter decays asymptotically and steady-state is reached. It is seen that the bottom permeability has a significant impact on surface elevation. The dispersion relation is also taken into account and the phase and group velocities for deep water and shallow water are analyzed for different values of the parameters in number of figures. The results for limiting case in which the current speed and the porosity vanishes are compared with the calculations investigated earlier.

Weighted log-rank tests for left-truncated data under Wei’s urn design: saddlepoint p-values and confidence intervals

ABSTRACT Left-truncated data are constructed from the time of an initial event and the time of the event of interest. That is to say, each individual represented by such data is exposed to an event prior to the event of interest, and both times are recorded. Weighted log-rank testing is commonly used for such data. In this paper, a saddlepoint approximation method is provided for computing p-values of the permutation distribution of tests from the weighted log-rank testing in the presence of left-truncated data and Wei’s urn design. A simulation study is used to assess the efficiency of the saddlepoint approximation. The accuracy of the saddlepoint approximation in comparison to the normal approximation enables us to compute accurate confidence intervals for the treatment effect.

On the asymptotic behavior of Jacobi polynomials with first varying parameter

We investigate the large n behavior of Jacobi polynomials with varying parameters Pn(an+α,bn+β)(1−2λ2) for a,b>−1 and λ∈(0,1). This is a well-studied topic in the literature but some of the published results appear to be discordant. To address this issue we provide an in-depth investigation of the case b=0, which is most relevant for our applications. Our approach is based on a new and surprisingly simple representation of Pn(an+α,β)(1−2λ2),a>−1 in terms of two integrals. The integrals’ asymptotic behavior is studied using standard tools of asymptotic analysis: one is a Laplace integral and the other is treated via the method of stationary phase. As a consequence we prove that if a∈(2λ1−λ,∞) then λanPn(an+α,β)(1−2λ2) shows exponential decay and we derive simple exponential upper bounds in this region. If a∈(−2λ1+λ,2λ1−λ) then the decay of λanPn(an+α,β)(1−2λ2) is O(n−1/2) and if a∈{−2λ1+λ,2λ1−λ} then λanPn(an+α,β)(1−2λ2) decays as O(n−1/3). A new phenomenon occurs in the parameter range a∈(−1,−2λ1+λ), where we find that the behavior depends on whether or not an+α is an integer: If a∈(−1,−2λ1+λ) and an+α is an integer then λanPn(an+α,β)(1−2λ2) decays exponentially. If a∈(−1,−2λ1+λ) and an+α is not an integer then λanPn(an+α,β)(1−2λ2) may increase exponentially depending on the proximity of the sequence (an+α)n to integers.

Elastic wave full-waveform inversion in the time domain by the trust region method

In this paper, we investigate the elastic wave full-waveform inversion (FWI) by the trust region method. FWI is an optimization problem of minimizing the misfit between the observed data and simulated data. Usually the gradient-based search methods such as the steepest descent method or quasi-Newton's methods are applied to update the model parameters iteratively. And the L-BFGS method with approximated Hessian information is preferred for its high computational efficiency and accuracy. At each iteration, a line search method computes a search direction and then finds a suitable step length or how far to move along the direction. The trust region method is an effective alternative to solve nonlinear optimization problems. In this paper, we investigate the performance of the trust region method in elastic time-domain FWI. In the trust region method, a trial step length called the trust-region radius is forced within a certain neighborhood of the current iterate point and a trust-region subproblem requires to solve. In order to solve the trust-region subproblem, the dogleg method and the two dimensional subspace method are adopted in this paper. In the trust region method, as long as the trust-region radius is well updated, the model updating can be performed for every iteration with the fixed trial step and there is no more extra computation for forward problem like the line search method. Numerical computations for the benchmark Marmousi model are given. The inversion results and comparisons show that the trust region method is very efficient and behaves better than the line search method such as the well-known L-BFGS method.

Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings

<p style='text-indent:20px;'>In this paper, for solving the variational inequality problem over the set of common fixed points of a finite family of demiclosed quasi-nonexpansive mappings in Hilbert spaces, we propose two new strongly convergent methods, constructed by specific combinations between the steepest-descent method and the block-iterative ones. The strong convergence is proved without the boundedly regular assumptions on the family of fixed point sets as well as the approximately shrinking property for each mapping of the family, that are usually assumed in recent literature for similar problems. Applications to the multiple-operator split common fixed point problem (MOSCFPP) and the problem of common minimum points of a finite family of lower semi-continuous convex functions with numerical experiments are given.</p>

Determination for 3D sedimentary basin depths from gravity anomaly data using steepest descent method

In solving the gravity exploration inverse problems, determining the depth of the sedimentary basin played an important role in the petroleum exploration. In this paper, the depth to the basement of 3D sedimentary basin was calculated from the gravity anomaly using the steepest descent method. This method allowed minimizing the objective function based on the first derivative and adjusting the step length by the iterative method; using the variable is the depth of adjacent rectangular prism in the x, y directions with parabolic function between density contrast and the depth. The parameters in the parabolic function were determined by the nonlinear regression method based on the information of deep borehole in the Mekong Delta. This parabolic function was used to calculate the theoretical gravity anomaly for the sedimentary basin and solve the 3D inverse gravity problem on the model and on the real data. The used objective function was the mean square error function between the measured gravity anomaly and the calculated anomaly. The proposed method was tested on the model, showing that the calculated sedimentary basin depth almost coincides with the original model depth; then the method was applied to calculate the depth of 3D sedimentary basin from the local gravity anomaly in Bac Lieu province. The analysis results were consistent with previous publications, but the calculation time was significantly shortened, so this method can be extended to analyze data on a large area.

Aproksimacija sedlaste tačke višeg reda

Introduction/purpose: Saddle point approximation has been considered in the paper Methods: The saddle point method is used in several different fields of mathematics and physics. Several terms of the expansion for the factorial function have been explicitely computed. Results: The integrals estimated in this way have values close to the exact one. Conclusions: Higher order corrections are not negligible even when requiring moderate levels of precision.

Theoretical study of the dynamics of the excitonic insulator order parameter of the Falicov–Kimball model excited by ultrashort laser pulse

The non-equilibrium dynamics of the excitonic insulator order parameter in the excitonic insulator phase of the Falicov–Kimball model excited by ultrashort laser pulse is studied. The Keldysh–Schwinger functional integral method with the saddle point approximation is employed. The numerical solution for the order parameter obtained from the saddle point equation is found to exhibit oscillatory behaviors, whose frequencies are consistent with the energy gap of the excitonic insulator. The fluctuations of the order parameter around the saddle point are identified with the collective modes in non-equilibrium states, and their Green’s functions are explicitly found in terms of the saddle point solution. Also, the interaction vertices responsible for the decay of the non-equilibrium order parameter is found, and the lowest order contribution for the decay process is expressed in terms of the vertices and the Green’s functions of the non-equilibrium collective modes. In the presence of a particular electron–phonon interaction, a hybrid mode of the excitonic order parameter and the phonon can be naturally defined in functional integral and its saddle point equation can be derived. It is shown that the saddle point solution with electron–phonon interaction is consistent with the experimental data and the existing theoretical result.

Излучение движущегося сгустка частиц с переменной величиной заряда

We study the electromagnetic radiation of a charged particle bunch with small size moving at a constant velocity and having a variable charge. The environment medium is considered to be isotropic and homogeneous, and it may have frequency dispersion, but not spatial dispersion. The general solution of the problem is obtained. The main attention is paid to the case when the bunch charge, starting from a certain moment, decreases exponentially with time. The saddle point method is used to obtain the approximate expressions for the field components that are valid in the wave zone. The energy characteristics of the excited spherical wave are studied and compared with the case of a decelerating charge. In the case of excitation of Vavilov-Cherenkov radiation, we obtain the asymptotics which are valid in the entire wave zone, including the region in which the field cannot be divided into the spherical and cylindrical waves.

Radiation from a Moving Bunch of Particles with a Variable Charge

We study the electromagnetic radiation of a charged particle bunch with small size moving at a constant velocity and having a variable charge. The environment medium is considered to be isotropic and homogeneous, and it may have frequency dispersion, but not spatial dispersion. The general solution of the problem is obtained. The main attention is paid to the case when the bunch charge, starting from a certain moment, decreases exponentially with time. The saddle point method is used to obtain the approximate expressions for the field components that are valid in the wave zone. The energy characteristics of the excited spherical wave are studied and compared with the case of a decelerating charge. In the case of excitation of Vavilov-Cherenkov radiation, we obtain the asymptotics which are valid in the entire wave zone, including the region in which the field cannot be divided into the spherical and cylindrical waves. Keywords: Charged particle bunch, spherical wave, cylindrical wave, Vavilov-Cherenkov radiation.

Interface reflection wave of axisymmetric directional spherical-wave

The sound radiation field of a circular piston source in an infinite plane rigid baffle can be approximated as an axisymmetric directional spherical-wave. The interface response expressions of the axisymmetric directional spherical-wave for the piston parallel with the interface has already been obtained in previous studies. On condition that the distance from the piston center to the interface is much greater than the piston radius, we first derive the conical wave expansion of the axisymmetric directional spherical-wave which is obtained by using the conical wave expansion of the homogeneous spherical-wave and the formula of the axisymmetric directional spherical-wave excited by a circular piston in an rigid infinite plane, and then derive the expression of the interface reflection wave of the axisymmetric directional spherical-wave for the piston non-parallel to the interface. The expression of the interface reflection wave is simplified into a simplified expression by saddle point method on condition that the source distance is much larger than the acoustic wavelength. The simplified expression is not only simple in the calculation, but also clear in the physical meaning. The simplified expression shows that the interface reflection wave of the axisymmetric directional spherical-wave can be regarded as the product of the axisymmetric directional spherical-wave excited by the piston mirror image and the reflection coefficient. The calculations show that when the piston radius is smaller than the acoustic wavelength, the reflected wave is less sensitive to the angle included between the piston and the interface and the azimuth of the receiving point, and the directivity of the reflected wave is weak. When the piston radius is greater than the acoustic wavelength, the reflected wave is very sensitive to the angle included between the piston and the interface and the azimuth of the receiving point, and the directivity of the reflected wave is strong. Increasing the angle included between the piston and the interface, the reflected wave and its directivity both first increase and then decrease. The reflected wave is largest and the directivity of the reflected wave is strongest when the angle included between the piston and the interface is equal to that between the interface normal and the connecting line between the mirror image of the piston center and the receiving point.