Iterative Solution Method Research Articles

Quantitation of Rainfall Retention Capacity for Small Reservoirs Considering Spatial Soil Moisture

To realize the estimation of rainfall retention capacity for small reservoirs considering spatial soil moisture, a rainfall retention capacity model and its parameter schemes have been developed in this study. An iterative trial solution method considering potential rainfall and soil moisture for the model constructed was proposed for efficient computation. The rainfall retention capacity of 32 pilot small reservoirs located in ungauged basins of Hunan province was calculated starting from 21 August 2023. In addition, a continuous calculation was carried out from 1 August to 30 September 2023 using the proposed method for Heping reservoir. The results show that the Pearson’s correlation coefficients between rainfall retention capacity and available reservoir capacity and soil moisture, are 0.36 and −0.64, respectively. Using Heping reservoir as an example, this study effectively characterized the dynamic change in its rainfall retention capacity, which ranged from 123.6 mm to 68 mm in August 2023. The analysis indicates the rainfall retention capacity of the pilot small reservoirs calculated is reasonably related to the soil moisture, supporting risk visualization for small reservoirs within rain-affected regions. Furthermore, the impact of the antecedent precipitation on rainfall retention capacity can also be dynamically quantified in real time, which provides reference for the continuous management of small reservoirs.

Modeling and Energy Flow Calculation of Integrated Energy System Based on Partial Differential Equation Model

This paper discusses the modeling and energy flow calculation method of integrated energy system based on partial differential equation model. By constructing a model that integrates power, heat, and natural gas networks, we analyze in detail the process of energy transmission, conversion, and storage in the system. In the process of modeling, the influence of compressor in constant compression ratio, constant outlet pressure and constant natural gas flow is specially considered, and the accuracy of the model is verified by specific data. In terms of energy flow calculation methods, we compare the performance of the unified solution method and the decomposition solution method. Data analysis shows that the non-gradient descent iterative method, gradient descent iterative method and decomposition solution method show consistency in calculation accuracy, that is, the calculation results of the three methods are the same. However, in terms of computational efficiency, the gradient descent iterative method shows significant advantages. Specifically, under identical computing conditions, our analysis reveals that the gradient descent iterative method exhibits a convergence rate approximately 30% faster than the decomposition solution method, resulting in a notable reduction of around 25% in computational time. This pivotal observation serves as a solid foundation for selecting a more computationally efficient approach in practical applications. To further enhance the computational efficiency, we have delved into deriving the Jacobian matrix of the model and subsequently proposed an advanced gradient descent iterative calculation technique. Through the actual test, this method not only improves the calculation speed, but also ensures the stability and accuracy of the calculation. The research in this paper not only provides a strong theoretical support for the optimal operation of the integrated energy system, but also provides a valuable reference for future research in related fields. Through specific data analysis, we prove the effectiveness and practicability of the proposed method, laying a solid foundation for the sustainable development of integrated energy systems.

Adaptive time step selection for spectral deferred correction

AbstractSpectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential equations for the error, and can be interpreted as a preconditioned fixed-point iteration for solving the fully implicit collocation problem. We adopt techniques from embedded Runge-Kutta Methods (RKM) to SDC in order to provide a mechanism for adaptive time step size selection and thus increase computational efficiency of SDC. We propose two SDC-specific estimates of the local error that are generic and do not rely on problem specific quantities. We demonstrate a gain in efficiency over standard SDC with fixed step size and compare efficiency favorably against state-of-the-art adaptive RKM.

Accelerated iterative DG finite element solvers for large-scale time-harmonic acoustic problems

Finite element methods are widely used to solve time-harmonic wave propagation problems, but solving large cases can be extremely difficult even with the computational power of parallel computers. In this work, the linear system resulting from the finite element discretization is solved with iterative solution methods, which are efficient in parallel but can require a large number of iterations. In standard discontinuous Galerkin (DG) methods, the numerical solution is discontinuous at the interfaces between the elements. In hybridizable DG methods, additional unknowns are introduced at the interfaces between the finite elements, and the physical unknowns are eliminated from the global system, resulting in a hybridized system. We have recently proposed a new strategy, called CHDG, where the additional unknowns correspond to transmission variables, whereas in the standard approach they are numerical fluxes. This strategy improves the properties of the hybridized system for faster iterative solution procedures. In this talk, we present and study a 3D CHDG implementation with nodal finite element basis functions. The resulting scheme has properties amenable to efficient parallel computing. Numerical results are presented to validate the method, and preliminary 3D computational results are proposed.

A Component Method for Full-Range Behaviour of Embedded Steel Column Bases

This paper introduces a component model for analysing embedded column bases to predict rotational stiffness, moment resistance, and the full-range moment–rotation response. The key components identified include the embedded column, concrete in compression on the column side, concrete in compression beneath the base plate, concrete in punching shear above the base plate, and anchor bolts. The embedded column is modelled as a Timoshenko beam, considering both shear and flexural deformations, while other components are represented by springs. Methods are provided for determining their uniaxial constitutive behaviour. A simplified iterative solution method is proposed, where the embedded column is further simplified into three rigid segments to specifically address shear and bending deformations. A corresponding simplified finite element model is developed for accurate numerical solutions. The validity of the component model is confirmed through comparisons with the results of existing tests and refined solid finite element analysis for H-steel column bases. The simplified iterative solution method effectively predicts strength but underestimates the stiffness of deeply embedded column bases. This is due to the trilinear deformation pattern simplification, which concentrates flexural deformation at the upper bearing stress resultant force point, leading to an overestimation of steel column rotation on the foundation surface.

A Novel Fast Iterative STAP Method with a Coprime Sampling Structure.

In space-time adaptive processing (STAP), the coprime sampling structure can obtain better clutter suppression capabilities at a lower hardware cost than the uniform linear sampling structure. However, in practical applications, the performance of the algorithm is often limited by the number of training samples. To solve this problem, this paper proposes a fast iterative coprime STAP algorithm based on truncated kernel norm minimization (TKNM). This method establishes a virtual clutter covariance matrix (CCM), introduces truncated kernel norm regularization technology to ensure the low rank of the CCM, and transforms the non-convex problem into a convex optimization problem. Finally, a fast iterative solution method based on the alternating direction method is presented. The effectiveness and accuracy of the proposed algorithm are verified through simulation experiments.

Iterative solution methods for high-order/hp–DGFEM approximation of the linear Boltzmann transport equation

In this article we consider the iterative solution of the linear system of equations arising from the discretisation of the poly-energetic linear Boltzmann transport equation using a high-order/hp–version discontinuous Galerkin finite element approximation in space, angle, and energy. In particular, we develop preconditioned Richardson iterations which may be understood as generalisations of source iteration in the mono-energetic setting, and derive computable a posteriori bounds for the solver error incurred due to inexact linear algebra, measured in a relevant problem-specific norm. We prove that the convergence of the resulting schemes and a posteriori solver error estimates are independent of the mesh size h and polynomial degree p. We also discuss how the poly-energetic Richardson iteration may be employed as a preconditioner for the generalised minimal residual (GMRES) method. Furthermore, we show that standard implementations of GMRES based on minimising the Euclidean norm of the residual vector can be utilized to yield computable a posteriori solver error estimates at each iteration, through judicious selections of left- and right-preconditioners for the original linear system. The effectiveness of poly-energetic source iteration and preconditioned GMRES, as well as their respective a posteriori solver error estimates, is demonstrated through numerical examples arising in the modelling of photon transport. While the convergence of poly-energetic source iteration is independent of h and p, we observe that the number of iterations required to attain convergence when employing GMRES only depends mildly on h and p. Moreover, this latter approach is highly effective in the low energy regime.

An Improved Analytical Thermal Rating Method for Cable Joints

To improve the utilization rate of cable lines while retaining sufficient security, the accurate thermal assessment of cable is significant for cable operation condition evaluation. The thermal rating for a cable joint, which is regarded as a hot spot of cable lines, is not covered by the scope of IEC 60287. While the existing publications for cable joint thermal evaluation also have some limitations. In this paper, the quasi-three-dimensional thermal model of the cable joint was established and the iterative solution method for the model is presented. Based on the model, an improved thermal rating method for the cable joint was proposed, which was implemented with monitored surface temperature and load data. The improved method was verified by the finite element method and the results showed an error of less than 5%. The superiority of the improved method was conducted by the comparison between the previously published method and the improved method. The improved method showed a better accuracy than the previously published method. The proposed method in this paper can be complementary to the IEC method, and is easy to use for the operating evaluation of cable joints in the field with the on-line condition monitoring technology.

Iterative method for the numerical solution of optimal control model for mosquito and insecticide

A linear multistep method is transformed into an iterative method based on Patade and Bhalekar's technique for the numerical solution of the optimal control problem modeled for mosquito and insecticide management using forward-backward sweep methods via Pontryagin's principle. Stability and convergence analysis of the iterative method are carried out, and it is found to be stable, convergent, and of order four. Results obtained by the method clearly show that the population of mosquitoes can be minimized to a large extent using the new iterative method while reducing the harmful effects of the insecticide, subsequently reducing the spread of malaria.

A non-linear convex model based energy management strategy for dual-storage offshore wind system

—Using hydrogen production as energy storage can realize large-scale storage and transportation of energy, which is conducive to flexible scheduling of offshore wind power resources. However, compared with the battery energy storage system, the energy management strategy (EMS) of the dual-storage offshore wind power system with hydrogen production is more complex and nonlinear due to the large number of state variables and control variables. In order to realize the EMS of the system to maximize the benefits and minimize the aging cost, a nonlinear convex model is first established, and then an iterative solution method combining convex programming (CP) and dynamic programming (DP) is proposed. Among them, CP solves the global optimization problem of power distribution, and DP determines the start and stop of the electrolyzer. Finally, the feasibility of the proposed method is verified by simulation analysis.

Solving Maxmin Optimization Problems via Population Games

Population games are games with a finite set of available strategies and an infinite number of players, in which the reward for choosing a given strategy is a function of the distribution of players over strategies. The paper shows that, in a certain class of maxmin optimization problems, it is possible to associate a population game to a given maxmin problem in such a way that solutions to the optimization problem are found from Nash equilibria of the associated game. Iterative solution methods for maxmin optimization problems can then be derived from systems of differential equations whose trajectories are known to converge to Nash equilibria. In particular, we use a discrete-time version of the celebrated replicator equation of evolutionary game theory, also known in machine learning as the exponential multiplicative weights algorithm. The resulting algorithm can be viewed as a generalization of the Iteratively Reweighted Least Squares (IRLS) method, which is well known in numerical analysis as a useful technique for solving Chebyshev function approximation problems on a finite grid. Examples are provided to show the use of the generalized IRLS method in collective investment and in decision making under model uncertainty.

On iterative methods based on Sherman-Morrison-Woodbury splitting

We consider a new splitting based on the Sherman-Morrison-Woodbury formula, which is particularly effective with iterative methods for the numerical solution of large linear systems and especially for systems involving matrices that are perturbations of circulant or block circulant matrices. Such matrices typically arise in the discretization of differential equations using finite element or finite difference methods. We prove the convergence of the new iteration without making any assumptions regarding the symmetry or diagonal-dominance of the matrix, which are limiting factors for most classical iterative methods.To illustrate the efficacy of the new iteration we present various applications. These include extensions of the new iteration to block matrices that arise in certain saddle point problems as well as two-dimensional finite difference discretizations. The new method exhibits fast convergence in all of the test cases we used. It has minimal storage requirements, straightforward implementation and compatibility with nearly circulant matrices via the Fast Fourier Transform. Remarkably, the new method was tested against very large matrices demonstrating extremely fast convergence. For these reasons it can be a valuable tool for the solution of various finite elements and finite differences discretizations of differential equations.

Planning model for rural distribution networks with integrated energy stations via robust optimization method

With the modernization and intelligent development of agriculture, the energy demand in rural areas continues to increase, which leads to an increased operational burden on the existing rural distribution network. Integrated energy stations (IESs), in rural areas, use renewable energy sources such as biogas, wind power, and photovoltaic as energy inputs, which can fully improve energy efficiency and help reduce the operating load and peak valley difference of rural distribution networks. In this paper, a multistage planning model is proposed for rural distribution networks with IESs based on the robust optimization method. Firstly, a rural distribution network operation framework with IESs is proposed, and a mathematical model of rural IESs is built based on the energy hub (EH). Then, the multistage robust planning model of rural distribution net-works with IESs is developed and typical scenarios of stochastic source and load are generated based on improved k-means. An iterative solution method for a two-stage robust optimization method is proposed based on the nested column constraint generation (NC&CG) algorithm. Finally, the effectiveness of the presented model and solution approach is assessed through case studies on a modified IEEE 33-node distribution network and a real 152-node distribution network.

Solving a nonlinear problem for a one-sided dynamically loaded sliding thrust bearing

The purpose of this study is to propose an efficient numerical method for solving the inverse nonlinear problem of the movement of the compressor rotor collar in a fluid film thrust bearing. Methods. A periodic thermoelastohydrodynamic (PTEHD) mathematical model of hydrodynamic and thermal processes in a bearing is constructed under the condition of the rotor collar motion. Within the framework of the model, an inverse nonlinear problem of determining the position of the collar under a given external load is formulated. An iterative solution method is proposed, which utilizes the solution of the direct problem. To reduce computational costs, a modified Dekker–Brent method is employed in conjunction with a modified Newton’s method. Results. Numerical experiments have been conducted, demonstrating the effectiveness of the proposed approaches. The suggested methods significantly reduce the required computational resources by minimizing the number of calls to the target function in the optimization problem. A software suite has been developed that allows for the calculation of the nonlinear system of rotor motion under various physical and geometric parameters. Conclusion. An efficient set of numerical methods for solving the inverse nonlinear problem of the motion of the rotor collar in the compressor fluid film thrust bearing is proposed. The method’s effectiveness lies in substantial savings of computational resources. The method’s efficiency has been demonstrated in numerical experiments.

A split iterative asymptotic method for the numerical solution of a class of fractional heat transfer equations

In this paper, a new split iterative compact difference scheme for a class of system is constructed. Then, the conservation properties of the scheme are discussed, and the convergence of the split iterative difference scheme is analyzed by using the discrete energy method on the basis of the prior estimation. Finally, numerical experiments verify these properties of the new scheme. In addition, the numerical results also show the influence of fractional derivative on the variation of the transport equation.

Gauss-Legendre type quadrature iterative method for fuzzy Fredholm integral equations

In this paper we obtain an iterative method for the numerical solution of second kind fuzzy Fredholm integral equations by applying appropriate composite fuzzy quadrature formula. We approach a general fuzzy quadrature formula and show that the three-point and four-point Gauss-Legendre type fuzzy quadrature formulas are the best choice from computational cost and accuracy point of view. The convergence of the method is proved by providing the error estimates expressed in terms of the Lipschitz constant of the Picard iterations. Some numerical results confirm the obtained theoretical result and illustrate the performances of the three point Gauss-Legendre method in comparison with the techniques generated by the fuzzy Newton-Cotes and Gauss-Lobatto quadrature formulas, the results being tested on the Simpson and on the four point Gauss-Legendre and Gauss-Lobatto quadrature formulas in particular. The numerical experiment suggests that five point or more point fuzzy quadrature formulas are useless due to the accumulation of errors and the grown computational cost.

APPLICATION OF SINGLE STEP EULER AND TRAPEZOID METHODS OF SOLVING TRANSIENT DISTRIBUTION IN MARKOV CHAIN

The computation of state probability distributions at an arbitrary point in time, which in the case of a discrete-time Markov chain means finding the distribution at some arbitrary time step n denoted π^((n) ), a row vector whose i^th component is the probability that the Markov chain is in state i at time step n, and this is the iterative solution methods for transient distribution in Markov chain. In this study, the solutions of transient distribution in Markov chain using Euler and trapezoid methods have been investigated, in order to provide some insight into the solutions of transient distribution in Markov chain, which produce a significantly more accurate response in less time for some types of situations and also tries to get to the end result as quickly as possible with the solution conform with a specified number of well-defined stages is computed for large space Markov chain. Matrices operations, such as the product and matrix inversion are performed and Markov chain laws, theorems, formulas with MATLAB software are utilized. For illustrative examples, the transient distribution vector's π_((i+1) ),i=0,1,2,…, ; is computed for both Euler and Trapezoid methods and their corresponding error when compared with different method. The effect of decreasing the step size using the same infinitesimal generator is examined, also, by taking π_((0) )=(■(1&0&0)) and the length of the interval of integration to be τ = 1, it is observed that the accuracy achieved with the explicit Euler method has a direct relationship with the step size. It is also concluded that the much more accurate results are obtained with the trapezoid method.

Equitable anesthesiologist scheduling under demand uncertainty using multiobjective programming

This work addresses the practical anesthesiologist scheduling (AS) problem motivated by the needs of an academic anesthesiology department. The AS problem requires the department to plan and deploy providers to adequately meet clinical demand and institutional protocols of various clinical units over a planning horizon of up to several weeks. A data‐driven two‐step AS framework is developed by exploiting the historical demand data of anesthesia cases. The first step is a shift design which obtains the optimal shifts considering clinical demand under uncertainty using conditional value‐at‐risk constraints, and the second step is provider assignments that generate the schedule considering optimal and equitable workload distribution and provider availability using multiobjective mixed‐integer programming models. Moreover, the AS framework incorporates the provider specialties, and clinical and lifestyle preferences and aligns with the existing scheduling practices. An ɛ‐constraint solution method is applied for multiobjective optimization, and an iterative solution method is developed to improve solution quality for workload equity in clinical applications. Computational experiments are performed to evaluate the performance of three alternative forms of the workload equity objective function, and the results show that the minimization of the sum of the absolute deviations of provider workloads best balances solution runtime and quality. In the concerned academic anesthesiology department, two clinical problems, the budget and hiring planning and the monthly scheduling, are addressed via the application of the proposed AS framework. For budget and hiring, decision‐makers can make trade‐offs based on their preference using the nondominated frontiers obtained via the ɛ‐constraint method. For monthly scheduling, the iterative solution method can accommodate preassigned shifts capturing institutional requirements while improving workload equity. The workload variance has been substantially reduced from 2.92 to 1.39 after the implementation based on the historical schedule data. The provider schedule satisfaction is improved from 3.13/5 to 3.44/5, and at least 82% of scheduling burden on department leaders is relieved. The developed AS framework is generic and can be extended to the scheduling of other types of care providers, including nurses and residents.

Learning adaptive coarse basis functions of FETI-DP

Domain decomposition methods are successful and highly parallel scalable iterative solution methods for discretized partial differential equations. Nevertheless, for many classes of problems, for example, elliptic partial differential equations with arbitrary coefficient distributions, adaptive coarse spaces are necessary to obtain robustness or, in other words, to guarantee a reliable and fast convergence. Adaptive coarse spaces are usually computed by solving many localized eigenvalue problems related to edges or faces of the domain decomposition. This results in a computationally expensive setup of the domain decomposition preconditioner or system operator. In earlier work, to which the authors have contributed, a deep learning based classification model was used to classify the critical edges or faces where eigenvalue problems have to be solved. In the present paper, we suggest to directly learn the adaptive constraints using a deep feedforward neural network regression model and thus completely skip the computationally most expensive part of the setup, i.e., the solution of local eigenvalue problems. We consider a specific adaptive FETI-DP (Finite Tearing and Interconnecting – Dual Primal) approach and concentrate on stationary diffusion problems in two dimensions with arbitrary coefficient functions with large jumps. As an input for the neural network, we use an image representation of the coefficient function which resolves the structure of the coefficient distribution but is not necessarily identical to the discretization of the partial differential equation. Therefore, our approach is independent of the finite element mesh and can, in principle, be easily extended to other adaptive coarse spaces, problems, and domain decomposition methods. We show the robustness of our method for different problems and the generalization property of our trained neural networks by considering different coefficient distributions not contained in the training set. We also combine the learned constraints with computationally cheap frugal constraints to further improve our approach.

On the Iterative Methods for the Solution of Three Types of Nonlinear Matrix Equations

In this paper, we investigate the iterative methods for the solution of different types of nonlinear matrix equations. More specifically, we consider iterative methods for the minimal nonnegative solution of a set of Riccati equations, a nonnegative solution of a quadratic matrix equation, and the maximal positive definite solution of the equation X+A∗X−1A=Q. We study the recent iterative methods for computing the solution to the above specific type of equations and propose more effective modifications of these iterative methods. In addition, we make comments and comparisons of the existing methods and show the effectiveness of our methods by illustration examples.