Introducing conjugate gradient optimization for modified HL-RF method

The gradient-based optimization methods are preferable for the large-scale three-dimensional (3D) magnetotelluric (MT) inverse problem. Compared with the popular nonlinear conjugate gradient (NLCG) method, however, the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) method is less adopted. This paper aims to implement a L-BFGS-based inversion algorithm for the 3D MT problem. And we develop our code on top of the ModEM package, which is highly extensible and popular among the MT community. To accelerate the convergence speed, the preconditioning technique by the affine linear transformation of the original model parameters is used. Two modifications of the conventional L-BFGS algorithm are also made to get a comparable convergence rate with the NLCG method. The impacts of the preconditioner parameters, the regularization parameters, the starting model, etc., on the inversion are evaluated by synthetic examples for both L-BFGS and NLCG methods. And the real MT Kayabe dataset is also inverted by the inversion algorithms. The synthetic tests show that through our L-BFGS inversion algorithm the similar resistivity models can be obtained with that from the NLCG method. For the real data inversion, the L-BFGS method performs more efficiently and reasonable results could be obtained by less iterations of the inversion process than the NLCG method. Thus, we suggest the common usage of the L-BFGS method for the 3D MT inverse problem.

We study the development of nonlinear conjugate gradient methods, Fletcher Reeves (FR) and Polak Ribiere (PR). FR extends the linear conjugate gradient method to nonlinear functions by incorporating two changes, for the step length αk a line search is performed and replacing the residual, rk (rk=b-Axk) by the gradient of the nonlinear objective function. The PR method is equivalent to FR method for exact line searches and when the underlying quadratic function is strongly convex. The PR method is basically a variant of FR and primarily differs from it in the choice of the parameter βk. On applying the nonlinear Rosenbrock function to the MATLAB code for the FR and the PR algorithms we observe that the performance of PR method (k=29) is far better than the FR method (k=42). But, we observe that when the MATLAB codes are applied to general nonlinear functions, specifically functions whose minimum is a large negative number not close to zero and the iterates too are large values far off from zero the PR algorithm does not perform well. This problem with the PR method persists even if we run the PR algorithm for more iterations or with an initial guess closer to the actual minimum. To improve the PR algorithm we suggest finding a better weighing parameter βk, using better line search method and/or using specific line search for certain functions and identifying specific restart criteria based on the function to be optimized.

In this chapter, we investigate recently proposed nonlinear conjugate gradient (NCG) methods for shape optimization problems. We briefly introduce the methods as well as the corresponding theoretical background and investigate their performance numerically. The obtained results confirm that the NCG methods are efficient and attractive solution algorithms for shape optimization problems.

Recently, similar to Hager and Zhang (SIAM J Optim 16:170–192, 2005), Yu (Nonlinear self-scaling conjugate gradient methods for large-scale optimization problems. Thesis of Doctors Degree, Sun Yat-Sen University, 2007) and Yuan (Optim Lett 3:11–21, 2009) proposed modified PRP conjugate gradient methods which generate sufficient descent directions without any line searches. In order to obtain the global convergence of their algorithms, they need the assumption that the stepsize is bounded away from zero. In this paper, we take a little modification to these methods such that the modified methods retain sufficient descent property. Without requirement of the positive lower bound of the stepsize, we prove that the proposed methods are globally convergent. Some numerical results are also reported.

A key component of conjugate gradient (CG) algorithms is the directions used in the CG approach. Because of their low memory requirements, these directed methods have proven effective in various applications, especially image processing. For this study, we have developed a new conjugate gradient coefficient based on the well-known quadratic model, that will accelerate the convergence and increase the accuracy of the CG approach. Because the new method uses the second order curvature and provides improved direction. So, the global convergence and critical descent characteristics of the final algorithm provide reliable performance in a range of scenarios. The results of extensive numerical testing on picture restoration show that the new formulae perform noticeably better than the current techniques. In particular, the performance of the suggested conjugate gradient (CG) scheme has been better than that of the conventional Fletcher-Reeves (FR) conjugate gradient approach. This development is a key tool in the field of computational imaging and optimization as it not only increases computing efficiency but also boosts picture restoration, image quality and its' accuracy.

In this paper, we propose a three-parameter family of conjugate gradient methods for unconstrained optimization. The three-parameter family of methods not only includes the already existing six practical nonlinear conjugate gradient methods, but subsumes some other families of nonlinear conjugate gradient methods as its subfamilies. With Powell’s restart criterion, the three-parameter family of methods with the strong Wolfe line search is shown to ensure the descent property of each search direction. Some general convergence results are also established for the three-parameter family of methods. This paper can also be regarded as a brief review on nonlinear conjugate gradient methods.

Optimum length of glass fiber-reinforced polymers (GFRP), as an advantageous type of composites, on the retrofitted columns of RC frames is obtained in this study by conducting seismic reliability analyses. To achieve this goal, maximum reliability index with different FRP lengths of columns of a six-story RC frame was calculated to find the optimal length for FRP-retrofitted columns. The seismic behavior of structure after verification was assessed by nonlinear time history analysis with OpenSees finite element software. Three scaled far-field earthquake records were selected for nonlinear time history analysis. The reliability analysis was performed by first-order reliability method and Monte Carlo simulation in RT software. The optimal length of FRP was obtained on the results of reliability analysis. The findings of the study showed that increased FRP material quantities do not affect the retrofitted reliability index. The findings of the study showed that increased FRP material quantities do not affect the retrofitted reliability index. The convergence of Monte Carlo simulation occurred after a variety of iterations and, consequently, was alternated with the first-order reliability analysis which has very short run time and is lower error methodology. Moreover, in all cases of reinforcement and analyses, the compressive strength of the concrete is extremely important.

High Quality 4-dimensional Cone Beam CT by Deforming Prior Images

Nonlinear conjugate gradient (CG) methods are widely used for solving large scale unconstrained optimization problems. Many studies have been devoted to modified and improve this method. In this paper, a new parameter of CG method that possesses global convergence properties using exact line search is proposed. Numerical results show that the new formula is best and more efficient when compared with the other classical CG methods

A study of Liu-Storey conjugate gradient methods for vector optimization

Nonlinear conjugate gradient methods are a class of techniques that are used for solving nonlinear optimisation problems frequently arising in many engineering applications such as machine learning, computer vision, least-square optimisations, to name a few. With so many surveys on the nonlinear conjugate gradient method (NLCG) available around, this paper addresses the current updates and sheds a new light on the evolution of hybrid conjugate gradient parameters with their global convergence properties.

Compared with the probability model, the convex model approach only requires the bound information on the uncertainty, and can make it possible to conduct the reliability analysis for many complex engineering problems with limited samples. Presently, by introducing the well-established techniques in probability-based reliability analysis, some methods have been successfully developed for convex model reliability. This paper aims to reveal some different phenomena and furthermore some severe paradoxes when extending the widely used first-order reliability method (FORM) into the convex model problems, and whereby provide some useful suggestions and guidelines for convex-model-based reliability analysis. Two FORM-type approximations, namely, the mean-value method and the design-point method, are formulated to efficiently compute the nonprobabilistic reliability index. A comparison is then conducted between these two methods, and some important phenomena different from the traditional FORMs are summarized. The nonprobabilistic reliability index is also extended to treat the system reliability, and some unexpected paradoxes are found through two numerical examples.

PreviousNext No AccessSEG Technical Program Expanded Abstracts 20013‐D magnetotelluric inversion for resource explorationAuthors: Randall L. MackieWilliam RodiM. Donald WattsRandall L. MackieGSY‐USA, Inc., William RodiMassachusetts Institute of Technology, and M. Donald WattsGeosystem srlhttps://doi.org/10.1190/1.1816392 SectionsAboutPDF/ePub ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InReddit Permalink: https://doi.org/10.1190/1.1816392FiguresReferencesRelatedDetailsCited by3D Step-by-step inversion strategy for audio magnetotellurics data based on unstructured mesh17 February 2022 | Applied Geophysics, Vol. 18, No. 3Three‐dimensional electrical structure of the northwestern margin of the Karamay region, China, revealed by magnetotelluric data14 September 2020 | Geophysical Prospecting, Vol. 68, No. 9Three-dimensional electrical conductivity in the mantle beneath the Payún Matrú Volcanic Field in the Andean backarc of Argentina near 36.5°S: evidence for decapitation of a mantle plume by resurgent upper mantle shear during slab steepening25 June 2014 | Geophysical Journal International, Vol. 198, No. 2NLCG and L-BFGS optimization for 3D HEM inversionYunhe Liu and Changchun Yin19 August 2013Methods and algorithms for reconstructing three-dimensional distributions of electric conductivity and polarization in the medium by finite-element 3D modeling using the data of electromagnetic sounding8 May 2013 | Izvestiya, Physics of the Solid Earth, Vol. 49, No. 3Multiple-domain, simultaneous joint inversion of geophysical data with application to subsalt imagingMichele De Stefano, Federico Golfré Andreasi, Simone Re, Massimo Virgilio, and Fred F. Snyder5 May 2011 | GEOPHYSICS, Vol. 76, No. 3A compressed implicit Jacobian scheme for 3D electromagnetic data inversionMaokun Li, Aria Abubakar, Jianguo Liu, Guangdong Pan, and Tarek M. Habashy28 April 2011 | GEOPHYSICS, Vol. 76, No. 3Electrical resistivity structure at the northern margin of the Tibetan Plateau and tectonic implications7 December 2011 | Journal of Geophysical Research, Vol. 116, No. B12Application of the 3D magnetotelluric inversion code in a geologically complex area9 June 2010 | Geophysical Prospecting, Vol. 58, No. 6Three‐dimensional regularized Gauss‐Newton inversion algorithm using a compressed implicit Jacobian calculation for electromagnetic applicationsMaokun Li, Aria Abubakar, Jianguo Liu, Guangdong Pan, and Tarek M. Habashy21 October 2010Three-dimensional inversion of magnetotelluric data for mineral exploration: An example from the McArthur River uranium deposit, Saskatchewan, CanadaJournal of Applied Geophysics, Vol. 68, No. 43D magnetotelluric inversion using a limited-memory quasi-Newton optimizationDmitry Avdeev and Anna Avdeeva27 April 2009 | GEOPHYSICS, Vol. 74, No. 3Advancing process‐based watershed hydrological research using near‐surface geophysics: a vision for, and review of, electrical and magnetic geophysical methods11 March 2008 | Hydrological Processes, Vol. 22, No. 18A survey of current trends in near-surface electrical and electromagnetic methodsEsben Auken, Louise Pellerin, Niels B. Christensen, and Kurt Sørensen7 September 2006 | GEOPHYSICS, Vol. 71, No. 5Three-Dimensional Electromagnetic Modelling and Inversion from Theory to ApplicationSurveys in Geophysics, Vol. 26, No. 6Mining, Environmental, Petroleum, and Engineering Industry Applications of Electromagnetic Techniques in GeophysicsSurveys in Geophysics, Vol. 26, No. 5Multi-dimensional electromagnetic modeling and inversion with application to near-surface earth investigationsComputers and Electronics in Agriculture, Vol. 46, No. 1-3Accelerated integral equation inversion of 3‐D magnetotelluric data in models with inhomogeneous backgroundNikolay Golubev and Michael S. Zhdanov7 December 20053D inversion of a scalar radio magnetotelluric field data setGregory A. Newman, Stephan Recher, Bülent Tezkan, and Fritz M. Neubauer29 May 2003 | GEOPHYSICS, Vol. 68, No. 32-D and 3-D interpretation of magnetotelluric data in the Bajawa geothermal field, central Flores, IndonesiaBULLETIN OF THE GEOLOGICAL SURVEY OF JAPAN, Vol. 53, No. 2-3 SEG Technical Program Expanded Abstracts 2001ISSN (print):1052-3812 ISSN (online):1949-4645Copyright: 2001 Pages: 2135 publication data© 2001 Copyright © 2001 Society of Exploration GeophysicistsPublisher:Society of Exploration Geophysicists HistoryPublished Online: 03 Jan 2005 CITATION INFORMATION Randall L. Mackie, William Rodi, and M. Donald Watts, (2001), "3‐D magnetotelluric inversion for resource exploration," SEG Technical Program Expanded Abstracts : 1501-1504. https://doi.org/10.1190/1.1816392 Plain-Language Summary PDF DownloadLoading ...

Both fully adaptive and partially adaptive processing algorithms can be implemented using linear and nonlinear conjugate gradient (CG) methods. This approach can be computationally attractive when a family of continuous time-varying adaptive problems must be solved. An iterative approach that starts with the previous solution can converge quickly to the new solution. This paper considers linear and nonlinear CG methods for adaptive processing. The example used is space-time adaptive processing (STAP) and clutter interference subspace tracking within the context of an airborne radar with a rotating array. It is seen that the linear CG method for adaptive filtering in this context can be several times less costly than sample matrix inversion if the interference subspace has sufficient eigenvalue structure, and that a nonlinear CG algorithm is capable of tracking the principal clutter interference subspace.

The nonlinear conjugate gradient method (CGM) is a very effective iterative method for solving large-scale optimal problems. In this paper, based on a variant of Polak–Ribiere–Polyak method, two modified CGMs with disturbance factors are proposed. By the disturbance factors, the two proposed methods not only generate sufficient descent direction at each iteration but also converge globally for nonconvex minimization if the strong Wolfe line search is used. Finally, elementary numerical experiment results are reported, which show that the proposed methods are promising.