Hessian Matrix Research Articles

Thermodynamic Aspects of Higher-Dimensional Black Holes in Einstein Gauss-Bonnet Gravity Through Exponential Entropy

Abstract We assume exponential corrections to the entropy of $5D$ charged AdS
black hole solutions, which are derived within the framework of
Einstein Gauss-Bonnet gravity and nonlinear electrodynamics.
Additionally, we consider two distinct versions of $5D$ charged AdS
black holes by setting the parameters $q\rightarrow0$ and
$k\rightarrow0$ (where $q$ represents charge, $k$ is the non-linear
parameter). We investigate these black holes in the extended phase
space, where the cosmological constant is interpreted as pressure,
and demonstrate the first law of black hole thermodynamics. The
focus extends to understanding the thermal stability or instability,
as well as identifying first and second-order phase transitions.
This exploration is carried out through the analysis of various
thermodynamic quantities, including heat capacity at constant
pressure, Gibbs free energy, Helmholtz free energy, and the trace of
the Hessian matrix. In order to visualize phase transitions,
identify critical points, analyzing stability and provide
comprehensive analysis, we have made the contour plot of the
mentioned thermodynamic quantities and observed that our results are
very much consistent. These investigations are conducted within the
context of exponentially corrected entropies, providing valuable
insights into the intricate thermodynamic behavior of these $5D$
charged AdS black holes under different parameter limits.

Uncertainty Based Machine Learning-DFT Hybrid Framework for Accelerating Geometry Optimization.

Geometry optimization is an important tool used for computational simulations in the fields of chemistry, physics, and material science. Developing more efficient and reliable algorithms to reduce the number of force evaluations would lead to accelerated computational modeling and materials discovery. Here, we present a delta method-based neural network-density functional theory (DFT) hybrid optimizer to improve the computational efficiency of geometry optimization. Compared to previous active learning approaches, our algorithm adds two key features: a modified delta method incorporating force information to enhance efficiency in uncertainty estimation, and a quasi-Newton approach based upon a Hessian matrix calculated from the neural network; the later improving stability of optimization near critical points. We benchmarked our optimizer against commonly used optimization algorithms using systems including bulk metal, metal surface, metal hydride, and an oxide cluster. The results demonstrate that our optimizer effectively reduces the number of DFT force calls by 2-3 times in all test systems.

Generalized Path Integral Energy and Heat Capacity Estimators of Quantum Oscillators and Crystals Using Harmonic Mapping.

Imaginary-time path integral (PI) is a rigorous tool to treat nuclear quantum effects in static properties. However, with its high computational demand, it is crucial to devise precise estimators. We introduce generalized PI estimators for the energy and heat capacity that utilize coordinate mapping. While it can reduce to the standard thermodynamic and centroid virial (CVir) estimators, the formulation can also take advantage of harmonic character of quantum oscillators and crystals to construct a coordinate mapping. The method is not applicable to fluids or systems with imaginary modes such as double-well potentials. This yields harmonically mapped averaging (HMA) estimators, with mappings that decouple (HMAc) or couple (HMAq) the centroid and internal modes. The HMAq is constructed with normal mode coordinates (HMAq-NM) with quadratic scaling of cost or harmonic oscillator staging (HMAq-SG) coordinates with linear scaling. The estimator performance is examined for a 1D anharmonic oscillator and a 3D Lennard-Jones crystal using path integral molecular dynamics (PIMD) simulation. The HMA estimators consistently provide more precise estimates compared to CVir, with the best performance obtained by HMAq-NM, followed by HMAq-SG, and then HMAc. We also examine the effect of anharmonicity (for AO), intrinsic quantumness, and Trotter number. The HMA formulation introduced assumes the availability of forces and Hessian matrix; however, an equally efficient finite difference alternative is possible when these derivatives are inaccessible. The remarkable improvement in precision offered by HMAq estimators provides a framework for efficient PI simulation of more challenging systems, such as those based on ab initio calculations.

Reducing Measurement Costs by Recycling the Hessian in Adaptive Variational Quantum Algorithms

Abstract Adaptive protocols enable the construction of more efficient state preparation circuits in variational quantum algorithms (VQAs) by utilizing data obtained from the quantum processor during the execution of the algorithm. This idea originated with ADAPT-VQE, an algorithm that iteratively grows the state preparation circuit operator by operator, with each new operator accompanied by a new variational parameter, and where all parameters acquired thus far are optimized in each iteration. In ADAPT-VQE and other adaptive VQAs that followed it, it has been shown that initializing parameters to their optimal values from the previous iteration speeds up convergence and avoids shallow local traps in the parameter landscape. However, no other data from the optimization performed at one iteration is carried over to the next. In this work, we propose an improved quasi-Newton optimization protocol specifically tailored to adaptive VQAs. The distinctive feature in our proposal is that approximate second derivatives of the cost function are recycled across iterations in addition to optimal parameter values. We implement a quasi-Newton optimizer where an approximation to the inverse Hessian matrix is continuously built and grown across the iterations of an adaptive VQA. The resulting algorithm has the flavor of a continuous optimization where the dimension of the search space is augmented when the gradient norm falls below a given threshold. We show that this inter-optimization exchange of second-order information leads the approximate Hessian in the state of the optimizer to be consistently closer to the exact Hessian. As a result, our method achieves a superlinear convergence rate even in situations where the typical implementation of a quasi-Newton optimizer converges only linearly. Our protocol decreases the measurement costs in implementing adaptive VQAs on quantum hardware as well as the runtime of their classical simulation.

Efficient non-probabilistic parallel model updating based on analytical correlation propagation formula and derivative-aware deep neural network metamodel

Non-probabilistic convex models are powerful tools for structural model updating with uncertain‑but-bounded parameters. However, existing non-probabilistic model updating (NPMU) methods often struggle with detecting parameter correlation due to limited prior information. Worth still, the unique core steps of NPMU, involving nested inner layer forward uncertainty propagation and outer layer inverse parameter updating, present challenges in efficiency and convergence. In response to these challenges, a novel and flexible NPMU scheme is introduced, integrating analytical correlation propagation and parallel interval bounds prediction to enable automatic detection of parameter correlations. In the forward uncertainty propagation phase, a linear coordinate transformation is applied to map the original parameter space to a standard hypercube space, simplifying correlation-involved bounds prediction into conventional interval bounds prediction. Moreover, an analytical correlation propagation formula is derived using a second-order response approximation to sidestep the complexities of geometry-based correlation calculations. To expedite forward propagation, a derivative-aware neural network model is employed to replace the physical solver, facilitating improved fitting capabilities and automatic differentiation, including the calculation of Jacobian and Hessian matrices essential for correlation propagation. The neural network's inherent parallelism accelerates interval bounds prediction through parallel computation of samples. In the inverse parameter updating phase, the block coordinate descent algorithm is embraced to narrow the search space and boost convergence capabilities, while the perturbation method is utilized to determine the optimal starting point for optimization. Two numerical examples illustrate the efficacy of the proposed method in updating structural models while considering correlations.

A novel branch and bound algorithm for solving the linear multiplicative programming problems

In this paper, we propose a novel branch and bound algorithm for solving the linear multiplicative programming (LMP) problem. A new quadratic relaxation is proposed. We first show that the Hessian matrix of the multiplication of two linear functions has a decomposition form, given by the difference of two rank-one matrices, resulting in a decomposition of each quadratic term of the objective function. And then the minus term is relaxed into a linear one. To find the global optimal solutions, branch and bound method is adopted. Theoretical analysis is provided that guarantees the convergence of our algorithm. Numerical experiments show that the new algorithm runs faster than several typical algorithms for certain types of LMP.

Morse index for solutions of a nonlinear Kirchhoff equation

In this paper, we study a perturbed nonlinear Kirchhoff equation with subcritical growth in R3. Although the existence of concentrated solutions with a single peak or multi peaks to the problem above has been obtained in Li et al. [J. Differ. Equations 268, 541–589 (2020)] and Luo et al. [Proc. R. Soc. Edinburgh, Sect. A 149, 1097–1122 (2019)], respectively, the Morse indices of them remain open. First, we compute the Morse index of single-peak solutions concentrated at a point P∈R3 by variational methods, which can also be applied to the case where P is a degenerate critical point of V. Then, we also study the Morse index of multi-peak solutions concentrated at the non-degenerate critical points of V. Here the main difficulty comes from the nonlocal term ∫R3|∇u|2dyΔu. In addition, since the estimates of the eigenvalues and the eigenfunctions of the linearized problem associated to the limit problem are unknown, we have to study them independently, which are quite interesting.

Nonlinear incompressible shear wave models in hyperelasticity and viscoelasticity frameworks, with applications to Love waves

General equations describing shear displacements in incompressible hyperelastic materials, holding for an arbitrary form of strain energy density function, are presented and applied to the description of nonlinear Love-type waves propagating on an interface between materials with different mechanical properties. The model is valid for a broad class of hyper-viscoelastic materials. For the Murnaghan constitutive model, shear wave equations contain cubic and quintic differential polynomial terms, including viscoelasticity contributions in terms of dispersion terms that include mixed derivatives uxxt of the material displacement. Full (2+1)-dimensional numerical simulations of waves propagating in the bulk of a two-layered solid are undertaken and analysed with respect to the source position and mechanical properties of the layers. Interfacial nonlinear Love waves and free upper surface shear waves are tracked; it is demonstrated that in the fully nonlinear case, the variable wave speed of interface and surface waves generally satisfies the linear Love wave existence condition c1<v<c2, while tending to the larger material wave speed c1 or c2 for large times.

An Improved Non-Monotonic Adaptive Trust Region Algorithm for Unconstrained Optimization

The trust region method is an effective method for solving unconstrained optimization problems. Incorrectly updating the rules of the trust region radius will increase the number of iterations and affect the calculation efficiency. In order to obtain an effective radius for the trust region, an adaptive radius updating criterion is proposed based on the gradient of the current iteration point and the eigenvalue of the Hessian matrix which avoids calculating the inverse of the Hessian matrix during radius updating. This approach reduces the computation time and enhances the algorithm’s performance. On this basis, we apply adaptive radius and non-monotonic techniques to the trust region algorithm and propose an improved non-monotonic adaptive trust region algorithm. Under proper assumptions, the convergence of the algorithm is analyzed. Numerical experiments confirm that the suggested algorithm is effective.

Decarbonization and waste reduction in a disrupted food supply chain: a step towards achieving food sustainability

PurposeThis study aims to reduce carbon emissions and minimize waste in the event of disruptions in a short and fast-food perishable such as fruits, vegetables, packaged food items, etc. supply chain through optimal investment in green and preservation technologies.Design/methodology/approach This study utilized a Hessian matrix approach to optimize decision variables with an objective to maximize the profit function.FindingsThe study demonstrates that investing in both green and preservation technology within a short and fast-food supply chain is highly beneficial for decarbonization and waste reduction and it leads to profit maximization. It has been shown with the help of a numerical experiments with investment in both green and preservation technology that total profit is 3.09% higher than without investment made in either technology.Practical implicationsThis study aids the industry in achieving food sustainability by minimizing waste of perishables and also minimizes carbon emissions which is essential for environmental protection. It assists industries in determining the optimal investment in preservation technology to minimize waste and in green technology to reduce emissions, thereby maximizing profits.Originality/valueThe current study formulates an inventory model that helps in decarbonization and waste reduction in food supply chain with the consideration of machine learning, demand disruption, preservation technology investment, screening of purchased items, waste disposal, a double triangular distribution deterioration rate, green technology investment, carbon emissions from various supply chain activities, carbon tax policy and fuel price variation over time for perishable food products in a two-warehouse system.

Distributed Optimization on Matrix‐Weighted Networks

ABSTRACTThis article is intended to tackle the optimization problems for continuous‐time first‐order and second‐order multi‐agent systems (MASs) operating over matrix‐weighted networks. A matrix‐weighted network serves as a powerful framework to model the interdependence among agents' multidimensional states, providing an effective approach to analyze smart grids, intelligent transportation systems, and so forth. Our optimization objective is to facilitate the convergence of all agents toward the optimal value of a global cost function, which is formed by a sum of local cost functions. To achieve this goal, distributed optimization algorithms based on Hessian matrix and gradient information are constructed. Additionally, an edge‐based event‐triggered mechanism is utilized to avoid communicating with all neighbors at the time of event triggering. It is proved that this mechanism theoretically excludes Zeno behavior. The results show that the proposed algorithms ensure that the agents can achieve the optimization goal while reducing energy consumption. Finally, an application is presented to substantiate the theoretical results.

Mathematical modeling for the efficiency function of the Retiro small hydroelectric power plant turbine-generator set

ABSTRACT Many authors use an efficiency constant for the turbine-generator set to calculate energy generation in hydroelectric projects. This often does not reflect reality because the efficiency of this system is relative and is related to the potential of the net head height and the flow of turbine discharge. The present work aims to perform a mathematical modeling for an analysis of the efficiency of the Turbine-Generator Small Hydroelectric Power Plant (SHPP) Retiro. To perform this analysis, a model based on a second-degree polynomial function was employed. A multivariate regression technique was applied by the Least Square Method (LSM) to obtain the coefficients of the function. To verify the behavior of the measured and adjusted data, the Determination Coefficient (R2) of Root Mean Square Error (RMSE) algorithm was applied. The RMSE Mean of 0.022 and the RMSE Std of 0.029 suggest that the second-degree polynomial function is more consistent and stable across different training and validation datasets. The R2 value of 0.993 indicates that 99.3% of the variability in the data is explained by the model. The Omnibus, Durbin-Watson, and Jarque-Bera tests were applied to diagnose potential issues related to normality, autocorrelation, and adequacy, thereby validating the accuracy and significance of the model. The Hessian matrix technique was also used to verify the critical points of the function. The critical point corresponding to a water head of 11.47 meters and a turbine flow of 145.1 m3/s presented the highest operational efficiency. This study allows presenting a turbine efficiency function, its Hill Curve and a generator efficiency function with its corresponding Efficiency Curve. For efficiency analysis in several flow, height and power settings. The technique has proved efficient and can be an applicable tool in energy parameter studies for other hydroelectric projects with the same characteristics as SHPP Retiro.

Assessing the Partial Hessian Approximation in QM/MM-Based Vibrational Analysis.

The partial Hessian approximation is often used in vibrational analysis of quantum mechanics/molecular mechanics (QM/MM) systems because calculating the full Hessian matrix is computationally impractical. This approach aligns with the core concept of QM/MM, which focuses on the QM subsystem. Thus, using the partial Hessian approximation implies that the main interest is in the local vibrational modes of the QM subsystem. Here, we investigate the accuracy and applicability of the partial Hessian vibrational analysis (PHVA) approach as it is typically used within QM/MM, i.e., only the Hessian belonging to the QM subsystem is computed. We focus on solute-solvent systems with small, rigid solutes. To separate two of the major sources of errors, we perform two separate analyses. First, we study the effects of the partial Hessian approximation on local normal modes, harmonic frequencies, and harmonic IR and Raman intensities by comparing them to those obtained using full Hessians, where both partial and full Hessians are calculated at the QM level. Then, we quantify the errors introduced by QM/MM used with the PHVA by comparing normal modes, frequencies, and intensities obtained using partial Hessians calculated using a QM/MM-type embedding approach to those obtained using partial Hessians calculated at the QM level. Another aspect of the PHVA is the appearance of normal modes resembling the translation and rotation of the QM subsystem. These pseudotranslational and pseudorotational modes should be removed as they are collective vibrations of the atoms in the QM subsystem relative to a frozen MM subsystem and, thus, not well-described. We show that projecting out translation and rotation, usually done for systems in isolation, can adversely affect other normal modes. Instead, the pseudotranslational and pseudorotational modes can be identified and removed.

Existence, uniqueness, stability results for tripled system which containing of mixed fractional derivatives with laplacian operator

Existence, uniqueness, stability results for tripled system which containing of mixed fractional derivatives with laplacian operator

The Behavior of Mixed Metal Based Copper–Organic Framework for Uptake of Chlorpyrifos Pesticide from Wastewater and its Antimicrobial Activity

Developing effective material for pesticide adsorption is a vital issue to protect the environment from their harmful effects. Copper-based metal–organic frameworks including Cu-BTC and its mixed metal derivatives (Fe-Cu-BTC, Co–Cu-BTC, and Mn-Cu-BTC) were successfully formed. Fe-Cu-BTC, Co–Cu-BTC and Mn-Cu-BTC MOFs were synthesized by direct substituting one Cu atom in Cu-BTC with Fe, Co, or Mn. Their structures were characterized using Powder X-ray diffraction (PXRD), Fourier-transform infrared spectroscopy FTIR, scanning electron microscopy with EDX, Transmission electron microscopy, BET surface area analysis, and Size distribution. Prepared MOFs adsorbed chlorpyrifos from wastewater and their adsorption capacities were compared. Pseudo-second-order kinetics and Langmuir isothermal models were the best to describe the adsorption of chlorpyrifos from water. The coordination bonding was the dominant mechanism; physical adsorption, π-π stacking interaction, and hydrogen bonding were also participated in the adsorption process. Cu-BTC, Fe-Cu-BTC, Co–Cu-BTC and Mn-Cu-BTC had elimination capacities of 379, 851, 683, and 762 mg/g, respectively. This study also investigates their antimicrobial activity against gram-positive and gram-negative bacteria and they exhibited a good inhibition effect. The inhibition zone of Co–Cu-BTC is greater than Cu-BTC with 1.44, 1.38, and 1.60 times for E. coli, Ps. Aeruginosa, S. aureus, respectively. The synthesized MOFs are promising materials for the removal of chlorpyrifos with effective antimicrobial agents.

Assessing the value of combined use of distributed acoustic sensing (DAS) and multicomponent vertical seismic profiling (VSP) data with network-based full waveform inversion and uncertainty quantification: A case study in Alberta, Canada

Distributed acoustic sensing (DAS) technology, deployed in a vertical seismic profiling (VSP) experimental configuration, has emerged as a candidate for non-disruptive and low-cost seismic monitoring of CO2 geostorage and plume evolution. Full waveform inversion (FWI) has likewise received significant attention because it uses relatively complete physical models of wave propagation and because of its sample-by-sample incorporation of data information. Recent artificial neural network-based FWI algorithms (built with, for instance, recursive neural networks, or RNNs) have added to FWI a range of flexible and efficient tools for gradient computation and options for uncertainty assessment and initial model proxies. An important current research area for the use of DAS data is to better understand how they change our confidence levels in models selected through FWI. In particular, we seek to understand whether either DAS data or conventional geophone data alone are optimal for FWI model selection in the CO2 problem, and if not, to what degree they complement each other. The Snowflake 4D VSP dataset, which includes multi-offset and multi-azimuth broadband sources illuminating both fiberoptic cable and densely sampled accelerometer phones in the borehole, was acquired by our group to directly address these questions. In this study, we quantify uncertainty (UQ) by sampling the posterior model covariance matrix from the inverse Hessian matrix at the end of RNN-FWI runs on the Snowflake baseline data, invoking a velocity-density parameterization, and involving mixtures of accelerometer and DAS data. In this UQ context, the complementary effect of combining accelerometer and DAS data is evident in the Vp and Vs models. Our confidence in the approach is bolstered by its independent prediction of a region of known high uncertainty. In the overall pursuit of reliable and low-cost monitoring tools, this supports continued consideration of a multicomponent sensors supported by DAS approach.

A Quasi-Newton Subspace Trust Region Algorithm for Nonmonotone Variational Inequalities in Adversarial Learning over Box Constraints

The first-order optimality condition of convexly constrained nonconvex nonconcave min-max optimization problems with box constraints formulates a nonmonotone variational inequality (VI), which is equivalent to a system of nonsmooth equations. In this paper, we propose a quasi-Newton subspace trust region (QNSTR) algorithm for the least squares problems defined by the smoothing approximation of nonsmooth equations. Based on the structure of the nonmonotone VI, we use an adaptive quasi-Newton formula to approximate the Hessian matrix and solve a low-dimensional strongly convex quadratic program with ellipse constraints in a subspace at each step of the QNSTR algorithm efficiently. We prove the global convergence of the QNSTR algorithm to an ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document}-first-order stationary point of the min-max optimization problem. Moreover, we present numerical results based on the QNSTR algorithm with different subspaces for a mixed generative adversarial networks in eye image segmentation using real data to show the efficiency and effectiveness of the QNSTR algorithm for solving large-scale min-max optimization problems.

Tuning the composition of mixed anthraquinone derivatives towards an affordable flow battery negolyte

Having a long lifespan and being capable of scaling capacity and power independently, redox flow batteries (RFB) offer great opportunities for energy storage. However, the challenge lies in finding an ideal electrolyte. The most mature version of RFB utilizes vanadium solutions and suffers from rising and highly volatile prices of this metal. To address this, organic electrolytes are gaining attention, as they can be obtained from abundant feedstocks. Among those, Anthraquinone-2,7-disulfonic acid (2,7-AQDS) solutions are particularly prominent, demonstrating reversible and fast redox kinetics coupled with reasonable solubility. This paper explores the possibility of synthesizing 2,7-AQDS together with other electroactive compounds (2,6-AQDS, 2-AQS) through the reaction of anthraquinone sulfonation. It shows that obtained mixtures act as electrolytes without any purification or separation, while synthesis conditions can adjust mixture composition and hence their redox behavior. Although the performance of anthraquinone-bromine RFB utilizing these mixtures exhibits a trade-off between power and stability, the best of them are comparable or even superior to 2,7-AQDS. For instance, RFB with a mixture free of 2-AQS demonstrates an energy efficiency of 76.4 % and a capacity fade rate of 0.04 %/cycle at a current density of 75 mA cm−2. The specific capacity of such mixtures can reach 70 Ah L−1, which makes them promising and affordable RFB negolyte.

SPIRF-CTA: Selection of parameter importance levels for reasonable forgetting in continuous task adaptation

Humans can adapt to changing environments and tasks and consolidate their previous knowledge by constantly learning and practicing new knowledge; however, it is extremely challenging for artificial intelligence to achieve this goal. With the development of the field of artificial intelligence, research on overcoming catastrophic forgetting in continuous learning is also faced with the challenges of focusing on the most important parameters of the current task, retaining knowledge from previous tasks and adapting it to new ones, and making better use of knowledge from previous and new tasks. To solve these problems, this article proposes a reasonable forgetting method, which is called selection of parameter importance levels for reasonable forgetting in continuous task adaptation (SPIRF-CTA). The SPIRF-CTA approach enables the constructed model to identify and focus on the most important parameters for the current task by designing a normalized parameter importance selection mechanism and a loss function with parameter importance penalties, and it adjusts the parameter updates by incorporating Hessian matrix information to achieve reasonable forgetting and prevent the new task from completely overwriting the knowledge of the previous task. Moreover, we design a model alignment loss function and a multitask loss function to use the knowledge of the new and previous tasks. We evaluate the SPIRF-CTA method on the Split CIFAR-10 Split CIFAR-100, and Split mini-ImageNet datasets, and the results show that the image classification accuracies of the proposed approach improve by 3.6%, 4.4%, and 3.36%, respectively; moreover, the SPIRF-CTA method exhibits excellent control of the degree of forgetting, with a forgetting rate of only 3.54%. Code is available at https://github.com/ybyangjing/CTA.

A Second-Order Numerical Method for a Class of Optimal Control Problems

The numerical solution of optimal control problems through second-order methods is examined in this paper. Controlled processes are described by a system of nonlinear ordinary differential equations. There are two specific characteristics of the class of control actions used. The first one is that controls are searched for in a given class of functions, which depend on unknown parameters to be found by minimizing an objective functional. The parameter values, in general, may be different at different time intervals. The second feature of the considered problem is that the boundaries of time intervals are also optimized with fixed values of the parameters of the control actions in each of the intervals. The special cases of the problem under study are relay control problems with optimized switching moments. In this work, formulas for the gradient and the Hessian matrix of the objective functional with respect to the optimized parameters are obtained. For this, the technique of fast differentiation is used. A comparison of numerical experiment results obtained with the use of first- and second-order optimization methods is presented.