Nonlinear Conjugate Gradient Algorithm Research Articles

Two Sets of High-Conductivity Systems with Different Scales Reveal the Seismogenic Mechanism of Earthquakes in the Songyuan Area, Northeastern China

To reveal the deep seismogenic environment and mechanism of earthquakes in Songyuan City, Northeastern China, 59 broadband magnetotelluric sites in the Songyuan area were arranged in this study at a spacing of 5 km. In addition, two intersecting magnetotelluric profiles, with a total of 23 measuring sites and a spacing of 2 km, were established near the Ningjiang earthquake swarm. Using a nonlinear conjugate gradient (NLCG) algorithm, resistivity structures in the lithosphere were obtained at different scales using three-dimensional (3D) inversion. The research results show that: a deep high-conductivity system (<10 Ω·m) was identified at 25–85 km depth in the lithosphere under Songyuan, corresponding closely to a region of high heat flow. It is inferred to be the molten material of mantle upwelling. In addition, a shallow high-conductivity system (<10 Ω·m) was identified beneath the Ningjiang earthquake swarm, which is interpreted to correspond to the Fuyu North fault. It is the main seismo-controlling structure of the Ningjiang earthquake swarm. The deep seismogenic environment and seismogenic mechanism of the Ningjiang earthquake swarm can be described as a deep upwelling of molten mantle material, which provides the power source. The deep magma intruded into the lower crust and accumulated, then intruded along faults and fissures, resulting in the activation of the North Fuyu fault and triggering the Ningjiang earthquake. It is attributed to the activation of shallow faults caused by the upwelling of molten mantle material.

A splitting scheme for the quantum Liouville-BGK equation

We introduce in this work an efficient numerical method for the simulation of the quantum Liouville-BGK equation, which models the diffusive transport of quantum particles. The corner stone to the model is the BGK collision operator, obtained by minimizing the quantum free energy under the constraint that the local density of particles is conserved during collisions. This leads to a large system of coupled nonlinear nonlocal PDEs whose resolution is challenging. We then define a splitting scheme that separates the transport and the collision parts, which, exploiting the local conservation of particles, leads to a fully linear collision step. The latter involves the resolution of a constrained optimization problem that is handled with the nonlinear conjugate gradient algorithm. We prove that the time semi-discrete scheme is convergent, and as an application of our numerical scheme, we validate the quantum drift-diffusion model that is obtained as the diffusive limit of the quantum Liouville-BGK equation.

Lithotectonic architecture of the Paleoproterozoic intracratonic Kaladgi rift basin, southwestern India: Inference from a magnetotelluric study

Peninsular India is traversed by several major Paleoproterozoic rift basins, among which the Kaladgi rift basin is a prominent one. Here, we present results from Magnetotelluric (MT) studies in the eastern section of this rift basin. The data were acquired along a ∼120 km long profile passing through Belavanik-Bagalkot-Ukkali regions covering the Paleoproterozoic intracratonic Kaladgi rift basin, the Cretaceous Deccan volcanic province and the Archean gneisses/granites/greenstones of the Dharwar craton. The lithospheric electrical structure was obtained from joint inversion of TE- and TM-modes data using a 2-D nonlinear conjugate gradient algorithm. The results reveal that the Kaladgi sediments are ∼500–1000 m in thickness overlying a highly fractured crystalline basement. Proterozoic sediments are exposed beneath the Deccan basalt cover and in the shallow fractured zones. The conductive-resistive transitions in the model at the crustal depths correspond to major tectonic faults and deformed crustal rocks. The transition boundary at ∼50 km depth defines the electrical Moho. At the same time, a ∼25 km wide, steep conductive feature from ∼75 km to upper mantle lithosphere depths (∼180–200 km) is interpreted as the trace of the Chitradurga Suture Zone (CSZ). The observed crustal heterogeneity, conductivity variations, and resistive lithosphere are attributed to the geological history of the region, including Archean collisional events and subsequent reactivation associated with the Reunion hotspot. The crustal conductors are associated with mafic intrusions from the underplated basalts, and also the high conductivity may be due to the sulfide mineralization in the fault zones. The MT study provides valuable insights into the deep tectonic framework of the region.

Missing data recovery of wind speed in wind farms: A spatial-temporal tensor decomposition approach

Missing data recovery plays a critical role in improving the data quality of wind speed in wind farms, and numerous methods have been proposed to address this issue. However, most of them suffer from the inability to fully use the information of known data, and thus, poor performance of recovery is usually achieved. In this paper, we propose a missing data recovery method based on spatial-temporal tensor decomposition. The proposed method rearranges the whole data based on discrete wavelet transform to construct a four-dimensional tensor of “site × week × scale × hour” for representing the spatial and temporal correlation of wind speed. A completeness tensor is estimated to impute missing data based on Tucker decomposition and the nonlinear conjugate gradient algorithm. The proposed method not only inherits the advantages of imputation methods based on the matrix pattern but also well mines the spatial and temporal inherent correlation of wind speed. Wind speed data of a wind farm are used to verify the effectiveness of the proposed method. The results show that the proposed method recovers missing data with much smaller mean absolute error and root mean square error and requires less effort for recovering missing data of fragmented or continuously, compared with the traditional methods.

A comparison between steepest descent and non-linear conjugate gradient algorithms for binding energy minimization of organic molecules

The main intention of optimization is to bring about the “best” of any model by prioritizing the needs along with a given set of constraints. There is a wide range of optimization problems, among which, unfortunately, the problems that are formulated from nature are not convex in nature. Solving non-convex optimization problems are quite trickier than the conventional method of derivatives. One such problem is Computing the minimum value of binding free energy of various molecules. Minimization of free energy of a molecule is highly significant in the field of Molecular mechanics which is the foundation of computational biology. For any molecule, free energy refers to the amount of energy needed to separate an individual particle from a system of particles or to disseminate all particles in a system. The significance of free energy is that it can be used to compute the lowest energy conformation, which corresponds to the least Steric energy. Hence, this paper aims at computing the minimum value of binding free energy of various organic molecules in isolated conditions using the steepest descent algorithm and conjugate gradient algorithm and comparing them.

On image restoration problems using new conjugate gradient methods

<span lang="EN-US">The nonlinear conjugate gradient algorithm is one of the effective algorithms for optimization since it has low storage and simple structure properties. The coefficient conjugate is the basis of conjugate gradient algorithms with the desirable conjugate property. In this manuscript, we have derived a new second order information for the Hessian from objective function, which can give a new search direction. Based on new search direction, we have proposed the update formula interesting and nonlinear conjugate gradient method. Under wolfe line search and mild assumptions on objective function, the method possess sufficient descent property and are always globally convergent. Numerical results show that the method is effective and competitive to recover the original image from an image corrupted by impulse noise.</span>

A modified nonlinear conjugate gradient algorithm for unconstrained optimization and portfolio selection problems

Conjugate gradient methods play a vital role in finding solutions of large-scale optimization problems due to their simplicity to implement, low memory requirements and as well as their convergence properties. In this paper, we propose a new conjugate gradient method that has a direction satisfying the sufficient descent property. We establish global convergence of the new method under the strong Wolfe line search conditions. Numerical results show that the new method performs better than other relevant methods in the literature. Furthermore, we use the new method to solve a portfolio selection problem.

Solving Electromagnetic Inverse Problem Using Adaptive Gradient Descent Algorithm

The nonlinear conjugate gradient (NLCG) algorithm is one of the popular linearized methods used to solve the frequency-domain electromagnetic (EM) geophysical inverse problem. During NLCG iterations, the model gradient guides the searching direction while the line-search algorithm determines the step length of each iteration. Normally, the line search requires solving the corresponding forward problem a few times. Since line search is usually computationally inefficient, we introduce the adaptive gradient descent (AGD) algorithm to accelerate solving the frequency-domain EM inverse problem within the linearized framework. The AGD algorithm is a variant of the classical gradient descent method and has been well-developed and widely used in deep learning. Rather than the time-consuming line search, its core idea is to algebraically manipulate the cumulative gradients and updates of the model from previous iterations to estimate the model parameter variables at the current iteration. For the inversion of magnetotelluric (MT) data, we here designed and implemented a framework using the AGD algorithm combined with the cool-down scheme to tune the regularization parameter. To improve the convergence performance of the AGD algorithm [specifying to Adam and root-mean-square propagation (RMSProp)], we proposed a tolerance strategy which has been tested numerically. To optimize the global learning rate, we carried out some comparative trials in the proposed inversion framework. The inverted results of synthetic and real-world data showed that both the AGD algorithms (Adam and RMSProp) can recover comparable results and save more than a third of CPU time compared with the NLCG algorithm.

A new three-term conjugate gradient algorithm with modified gradient-differences for solving unconstrained optimization problems

<abstract><p>Unconstrained optimization problems often arise from mining of big data and scientific computing. On the basis of a modified gradient-difference, this article aims to present a new three-term conjugate gradient algorithm to efficiently solve unconstrained optimization problems. Compared with the existing nonlinear conjugate gradient algorithms, the search directions in this algorithm are always sufficiently descent independent of any line search, as well as having conjugacy property. Using the standard Wolfe line search, global and local convergence of the proposed algorithm is proved under mild assumptions. Implementing the developed algorithm to solve 750 benchmark test problems available in the literature, it is shown that the numerical performance of this algorithm is remarkable, especially in comparison with that of the other similar efficient algorithms.</p></abstract>

Hybrid spectrum conjugate gradient algorithm in electromagnetic tomography

Electromagnetic tomography is a process detection technology based upon the principles of electromagnetic induction. The forward problem model and sensitivity distribution matrix of electromagnetic tomography are introduced as the basis of the inverse problem. The search direction and iterative parameters of the conjugate gradient algorithm are modified to improve the quality and convergence of image reconstruction. A new spectral parameter conjugate gradient algorithm is described to modify the search direction, which is used to control the angle between the old and new search directions. The search direction is determined according to the iteration of each step in order to find the optimal solution. Combining the advantages of the Fletcher-Reeves and Polak-Ribiere-Polyak algorithms in the nonlinear conjugate gradient algorithm, they are mixed in a specific proportion to obtain a new hybrid conjugate gradient algorithm. In order to verify the effectiveness of the modified conjugate gradient algorithm, three physical models of electromagnetic tomography system are constructed, and the modified conjugate gradient algorithm is compared with the traditional algorithm. The experimental results show that the reconstructed image quality of the modified spectral conjugate gradient algorithm is higher and has better numerical performance. The hybrid conjugate gradient algorithm highlights the advantages of the Fletcher-Reeves and Polak-Ribiere-Polyaks algorithms. The convergence speed is faster than the Polak-Ribiere-Polyak method, and the imaging quality is higher than the other algorithms.

An efficient new hybrid CG-method as convex combination of DY and CD and HS algorithms

In this paper, we proposed a new hybrid conjugate gradient algorithm for solving unconstrained optimization problems as a convex combination of the Dai-Yuan algorithm, conjugate-descent algorithm, and Hestenes-Stiefel algorithm. This new algorithm is globally convergent and satisfies the sufficient descent condition by using the strong Wolfe conditions. The numerical results show that the proposed nonlinear hybrid conjugate gradient algorithm is efficient and robust.

Modification of the new conjugate gradient algorithm to solve nonlinear fuzzy equations

The conjugate gradient approach is a powerful tool that is used in a variety of areas to solve problems involving large-scale reduction. In this paper, we propose a new parameter in nonlinear conjugate gradient algorithms to solve nonlinear fuzzy equations based on Polak and Ribiere (PRP) method, where we prove the descent and global convergence properties of the proposed algorithm. In terms of numerical results, the new method has been compared with the methods of Fletcher (CD), Fletcher and Reeves (FR), and Polak and Ribiere (PRP). The proposed algorithm has outperformed the rest of the algorithms in the number of iterations and in finding the best value for the function and the best value for the variables.

Magnetotelluric data analysis using 2D resistivity modelling in Gondang region, Bojonegoro

Previous studies in Gondang Subdistricthave identified underground mud reservoirnear a rock intrusion, while another study near Gondang Subdistrict has identified deep fault structures. This study will identify the distribution and characteristics of possible geological features in the area using magnetotelluric (MT) method to further describe the relationship between the geological features related to local geology. MT data measurements were conducted on 7 stations alonga north-south line, then modelled in 2D using nonlinear conjugate gradient algorithm. The model was used to describe the subsurface resistivity distribution and to identify the geological features. The results show 5 resistive zones (20–1250 Ω⋅m) and 4 conductive zones (≤10 Ω⋅m). The former consist of 4 vertical zones, 1 vertical zone deeper than 5 km, and 1 horizontal zone near the surface. 2 conductive zones surround a resistive zone, while 2 others stretch below the horizontal resistive zone. The vertical resistive zones are interpreted as andesite intrusions, and the horizontal one as volcanic breccia. The conductive zones are interpreted consisting of tuff and marl with possible saline water content. 3 vertical intrusions are thought to have the same source, and all vertical intrusions are suspected to co-occurr with the Pandan Volcano intrusion.

A new family of conjugate gradient methods to solve unconstrained optimization problems

A conjugate optimal coefficient is a crucial characteristic of conjugate gradient algorithms. The idea of accelerating the conjugate gradient by utilizing the conjugacy condition information and quadratic model. The gradient of the objective function is used to specify the search directions for traditional techniques. This work provides nonlinear conjugate gradient algorithms that primarily consider objective function information. One of the ways is as efficient as or more efficient than the conventional methods, according to numerical examples.

Cascaded Metasurface Design Using Electromagnetic Inversion With Gradient-Based Optimization

This article presents an electromagnetic inversion algorithm for the design of cascaded metasurfaces that enables the design process to begin from more practical output field specifications, such as a desired power pattern or far-field (FF) performance criteria. Thus, this method combines the greater field transformation support of multiple metasurfaces with the flexibility of the electromagnetic inverse source framework. To this end, two optimization problems are formed: one associated with the interior space between two metasurfaces and the other for the exterior space. The cost functionals corresponding to each of these two optimization problems are minimized using the nonlinear conjugate gradient (CG) algorithm with analytic expressions for the gradient operators. The numerical implementation of the developed design procedure is presented in detail, including a total variation (TV) regularizer that is incorporated into the optimization procedure to favor smooth field variations from one unit cell to the next. The capabilities of the method are demonstrated by converting the produced surface susceptibilities into three-layer admittance sheet models, which are simulated in several 2-D examples.

Application of geophysical and hydrogeological analyses to predict water and hydrocarbon entry to the T5 Tunnel, west Iran

The Garmsiri Project, including the 4.5 km long T5 Tunnel, is under construction in western Iran. The T5 Tunnel passes through the NW–SE-trending Emam Hasan Anticline (EHA), perpendicular to the fold axis. The fold is mainly composed of marlstone and argillaceous limestone layers of Cretaceous to Miocene age, incorporating the Pabdeh–Gurpi Formation, karst limestone of the Asmari Formation, and marlstone and gypsum of the Gachsaran Formation. There was a risk of water entry into the tunnel as it was constructed below the regional groundwater table elevation. In addition, the entry of hydrocarbons, in either liquid or vapour phase, into the tunnel was possible owing to the presence of numerous active bitumen mines in the vicinity of the anticline. To predict the risk of water or hydrocarbon entry geological and hydrogeological analyses together with the audio magnetotelluric (AMT) method were applied. Based on the field work, resistivity and geological cross-sections were provided along the tunnel path. Several boreholes were drilled along the tunnel route and water-table elevation, rock quality designation (RQD) and permeability values were measured. To determine a broad range of features related to the anticline, 55 AMT stations were positioned along the tunnel route. Dimensionality analysis confirmed 2D dimensionality of the AMT transfer functions, which allowed us to apply the 2D bimodal inversion using a non-linear conjugate gradient algorithm. Integration of the geological and hydrogeological data with the resistivity model resulted in a more detailed geological section along the tunnel, including water-table elevation and identification of highly conductive zones in which bitumen migrated. It was predicted that water entry would be observed through the Asmari Formation and also that elevated H 2 S concentrations would be encountered during drilling in the anomalous conductive zones. Monitoring results and field observations gained during the tunnel construction were compared with the predictions.

Spectral CG Algorithm for Solving Fuzzy Nonlinear Equations

The nonlinear conjugate gradient method is an effective technique for solving large-scale minimizations problems, and has a wide range of applications in various fields, such as mathematics, chemistry, physics, engineering and medicine. This study presents a novel spectral conjugate gradient algorithm (non-linear conjugate gradient algorithm), which is derived based on the Hisham–Khalil (KH) and Newton algorithms. Based on pure conjugacy condition The importance of this research lies in finding an appropriate method to solve all types of linear and non-linear fuzzy equations because the Buckley and Qu method is ineffective in solving fuzzy equations. Moreover, the conjugate gradient method does not need a Hessian matrix (second partial derivatives of functions) in the solution. The descent property of the proposed method is shown provided that the step size at meets the strong Wolfe conditions. In numerous circumstances, numerical results demonstrate that the proposed technique is more efficient than the Fletcher–Reeves and KH algorithms in solving fuzzy nonlinear equations.

Paleozoic suture and Mesozoic tectonic evolution of the lithosphere between the northern section of the Xing'an Block and the Songnen Block: Evidence from three-dimensional magnetotelluric detection

The Xing'an and Songnen Blocks are microcontinents in the eastern Central Asian Orogenic Belt (CAOB). The Heihe–Hegenshan suture has been interpreted to represent the boundary between the blocks. However, the lack of ophiolite exposure and deformation towards the northern extent of the suture mean that its precise nature and location remain controversial. To better understand the position and structure of the suture, magnetotelluric data were acquired at 104 stations along five profiles in the northern Songnen and Xing'an Blocks. A three-dimensional nonlinear conjugate gradient algorithm was used to image the resistivity structure of the lithosphere. Our resistivity models indicate that the lithospheres of both the Xing'an and Songnen Blocks are characterized by overall high resistivity and have heterogeneous electrical structures. We define a highly conductive anomaly (C2) to the east of the Duobaoshan Island Arc that divides lithosphere of high resistivity and controls the distribution of Carboniferous magmatic rocks related to collision. We suggest that C2 is the northern extension of the Heihe–Hegenshan suture. The highly conductive anomaly (C1) is located in the Xing'an Block on the west side of the island arc. While the exposed lithology on the surface is Jurassic intrusions, usually with high resistivity. We suggest that C1 may be an early structure, which was transformed by the closure of the Mongolia–Okhotsk Ocean. No large-scale Mesozoic intrusive bodies occur above C3, a highly conductive anomalous region in the Songnen Block. And the seismic reflection characteristics have changed significantly in the region C3. Therefore, we interpret C3 as a shear zone in the crust related to a change in the subduction direction of the paleo-Pacific plate. The resistivity model provides new geophysical evidence for determining the suture and the evolutionary traces of the Mongolia-Okhotsk Ocean and the Paleo-Pacific Ocean.

Lithospheric structure and transformation of the Songnen and Jiamusi blocks in the eastern segment of the Central Asian Orogenic Belt revealed by 3D magnetotelluric inversion

The Central Asian Orogenic Belt (CAOB) is the largest accretionary orogenic belt in the world. The Songnen and Jiamusi blocks, which are located along the eastern segment of the CAOB, form a complex structure influenced by the Mudanjiang ocean closure and the Paleo-Pacific plate subduction. Constraints on the deep structure of the Songnen and Jiamusi blocks are critical for studying the evolution, transformation and current activity of the CAOB. This study presents an analysis and interpretation of a 450-km-long magnetotelluric profile that imaged the deep lithospheric structure of these two blocks. The best-fit three-dimensional resistivity model, based on the nonlinear conjugate gradient algorithm, highlights the key lithospheric structures. The results show that the lithosphere beneath the Lesser Xing'an Range in the Songnen block is characterized by a high-resistivity structure, which is mainly related to the granite formed in multiple stages. The crust forming the Sanjiang basin in the Jiamusi block possesses a high–low–high resistivity structure in the transverse direction, corresponding well to the tectonic units. The low-resistivity structure of the Fujin uplift extends to the mantle, which represents mantle material that intruded the crust. There is a low-resistivity structure beneath the suture zone of the two blocks that is homologous with the large-scale low-resistivity anomaly beneath the basin and is related to the subducted Paleo-Pacific plate. Strong dehydration at the inflection point of the plate reduces the solidus temperature of the mantle material, resulting in melt and upwelling of mantle material. The Jiamusi block can be divided into southern and northern parts, possessing different structures. The southern part represents an original part of either the Gondwana continent or Tarim craton. The Sanjiang basin, in the northern part, has been heavily influenced by the subducted Paleo-Pacific plate, and is a region of strong activity.

A Modified Hybrid Conjugate Gradient Method for Unconstrained Optimization

The nonlinear conjugate gradient algorithms are a very effective way in solving large-scale unconstrained optimization problems. Based on some famous previous conjugate gradient methods, a modified hybrid conjugate gradient method was proposed. The proposed method can generate decent directions at every iteration independent of any line search. Under the Wolfe line search, the proposed method possesses global convergence. Numerical results show that the modified method is efficient and robust.