Generalized Gradient Method Research Articles

Feeling the strain: Enhancing the thermodynamics characteristics of magnesium nickel hydride Mg2NiH4 for hydrogen storage applications through strain engineering

This work is based on density functional theory (DFT) and studies the structural, electronic, thermodynamic properties and dehydrogenation kinetics of the hydride Mg2NiH4 under different uniaxial and biaxial strain applied to Mg2Ni or Mg2NiH4. structural properties are first studied based on generalized gradient (GGA) and local density (LDA). Thermodynamic and electronic properties and dehydrogenation kinetics are then studied using the GGA method based on the Perdew-Burke-Ernzerhor (PPE) approximation. The results obtained show that uniaxial/biaxial tensile and compressive strains are capable of inducing structural deformation by increasing the value of the strain step on both Mg2Ni and Mg2NiH4, and that these strains applied to Mg2NiH4 are capable of reducing the stability of the system more than strains applied to Mg2Ni, due to the contribution of strain energy. More specifically, it lowers the enthalpy of formation to −40 kJ/mol.H2 under compressive strain ε = −7.5% or biaxial tensile strain ε = +7.8%, which corresponds exactly to the ideal value suggested by the US Department of Energy (DOE) for materials suitable for hydrogen storage. And the decomposition temperature fell within the fuel cell operating range (PEM) of 289–393 K. On the other hand, under uniaxial and biaxial strains, the activation energy of hydrogen atom diffusion in Mg2NiH4 varied between 0.40 eV and 0.22 eV, resulting in a remarkable improvement in dehydrogenation properties. Analysis of the total (TDOS) and partial (PDOS) density curves shows that the Mg2NiH4 system has a metallic characteristic, and that this characteristic is maintained by increasing the strain pitch, with a variation in the energy width of the valence band.

Methods for Minimizing the Savage Function with Various Constraints

Introduction. When making decisions under uncertainty, Savage's criterion is sometimes used, or the criterion of minimizing regrets [1]. Usually, in the literature, this decision situation is described in matrix language. In other words, both the number of decision alternatives and the number of states of nature are finite. Of particular interest are situations where the admissible domain of decision variants is a convex set, and the regrets with respect to each state of nature are expressed by convex functions. In this paper, we propose numerical methods for minimizing Savage's regret function, constructed based on the subgradient projection method with automatic step size adjustment [2, 3]. The convergence of these methods is demonstrated. Goal. In the article, the Savage function is defined as a function that expresses the maximum regret value, assumed to be a convex function with respect to the decision factors. This function measures the effectiveness of each decision relative to the set of states of nature. It is important to note that computing the values of these functions is complex because of the need to know the optimal solution for each state of nature. This difficulty is successfully overcome in the process of solving the problem of minimizing functions on convex sets, thanks to parallel solutions of m "internal" algorithms based on the number of states of nature, and one external algorithm, aimed at minimizing the Savage function. Each of the m+1 algorithms represent modifications of the subgradient projection method with a programmable way of adjusting the step size. Depending on the complexity of the constraints and the required precision, three theorems have been proven, confirming the convergence of the investigated methods. Results obtained. Constructive numerical algorithms have been developed for determining optimal decision alternatives under uncertainty, when the number of states of nature is finite, the admissible domain of control factors is convex and compact, and the Savage regret function serves as a decision criterion. The convergence of the corresponding algorithms to the set of optimal solutions has been proven, without knowing the exact values of the Savage function. Instead, estimates obtained from parallel runs of algorithms were used, aimed at determining optimal solutions for each state of nature. Conclusions. Uncertainty poses significant difficulties in designing and making decisions. Any decision made under uncertainty represents a certain risk or a certain regret. In cases where the number of states of nature is finite, the decision domain is convex, the target function with respect to each state of nature is convex, and the Savage regret function is adopted as the decision criterion, the decision-making problem can be successfully solved using numerical algorithms based on the generalized gradient method. The implementation of the algorithm is relatively simple, and the fields of application can be very diverse. Keywords: uncertainty, decisions, Savage function, optimization.

First-principles study on adsorption of gas molecules by metal Sc modified Ti<sub>2</sub>CO<sub>2</sub>

MXene materials have received increasing attention due to their unique properties and potential applications. Ti<sub>2</sub>CO<sub>2</sub>, as a typical MXene material that has been prepared, has been widely studied. The adsorption characteristics of two-dimensional materials for gas molecules can be significantly improved through transition metal modification. However, there are few studies on the use of transition metals to modify Ti<sub>2</sub>CO<sub>2</sub>. In this work, the adsorption processes of different harmful gases (CO, NH<sub>3</sub>, NO, SO<sub>2</sub>, CH<sub>4</sub>, H<sub>2</sub>S) on the surfaces of these two materials, i.e. Ti<sub>2</sub>CO<sub>2</sub> and metal Sc modified Ti<sub>2</sub>CO<sub>2,</sub> are studied and analyzed based on first-principles density functional theory and generalized gradient method. The geometric optimization calculation of the metal-modified adsorption harmful gas structure is carried out, and the kinetic energy cutoff energy of the plane wave basis set is taken as 450 eV. The calculation results show that the structure in which Sc atoms are located above the C atoms in the hollow position has a large binding energy, but it is smaller than the experimental value of the cohesive energy of solid Sc (3.90 eV). Sc atoms can effectively avoid clustering. Surface Sc metal provides active sites for gas adsorption. By analyzing the optimal adsorption points, adsorption energy and other parameters of different gases, the adsorption effects of metal Sc-modified Ti<sub>2</sub>CO<sub>2</sub> on these gases are analyzed. Among them, the adsorption effect of SO<sub>2</sub> is better, the adsorption energy is increased from –0.314 eV to –2.043 eV, and the adsorption effects of other gases are improved. Due to the introduction of new atoms on the surface of Ti<sub>2</sub>CO<sub>2</sub>, the carrier density and carrier mobility of the material are increased, thereby improving the charge transfer on the surface of the material, which is beneficial to its sensitivity to gas molecules. The results of density of states and work function further verify that the carrier density and carrier mobility of Sc-Ti<sub>2</sub>CO<sub>2</sub> are increased, which is beneficial to gas adsorption. It is expected that the metal Sc-modified Ti<sub>2</sub>CO<sub>2</sub> becomes an excellent gas-sensing material for the detection of CO, NH<sub>3</sub>, NO, SO<sub>2</sub>, CH<sub>4</sub> and H<sub>2</sub>S, and the present work can provide a reference for theoretically studying the gas-sensing performance of metal Sc-modified Ti<sub>2</sub>CO<sub>2</sub> materials.

Improvement of density inversion efficiency with projected gradient method: Application to FTG data

With the development of instruments for measuring Earth's gravity gradient tensor, full tensor gravity (FTG) data can be obtained validly and accurately. The highly precise performance of FTG data ensures that they can play an important role in many domains. Generally, FTG data can be acquired from airborne platform, and large-scale data can be obtained by long term observations. For this type of data, research methods for large data processing are very important. Three-dimensional data inversion is an effective and significant method for the quantitative interpretation of FTG data, which has received much attention. Large-scale data inversion is inevitable, and may consume large amounts of time with increase in the amount of data and model parameters. The main aim of this study is to improve the calculation efficiency of FTG data inversion compared with general conjugate gradient (CG) method. Accordingly, an improved gradient method with projections onto a convex set (the projected gradient method) was applied to accelerate the recovery rate of the sub-surface rock density, and a corresponding preconditioning factor was introduced to further improve the calculation efficiency of inversion. This new projected gradient method can improve the calculation efficiency of large amounts of FTG data, which is very necessary for practical application of FTG data. The application of synthetic FTG data with Gaussian noise and the real data demonstrated the feasibility and application value of the proposed inversion method.

Privacy-Preserving and Secure Cloud Computing: A Case of Large-Scale Nonlinear Programming

The volume of data is increasing rapidly currently, which poses a great challenge for resource-constrained clients to process and analyze. A promising approach for solving computation-intensive tasks is to outsource them to the cloud to take advantage of the clouds powerful computing ability. However, it also brings privacy and security issues since the data uploaded to the cloud may contain sensitive and private information which should be protected. In this paper, we address this problem and focus on the privacy-preserving and secure outsourcing of large-scale nonlinear programming problems (NLPs) subject to both linear constraints and nonlinear constraints. Large-scale NLPs play an important role in the eld of data analytics but have not received enough attention. In our outsourcing protocol, we rst apply a secure and efcient transformation scheme at the client side to encrypt its private information. Then, we use the reduced gradient method and generalized gradient method at the server side to solve the transformed large-scale NLPs under linear constraints and nonlinear constraints, respectively. We provide security analysis of the proposed protocol, and evaluate its performance. The experimental results show that we can efciently solve the large-scale NLPs and save huge amount of time for the clients.

Investigation of SGS alloys CoNbMnZ (Z = As, Sb) suitable for dissipationless spintronic devices and thermoelectric technology

AbstractThe current study examines two quaternary Heusler alloys, CoNbMnZ (Z = As, Sb). The structural optimization displays a Y2‐type structure with a ferromagnetic feature as a stable state. Mechanical characteristics, cohesive, and formation energies further support the stability of these materials. The generalized gradient method band structure depicts the metallic character of both alloys. Upon integrating the modified Becke–Johnson potential to the generalized gradient method, the alloys signify a spin gapless semiconducting nature (SGS). The origin of the spin gapless property is explained using constituent hybridization through schematic representation. Since SGS alloys have zero spin‐flip energy, both holes and electrons display 100% spin polarization, making them ideal candidates for spintronics applications, which is a unique property of these alloys. Both alloys have a magnetic moment 2μB following the Slater–Pauling rule MT = ZT − 24. Furthermore, the computed thermoelectric properties suggest that these alloys are thermo‐efficient, might generate thermoelectric power, and act as green energy materials.

材料パラメータ同定問題におけるガウスニュートン法の適用

In this study, the application of the Gauss-Newton method to several material parameter identification problems was investigated. The parameters of materials with complicated constitutive laws are determined to reproduce known physical quantities such as stiffness and natural frequency. The identification problem is usually formulated as a squared error minimization problem for the physical quantity. Gauss-Newton method is one of the important solution methods. In this method, the Hessian matrix is approximated using the first order differentiation of the physical quantity with respect to the parameters. Although this method obtains good convergence in the initial stage, it is observed that it becomes unstable in the final stage. We introduced a line search technique based on the Marquardt method to avoid the instability. We compared this method with the general gradient method and confirmed good convergence of this method.

Theoretical Basis of Quantum-Mechanical Modeling of Functional Nanostructures

The paper presents an analytical review of theoretical methods for modeling functional nanostructures. The main evolutionary changes in the approaches of quantum-mechanical modeling are described. The foundations of the first-principal theory are considered, including the stationery and time-dependent Schrödinger equations, wave functions, the form of writing energy operators, and the principles of solving equations. The idea and specifics of describing the motion and interaction of nuclei and electrons in the framework of the theory of the electron density functional are presented. Common approximations and approaches in the methods of quantum mechanics are presented, including the Born–Oppenheimer approximation, the Hartree–Fock approximation, the Thomas–Fermi theory, the Hohenberg–Kohn theorems, and the Kohn–Sham formalism. Various options for describing the exchange–correlation energy in the theory of the electron density functional are considered, such as the local density approximation, generalized and meta-generalized gradient approximations, and hybridization of the generalized gradient method. The development of methods of quantum mechanics to quantum molecular dynamics or the dynamics of Car–Parrinello is shown. The basic idea of combining classical molecular modeling with calculations of the electronic structure, which is reflected in the potentials of the embedded atom, is described.

On Optimization Techniques for the Construction of an Exponential Estimate for Delayed Recurrent Neural Networks

This work is devoted to the modeling and investigation of the architecture design for the delayed recurrent neural network, based on the delayed differential equations. The usage of discrete and distributed delays makes it possible to model the calculation of the next states using internal memory, which corresponds to the artificial recurrent neural network architecture used in the field of deep learning. The problem of exponential stability of the models of recurrent neural networks with multiple discrete and distributed delays is considered. For this purpose, the direct method of stability research and the gradient descent method is used. The methods are used consequentially. Firstly we use the direct method in order to construct stability conditions (resulting in an exponential estimate), which include the tuple of positive definite matrices. Then we apply the optimization technique for these stability conditions (or of exponential estimate) with the help of a generalized gradient method with respect to this tuple of matrices. The exponential estimates are constructed on the basis of the Lyapunov–Krasovskii functional. An optimization method of improving estimates is offered, which is based on the notion of the generalized gradient of the convex function of the tuple of positive definite matrices. The search for the optimal exponential estimate is reduced to finding the saddle point of the Lagrange function.

Application of fuzzy programming techniques to solve solid transportation problem with additional constraints

An innovative, real-life solid transportation problem is explained in a non-linear form. As in real life, the total transportation cost depends on the procurement process or type of the items and the distance of transportation. Besides, an impurity constraint is considered here. The proposed model is formed with fuzzy imprecise nature. Such an interesting model is optimised through two different fuzzy programming techniques and fractional programming methods, using LINGO-14.0 tools followed by the generalized gradient method. Finally, the model is discussed concerning these two different methods.

Design of unimodular sequence train with low central and recurrent autocorrelations

This study considers the unimodular sequence train design with low central and recurrent autocorrelation sidelobes. The authors attain the design goal by minimising the integrated energy in central and recurrent autocorrelations with user-supplied weights. They devise a way to employ a fast Fourier transform in calculating both the objective function and its gradient, such that the considered design problem can be efficiently solved using the general gradient method. Numeric results indicate that the proposed method can effectively suppress the range sidelobes concerned and it can design sequence train of very large sequence length and set cardinality.

Methodology for Identifying the Differentiated Mineral Extraction Tax Rates Relating to the Recovery of Solid Minerals

The paper presents the methodology for identifying the differentiated Mineral Extraction Tax (MET) rates relating to the recovery of solid minerals. Experiments, aimed to find solutions to these tasks by using generalized gradient method, have shown that the objective functions may have more than one local extremum. In this respect, to calculate ad valorem MET rates, it is suggested to use evolutionary algorithms. The technogenic raw materials are currently of economic interest for the purpose of extraction of valuable components and production of finished goods. Moreover, sometimes the content of valuable components in technogenic deposits (TD) exceeds their content in natural fields to be processed. Secondary mineral resources bring harm to the ecosystem, yet it is impossible to insure the environmental risks due to the lack of objects of subsoil use. The differentiated rates are selected on the basis of maximum MET capacity on all the valuable components extracted from deposits provided that each deposit is considered to be an investment project for the stated problem.

A projection gradient method for computing ground state of spin-2 Bose–Einstein condensates

In this paper, a projection gradient method is presented for computing ground state of spin-2 Bose–Einstein condensates (BEC). We first propose the general projection gradient method for solving energy functional minimization problem under multiple constraints, in which the energy functional takes real functions as independent variables. We next extend the method to solve a similar problem, where the energy functional now takes complex functions as independent variables. We finally employ the method into finding the ground state of spin-2 BEC. The key of our method is: by constructing continuous gradient flows (CGFs), the ground state of spin-2 BEC can be computed as the steady state solution of such CGFs. We discretized the CGFs by a conservative finite difference method along with a proper way to deal with the nonlinear terms. We show that the numerical discretization is normalization and magnetization conservative and energy diminishing. Numerical results of the ground state and their energy of spin-2 BEC are reported to demonstrate the effectiveness of the numerical method.

Audio classification with low-rank matrix representation features

In this article, a novel framework based on trace norm minimization for audio classification is proposed. In this framework, both the feature extraction and classification are obtained by solving corresponding convex optimization problem with trace norm regularization. For feature extraction, robust principle component analysis (robust PCA) via minimization a combination of the nuclear norm and the ℓ 1 -norm is used to extract low-rank matrix features which are robust to white noise and gross corruption for audio signal. These low-rank matrix features are fed to a linear classifier where the weight and bias are learned by solving similar trace norm constrained problems. For this linear classifier, most methods find the parameters, that is the weight matrix and bias in batch-mode, which makes it inefficient for large scale problems. In this article, we propose a parallel online framework using accelerated proximal gradient method. This framework has advantages in processing speed and memory cost. In addition, as a result of the regularization formulation of matrix classification, the Lipschitz constant was given explicitly, and hence the step size estimation of the general proximal gradient method was omitted, and this part of computing burden is saved in our approach. Extensive experiments on real data sets for laugh/non-laugh and applause/non-applause classification indicate that this novel framework is effective and noise robust.

Structural and elastic properties of ternary metal nitrides TixTa1 − xN alloys: First-principles calculations versus experiments

First-principles pseudopotential calculations of the lattice constants and of the single-crystal elastic constants for TixTa1−xN (0≤x≤1) alloys with B1-rocksalt structure were first carried out. These calculations were performed using density functional perturbation theory (DFPT) within the virtual crystal approximation (VCA) for the disordered alloys and the supercell method (SC) for the ordered alloys, with the ABINIT program. For disordered structures, partial comparisons of the lattice constants and of the bulk modulus are provided with calculations that used the coherent potential approximation (CPA) with the exact muffin-tin orbitals (EMTO). For the exchange-correlation potential we used the generalized gradient methods (GGA). The calculated equilibrium lattice parameters by VCA are in good agreement with the stress-free lattice parameters a0 and exhibit a positive deviation from Vegard's rule corresponding to a positive bowing parameter while the calculated single-crystal stiffness c12 and c44, gradually increase when c11 decreases from TaN to TiN. In a second stage, we have estimated by homogenization methods the averaged stiffnesses <cij>, the Young modulus and Poisson ratio of polycrystalline TixTa1−xN (0≤x≤1) alloys considering a random orientation of crystallites. Finally, comparisons are made with the experimental effective out-of-plane shear elastic modulus C44 and the effective out-of-plane longitudinal elastic constant C33 measured by Brillouin light scattering and picosecond ultrasonics, respectively, on thin films elaborated by magnetron sputtering.

Structural and elastic properties of single-crystal and polycrystalline Ti1−xZrxN alloys: A computational study

First-principles pseudopotential calculations of the lattice constants and of the single-crystal elastic constants of Ti1−xZrxN (0≤x≤1) alloys were carried out. These calculations were performed using density functional perturbation theory (DFPT) within the virtual crystal approximation (VCA) for the disordered alloys and the supercell method (SC) for the ordered alloys. For the exchange-correlation potential we used both the local density (LDA) and the generalized gradient methods (GGA). The calculated equilibrium lattice parameters exhibit a positive deviation from Vegard's rule corresponding to a positive bowing parameter, while the calculated single-crystal stiffnesses, namely C11, C12 and C44, gradually decrease from TiN to ZrN. In a second stage, in the frame of anisotropic elasticity, we have estimated by homogenization methods the averaged stiffnesses 〈Cij〉, direction dependent Young's moduli and Poisson's ratios of polycrystalline Ti1−xZrxN (0≤x≤1) alloys considering a {111}-fiber texture.

Eshelby’s inclusion problem in the gradient theory of elasticity: Applications to composite materials

We extend Eshelby’s integral representations for elastic inclusion problems to the case of gradient theory of elasticity. Gradient elastic effects associated with the existence of an interphase layer, within a simple and robust gradient model whose properties are described by the harmonic equation, are discussed. The decomposition of the corresponding solution into “classical” and “gradient” components is established. It is shown that the aforementioned Eshelby-type integral formulas for gradient elasticity can be expressed in the same form as in the standard theory of elasticity, but only for the “classical” part of the solution. The implementation of Eshelby’s approach in determining the effective properties of composites by the three-phase method requires the derivation of a complete solution for the gradient model. An example of application of the so-obtained generalized gradient method for determining the effective properties of composites with size effects due to cohesion and surface forces is given.

Linear Convergence of Iterative Soft-Thresholding

In this article a unified approach to iterative soft-thresholding algorithms for the solution of linear operator equations in infinite dimensional Hilbert spaces is presented. We formulate the algorithm in the framework of generalized gradient methods and present a new convergence analysis. As main result we show that the algorithm converges with linear rate as soon as the underlying operator satisfies the so-called finite basis injectivity property or the minimizer possesses a so-called strict sparsity pattern. Moreover it is shown that the constants can be calculated explicitly in special cases (i.e. for compact operators). Furthermore, the techniques also can be used to establish linear convergence for related methods such as the iterative thresholding algorithm for joint sparsity and the accelerated gradient projection method.

A General Projection Gradient Method for Linear Constrained Optimization with Superlinear Convergence

A General Projection Gradient Method for Linear Constrained Optimization with Superlinear Convergence

ELECTRON DENSITY FUNCTIONAL THEORY

A brief summary is given of electronic density functional theory, including recent developments: generalized gradient methods, hybrid functionals, time dependent density functionals and excited states, van der Waals energy functionals.