Conjugate Gradient Research Articles

Slab waveguide communication study using Finite Difference Method (FDM) with fourth-order compact scheme

An accurate and efficient semi-vectorial mode solver, examining Electric (E) and Magnetic (H) fields, is utilized to explore the communication characteristics of various slab waveguides. This approach relies on the Finite Difference Method (FDM) using uniform grid points, employing a Higher (Fourth) Order Compact type technique coupled with Conjugate Gradient (CG) iteration. The present study is an extension of the previous work where various aspects of E- and H-field propagations through rib-strucured waveguide was analyzed successfully. By incorporating the refractive index profile of the slab structure, excellent functionality of the waveguide is demonstrated here in Electric field by utilizing the concepts of normalized index and effective or modal index, confinement factor, and the modal birefringence properties which proves to be critical for potential applications in photonic integrated circuits. The materials chosen for the purpose are AlGaAs-GaAs and GeSi which play a vital role in characterizing the waveguides’performances through simulation using MATLAB. The use of surface and contour plots aids in visualizing the distribution of corresponding field within the guided modes. This analysis significantly contributes in understanding the waveguide's behavior and confinement capability during the field propagation. Most importantly, the variations of the waveguides’performance indices with their structure parameters help to identify their optimized values for fabrication to enhance the transmission efficiency of the waveguide.

Deep learning for kinetics parameters identification: A novel approach for multi-variate optimization

This work presents an innovative deep-learning approach for multi-variate optimization, focusing on the identification of Mg(OH)2 precipitation kinetics parameters. The study employs three distinct experimental datasets, one for the Population Balance Model (PBM) fitting and two for validation. These datasets explore the impact on Particle Size Distributions (PSDs) of (i) increasing the initial reactant concentrations from 0.125 to 1 M and (ii) decreasing the flow rate from 12 to 4 m/s, both in a T-mixer, (iii) increasing the initial reactant concentration over a wider concentration range from 0.01 to 1 M in a more complex Y-mixer system. Leveraging PBM, we create a dataset to train a Neural Network (NN), referred to as the ‘mirror model,’ which predicts kinetics parameters based on experimental sizes. Notably, the PBM, fitted with dataset (i), excels at describing changes in flow rate (dataset (ii)) and substantial reductions in reactant concentrations in the Y-mixer (dataset (iii)), even though these conditions were not encountered during the fitting step. Key Performance Indicators (KPIs) reveal that the mirror model consistently outperforms two widely used algorithms, Conjugate Gradient (CG) and Particle Swarm Optimization (PSO), highlighting its remarkable potential for practical applications.

Bayesian reconstruction of Cartesian product graph signals with general patterns of missing data

In this paper, we address the challenge of signal reconstruction on Cartesian product graphs, which frequently arises in applications where node-level data may be corrupted by equipment failure or difficult to reliably collect. A recent approach to Graph Signal Reconstruction (GSR) is to formulate it as a Bayesian inference problem, where the underlying signal is presumed to be smooth with respect to the graph topology. However, the resulting linear system derived from this model specification is often intractable to solve directly for general Cartesian product graphs. To overcome this limitation, we introduce two robust and efficient iterative procedures for computing the posterior mean. The first method is a stationary iterative technique based on matrix splitting, while the second employs a conjugate gradient approach with a symmetric graph-spectral preconditioner. We provide a theoretical and numerical comparison of the convergence behaviour of these algorithms, demonstrating that the optimal choice depends on the model hyperparameters. Finally, we propose a novel supervised learning-based technique, enhanced by an active learning strategy, for estimating the posterior marginal variance. This circumvents the need to instantiate, invert or decompose large matrices directly in memory and results in an R2 statistic of over 0.95 on a synthetic test data set. This paper contributes to the development of advanced signal reconstruction techniques applicable in various domains, including large-scale sensor networks and social network analysis.

Integration of Modified Classical Conjugate Gradient Methods for Unconstrained Optimization

The integration of modified classical conjugate gradient methods (CGMs) for unconstrained optimization represents a crucial and evolving area of research within the field of optimization algorithms. Over time, numerous studies have put forth diverse modifications and novel approaches to enhance the effectiveness of classical CGMs. These modifications aim to address specific challenges and improve the overall performance of optimization algorithms in unconstrained scenarios. In order to tackle unconstrained optimization challenges and improve our understanding of their synergies, this ongoing study aims to unify different modified classical CGMs. Conventional CGMs have proven effective for optimization tasks, and a range of different approaches have been produced by carefully modifying these techniques. The main goal of this paper is to combine these modified versions, with particular attention to those that have similar numerators. The integration process involves systematically merging the advantageous aspects of these modified methods to develop not only innovative but also more resilient approaches to unconstrained optimization problems. The ultimate goal of this unification effort is to capitalize on the strengths inherent in different approaches to create a cohesive framework that significantly improves overall optimization performance. To thoroughly assess the efficacy of the integrated methods, a series of comprehensive performance tests are conducted. These tests include a meticulous comparison of outcomes with those of classical CGMs, providing valuable insights into the relative strengths and weaknesses of the modified approaches across diverse optimization scenarios. The evaluation criteria encompass convergence rates, solution accuracy, and computational efficiency. The conclusive outcome demonstrates that the unified approaches consistently outperform individual methods across all three crucial evaluation criteria.

A family of limited memory three term conjugate gradient methods

Based on several modified Hestenes–Stiefel and Polak–Ribière–Polyak nonlinear conjugate gradient methods, a family of three term limited memory CG methods are developed. When the current search direction falls into the subspace spanned by the previous m directions, the algorithm branches to optimize the objective function on the subspace by using L-BFGS method. We use this strategy to avoid potential local loops and accelerate the convergence. The proposed methods are sufficient descent. When the steplength is determined by Wolfe or Armijo line search, we establish the global convergence of the methods in a concise way. Numerical results verify the efficiency of the proposed method for the unconstrained optimization problems in the CUTEst library.

Inverse Estimation of Thermal Contact Resistance Between Two Layers of Cylindrical Wall

This study presents a method for estimating the space-dependent thermal contact resistance between the two-layer walls of a furnace using the boundary element method (BEM) and conjugate gradient method (CGM) for the heat conduction problem. The global solution equation in matrix form is derived using the interface conditions, and the BEM is used to solve the direct problem. The CGM minimizes the objective function and calculates the sensitivity coefficients with the complex variable derivation method (CVDM). Comparative results show that the present approach is more accurate, stable, and efficient than the conventional CGM, which is attributed to the calculation of the sensitivity coefficients by CVDM. The effects of the value of thermal contact resistance, thermal conductivity ratio, Biot number, initial guess, measurement error, and the number and position of measurement points on the inversion results are also analyzed. Finally, the effectiveness of this approach is demonstrated through numerical examples, and the inversion results show its stability, efficiency, and accuracy in identifying different and complex distributions of thermal contact resistance. Furthermore, this approach is feasible for nonintrusive measurement, which is very meaningful in practical applications.

A hybrid complex spectral conjugate gradient learning algorithm for complex-valued data processing

Complex-valued neural networks (CVNNs) have become a powerful modelling tool for complex-valued data processing. Because most of the critical points of CVNNs are saddle points, the gradient-based learning algorithms for CVNNs enjoy more chances to reach the global minima while suffering from slow convergence. To this end, we propose a hybrid complex spectral conjugate gradient learning algorithm for fast training CVNNs in this paper. The proposed algorithm combines the scaled negative gradient with a Barzilai–Borwein stepsize and an optimized conjugate term to define a new training direction, thus providing an accurate approximation of the second-order curvature of the objective function. The complex Wolfe conditions are employed to adaptively determine the optimal training stepsize. Under mild conditions, the descent property of the training direction and the convergence of the proposed algorithm are theoretically established. Simulation results on a number of benchmark complex-valued data processing problems demonstrate the efficiency of the proposed algorithm.

An inertial Fletcher–Reeves-type conjugate gradient projection-based method and its spectral extension for constrained nonlinear equations

An inertial Fletcher–Reeves-type conjugate gradient projection-based method and its spectral extension for constrained nonlinear equations

Simultaneous estimation of heat source strength of hot wire and thermal conductivity of material

The hot-wire (HW) technique is a widely used method for measuring the thermal conductivity (k) of insulation materials. However, there are some limitations to the applicability. For instance, inaccuracies may arise when the wire is assumed to be infinitely thin and long, or when the sample material is assumed to be an infinite medium. Additionally, the accuracy of k measurements relies on accurately specifying the heat source strength (QW ) generated by the HW. The modified conjugate gradient method (CGM) provides a solution to these limitations of the HW technique. In the modified CGM technique, a scaling factor based on the Levenberg-Marquardt method is utilized to address the low derivative and divergence issues of the traditional CGM. The current work simultaneously estimates QW generated by the HW and k of the sample materials by using modified CGM. The investigation looked into the effect of sensor positioning relative to the HW and the introduction of artificial Gaussian error (noise) with a standard deviation of 0.01 °C, 0.02 °C and 0.1 °C into the simulated temperature measurements. The modified CGM estimates the unknown parameters accurately compared to the traditional CGM. The measurement location should be closer to the HW to obtain accurate estimation results. The noise level of 0.1 °C results in low accuracy of estimation of k and QW for Plexiglas with one sensor, which can be enhanced by employing two sensors. Overall, the modified CGM algorithm proved to be a robust inverse algorithm that could be employed to estimate unknown parameters.

A conjugate gradient projection method with restart procedure for solving constraint equations and image restorations

A conjugate gradient projection method with restart procedure for solving constraint equations and image restorations

A New Dai-Liao Conjugate Gradient Method based on Approximately Optimal Stepsize for Unconstrained Optimization

Conjugate gradient methods are a class of very effective iterative methods for large-scale unconstrained optimization. In this paper, a new Dai-Liao conjugate gradient method for solving large-scale unconstrained optimization problem is proposed. Based on the approximately optimal stepsize for the gradient method, we derive three new choices for the important parameters tk in Dai-Liao conjugate gradient method. The search direction satisfies the sufficient descent condition, and the global convergences of the proposed method for uniformly convex and general functions are proved under some mild conditions. Numerical experiments on a set of test problems from the CUTEst library show that the proposed method is superior to some well-known conjugate gradient methods.

PAPR reduction techniques optimization-based OFDM signal for wireless communication systems

OFDM (Orthogonal Frequency Division Multiplexing) is a widely used modulation technique in many standards such as Wi-Fi (Wireless Fidelity), LTE (Long Term Evolution), and 4 G. It allows for high transmission data rates and excellent spectral efficiency. However, one of the main limitations of OFDM is its high PAPR (Peak-to-Average Power Ratio). In this paper, we will first describe some methods to reduce the PAPR associated with OFDM signal. We will discuss some methods such as the clipping, the Selective mapping SLM, the Partial Transmit Sequence (PTS), and the Tone Reservation (TR). Let's list their advantages and disadvantages before focusing on the TR method which we will detail. This work aims to improve the performance of the TR method in terms of speed of convergence and reduction of the PAPR, using the conjugate different algorithms while respecting the frequency specifications of the IEEE 802.11a standard. The proposed algorithm which is based on the gradient method, Conjugate gradient methods, and Quasi-Newton method shows good performance regarding PAPR reduction and speed of convergence. The choice of an appropriate technique depends on several factors: required PAPR reduction, acceptable complexity, and performances (spectral efficiency, latency, etc.). A trade-off must be found in practice between these different criteria. It has been shown that the Quasi-Newton method algorithm with its two variants converges more quickly.

Numerical modeling of edge inhomogeneity in a macro cantilever beam

This study explores the inhomogeneity problem near the edge of a cantilever beam. Common in gears and turbine blades, inhomogeneities under local edge contact, coupled with macroscale bending effects, result in extreme and complex stress concentration. This model employs the Conjugate Gradient Method to combines the macroscopic beam theory and microscopic elasticity. The Eshelby inclusion method for simulating inhomogeneities. The Hetényi image method is used for address edge contacts in the beam. Enhanced by the Discrete Convolution fast Fourier Transform algorithm, the model’s accuracy and efficiency are validated with the Finite element method. Parametric analysis highlights the influence of distance between indenter and edge, stress components distribution and potential failure in cross-scale challenges.

Reconstruction of Cosmological Initial Density Field with Observations from the Epoch of Reionization

The initial density distribution provides a basis for understanding the complete evolution of cosmological density fluctuations. While reconstruction in our local Universe exploits the observations of galaxy surveys with large volumes, observations of high-redshift galaxies are performed with a small field of view and therefore can hardly be used for reconstruction. Here, we propose reconstructing the initial density field using the H i 21 cm and CO line intensity maps from the epoch of reionization. Observations of these two intensity maps provide complementary information on the density field—the H i 21 cm field is a proxy of matter distributions in the neutral regions, while the CO line intensity maps are sensitive to the high-density, star-forming regions that host the sources for reionization. Technically, we employ the conjugate gradient method and develop the machinery for minimizing the cost function for the intensity mapping observations. Analytical expressions for the gradient of cost function are derived explicitly. We show that the resimulated intensity maps match the input maps of mock observations using semi-numerical simulations of reionization with an rms error ≲7% at all stages of reionization. This reconstruction is also robust with an rms error of ∼10% when an optimistic level of shot noise is applied to the CO map or white noise at the level of ≲10% of the standard deviation is applied to each map. Our proof-of-concept work demonstrates the robustness of the reconstruction method, thereby providing an effective technique for reconstructing the cosmological initial density distribution from high-redshift observations.

Enhancing CMB map reconstruction and power spectrum estimation with convolutional neural networks

The accurate reconstruction of Cosmic Microwave Background (CMB) maps and the measurement of its power spectrum are crucial for studying the early universe. In this paper, we implement a convolutional neural network to apply the Wiener Filter to CMB temperature maps, and use it intensively to compute an optimal quadratic estimation of the power spectrum. Our neural network has a UNet architecture as that implemented in WienerNet, but with novel aspects such as being written in python 3 and TensorFlow 2. It also includes an extra channel for the noise variance map, to account for inhomogeneous noise, and a channel for the mask. The network is very efficient, overcoming the bottleneck that is typically found in standard methods to compute the Wiener Filter, such as those that apply the conjugate gradient. It scales efficiently with the size of the map, making it a useful tool to include in CMB data analysis. The accuracy of the Wiener Filter reconstruction is satisfactory, as compared with the standard method. We heavily use this approach to efficiently estimate the power spectrum, by performing a simulation-based analysis of the optimal quadratic estimator. We further evaluate the quality of the reconstructed maps in terms of the power spectrum and find that we can properly recover the statistical properties of the signal. We find that the proposed architecture can account for inhomogeneous noise efficiently. Furthermore, increasing the complexity of the variance map presents a more significant challenge for the convergence of the network than the noise level does.

A New Conjugate Gradient Method for Impact Load Identification of Stochastic Composite Structures

This paper presents a novel conjugate gradient method to identify the impact force of stochastic composite structure. Firstly, the proposed method is established based on constructing a new gradient regularization operator, and its stability and global convergence are strictly proven. Moreover, the proposed method solves the deterministic inverse problem of composite laminated cylindrical shells. Then, using the matrix perturbation method, the uncertain inverse problem of the impact force reconstruction of stochastic structure is converted to definite inverse problems. At last, the statistical properties of the reconstructed impact load are also analyzed. Numerical simulations show that the proposed method performs well in identifying the impact force of composite laminated cylindrical shells with stochastic and non-stochastic properties.

Estimation of soil erodibility in Peninsular Malaysia: A case study using multiple linear regression and artificial neural networks

Soil erodibility (K) is an essential component in estimating soil loss indicating the soil's susceptibility to detach and transport. Data Computing and processing methods, such as artificial neural networks (ANNs) and multiple linear regression (MLR), have proven to be helpful in the development of predictive models for natural hazards. The present case study aims to assess the efficiency of MLR and ANN models to forecast soil erodibility in Peninsular Malaysia. A total of 103 samples were collected from various sites and K values were calculated using the Tew equation developed for Malaysian soil. From several extracted parameters, the outcomes of correlation and principal component analysis (PCA) revealed the influencing factors to be used in the development of ANN and MLR models. Based on the correlation and PCA results, two sets of influencing factors were employed to develop predictive models. Two MLR (MLR-1 and MLR-2) models and four neural networks (NN-1, NN-2, NN-3, and NN-4) optimized using Levenberg-Marquardt (LM) and scaled conjugate gradient (SCG) were developed and evaluated. The model performance validation was conducted using the coefficient of determination (R2), mean squared error (MSE), root mean squared error (RMSE), and Nash-Sutcliffe efficiency coefficient (NSE). The analysis showed that ANN models outperformed MLR models. The R2 values of 0.446 (MLR-1), 0.430 (MLR-2), 0.894 (NN-1), 0.855 (NN-2), 0.940 (NN-3), and 0.826 (NN-4); MSE values of 0.0000306 (MLR-1), 0.0000315 (MLR-2), 0.0000158 (NN-1), 0.0000261 (NN-2), 0.0000318 (NN-3), and 0.0000216 (NN-4) suggested the higher accuracy and lower modelling error of ANN models as compared with MLR. This study could provide an empirical basis and methodological support for K factor estimation in the region.

A domain decomposition solver for spectral stochastic finite element: an approximate sparse expansion approach

Spectral stochastic finite element (SSFE) has been widely used in the uncertainty quantification of real-life problems. However, the prohibitive computational burden prevents the application of the method in practical engineering systems because an enormous augmented system has to be solved. Although the domain decomposition method has been introduced to SSFE to improve the efficiency for the solution of the augmented system, there still exist significant challenges in solving the extended Schur complement (e-SC) system from domain decomposition method. In this paper, we develop an approximate sparse expansion-based domain decomposition solver to generalize the application of SSFE. An approximate sparse expansion is first presented for the subdomain-level augmented matrix so that the computational cost in each iteration of the preconditioned conjugate gradient is greatly alleviated. Based on the developed sparse expansion, we further establish an approximate sparse preconditioner to accelerate the convergence of the preconditioned conjugate gradient. The developed approximate sparse expansion-based domain decomposition solver is then incorporated in the context of SSFE. Since the difficulties of solving the e-SC system have been overcome, the developed approximate sparse expansion-based solver greatly improves the computational efficiency of the solution of the e-SC system, and thereby, the SSFE is capable of dealing with large-scale engineering systems. Two numerical examples demonstrate that the developed method can significantly enhance the efficiency for the stochastic response analysis of practical engineering systems.

A New Hybrid Three-Term LS-CD Conjugate Gradient In Solving Unconstrained Optimization Problems

The Conjugate Gradient (CG) method is renowned for its rapid convergence in optimization applications. Over the years, several modifications to CG methods have emerged to improve computational efficiency and tackle practical challenges. This paper presents a new three-term hybrid CG method for solving unconstrained optimization problems. This algorithm utilizes a search direction that combines Liu-Storey (LS) and Conjugate Descent (CD) CG coefficients and standardizes it using a spectral which acts as a scheme for the choices of the conjugate parameters. This resultant direction closely approximates the memoryless Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton direction, known for its bounded nature and compliance with the sufficient descent condition. The paper establishes the global convergence under standard Wolfe conditions and some appropriate assumptions. Additionally, the numerical experiments were conducted to emphasize the robustness and superior efficiency of this hybrid algorithm in comparison to existing approaches.

Stochastic three-term conjugate gradient method with variance technique for non-convex learning

Stochastic three-term conjugate gradient method with variance technique for non-convex learning