BFGS Research Articles

Multispectral two-dimensional temperature field reconstruction using particle swarm optimization and the Broyden–Fletcher–Goldfarb–Shanno algorithm with multiple extreme value optimization

The flame temperature is a crucial parameter in combustion, indicating heat generation and transfer. However, determining the correct temperature becomes challenging when flame emissivity is unknown. This article presents a multispectral 2D temperature field reconstruction method using particle swarm optimization and the Broyden–Fletcher–Goldfarb–Shanno algorithm, combined with multivariate extreme-value optimization. This method establishes an objective function based on the correlation between the true temperature and spectral data. The objective function was minimized by adjusting the emissivity, enabling the reconstruction of 2D temperature and emissivity images without assuming an emissivity model. Calibration with a multispectral camera yielded the spectral response coefficients, and three-time spline interpolation was used to verify the calibration data. The reconstructed 2D temperature field showed a mean absolute error (MAE) of 4.61 and structural similarity (SSIM) of 0.995. The reconstructed emissivity image had an MAE range of 0.04 to 0.053 and an SSIM range of 0.897 to 0.915. The system, tested on a butane flame, showed a temperature range of 1001 K to 1356 K, with an average temperature of 1194 K and emissivity values ranging from 0.3166 to 0.8621. The results align with the expected combustion process and radiation characteristics distribution of the butane flame.

CSEM Optimization Using the Correspondence Principle

Traditionally, 3D modeling of marine controlled-source electromagnetic (CSEM) data (in the frequency domain) involves high-memory demand, requiring solving a large linear system for each frequency. To address this problem, we propose to solve Maxwell’s equations in a fictitious dielectric medium with time-domain finite-difference methods, with the support of the correspondence principle. As an advantage of this approach, we highlight the possibility of its implementation for execution with GPU accelerators, in addition to multi-frequency data modeling with a single simulation. Furthermore, we explore using the correspondence principle to the inversion of CSEM data by calculating the gradient of the least-squares objective function employing the adjoint-state method to establish the relationship between adjoint fields in a conductive medium and their counterparts in the fictitious dielectric medium, similar to the approach used in forward modeling. We validate this method through 2D inversions of three synthetic CSEM datasets, computed for a simple model consisting of two resistors in a conductive medium, a model adapted from a CSEM modeling and inversion package, and the last one based on a reference model of turbidite reservoirs on the Brazilian continental margin. We also evaluate the differences between the results of inversions using the steepest descent method and our proposed momentum method, comparing them with the limited-memory BFGS (Broyden–Fletcher–Goldfarb–Shanno) algorithm (L-BFGS-B). In all experiments, we use smoothing by model reparameterization as a strategy for regularizing and stabilizing the iterations throughout the inversions. The results indicate that, although it requires more iterations, our modified momentum method produces the best models, which are consistent with results from the L-BFGS-B algorithm and require less storage per iteration.

Regularized step directions in nonlinear conjugate gradient methods

Conjugate gradient minimization methods (CGM) and their accelerated variants are widely used. We focus on the use of cubic regularization to improve the CGM direction independent of the step length computation. In this paper, we propose the Hybrid Cubic Regularization of CGM, where regularized steps are used selectively. Using Shanno’s reformulation of CGM as a memoryless BFGS method, we derive new formulas for the regularized step direction. We show that the regularized step direction uses the same order of computational burden per iteration as its non-regularized version. Moreover, the Hybrid Cubic Regularization of CGM exhibits global convergence with fewer assumptions. In numerical experiments, the new step directions are shown to require fewer iteration counts, improve runtime, and reduce the need to reset the step direction. Overall, the Hybrid Cubic Regularization of CGM exhibits the same memoryless and matrix-free properties, while outperforming CGM as a memoryless BFGS method in iterations and runtime.

Efficient BFGS quasi-Newton method for large deformation phase-field modeling of fracture in hyperelastic materials

The prediction of crack propagation in materials is a crucial problem in solid mechanics, with many practical applications ranging from structural integrity assessment to the design of advanced materials. The phase-field method has emerged as a powerful tool for modeling crack propagation in materials, due to its ability to accurately capture the propagation of cracks. However, current phase-field algorithms suffer from the elevated computational cost associated with the so-called staggered solution scheme, which requires extremely small time increments to advance the crack due to its inherent conditional stability. In this paper, we present, for the first time, a quantitative analysis detailing the numerical implementation and comparison of two common solution strategies for the coupled large-deformation solid-mechanics-phase-field problem, namely the quasi-Newton based Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGS) and the full Newton based alternating minimization (or staggered) (AM/staggered) algorithm. We demonstrate that the BFGS algorithm is a more efficient and advantageous alternative to the traditional AM/staggered approach for solving coupled large-deformation solid-mechanics-phase-field problems. Our results highlight the potential of the quasi-Newton BFGS algorithm to significantly reduce the computational cost of predicting crack propagation in hyperelastic materials while maintaining the accuracy and robustness of the phase-field method.

Numerical implementation and comparison study on simulating thermo-elastic fracture using adaptive phase-field method combined with BFGS algorithm and AM algorithm

In this paper, the computational performance of alternate minimization (AM) and Broyden–Fletcher–Goldfarb–Shanno (BFGS) to solve the coupled partial differential equations arising within the framework of the phase field method is numerically studied. Both the numerical approaches are employed on an adaptive phase field method. In the case of BFGS, a double-loop recursive algorithm in combination with a line search is implemented. The local refinement is based on a pre-defined threshold on the damage variable and element size. The incompatible nodes due to adaptive refinement are handled by variable-node elements. The performance of the solution algorithm is numerically studied for a few problems with different loading conditions, viz., pure mechanical, pure thermal and combined thermo-mechanical. From the study, it is opined that the BFGS algorithm is 2∼5 times faster than the AM approach, which is typically used for adaptive phase field framework. In addition, the detailed numerical implementation is presented.

Enhancements Self-Scaling Quasi-Newton for Unconstrained Optimization

A self-scaling for the quasi-Newton tecnique is derive by using a second_order Taylor's expansion to achieve optimal computational performance. Following this, new updating formulas for the quasi-Newton method are introduced based on the newly derived self-scaling equation. The numerical results confirm this derivation and suggest that the new method could potentially rival the BFGS method in terms of performance.

Enhancements Self-Scaling Quasi-Newton for Unconstrained Optimization

A self-scaling for the quasi-Newton tecnique is derive by using a second_order Taylor's expansion to achieve optimal computational performance. Following this, new updating formulas for the quasi-Newton method are introduced based on the newly derived self-scaling equation. The numerical results confirm this derivation and suggest that the new method could potentially rival the BFGS method in terms of performance.

A novel phase‐field monolithic scheme for brittle crack propagation based on the limited‐memory BFGS method with adaptive mesh refinement

A novel phase‐field monolithic scheme for brittle crack propagation based on the limited‐memory BFGS method with adaptive mesh refinement

Efficient Parameterization of Density Functional Tight-Binding for 5f-Elements: A Th-O Case Study.

Density functional tight binding (DFTB) models for f-element species are challenging to parametrize owing to the large number of adjustable parameters. The explicit optimization of the terms entering the semiempirical DFTB Hamiltonian related to f orbitals is crucial to generating a reliable parametrization for f-block elements, because they play import roles in bonding interactions. However, since the number of parameters grows quadratically with the number of orbitals, the computational cost for parameter optimization is much more expensive for the f-elements than for the main group elements. In this work we present a set of efficient approaches for mitigating the hurdle imposed by the large size of the parameter space. A novel group-by-orbital correction functions for two-center bond integrals was developed. With this approach the number of parameters is reduced, and it grows linearly with the number of elements, maintaining the accuracy and the number of parameters, in the case of f elements, by more than 40%. The parameter optimization step was accelerated by means of the mini-batch BFGS method. This method allows parameter optimizations with much larger training sets than other single batch methods. A stochastic optimizer was employed that helped overcome shallow local minima in the objective function. The proposed algorithm was used to parametrize the DFTB Hamiltonian for the Th-O system, which was subsequently applied to the study of ThO2 nanoparticles. The training set consisted of 6322 unique structures, which is barely feasible with conventional optimization methods. The optimized parameter set, LANL-ThO, displays good agreement with DFT-calculated properties such as energies, forces, and structures for both clusters and bulk ThO2. Benefiting from the fewer number of parameters and lower computational costs for objective function evaluations, this new approach shows its potential applications in DFTB parametrization for elements with high angular momentum, which present a challenge to conventional methods.

Portfolio Optimization with Multi-Trend Objective and Accelerated Quasi-Newton Method

We propose a portfolio optimization method with a multi-trend objective and an accelerated quasi-Newton method (MTO-AQNM). It leverages a BFGS-based quasi-Newton algorithm and incorporates an ℓ1 regularization term and the self-funding constraint. The MTO is designed to identify multiple trend reversals. Different trend reversals are asymmetric, and we hoped to extract rich and effective information from them. The AQNM adopts the BFGS method with the Wolfe conditions, which reduces computational complexity and improves convergence speed. We wanted to evaluate the performance of our algorithm through financial markets that were asymmetric in all respects. To this end, we conducted comprehensive experimental approaches on six benchmark data sets of real-world financial markets that were asymmetric in time, frequency, and asset type. Our method demonstrated superior performance over other state-of-the-art competitors across several mainstream evaluation metrics.

A Combined Approach for Predicting the Distribution of Harmful Substances in the Atmosphere Based on Parameter Estimation and Machine Learning Algorithms

This paper proposes a new approach to predicting the distribution of harmful substances in the atmosphere based on the combined use of the parameter estimation technique and machine learning algorithms. The essence of the proposed approach is based on the assumption that the concentration values predicted by machine learning algorithms at observation points can be used to refine the pollutant concentration field when solving a differential equation of the convection-diffusion-reaction type. This approach reduces to minimizing an objective functional on some admissible set by choosing the atmospheric turbulence coefficient. We consider two atmospheric turbulence models and restore its unknown parameters by using the limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm. Three ensemble machine learning algorithms are analyzed for the prediction of concentration values at observation points, and comparison of the predicted values with the measurement results is presented. The proposed approach has been tested on an example of two cities in the Republic of Kazakhstan. In addition, due to the lack of data on pollution sources and their intensities, an approach for identifying this information is presented.

Parallel-in-time multiple shooting for optimal control problems governed by the Navier–Stokes equations

In the past decades, multiple shooting methods have proven to be a promising direction to speed up the optimization process, especially in the context of ODE-based optimization. Very recently, Fang et al. (Journal of Computational Physics, vol. 452, 110926, 2022) proposed a multiple shooting algorithm for large-scale PDE-constrained optimization. The current paper continues this line of work and explores the potential of multiple shooting methods for optimal control problems governed by the three-dimensional Navier–Stokes equations. Similar to Fang et al., the augmented Lagrangian (AL) method is used to solve the resulting equality-constrained optimization problem, and we employ the classical limited-memory BFGS method for the unconstrained subproblems inside the AL loop. In the current work, we exploit the multiple shooting paradigm in full by processing the shooting windows parallel-in-time, allowing for significant parallel speed-ups compared to single shooting. The proposed method is validated on a velocity tracking case, using up to 100 windows. Our analysis shows that the multiple shooting algorithm allows for considerable algorithmic and parallel speed-ups. While algorithmic speed-up depends on the exact tracking case and initialization of the shooting windows, the multiple shooting algorithm always outperforms single shooting in terms of computational time (if the number of windows is sufficiently high) due to the parallel-in-time implementation. For a given amount of resources, we also show that the proposed parallel-in-time strategy can outperform spatial parallelization alone, especially when the spatial parallelization is saturated.

Three-Dimensional Limited-Memory BFGS Inversion of Magnetic Data Based on a Multiplicative Objective Function

At present, the traditional magnetic three-dimensional inversion method has been fully developed and is widely used. Magnetic exploration is a kind of geophysical exploration method that uses the magnetic field changes (magnetic anomalies) caused by the magnetic differences between various rocks in the crust to find useful mineral resources and study the underground structure. Traditional magnetic three-dimensional inversion is relatively inefficient. Moreover, the traditional additive objective function (data fitting difference term plus regularization term and logarithmic obstacle term), which causes the regularization factor selection to be more complicated, is implemented in this method. Therefore, it is necessary to establish a new efficient three-dimensional magnetic inversion algorithm and optimize the selection of regularization factors. In this paper, based on the limited-memory BFGS (L-BFGS) method, the three-dimensional magnetic inversion of a multiplicative objective function is realized. The inversion test is conducted in this paper using both theoretical synthesis data and measured data. The results demonstrate that the limited-memory BFGS method significantly enhances the inversion efficiency and yields superior inversion outcomes compared to traditional magnetic three-dimensional inversion methods. Additionally, the multiplicative objective function-based three-dimensional magnetic inversion method simplifies the process of selecting weight factors for regularization terms.

Prediction Parameters for Mining Subsidence Based on Interferometric Synthetic Aperture Radar and Unmanned Aerial Vehicle Collaborative Monitoring

Coal mining induces surface subsidence, making rapid and precise monitoring of this subsidence a key area of current research. To address the limitations of D-InSAR technology in capturing large-gradient deformations in the central subsidence basin and the challenges facing UAVs in accurately monitoring small deformations at the basin’s edge, we propose a method for inverting the expected parameters of surface subsidence by synergistically integrating InSAR and UAV monitoring. We determined the cumulative subsidence of monitoring points along the dip and strike observation line of the Banji 110,801 working face between 10 April 2021 and 28 June 2022, employing D-InSAR and UAV techniques. By leveraging the complementary strengths of both monitoring techniques, we fused the two types of monitoring data and verified the error of the fusion data to be within 10 cm through leveling data verification. Simulation experiments utilizing the probability integration method and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimization algorithm confirmed that the 10 cm data source error remains within the required limits for probability integration parameter inversion. Finally, the BFGS algorithm was employed to invert the parameters of the probability integration method based on the fusion data results. Subsequently, these inversion parameters were used to predict the subsidence at the monitoring point and were compared with the level measured data. The results demonstrate that the use of collaborative InSAR and UAV monitoring technology for inverting the expected parameters of surface subsidence in the mining area yields superior results, aligning with the actual patterns of ground surface movement and deformation. This study addresses the global need for unmanned monitoring of mining-related subsidence. It employs InSAR and UAV technologies in a synergistic approach to monitor surface subsidence in mining regions. This approach harnesses the strengths of multiple data sources and presents a novel concept for the unmanned monitoring of surface subsidence in mining areas, contributing to environmental protection efforts.

Quasi-Newton algorithms for solving interval-valued multiobjective optimization problems by using their certain equivalence

In this paper, we investigate a class of interval-valued multiobjective optimization problems (IVMOP), and formulate a certain equivalent multiobjective optimization problem (MOP). We establish that any Pareto optimal solution of the equivalent MOP is also an effective solution of IVMOP. We introduce two variants of the quasi-Newton method to solve IVMOP when every component of the objective function has second-order partial generalized Hukuhara derivatives. In the first variant, we assume that the objective function of IVMOP is strongly convex, and approximate Hessian matrix of each component of the objective function of MOP by using the well-known BFGS method. The algorithm generated by this variant exhibits superlinear convergence to a locally effective solution of IVMOP. In the second variant, we approximate the Hessian matrix of each component of MOP by applying the Levenberg–Marquardt method. Moreover, this variant can be applied to non-convex functions, and the sequence generated by this variant converges to a locally effective solution of IVMOP as well. Finally, some numerical examples are furnished to illustrate the effectiveness of our developed methodology.

Nonlinear and infinite gain scheduling neural predictive control of the outlet temperature in a parabolic trough solar field: A comparative study

Solar thermal plants have high nonlinearities and non-manipulated energy source which make their control task a very challenging work. Linear controllers cannot cope with undesirable deviations of the outlet temperature over all the operation range of the dynamics of this type of plants. Moreover, nonlinear predictive control relying on online nonlinear optimization have the drawback of time consuming and numerical calculus issues. In this paper, an infinite gain scheduling neural predictive control is designed and applied to control the temperature in a distributed parabolic trough solar collector field. The performance of both tracking and disturbance rejection of the proposed controller is compared to those of four nonlinear predictive control variants: Two unconstrained neural predictive control using the Levenberg–Marquardt and the Broyden–Fletcher–Goldfarb–Shanno algorithms, and two constrained nonlinear predictive control using interior point algorithm, one is based on a neural network model and the other one is based on a first principal model. The superiority of the proposed control strategy is well demonstrated through simulation results.

Multiobjective BFGS method for optimization on Riemannian manifolds

Multiobjective BFGS method for optimization on Riemannian manifolds

Global Convergence of a Modified BFGS Method with Simultaneous Correction of Direction and Step Size

Global Convergence of a Modified BFGS Method with Simultaneous Correction of Direction and Step Size

An adaptive projection BFGS method for nonconvex unconstrained optimization problems

An adaptive projection BFGS method for nonconvex unconstrained optimization problems

An Evaluation of ANN Algorithm Performance for MPPT Energy Harvesting in Solar PV Systems

In this paper, the Levenberg–Marquardt (LM), Bayesian regularization (BR), resilient backpropagation (RP), gradient descent momentum (GDM), Broyden–Fletcher–Goldfarb–Shanno (BFGS), and scaled conjugate gradient (SCG) algorithms constructed using artificial neural networks (ANN) are applied to the problem of MPPT energy harvesting in solar photovoltaic (PV) systems for the purpose of creating a comparative evaluation of the performance of the six distinct algorithms. The goal of this analysis is to determine which of the six algorithms has the best overall performance. In the study, the performance of managing the training dataset is compared across the algorithms. The maximum power point tracking energy harvesting system is created using the environment of MATLAB or Simulink, and the produced model is examined using the artificial neural network toolkit. A total of 1000 datasets of solar irradiance, temperature, and voltage were used to train the suggested model. The data are split into three categories: training, validation, and testing. Eighty percent of the total data is used for training the model, and the remaining twenty percent is divided equally for testing and validation. According to the results, the regression values of LM, RP, BR, and BFGS are 1, whereas the regression values for SCG and GDM are less than 1. The gradient values for LM, RP, BFGS, SCG, BR, and GDM are 7.983 × 10−6, 0.033415, 1.0211 × 10−7, 0.14161, 0.00010493, and 11.485, respectively. Similarly, the performance values for these algorithms are 2.0816 × 10−10, 2.8668 × 10−6, 9.98 × 10−17, 0.052985, 1.583 × 10−7, and 0.15378. Overall, the results demonstrate that the LM and BFGS algorithms exhibit superior performance in terms of gradient and overall performance. The RP and BR algorithms also perform well across various metrics, while the SCG and GDM algorithms show comparatively less effectiveness in addressing the proposed problem. These findings provide valuable insights into the relative performance of the six evaluated algorithms for MPPT energy harvesting in solar PV systems.