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Consider the nonlinear pseudo-monotone equations over a nonempty closed convex set. A spectral conjugate gradient projection method with the inertial factor is proposed for solving the problem under discussion. Following the projection strategy, we prove that the sequence of spectral parameters is bounded. The search direction generated by the algorithm satisfies the sufficient descent condition and possesses trust region property at each iteration. Under some mild conditions, the global convergence of the proposed method is established without the Lipschitz continuity assumption. Under some standard assumptions, we also establish the linear convergence rate of our method. Preliminary numerical results on constrained nonlinear monotone and pseudo-monotone equations demonstrate the efficiency of the proposed method. Furthermore, to highlight its applicability, we extend our method to deal with logistic regression problems. [ABSTRACT FROM AUTHOR]

A modified Polak Ribiere Polyak(PRP) conjugate gradient(CG) method is proposed for solving unconstrained optimization problems. The search direction generated by this method satisfies sufficient descent condition at each iteration and this method inherits one remarkable property of the standard PRP method. Under the standard Armijo line search, the global convergence and the linearly convergent rate of the presented method is established. Some numerical results are given to show the effectiveness of the proposed method by comparing with some existing CG methods. [ABSTRACT FROM AUTHOR]

The conjugate gradient method plays a very important role in several fields, to solve problems of large sizes. To improve the efficiency of this method, a lot of works have been done; in this paper we propose a new modification of PRP method to solve a large scale unconstrained optimization problems in relation with strong Wolf Powell Line Search property, when the latter was used under some conditions, a global convergence result was proved. In comparison with other known methods the efficiency of this method proved that it is better in the number of iterations and in time on 90 proposed problems by use of Matlab. [ABSTRACT FROM AUTHOR]

Recent advancements in quantum computing and quantum-inspired algorithms have sparked renewed interest in binary optimization. These hardware and software innovations promise to revolutionize solution times for complex problems. In this work, we propose a novel method for solving linear systems. Our approach leverages binary optimization, making it particularly well-suited for problems with large condition numbers. We transform the linear system into a binary optimization problem, drawing inspiration from the geometry of the original problem and resembling the conjugate gradient method. This approach employs conjugate directions that significantly accelerate the algorithm's convergence rate. Furthermore, we demonstrate that by leveraging partial knowledge of the problem's intrinsic geometry, we can decompose the original problem into smaller, independent sub-problems. These sub-problems can be efficiently tackled using either quantum or classical solvers. Although determining the problem's geometry introduces some additional computational cost, this investment is outweighed by the substantial performance gains compared to existing methods. [ABSTRACT FROM AUTHOR]

AbstractLatent Gaussian process (GP) models are flexible probabilistic non-parametric function models. Vecchia approximations are accurate approximations for GPs to overcome computational bottlenecks for large data, and the Laplace approximation is a fast method with asymptotic convergence guarantees to approximate marginal likelihoods and posterior predictive distributions for non-Gaussian likelihoods. Unfortunately, the computational complexity of combined Vecchia-Laplace approximations grows faster than linearly in the sample size when used in combination with direct solver methods such as the Cholesky decomposition. Computations with Vecchia-Laplace approximations can thus become prohibitively slow precisely when the approximations are usually the most accurate, i.e., on large data sets. In this article, we present iterative methods to overcome this drawback. Among other things, we introduce and analyze several preconditioners, derive new convergence results, and propose novel methods for accurately approximating predictive variances. We analyze our proposed methods theoretically and in experiments with simulated and real-world data. In particular, we obtain a speed-up of an order of magnitude compared to Cholesky-based calculations and a threefold increase in prediction accuracy in terms of the continuous ranked probability score compared to a state-of-the-art method on a large satellite data set. All methods are implemented in a free C++ software library with high-level Python and R packages. 4 [ABSTRACT FROM AUTHOR]

Conjugate gradient methods are widely used and attractive for large-scale unconstrained smooth optimization problems, with simple computation, low memory requirements, and interesting theoretical information on the features of curvature. Based on the strongly convergent property of the Dai–Yuan method and attractive numerical performance of the Hestenes–Stiefel method, a new hybrid descent conjugate gradient method is proposed in this paper. The proposed method satisfies the sufficient descent property independent of the accuracy of the line search strategies. Under the standard conditions, the trust region property and the global convergence are established, respectively. Numerical results of 61 problems with 9 large-scale dimensions and 46 ill-conditioned matrix problems reveal that the proposed method is more effective, robust, and reliable than the other methods. Additionally, the hybrid method also demonstrates reliable results for some image restoration problems. [ABSTRACT FROM AUTHOR]

Least squares models appear frequently in many fields, such as data fitting, signal processing, machine learning, and especially artificial intelligence. Nowadays, the model is a popular and sophisticated way to make predictions about real-world problems. Meanwhile, conjugate gradient methods are traditionally known as efficient tools to solve unconstrained optimization problems, especially in high-dimensional cases. This paper presents a new structured spectral conjugate gradient method based on a modification of the modified structured secant equation of Zhang, Xue, and Zhang. The proposed method uses a novel appropriate spectral parameter. It is proved that the new direction satisfies the sufficient descent condition regardless of the line search. The global convergence of the proposed method is demonstrated under some standard assumptions. Numerical experiments show that our proposed method is efficient and can compete with other existing algorithms in this area. [ABSTRACT FROM AUTHOR]

The purpose of this article is to investigate the effects of the use of mass‐lumping in the finite element discretization with mesh size h$$ h $$ of the reduced first‐order optimality system arising from a standard tracking‐type, distributed elliptic optimal control problem with L2$$ {L}_2 $$ regularization, involving a regularization (cost) parameter ϱ$$ \varrho $$ on which the solution depends. We show that mass‐lumping will not affect the L2$$ {L}_2 $$ error between the desired state yd$$ {y}_d $$ and the computed finite element state ŷϱh$$ {\hat{y}}_{\varrho h} $$, but will lead to a Schur‐complement system that allows for a fast matrix‐by‐vector multiplication. We show that the use of the Schur‐complement preconditioned conjugate gradient method in a nested iteration setting leads to an asymptotically optimal solver with respect to the complexity. While the proposed approach is applicable independently of the regularity of the given target, our particular interest is in discontinuous desired states yd$$ {y}_d $$ that do not belong to the state space. However, the corresponding control ûϱh$$ {\hat{u}}_{\varrho h} $$ belongs to L2$$ {L}_2 $$ whereas the cost ‖ûϱh‖L2→∞$$ {\left\Vert {\hat{u}}_{\varrho h}\right\Vert}_{L_2}\to \infty $$ as ϱ→0$$ \varrho \to 0 $$. This motivates to use ϱ=h4$$ \varrho ={h}^4 $$ in order to balance the error ‖ŷϱh−yd‖L2$$ {\left\Vert {\hat{y}}_{\varrho h}-{y}_d\right\Vert}_{L_2} $$ and the maximal costs ‖ûϱh‖L2$$ {\left\Vert {\hat{u}}_{\varrho h}\right\Vert}_{L_2} $$ we are willing to accept. This can be embedded into a nested iteration process on a sequence of refined finite element meshes in order to control the error ‖ŷϱh−yd‖L2$$ {\left\Vert {\hat{y}}_{\varrho h}-{y}_d\right\Vert}_{L_2} $$ and the cost ‖ûϱh‖L2$$ {\left\Vert {\hat{u}}_{\varrho h}\right\Vert}_{L_2} $$ simultaneously. [ABSTRACT FROM AUTHOR]

In this study, we introduce a novel hybrid conjugate gradient method with an adaptive strategy called asHCG method. The asHCG method exhibits the following characteristics. (i) Its search direction guarantees sufficient descent property without dependence on any line search. (ii) It possesses strong convergence for the uniformly convex function using a weak Wolfe line search, and under the same line search, it achieves global convergence for the general function. (iii) Employing the Armijo line search, it provides an approximate guarantee for worst-case complexity for the uniformly convex function. The numerical results demonstrate promising and encouraging performances in both unconstrained optimization problems and image restoration problems. [ABSTRACT FROM AUTHOR]

In this work we present a new stochastic framework for obtaining optimal treatment regimes in prostate cancer. We model the realistic scenario of randomized clinical trials for incorporating randomness related to interaction between a prostate cancer cell and androgen cell quota, due to cancer heterogeneities, across different patients in a given group, using a Liouville partial differential equation. We then solve two optimization problems: one for determining the model parameters to fit the measured data and the second to determine the optimal androgen deprivation therapy. The optimization problems are implemented using a positive, stable, and conservative finite volume solver for the Liouville equations and the projected non-linear conjugate gradient method. Several numerical results, including comparison with ordinary differential equations modeling framework, demonstrate the robustness and accuracy of our proposed framework to obtain optimal treatment regimes in real time. • This work presents a new stochastic modeling and control framework of prostate cancer dynamics using Liouville equations. • Such a framework can accurately represent the setup of randomized clinical trials to account for cancer heterogeneities. • Two optimization problems are implemented for model fitting and optimal treatment regimes. • Numerical experiments and comparison with a traditional modeling setup demonstrate the efficiency of the proposed framework. [ABSTRACT FROM AUTHOR]

Let H be an infinite-dimensional complex Hilbert space and let G ∞ (H) be the set of all closed subspaces of H whose dimension and codimension both are infinite. We investigate (not necessarily surjective) transformations of G ∞ (H) sending every pair of subspaces to an equivalent pair of subspaces; two pairs of subspaces are equivalent if there is a linear isometry sending one of these pairs to the other. Let f be such a transformation. We show that there is up to a scalar multiple a unique linear or conjugate-linear isometry L : H → H such that for every X ∈ G ∞ (H) the image f (X) is the sum of L (X) and a certain closed subspace O (X) orthogonal to the range of L. In the case when H is separable, we give the following sufficient condition to assert that f is induced by a linear or conjugate-linear isometry: if O (X) = 0 for a certain X ∈ G ∞ (H) , then O (Y) = 0 for all Y ∈ G ∞ (H). [ABSTRACT FROM AUTHOR]

In addressing the multifaceted problem of multiple-input multiple-output (MIMO) detection in wireless communication systems, this work highlights the pressing need for enhanced detection reliability under variable channel conditions and MIMO antenna configurations. We propose a novel method that sets a new standard for deep unfolding in MIMO detection by integrating the iterative conjugate gradient method with Tikhonov regularization, combining the adaptability of modern deep learning techniques with the robustness of classical regularization. Unlike conventional techniques, our strategy treats the Tikhonov regularization parameter, as well as the step size and search direction coefficients of the conjugate gradient (CG) method, as trainable parameters within the deep learning framework. This enables dynamic adjustments based on varying channel conditions and MIMO antenna configurations. Detection performance is significantly improved by the proposed approach across a range of MIMO configurations and channel conditions, consistently achieving lower bit error rate (BER) and normalized minimum mean square error (NMSE) compared to well-known techniques like DetNet and CG. The proposed method has superior performance over CG and other model-based methods, especially with a small number of iterations. Consequently, the simulation results demonstrate the flexibility of the proposed approach, making it a viable choice for MIMO systems with a range of antenna configurations, modulation orders, and different channel conditions. [ABSTRACT FROM AUTHOR]

The airborne distributed position and orientation system (ADPOS), which integrates multi-inertia measurement units (IMUs), a data-processing computer, and a Global Navigation Satellite System (GNSS), serves as a key sensor in new higher-resolution airborne remote sensing applications, such as array SAR and multi-node imaging loads. ADPOS can provide reliable, high-precision and high-frequency spatio-temporal reference information to realize multinode motion compensation with the various nonlinear filter estimation methods such as Central Difference Kalman Filtering (CDKF), and modified CDKF. Although these known nonlinear models demonstrate good performance, their noise estimation performance with its linear minimum variance estimation criterion is limited for ADPOS. For this reason, in this paper, Central Difference Variational Filtering (CDVF) based on the variational optimization process is presented. This method adopts the conjugate gradient algorithm to enhance the estimation performance for mean correction in the filtering update stage. On one hand, the proposed method achieves adaptability by estimating noise covariance through the variational optimization method. On the other hand, robustness is implemented under the minimum variance estimation criterion based on the conjugate gradient algorithm to suppress measurement noise. We conducted a real ADPOS flight test, and the experimental results show that the accuracy of the slave motion parameters has significantly improved compared to the current CDKF. Moreover, the compensation performance shows a clear enhancement. [ABSTRACT FROM AUTHOR]

Forward jammers replicate and retransmit radar signals back to generate coherent jamming signals and false targets, making anti-jamming an urgent issue in electronic warfare. Jamming transmitters work at saturation to maximize the retransmission power such that only the phase information of the angular waveform at the designated direction of arrival (DOA) is retained. Therefore, amplitude modulation of MIMO radar angular waveforms offers an advantage in combating forward jamming. We address both the design of unimodular MIMO waveforms and their associated mismatch filters to confront mainlobe jamming in this paper. Firstly, we design the MIMO waveforms to maximize the discrepancy between the retransmitted jamming and the spatially synthesized radar signal. We formulate the problem as unconstrained non-linear optimization and solve it using the conjugate gradient method. Particularly, we introduce fast Fourier transform (FFT) to accelerate the numeric calculation of both the objection function and its gradient. Secondly, we design a mismatch filter to further suppress the filtered jamming through convex optimization in polynomial time. The simulation results show that for an eight-element MIMO radar, we are able to reduce the correlation between the angular waveform and saturated forward jamming to −6.8 dB. Exploiting this difference, the mismatch filter can suppress the jamming peak by 19 dB at the cost of an SNR loss of less than 2 dB. [ABSTRACT FROM AUTHOR]

ABSTRACT This article presents and analyzes a family of three‐term conjugate gradient projection methods with the inertial technique for solving large‐scale nonlinear pseudo‐monotone equations with convex constraints. The generated search direction exhibits good properties independent of line searches. The global convergence of the family is proved without the Lipschitz continuity of the underlying mapping. Furthermore, under the locally Lipschitz continuity assumption, we conduct a thorough analysis related to the asymptotic and non‐asymptotic global convergence rates in terms of iteration complexity. To our knowledge, this is the first iteration‐complexity analysis for inertial gradient‐type projection methods, in the literature, under such a assumption. Numerical experiments demonstrate the computational efficiency of the family, showing its superiority over three existing inertial methods. Finally, we apply the proposed family to solve practical problems such as ℓ2$$ {\ell}_2 $$‐regularized logistic regression, sparse signal restoration and image restoration problems, highlighting its effectiveness and potential for real‐world applications. [ABSTRACT FROM AUTHOR]

The one parameter conjugate gradient method by Hager and Zhang (Pac J Optim, 2(1):35–58, 2006) represents a family of descent iterative methods for solving large‐scale minimization problems. The nonnegative parameter of the scheme determines the weight of conjugacy and descent, and by extension, the numerical performance of the method. The scheme, however, does not converge globally for general nonlinear functions, and when the parameter approaches 0, the scheme reduces to the conjugate gradient method by Hestenes and Stiefel (J Res Nat Bur Stand, 49:409–436, 1952), which in practical sense does not perform well due to the jamming phenomenon. By carrying out eigenvalue analysis of an adaptive two parameter Hager–Zhang type method, a new scheme is presented for system of monotone nonlinear equations with its application in compressed sensing. The proposed scheme was inspired by nice attributes of the Hager–Zhang method and the various schemes designed with double parameters. The scheme is also applicable to nonsmooth nonlinear problems. Using fundamental assumptions, analysis of the global convergence of the scheme is conducted and preliminary report of numerical experiments carried out with the scheme and some recent methods indicate that the scheme is promising. [ABSTRACT FROM AUTHOR]

Multiple kernel k‐means clustering (MKKC) can efficiently incorporate multiple base kernels to generate an optimal kernel. Many existing MKKC methods all need two‐step operation: learning clustering indicator matrix and performing clustering on it. However, the optimal clustering results of two steps are not equivalent to those of original problem. To address this issue, in this paper we propose a novel method named one‐step multiple kernel k‐means clustering based on block diagonal representation (OS‐MKKC‐BD). By imposing a block diagonal constraint on the product of indicator matrix and its transpose, this method can encourage the indicator matrix to be block diagonal. Then the indicator matrix can produce explicit clustering indicator, so as to implement one‐step clustering, which avoids the disadvantage of two‐step operation. Furthermore, a simple kernel weighting strategy is used to obtain an optimal kernel, which boosts the quality of optimal kernel. In addition, a three‐step iterative algorithm is designed to solve the corresponding optimization problem, where the Riemann conjugate gradient iterative method is used to solve the optimization problem of the indicator matrix. Finally, by extensive experiments on eleven real data sets and comparison of clustering results with 10 MKC methods, it is concluded that OS‐MKKC‐BD is effective. [ABSTRACT FROM AUTHOR]

The conjugate gradient (CG) directions are among the important components of the CG algorithms. These directions have proven their effectiveness in many applications—more specifically, in image processing due to their low memory requirements. In this study, we derived a new conjugate gradient coefficient based on the famous quadratic model. The derived algorithm is distinguished by its global convergence and essential descent properties, ensuring robust performance across diverse scenarios. Extensive numerical testing on image restoration and unconstrained optimization problems have demonstrated that the new formulas significantly outperform existing methods. Specifically, the proposed conjugate gradient scheme has shown superior performance compared to the traditional Fletcher–Reeves (FR) conjugate gradient method. This advancement not only enhances computational efficiency on unconstrained optimization problems, but also improves the accuracy and quality of image restoration, making it a highly valuable tool in the field of computational imaging and optimization. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study the superconvergence of nonconforming Rannacher-Turek finite element for elliptic interface problems under unfitted square meshes. In particular, we analyze its superclose property between the gradient of the numerical solution and the gradient of the interpolation of the exact solution. Moreover, we introduce a postprocessing interpolation operator which is applied to numerical solution, and we prove that the postprocessed gradient converges to the exact gradient with a superconvergent rate O (h 3 2 ). Finally, numerical results coincide with our theoretical analysis, and they show that the error estimates do not depend on the ratio of the discontinuous coefficients. [ABSTRACT FROM AUTHOR]

Over the years, a considerable number of conjugate gradient methods have been proposed based on modifications on the well-known classical conjugate gradient methods. These methods were shown to have satisfied descent condition taking into consideration the strong Wolfe line search and other line search schemes. Convergence of objective functions were also guarantied. In this study, a decent conjugate gradient method for solving unconstrained non-linear optimization problems is developed. Algorithm of the proposed method was well developed by constructing its update parameter. Descent properties of the method based on some assumptions on the objective function were established. The convergence analysis of the method showed that it converges globally taking into consideration the strong Wolfe conditions . Dolan and More performance profile was used to compare the numerical strength of this method with other methods, showing clear evidence of better performance of the new method in the profiles tested. [ABSTRACT FROM AUTHOR]

This paper describes a method for solving smooth nonconvex minimization problems subject to bound constraints with good worst-case complexity guarantees and practical performance. The method contains elements of two existing methods: the classical gradient projection approach for bound-constrained optimization and a recently proposed Newton-conjugate gradient algorithm for unconstrained nonconvex optimization. Using a new definition of approximate second-order optimality parametrized by some tolerance ϵ (which is compared with related definitions from previous works), we derive complexity bounds in terms of ϵ for both the number of iterations required and the total amount of computation. The latter is measured by the number of gradient evaluations or Hessian-vector products. We also describe illustrative computational results on several test problems from low-rank matrix optimization. [ABSTRACT FROM AUTHOR]

Metric learning focuses on finding similarities between data and aims to enlarge the distance between the samples with different labels. This work proposes a semi-supervised metric learning method based on the point-to-class structure of the labeled data, which is computationally less expensive, especially than using point-to-point structure. Specifically, the point-to-class structure is formulated into a new triplet constraint, which could narrow the distance of inner-class data and enlarge the distance of inter-class data simultaneously. Moreover, for measuring dissimilarity between different classes, weights are introduced into the triplet constraint and forms the weighted triplet constraint. Then, two kinds of regularizers such as spatial regularizer are rationally incorporated respectively in this model to mitigate the overfitting phenomenon and preserve the topological structure of the data. Furthermore, Riemannian gradient descent algorithm is adopted to solve the proposed model, since it can fully exploit the geometric structure of Riemannian manifolds and the proposed model can be regarded as a generalization of the unconstrained optimization problem in Euclidean space on Riemannian manifold. By introducing such solution strategy, the variables are constrained to a specific Riemannian manifold in each step of the iterative solution process, thereby enabling efficient and accurate model resolution. Finally, we conduct classification experiments on various data sets and compare the classification performance to state-of-the-art methods. The experimental results demonstrate that our proposed method has better performance in classification, especially for hyperspectral image data. [ABSTRACT FROM AUTHOR]

A boundary shape estimation problem for the furnace's inner surface is solved using the dual reciprocity boundary element method (DRBEM) and the conjugate gradient method (CGM) under transient heat conduction. The DRBEM is utilized to eliminate the drawbacks of traditional numerical methods and classical boundary element method of discretizing the entire computational domain, with only boundary discretization. The inversion results are obtained by applying the CGM to minimize the objective function, in which the sensitivity coefficients are calculated with the complex variable derivation method (CVDM), making the calculation precise and independent of step size. To verify the accuracy of the DRBEM in solving the transient heat conduction problem, the influencing factors including radial-basis function, the number of internal collocation points, and time step size are investigated. The influences of measurement time interval, future time step, initial guess, measurement error, and the number and position of measurement points on the inversion results are analyzed. Meanwhile, the effectiveness of the proposed approach is tested by numerical examples, and the inversion results show that it is stable, accurate, and efficient for identifying different and complicated unknown boundary shapes of the furnace. The boundary shape identification problem of the furnace wall is solved under transient heat conduction. The dual reciprocity boundary element method (DRBEM) is applied to solve the transient direct heat conduction problem, which retains the advantages of pure boundary discretization. In the conjugate gradient method (CGM), the sensitivity coefficients are calculated by the complex variable derivation method (CVDM) which is accurate and independent of the step size. The inner boundary shapes of the furnace with different complicated function forms are identified successfully. [ABSTRACT FROM AUTHOR]

The Delayed Weighted Gradient Method (DWGM) is a two-step gradient algorithm that is efficient for the minimization of large scale strictly convex quadratic functions. It has orthogonality properties that make it to compete with the Conjugate Gradient (CG) method. Both methods calculate in sequence two step-sizes, CG minimizes the objective function and DWGM the gradient norm, alongside two search directions defined over first order current and previous iteration information. The objective of this work is to accelerate the recently developed extension of DWGM to nonquadratic strongly convex minimization problems. Our idea is to define the step-sizes of DWGM in a unique two dimensional convex quadratic optimization problem, calculating them simultaneously. Convergence of the resulting algorithm is analyzed. Comparative numerical experiments illustrate the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

Abstract This research focuses on the identification of failure times in thermal systems governed by partial differential equations, a task known for its complexity. A new model‐based diagnostic approach is presented that aims to accurately identify failing heat sources and accurately determine their failure times, which is crucial when multiple heat sources fail and there is a delay in detection by distant sensors. To validate the effectiveness of the approach, a comparative analysis is carried out with an established method based on a Bayesian filter, the Kalman filter. The aim is to provide a comprehensive analysis, highlighting the advantages and potential limitations of the methodology. In addition, a Monte Carlo simulation is implemented to assess the impact of sensor measurements on the performance of this new approach.

We propose two hybrid methods for solving large-scale monotone systems, which are based on derivative-free conjugate gradient approach and hyperplane projection technique. The conjugate gradient approach is efficient for large-scale systems due to low memory, while projection strategy is suitable for monotone equations because it enables simply globalization. The derivative-free function-value-based line search is combined with Hu-Storey type search directions and projection procedure, in order to construct globally convergent methods. Furthermore, the proposed methods are applied into solving a number of large-scale monotone nonlinear systems and reconstruction of sparse signals. Numerical experiments indicate the robustness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this work, for unconstrained optimization, we introduce an improved Dai‐Liao‐style hybrid conjugate gradient method based on the hybridization‐based self‐adaptive technique, and the search direction generated fulfills the sufficient descent and trust region properties regardless of any line search. The global convergence is established under standard Wolfe line search and common assumptions. Then, combining the hyperplane projection technique and a new self‐adaptive line search, we extend the proposed conjugate gradient method and obtain an improved Dai‐Liao‐style hybrid conjugate gradient projection method to solve constrained nonlinear monotone equations. Under mild conditions, we obtain its global convergence without Lipschitz continuity. In addition, the convergence rates for the two proposed methods are analyzed, respectively. Finally, numerical experiments are conducted to demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper presents a fast focusing inversion algorithm of magnetic data based on the conjugate gradient method, which can be used to describe the underground target geologic body efficiently and clearly. The proposed method realizes an effect similar to matrix compression by changing the computation order, calculating the inner product of vectors and equivalent expansion of expressions. Model tests show that this strategy successfully reduces the computation time of a single iteration of the conjugate gradient method, so the three-dimensional magnetic data inversion is realized under a certain number of iterations. In this paper, the detailed calculation steps of the proposed inversion method are given, and the effectiveness and high efficiency of the proposed fast focusing inversion method are verified by three theoretical model tests and a set of measured data. Finally, the fast focus inversion algorithm is applied to the magnetic data of Gonghe Basin, Qinghai Province, to describe the spatial distribution range of deep hot dry rock, which provides a direction for the continuous exploration of geothermal resources in this area. [ABSTRACT FROM AUTHOR]

This paper proposes an improved three-term conjugate gradient algorithm designed to solve nonlinear equations with convex constraints. The key features of the proposed algorithm are as follows: (i) It only requires that nonlinear equations have continuous and monotone properties; (ii) The designed search direction inherently ensures sufficient descent and trust-region properties, eliminating the need for line search formulas; (iii) Global convergence is established without the necessity of the Lipschitz continuity condition. Benchmark problem numerical results illustrate the proposed algorithm's effectiveness and competitiveness relative to other three-term algorithms. Additionally, the algorithm is extended to effectively address the image denoising problem. [ABSTRACT FROM AUTHOR]

This paper presents a new conjugate gradient method on Riemannian manifolds and establishes its global convergence under the standard Wolfe line search. The proposed algorithm is a generalization of a Wei-Yao-Liu-type Hestenes-Stiefel method from Euclidean space to the Riemannian setting. We prove that the new algorithm is well-defined, generates a descent direction at each iteration, and globally converges when the step lengths satisfy the standard Wolfe conditions. Numerical experiments on the matrix completion problem demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Solving large-scale systems of nonlinear equations (SoNE) is a central task in mathematics that traverses different areas of applications. There are several derivative-free methods for finding SoNE solutions. However, most of the methods contributed to find SoNE solutions involve a monotone cost function. Methods dealing with pseudomonotone cost function remain rare. In this paper, we introduce two classes of derivative-free spectral three-term methods to solve large-scale continuous pseudomonotone SoNE. We combine the projection method of Solodov and Svaiter with the structure of the recently developed spectral three-term conjugate gradient method for unconstrained optimization by Amini and Faramarzi. We prove that the proposed methods possess sufficient descent property, trust region property, and global convergence without relying on Lipschitz continuity. Numerical experiments show that the proposed methods are efficient and competitive with existing methods. Finally, the proposed methods have been successfully applied to recover a sparse signal from incomplete and contaminated sampling measurements, yielding promising results. [ABSTRACT FROM AUTHOR]

This paper presents modified memoryless quasi-Newton methods based on the spectral-scaling Broyden family on Riemannian manifolds. The method involves adding one parameter to the search direction of the memoryless self-scaling Broyden family on the manifold. Moreover, it uses a general map instead of vector transport. This idea has already been proposed within a general framework of Riemannian conjugate gradient methods where one can use vector transport, scaled vector transport, or an inverse retraction. We show that the search direction satisfies the sufficient descent condition under some assumptions on the parameters. In addition, we show global convergence of the proposed method under the Wolfe conditions. We numerically compare it with existing methods, including Riemannian conjugate gradient methods and the memoryless spectral-scaling Broyden family. The numerical results indicate that the proposed method with the BFGS formula is suitable for solving an off-diagonal cost function minimization problem on an oblique manifold. [ABSTRACT FROM AUTHOR]

Due to their simplicity and global convergence properties, the conjugate gradient (CG) methods are widely used for solving unconstrained optimization problems, especially those of large scale. To establish the global convergence and to obtain better numerical performance in practice, much effort has been devoted to develop new CG methods or even to modify well- known methods. In 2012, Rivaie et al., have proposed a new CG method, called RMIL which has good numerical results and globally convergent under the exact line search. However, in 2016, Dai has pointed out a mistake in the steps of the proof of global convergence of RMIL and hence to guarantee the global convergence he suggested a modified version of RMIL, called RMIL+. In this paper, we present another modified version of RMIL, which is globally convergent via the exact line search. Furthermore, to support the theoretical proof of the global convergence of the modified version in practical computation, a numerical experiment based on comparing it with RMIL, RMIL+, and CG-DESCENT was done. [ABSTRACT FROM AUTHOR]

Characterizing the fluid-driven fracture-tip advancing process presents a significant challenge due to the difficulty of replicating real-world conditions in laboratory experiments and the lack of precise field measurements. However, recent advances in low-frequency distributed acoustic sensing (LF-DAS) technology offer new opportunities to investigate the dynamics of propagating hydraulic fractures. In this study, we propose an iterative inversion method to characterize fracture-tip advancing behaviors using LF-DAS data. A forward geomechanical model is developed using the three-dimensional displacement discontinuity method, and the optimization is realized by a conjugate gradient method. The performance of the inversion algorithm is demonstrated using a synthetic case, in which the fracture half-length evolution and propagation velocity match well with the reference solutions. In addition, the averaged fracture cross-section area, fracture volume, and fracturing fluid efficiency can also be estimated, showing good agreements with true values of the synthetic case under reasonable assumptions. Then, a field case with a single-cluster hydraulic fracturing treatment from the Hydraulic Fracturing Test Site 2 project (HFTS-2) is studied. Our analysis of the inversion results reveals that the fracture propagates intermittently, as evidenced by the fracture half-length evolution. This unique field evidence can guide modeling efforts to incorporate this important physical behavior into fracture models, and the secondary information gathered from the study, including fracture cross-section area and volume, can help evaluate and optimize fracturing efficiency. Highlights: Low-frequency distributed acoustic sensing provides a unique dataset to characterize the fracture propagation process. A gradient-based inversion algorithm is developed and validated using a synthetic case to estimate the fracture tip advancing process. In the presented field case, fracture propagates continuously in the beginning, followed by an intermittent advancement pattern [ABSTRACT FROM AUTHOR]

The Fletcher-Reeves (FR) method is widely recognized for its drawbacks, such as generating unfavorable directions and taking small steps, which can lead to subsequent poor directions and steps. To address this issue, we propose a modification to the FR method, and then we develop it into the three-term conjugate gradient method in this paper. The suggested methods, named "HZF" and "THZF", preserve the descent property of the FR method while mitigating the drawbacks. The algorithms incorporate strong Wolfe line search conditions to ensure effective convergence. Through numerical comparisons with other conjugate gradient algorithms, our modified approach demonstrates superior performance. The results highlight the improved efficacy of the HZF algorithm compared to the FR and three-term FR conjugate gradient methods. The new algorithm was applied to the problem of image restoration and proved to be highly effective in image restoration compared to other algorithms. [ABSTRACT FROM AUTHOR]

In this article, we consider parabolic partial differential equations in (2+1) dimensions, with a focus on optimal convergence for solutions that are smooth in time but have corner singularities in space owing to the non-smoothness of the spatial boundary. The method employs geometric mesh along with a nonconforming least squares spectral element approximation in space coupled with the implicit Crank-Nicolson finite difference scheme in time. The unconditional stability of the scheme is rigorously proved and the error estimate which is almost optimal second order accurate in time and exponentially accurate in space is established. At each time instant, the subdivided problem is solved in parallel via the preconditioned conjugate gradient method in different processors by using an almost optimal preconditioner, in the sense that the condition number of the resulting system is O ((l n N) 2) (N being the degree of spatial approximation), without having to store any stiffness (or mass) matrix or load vector and the inter-element communication is established through Message Passing Interface (MPI). Specific numerical examples are presented with regard to the performance of the scheme. [ABSTRACT FROM AUTHOR]

Kernel methods are popular in nonlinear and nonparametric regression due to their solid mathematical foundations and optimal statistical properties. However, scalability remains the primary bottleneck in applying kernel methods to large-scale data regression analysis. This paper aims to improve the scalability of kernel methods. We combine Nyström subsampling and the preconditioned conjugate gradient method to solve regularized kernel regression. Our theoretical analysis indicates that achieving optimal convergence rates requires only (n) memory and (n n) time (up to logarithmic factors). Numerical experiments show that our algorithm outperforms existing methods in time efficiency and prediction accuracy on large-scale datasets. Notably, compared to the FALKON algorithm [A. Rudi, L. Carratino and L. Rosasco, Falkon: An optimal large scale kernel method, in Advances in Neural Information Processing Systems (Curran Associates, 2017), pp. 3891–3901], which is known as the optimal large-scale kernel method, our method is more flexible (applicable to non-positive definite kernel functions) and has a lower algorithmic complexity. Additionally, our established theoretical analysis further relaxes the restrictive conditions on hyperparameters previously imposed in convergence analyses. [ABSTRACT FROM AUTHOR]

The problem of generalized tensor eigenvalue is the focus of this paper. To solve the problem, we suggest using the normalized Newton generalized eigenproblem approach (NNGEM). Since the rate of convergence of the spectral gradient projection method (SGP), the generalized eigenproblem adaptive power (GEAP), and other approaches is only linear, they are significantly improved by our proposed method, which is demonstrated to be locally and cubically convergent. Additionally, the modified normalized Newton method (MNNM), which converges to symmetric tensors Z‐eigenpairs under the same γ$$ \gamma $$‐Newton stability requirement, is extended by the NNGEM technique. Using a Gröbner basis, a polynomial system solver (NSolve) generates all of the real eigenvalues for us. To illustrate the efficacy of our methodology, we present a few numerical findings. [ABSTRACT FROM AUTHOR]

Context: The paper considers the features of the structure and dipole moments of several amino acids and their dipeptides which play an important role in the formation of the peptide nanotubes based on them. The influence of the features of their chirality (left L and right D) and the alpha-helix conformations of amino acids are taken into account. In particular, amino acids with aromatic rings, such as phenylalanine (Phe/F), and branched-chain amino acids (BCAAs)—leucine (Leu/L) and isoleucine (Ile/I)—as well as corresponding dipeptides (diphenylalanine (FF), dileucine (LL), and diisoleucine (II)) are considered. The main features and properties of these dipeptide structures and peptide nanotubes (PNTs), based on them, are investigated using computational molecular modeling and quantum-chemical semi-empirical calculations. Their polar, piezoelectric, and photoelectronic properties and features are studied in detail. The results of calculations of dipole moments and polarization, as well as piezoelectric coefficients and band gap width, for different types of helical peptide nanotubes are presented. The calculated values of the chirality indices of various nanotubes are given, depending on the chirality of the initial dipeptides—the results obtained are consistent with the law of changes in the type of chirality as the hierarchy of molecular structures becomes more complex. The influence of water molecules in the internal cavity of nanotubes on their physical properties is estimated. A comparison of the results of these calculations by various computational methods with the available experimental data is presented and discussed. Method: The main tool for molecular modeling of all studied nanostructures in this work was the HyperChem 8.01 software package. The main approach used here is the Hartree–Fock (HF) self-consistent field (SCF) with various quantum-chemical semi-empirical methods (AM1, PM3, RM1) in the restricted Hartree–Fock (RHF) and in the unrestricted Hartree–Fock (UHF) approximations. Optimization of molecular systems and the search for their optimal geometry is carried out in this work using the Polak–Ribeire algorithm (conjugate gradient method), which determines the optimized geometry at the point of their minimum total energy. For such optimized structures, dipole moments D and electronic energy levels (such as EHOMO and ELUMO), as well as the band gap Eg = ELUMO − EHOMO, were then calculated. For each optimized molecular structure, the volume was calculated using the QSAR program implemented also in the HyperChem software package. [ABSTRACT FROM AUTHOR]

Trust-region methods have received massive attention in a variety of continuous optimization. They aim to obtain a trial step by minimizing a quadratic model in a region of a certain trust-region radius around the current iterate. This paper proposes an adaptive Riemannian trust-region algorithm for optimization on manifolds, in which the trust-region radius depends linearly on the norm of the Riemannian gradient at each iteration. Under mild assumptions, we establish the liminf-type convergence, lim-type convergence, and global convergence results of the proposed algorithm. In addition, the proposed algorithm is shown to reach the conclusion that the norm of the Riemannian gradient is smaller than ϵ within O (1 ϵ 2) iterations. Some numerical examples of tensor approximations are carried out to reveal the performances of the proposed algorithm compared to the classical Riemannian trust-region algorithm. [ABSTRACT FROM AUTHOR]

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It is known (see, e.g., [SIAM J. Matrix Anal. Appl. 2014;35(1):143‐173]) that the performance of iterative methods for solving the Stokes problem essentially depends on the quality of the preconditioner for the Schur complement matrix, S$$ S $$. In this paper, we consider two preconditioners for S$$ S $$: the identity one and the SIMPLE one, and numerically study their performance for solving the Stokes problem in tight geometries. The latter are characterized by a high surface‐to‐volume ratio. We show that for such geometries, S$$ S $$ can become severely ill‐conditioned, having a very large condition number and a significant portion of non‐unit eigenvalues. As a consequence, the identity matrix, which is broadly used as a preconditioner for solving the Stokes problem in simple geometries, becomes very inefficient. We show that there is a correlation between the surface‐to‐volume ratio and the condition number of S$$ S $$: the latter increases with the increase of the former. We show that the condition number of the diffusive SIMPLE‐preconditioned Schur complement matrix remains bounded when the surface‐to‐volume ratio increases, which explains the robust performance of this preconditioner for tight geometries. Further on, we use a direct method to calculate the full spectrum of S$$ S $$ and show that there is a correlation between the number of its non‐unit eigenvalues and the number of grid points at which no‐slip boundary conditions are prescribed. To illustrate the above findings, we examine the Pressure Schur Complement formulation for staggered finite‐difference discretization of the Stokes equations and solve it with the preconditioned conjugate gradient method. The practical problem which is of interest to us is computing the permeability of tight rocks. [ABSTRACT FROM AUTHOR]

A new spectral conjugate subgradient method is presented to solve nonsmooth unconstrained optimization problems. The method combines the spectral conjugate gradient method for smooth problems with the spectral subgradient method for nonsmooth problems. We study the effect of two different choices of line search, as well as three formulas for determining the conjugate directions. In addition to numerical experiments with standard nonsmooth test problems, we also apply the method to several image reconstruction problems in computed tomography, using total variation regularization. Performance profiles are used to compare the performance of the algorithm using different line search strategies and conjugate directions to that of the original spectral subgradient method. Our results show that the spectral conjugate subgradient algorithm outperforms the original spectral subgradient method, and that the use of the Polak–Ribière formula for conjugate directions provides the best and most robust performance. [ABSTRACT FROM AUTHOR]

AbstractConjugate gradient methods are probably the most used methods in solving large scale unconstrained optimization problems. They have become popular because of their simplicity and low memory requirements. In this paper, we propose a hybrid conjugate gradient method based on the improved Fletcher-Reeves (IFR) and Rivaie-Mustafa-Ismail-Leong+ (RMIL+) methods and establish its global convergence under the strong Wolfe line search conditions. The new conjugate gradient direction satisfies the sufficient descent condition. The method is then compared to other methods in the literature and numerical experiments show that it is competent when solving large scale unconstrained optimization problems. Furthermore, the method is applied to solve a problem in portfolio selection. [ABSTRACT FROM AUTHOR]

The conjugated gradient methods can solve smooth functions with large-scale variables in the specified number of iterations for that they are highly important methods compared to concerning other iterative methods. In this paper, we propose two new conjugate gradient methods, namely the PMDL-1 and PMDL-2. However, for non-smooth functions, which are called conjugate gradient-free derivative methods depending on the projection technique. The two methods give great results compared to the basic PDL method. Moreover, we provide theorems that prove the global convergence between these two methods. [ABSTRACT FROM AUTHOR]

The Vlasov--Poisson system describes the time evolution of a plasma in the so-called collisionless regime. The investigation of a high-temperature plasma that is influenced by an exterior magnetic field is one of the most significant aspects of thermonuclear fusion research. In this paper, we formulate and analyze a kinetic optimal control problem for the Vlasov--Poisson system where the control is represented by an external magnetic field. The main goal of such optimal control problems is to confine the plasma to a certain region in phase space. We first investigate the optimal control problem in terms of mathematical analysis, i.e., we show the existence of at least one global minimizer and rigorously derive a first-order necessary optimality condition for local minimizers by the adjoint approach. Then we build a Monte Carlo framework to solve the state equations as well as the adjoint equations by means of a particle-in-cell method, and we apply a nonlinear conjugate gradient method to solve the optimization problem. Eventually, we present numerical experiments that successfully validate our optimization framework. [ABSTRACT FROM AUTHOR]

In this paper, we describe an upgrade of the Alya code with up-to-date parallel linear solvers capable of achieving reliability, efficiency and scalability in the computation of the pressure field at each time step of the numerical procedure for solving a Large Eddy Simulation formulation of the incompressible Navier–Stokes equations. We developed a software module in the Alya's kernel to interface the libraries included in the current version of PSCToolkit, a framework for the iterative solution of sparse linear systems, on parallel distributed-memory computers, by Krylov methods coupled to Algebraic MultiGrid preconditioners. The Toolkit has undergone various extensions within the EoCoE-II project with the primary goal of facing the exascale challenge. Results on a realistic benchmark for airflow simulations in wind farm applications show that the PSCToolkit solvers significantly outperform the original versions of the Conjugate Gradient method available in the Alya's kernel in terms of scalability and parallel efficiency and represent a very promising software layer to move the Alya code toward exascale. [ABSTRACT FROM AUTHOR]