Steepest Decent Method Research Articles

On fast multipole methods for Fredholm integral equations of the second kind with singular and highly oscillatory kernels

ABSTRACTThis paper considers a special boundary element method for Fredholm integral equations of the second kind with singular and highly oscillatory kernels. To accelerate the resolution of the linear system and the matrix-vector multiplication in each iteration, the fast multipole method (FMM) is applied, which reduces the complexity from to . The oscillatory integrals are calculated by the steepest decent method, whose accuracy becomes more accurate as the frequency increases. We study the role of the high-frequency w in the FMM, showing that the discretization system is more well conditioned as high-frequency w increase. Moreover, the larger w may reduce rank expressions from the kernel function, and decrease the absolute errors. At last, the optimal convergence rate of truncation is also represented in this paper. Numerical experiments and applications support the claims and further illustrate the performance of the method.

Synthesis and Characteristic Evaluation of Convex Metallic Reflectarray Antennas to Radiate Relatively Orthogonal Multibeams

In this paper, we present the design of metallic reflectarray antennas consisting of convex nonresonant elements to radiate multiple beams with optimum radiation gains and relatively orthogonal overlapping. In particular, an antenna is realized at 38 GHz to provide sufficient gains and beam widths for simultaneous link-budget and radio coverage satisfaction in the applications of fifth-generation mobile communications. The pattern optimization employs the steepest decent method (SDM) to optimize the phases of reflecting elements for gain tradeoff. The SDM procedure is very efficient because relatively simple formulations of analysis in closed form are developed to subsequently update the phases in each iteration. This procedure avoids the cumbersome and numerous full-wave numerical simulations of scattering fields. Design examples and their radiation characteristics are presented to demonstrate the concepts and validate the feasibility.

Source model of 2016 Mw6.6 Aketao earthquake, Xinjiang derived from Sentinel-1 InSAR observation

On 25 November 2016 (14:24:30 UTC), an Mw6.6 earthquake occurred in Aketao county of Xinjiang. We derived the coseismic deformation field of the earthquake from ESA's Sentinel-1B images and inverted the fault geometry and slip distribution by using a homogenous half-space elastic model and the Steepest Decent Method (SDM) program. The slip model shows that the rupture plane is about 68 km in length and 40 km in width and has an average strike of 105.2°SE and an average dip of 76.0°S. The average amount of slip is ∼0.16 m and the maximum amount of slip is 0.59 m. The estimated seismic moment is 1.31 × 1019 N·m and the corresponding moment magnitude is Mw6.68. The dislocation is mainly distributed in the depth of 5–22 km, where the rupture center is located at 39.17°E, 74.38°N, and a depth of ∼8.8 km, so this earthquake is a shallow earthquake. In terms of the tectonic settings, the earthquake was triggered by the release of residual stresses on the seismogenic Muji fault. The ruptures were stimulated simultaneously by the main shock, which made the slip distribution pattern of a strike-slip fault with a double fracture. Due to the tectonic setting and the seismic gap between the ruptures, the Muji fault would have a risk of rupture to some extent and trigger an earthquake in future.

Chebyshev wavelet method to nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions

This research work addresses the numerical solutions of nonlinear fractional integro-differential equations with mixed boundary conditions, using Chebyshev wavelet method. The basic idea of this work started from the Caputo definition of fractional differential operator. The fractional derivatives are replaced by Caputo operator, and the solution is approximated by wavelet family of functions. The numerical scheme by Chebyshev wavelet method is constructed through a very simple and straightforward way. The numerical results of the current method are compared with the exact solutions of the problems, which show that the proposed method has a strong agreement with the exact solutions of the problems. The numerical solutions of the present method are also compared with steepest decent method and Adomian decomposition method. The comparison with other methods reveals that this method has the highest degree of accuracy than those methods.

SLIP MODEL OF THE 2001 KUNLUN MOUNTAIN MS8.1 EARTHQUAKE BY SDM: JOINT INVERSION FROM GPS AND INSAR DATA

AbstractFor the 2001 MS8.1 Kunlun earthquake, which was one of the largest events occurred around the Tibet plateau, a large controversy still exists about its rupture detail. In this paper, we invert the co‐seismic GPS and InSAR data for a robust finite‐fault model of the Kunlun earthquake based on a realistic fault geometry buried in a layered earth structure. The inversion is based on the constrained least‐squares principle and realized using the steepest decent method (SDM). The different data sets are weighted according to their variance and spatial coverage. The results show that the slip maximum can reach up to ∼6.9 m and is located at 35.76° N and 93.40° E. The main rupture area is located at the shallow depth above 20 km. The inverted shallow slip agrees with the surface rupture observed by the field survey, and the whole slip pattern appears generally consistent with the results obtained from previous geological and seismic wave studies.

Barycentre method for solving distance equations

When iteratively solving the distance equations, the Newton's method has quadratic convergence but it requires the second-order derivatives. The Gauss–Newton's method needs no information about the second-order derivatives but it may fail without the line search strategy. A simple method called barycentre method is proposed to locally solving the distance equations without the Hessian matrix, the matrix inversion and the line search strategy. The geometrical meaning of the non-linear least-squares solution of the distance equations is revealed that it is the barycentre of a particle system composed of the observational weights at the endpoints of observed distance vectors. By the geometrical meaning of the non-linear least squares, the authors structure an iterative equation to solve the distance equation, and then the convergence and the computational complexity of the method proposed is discussed. It shows that the barycentre method is a conservatively steepest decent method to guarantee the convergence and this method has good performances for solving well-conditioned problems. Ultimately, the method proposed is applied to well-condition and ill-posed positioning equations and the main results are verified.

A Modified Algorithm of Steepest Descent Method for Solving Unconstrained Nonlinear Optimization Problems

The steepest descent method (SDM), which can be traced back to Cauchy (1847), is the simplest gradient method for unconstrained optimization problem. The SDM is effective for well-posed and low-dimensional nonlinear optimization problems without constraints; however, for a large-dimensional system, it converges very slowly. Therefore, a modified steepest decent method (MSDM) is developed to deal with these problems. Under the MSDM framework, the original global minimization problem is transformed into a quadratic-form minimization based on the SDM and the current iterative point. Our starting point is a manifold defined in terms of the quadratic function and a fictitious time variable. Thereafter, we can derive an iterative algorithm by including a parameter in the final stage. Through a Hopf bifurcation, this parameter indeed plays a major role to switch the situation of slow convergence to a new situation that the new algorithm converges faster. Several numerical examples are examined and compared with exact solutions. It is found that the new algorithm of the MSDM has better computational efficiency and accuracy, even for a large-dimensional non-convex minimization problem of the generalized Rosenbrock function.

GLOBAL CONVERGENCE METHODS FOR NONSMOOTH EQUATIONS WITH FINITELY MANY MAXIMUM FUNCTIONS AND THEIR APPLICATIONS

Nonsmooth equations with finitely many maximum functions is often used in the study of complementarity problems, variational inequalities and many problems in engineering and mechanics. In this paper, we consider the global convergence methods for nonsmooth equations with finitely many maximum functions. The steepest decent method and the smoothing gradient method are used to solve the nonsmooth equations with finitely many maximum functions. In addition, the convergence analysis and the applications are also given. The numerical results for the smoothing gradient method indicate that the method works quite well in practice.

Fast source optimization involving quadratic line-contour objectives for the resist image

In Abbe's formulation, source optimization (SO) is often formulated into a linear or quadratic problem, depending on the choice of objective functions. However, the conventional approach for the resist image, involving a sigmoid transformation of the aerial image, results in an objective with a functional form. The applicability of the resist-image objective to SO or simultaneous source and mask optimization (SMO) is therefore limited. In this paper, we present a linear combination of two quadratic line-contour objectives to approximate the resist image effect for fast convergence. The line-contour objectives are based on the aerial image on drawn edges using a constant threshold resist model and that of pixels associated with an intensity minimum for side-lobe suppression. A conjugate gradient method is employed to assure the convergence to the global minimum within the number of iterations less than that of source variables. We further compare the optimized illumination with the proposed line-contour objectives to that with a sigmoid resist-image using a steepest decent method. The results show a 100x speedup with comparable image fidelity and a slightly improved process window for the two cases studied.

Numerical Synthesis of Dual-Band Reflectarray Antenna for Optimum Near-Field Radiation

A dual-band reflectarray antenna with a double layer, employing concentric square rings as reflecting elements, is proposed. With a procedure of the steepest decent method (SDM), the impressed phases of the reflecting elements are synthesized for the proposed antenna to radiate an optimum field pattern in the near zone of array aperture. The reflectarray antenna is designed for two frequency bands, covering 915 MHz and 2.4 GHz, which both can be used in RFID applications.

SU-GG-T-601: Design and Development of a New Micro Beam Treatment Planning System

Purpose: To develop the Treatment Planning System for a new microbeam treatment system. Method and Materials: The microbeam treatment system was composed of an x‐band linac with a robot arm, the real‐time tracking system using x‐ray fluoroscopy, and the treatment planning system. The system realizes 4 . irradiation usingmany microbeams of which the diameter is about 1mm. We developed the software for the optimization of beam directions/beam intensities and dose calculation. We used the pseudo Beam's Eye Beam (pBEV) method in the beam direction optimization and the steepest decent method and the objective function in the beam intensity optimization. We employed the superposition/convolution algorithm and the density scaling method (developed by Siddon) in the dose calculation. The dose kernel was calculated by using the EGS5 Monte Carlo simulation code. The calculated result was compared with the Monte Carlo‐calculated PDD curve in water. Finally, the microbeam irradiation for a small tumor in lung was simulated using the treatment planning system. Results: One hundred beams were selected using the pBEV scores from 408 beams. The beam intensities of each beam were determined by the steepest decent method. The dose distribution for a small lungtumor was calculated using the beam direction/intensity. The dose was concentrated on the PTV and the OARs were spared effectively. The dose difference between the calculated PDD curve with our system and the Monte‐Carlo calculated curve in water was within 1%. The calculation time of 1 port was about 15sec. Conclusion: We developed the treatment planning system for a new microbeam treatment system. The dose distribution for a small lungtumor was simulated using the software. Conflict of Interest Statement: Research sponsored by New Energy and Industrial Technology Development Organization, Japan.

SU-GG-T-614: The Implementation of Auto-Optimization Method to Determine Photon Energy Spectrums and Dose Profiles with Various Field Sizes for Collapsed Cone Convolution Algorithm

Purpose: As a part of developing treatment planning system, auto-optimization method was designed to determine photon energy spectrum and the shape of dose profile with various field sizes. Method and Materials: The initial distribution of energy spectrum was designed using approximate formulation. Multi-spectrum kernels were constructed with the determine photon energy fluence using published mono-energy kernel, and collapse cone convolution algorithm was implemented to calculate PDDs and dose profiles by considering the effects of beam hardening with depth, kernel tilting, beam softening with off axis distance, off axis ratio, and finite beam source size. Objective function was defined with the sums of differences between measured PDD or dose profiles and calculated PDD or dose profiles. Auto-optimization method based on steepest decent method was used to minimize objective function by calculating gradient values at each iteration. Results: Determined energy spectrums were not similar to rear photon spectrums, but calculated PDDs showed good agreements with measured data. The dose error beyond build-up depth was less than 2 % at any depth. Some under-estimated results (>5%) were showed at the buildup regions (<2cm depth) at large field (>30×30 cm2) because not considering electron contamination. Calculated dose profiles were also well agreed with measured dose profiles (<3% errors) up to 10×10 cm2, but some discrepancies at penumbra region were appeared at large field. Conclusion: Determined energy spectrums of photon beam with various field sizes were useful to simulate the variation of PDDs. Multi-spectrum model with field sizes based on auto-optimization was good approximated method to simulate measured dose distribution, and it seemed to compensate the limitation of classical convolution/superposition algorithm using finite radius of kernel.

FAST SDM FOR SHAPED REFLECTOR ANTENNA SYNTHESIS VIA PATCH DECOMPOSITIONS IN PO INTEGRALS

This paper presents an approach of a shaped reflector antenna synthesis using a steepest decent method (SDM). It first discretizes the reflector surface into small patches and then uses grid nodes as variables in the synthesis procedure. Even though the number of variables can be very large for a large reflector antenna, the advantage of providing closed-form solutions for the derivatives of a cost function potentially makes this approach very efficient. The large number of variables also assists this procedure to reach a more global optimum as usually met in ordinary SDM. Numerical examples are presented to validate this approach.

Design of a high power millimeter wave launcher for EC H&CD system on ITER

An optimization procedure of the quasi-optical system for a millimeter wave launcher is developed for the ITER electron cyclotron heating and current drive (EC H&CD) launcher. In the launcher, the radiated RF beams from eight corrugated waveguides are reflected sequentially by two mirrors and injected into a plasma through a small aperture in the blanket shield module on the top of the launcher. Using a steepest decent method, the heat load on the mirrors is successfully reduced to the acceptable level by flattening the RF power profile on the mirrors keeping the scattering of the RF power to a minimum from the mirrors. It is found that 20 MW injection will be acceptable even when the resistivity of 2.64 × 10 −8 Ωm for the surface of the mirror (dispersion strengthened copper, 151 °C assumed) is increased by a factor of ∼10 with a contamination.

Diffraction of sound waves by a finite barrier in a moving fluid

The diffraction of a line source by an absorbing finite barrier, satisfying Myers' impedance condition [M.K. Myers, On the acoustic boundary condition in the presence of flow, J. Sound Vibration 71 (1980) 429–434] in the presence of a subsonic flow is studied. The problem is solved analytically by using Integral transforms, Wiener–Hopf technique and the asymptotic methods. The expression for the diffracted field is shown to be the sum of the fields produced by the two edges of the strip and a field due to the interaction of the two edges. The diffracted field in the far zone is determined by the method of steepest decent.

A note on plane wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium

We studied the problem of diffraction of an electromagnetic plane wave by a perfectly conducting finite strip in a homogeneous bi-isotropic medium and obtained some improved results which were presented both mathematically and graphically. The problem was solved by using the Wiener-Hopf technique and Fourier transform. The scattered field in the far zone was determined by the method of steepest decent. The significance of present analysis was that it recovered the results when a strip was widened into a half plane.

Simulation of a regression-model and PCA based searching method developed for setting the robust injection molding parameters of multi-quality characteristics

This article proposes an advanced searching method for setting the robust process parameters for injection molding based on the principal component analysis (PCA) and a regression model-based searching method. This method could effectively reduce the influence of environmental noise on molded parts’ multi-quality characteristics in the injection molding process. Firstly, the PCA is utilized to construct a composite quality indicator to represent the quality loss function of multi-quality characteristics. The design of experiment and ANOVA methods are then used to choose the major parameters, which affect parts quality and are called as adjustment factors. Secondly, a two-level statistically designed experiment with the least squared error method was used to generate a regression model between part quality and adjustment factors. Based on this mathematical model, the steepest decent method is used to search for the optimal process parameters. To verify the performance, computer simulations and experiment of the light-guided plate molding were investigated in this work. By comparing the robust qualities using Taguchi method and our proposed method, it is found that our proposed method yields a better uniform production quality.

Adjoint modeling for Acoustic Inversion based on an Adjoint Parabolic Equation

In this paper the acoustic propagation problem is modeled by the wide angle parabolic equation and the bottom boundary condition is in the form of a Neumann to Dirichlet map. We formulate the inversion as an optimal control problem, the control parameters being the sound speed in the water and the sound speed, density and attenuation in the bottom. Using the adjoint operator to the wide angle parabolic equation, an inversion scheme is derived for the acoustic parameters of the water column and the bottom region. The optimization approach used is the method of steepest decent. Several test cases are exhibited.

A new TDOA algorithm based on Taylor series expansion in cellular networks

Time difference of arrival (TDOA) is the positioning technique with the most potential in cellular mobile telecommunication systems. The Taylor series expansion method has been widely used in solving nonlinear equations for its high accuracy and good robustness. However, the performance of the Taylor’s method depends highly on the initial estimation. Therefore, one new algorithm, hybrid optimizing algorithm (HOA) was proposed, which combines the Taylor series expansion method with the steepest decent method. The steepest decent method features fast convergence at the initial iteration and small computation complexity. HOA takes great advantage of both methods. Simulation results show that HOA achieves better performance on positioning accuracy and efficiency.

Identification of Material Properties of PZT Single Crystals through Crystallographic Homogenization Method

Single crystals of lead zirconium titanate (PZT) are difficult to fabricate. Thus, not all material properties of PZT have been fully characterized. In this paper, the mechanical and electrical properties of a PZT single crystal, which can be assumed to be identical to those of a crystal grain in a polycrystal, have been computed from those of a polycrystalline PZT ceramic by the steepest decent method and multiscale finite element modeling based on crystallographic homogenization method. Crystallographic homogenization enables us to predict macroscopic properties of ceramics taking into account the inhomogeneous microstructure of an aggregate of crystal grains. The crystal morphology of the PZT ceramic was measured by the SEM·EBSD technique, and the result was used in the microscopic finite element model. Then, the mechanical and electrical properties of the crystal grain were derived by the steepest decent method so that its macroscopic properties would correspond to the measured properties of the PZT ceramic. The proposed computational method was applied to barium titanate (BT) and validated by comparison of the computed material properties with known properties of the BT single crystal. Finally, the computed material properties, such as the elastic compliance, and the dielectric and piezoelectric constants, were presented for the PZT single crystal.