Search Results (29)

In this paper, a pattern for visualizing fractals, namely Julia and Mandelbrot sets for complex functions of the form $T\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }\mathrm{for}\mathrm{all}u\in \mathbb{C}$ and $a\in \mathbb{N}\setminus \left\{1\right\}$, $\xi \in \mathbb{C}$, $r,\rho \in \mathbb{C}\setminus \left\{0\right\}$ are created using novel fast convergent iterative techniques. The new iteration scheme discussed in this study uses s-convexity and improves earlier approaches, including the Mann and Picard–Mann schemes. Further, the proposed approach is amplified by unique escape conditions that regulate the convergence behavior and generate Julia and Mandelbrot sets. This new technique allows greater versatility in fractal design, influencing the shape, size, and aesthetic structure of the designs created. By modifying various parameters in the suggested scheme, a significant number of visually interesting fractals can be generated and evaluated. Furthermore, we provide numerical examples and graphic demonstrations to demonstrate the efficiency of this novel technique. Full article

Massive multiple input–multiple output (M-MIMO) is a promising and pivotal technology in contemporary wireless communication systems that can effectively enhance link reliability and data throughput, especially in uplink scenarios. Even so, the receiving end requires more computational complexity to reconstitute the signal. This problem has emerged in fourth-generation (4G) MIMO system; with the dramatic increase in demand for devices and data in beyond-5G (B5G) systems, this issue will become yet more obvious. To take into account both complexity and signal-revested capability at the receiver, this study uses the matrix iteration method to avoid the staggering amount of operations produced by the inverse matrix. Then, we propose a highly efficient multi-user detector (MUD) named hybrid SOR-based Chebyshev acceleration (CHSOR) for the uplink of M-MIMO orthogonal frequency-division multiplexing (OFDM) and universal filtered multi-carrier (UFMC) waveforms, which can be promoted to B5G developments. The proposed CHSOR scheme includes two stages: the first consists of successive over-relaxation (SOR) and modified successive over-relaxation (MSOR), combining the advantages of low complexity of both and generating a better initial transmission symbol, iteration matrix, and parameters for the next stage; sequentially, the second stage adopts the low-cost iterative Chebyshev acceleration method for performance refinement to obtain a lower bit error rate (BER). Under constrained evaluation settings, Section (Simulation Results and Discussion) presents the results of simulations performed in MATLAB version R2022a. Results show that the proposed detector can achieve a 91.624% improvement in BER performance compared with Chebyshev successive over-relaxation (CSOR). This is very near to the performance of the minimum mean square error (MMSE) detector and is achieved in only a few iterations. In summary, our proposed CHSOR scheme demonstrates fast convergence compared to previous works and as such possesses excellent BER and complexity performance, making it a competitive solution for uplink M-MIMO B5G systems. Full article

In this paper, we introduce a fast iterative scheme and establish its convergence under a contractive condition. This new scheme can be viewed as an extension and generalization of existing iterative schemes such as Picard–Noor and UO iterative schemes for solving nonlinear equations. We demonstrate theoretically and numerically that the new scheme converges faster than several existing iterative schemes with the fastest known convergence rates for contractive mappings. We also analyze the stability of the new scheme and provide numerical computations to validate the analytic results. Finally, we implement the new scheme in MATLAB R2023b to simulate the dynamics of the Ebola virus disease. Full article

Eigenvalue problems play a fundamental role in many scientific and engineering disciplines, including structural mechanics, quantum physics, and control theory. In this paper, we propose a fast and stable fractional-order parallel algorithm for solving eigenvalue problems. The method is implemented within a parallel computing framework, allowing simultaneous computations across multiple processors to improve both efficiency and reliability. A theoretical convergence analysis shows that the scheme achieves a local convergence order of $6\kappa +4$, where $\kappa \in (0,1]$ denotes the Caputo fractional order prescribing the memory depth of the derivative term. Comparative evaluations based on memory utilization, residual error, CPU time, and iteration count demonstrate that the proposed parallel scheme outperforms existing methods in our test cases, exhibiting faster convergence and greater efficiency. These results highlight the method’s robustness and scalability for large-scale eigenvalue computations. Full article

This paper presents a dynamic homotopy technique that can be used to calculate a preliminary result for a power flow problem (PFP). This result can then be used as an initial estimate to efficiently solve the PFP using either the classical Newton-Raphson (NR) method or its fast decoupled version (FDXB) while still maintaining high accuracy. The preliminary stage for the dynamic homotopy problem is formulated and solved by employing integration techniques, where implicit and explicit schemes are studied. The dynamic problem assumes an initial condition that coincides with the initial estimate for a traditional iterative method such as NR. In this sense, the initial guess for the FPF is adequately set as a flat start, which is a starting for the case when this initialization is of difficult assignment for convergence. The static homotopy method requires a complete solution of a PFP per homotopy pathway point, while the dynamic homotopy is based on numerical integration methods. This approach can require only one LU factorization at each point of the pathway. Allocating these points properly helps avoid several PFP resolutions to build the pathway. The hybrid technique was evaluated for large-scale systems with poor conditioning, such as a 109,272-bus model and other test systems under stressed conditions. A scheme based on the implicit backward Euler scheme demonstrated the best performance among other numerical solvers studied. It provided reliable partial results for the dynamic homotopy problem, which proved to be suitable for achieving fast and highly accurate solutions using both the NR and FDXB solvers. Full article

Waveform design is a crucial factor in electronic surveillance (ES) systems. In this paper, we introduce an algorithm that designs a low probability of intercept (LPI) radar waveform. Our approach directly minimizes the detection probability of summation detectors based on FFT filter banks. The algorithm is derived from the general quadratic optimization framework, which inherits the monotonic properties of such methods. To expedite overall convergence, we have integrated acceleration schemes based on the squared iterative method (SQUAREM). Additionally, the proposed algorithm can be executed through fast Fourier transform (FFT) operations, enhancing computational efficiency. With some modifications, the algorithm can be adjusted to incorporate spectral constraints, increasing its flexibility. Numerical experiments indicate that our proposed algorithm outperforms existing ones in terms of both intercept properties and computational complexity. Full article

To better simulate the prices of underlying assets and improve the accuracy of pricing financial derivatives, an increasing number of new models are being proposed. Among them, the Lévy process with jumps has received increasing attention because of its capacity to model sudden movements in asset prices. This paper explores the Hamilton–Jacobi–Bellman (HJB) equation with a fractional derivative and an integro-differential operator, which arise in the valuation of American options and stock loans based on the Lévy-$\alpha$-stable process with jumps model. We design a fast solution strategy that includes the policy iteration method, Krylov subspace method, and banded preconditioner, aiming to solve this equation rapidly. To solve the resulting HJB equation, a finite difference method including an upwind scheme, shifted Grünwald approximation, and trapezoidal method is developed with stability and convergence analysis. Then, an algorithmic framework involving the policy iteration method and the Krylov subspace method is employed. To improve the performance of the above solver, a banded preconditioner is proposed with condition number analysis. Finally, two examples, sugar option pricing and stock loan valuation, are provided to illustrate the effectiveness of the considered model and the efficiency of the proposed preconditioned policy–Krylov subspace method. Full article

Conventional maximum likelihood-based algorithms for 3D Compton image reconstruction are often stuck with slow convergence and large data volume, which could be unsuitable for some practical applications, such as nuclear engineering. Taking advantage of the Bayesian framework, we propose a fast-converging iterative maximum a posteriori reconstruction algorithm under the assumption of the Poisson data model and Markov random field-based convex prior in this paper. The main originality resides in developing a new iterative maximization scheme with simultaneous updates following the line search strategy to bypass the spatial dependencies among neighboring voxels. Numerical experiments on real datasets conducted with hand-held Temporal Compton cameras developed by Damavan Imaging company and punctual 0.2 MBq ²²Na sources with zero-mean Gaussian Markov random field confirm the outperformance of the proposed maximum a posteriori algorithm over various existing expectation–maximization type solutions. Full article

Recently, when modeling transient problems of conjugate heat transfer, the independent construction of grid models for fluid and solid subdomains is increasingly being used. Such grid models, as a rule, are unmatched and require the development of special grid interfaces that match the heat fluxes at the interface. Currently, the most common sequential approach to modeling problems of conjugate heat transfer requires the iterative matching of boundary conditions, which can significantly slow down the process of the convergence of the solution in the case of modeling transient problems with fast processes. The present study is devoted to the development of a direct method for solving conjugate heat transfer problems on grid models consisting of inconsistent grid fragments on adjacent boundaries in which, in the general case, the number and location of nodes do not coincide. A conservative method for the discretization of the heat transfer equation by the direct method in the region of inconsistent interface boundaries between liquid and solid bodies is proposed. The proposed method for matching heat fluxes at mismatched boundaries is based on the principle of forming matched virtual boundaries, proposed in the GGI (General Grid Interface) method. A description of a numerical scheme is presented, which takes into account the different scales of cells and the sharply different thermophysical properties at the interface between liquid and solid media. An algorithm for constructing a conjugate matrix, the form of matrix coefficients responsible for conjugate heat transfer, and methods for calculating them are described. The operability of the presented method is demonstrated by the example of calculating conjugate heat transfer problems, the grid models of which consist of inconsistent grid fragments. The use of the direct conjugation method makes it possible to effectively solve both stationary and non-stationary problems using inconsistent meshes, without the need to modify them in the conjugation region within a single CFD solver. Full article

For the purpose of improving the scientific nature, reliability, and accuracy of flood forecasting, it is an effective and practical way to construct a flood forecasting scheme and carry out real-time forecasting with consideration of different rain patterns. The technique for rain pattern classification is of great significance in the above-mentioned technical roadmap. With the rapid development of artificial intelligence technologies such as machine learning, it is possible and necessary to apply these new methods to assist rain classification applications. In this research, multiple machine learning methods were adopted to study the time-history distribution characteristics and conduct rain pattern classification from observed rainfall time series data. Firstly, the hourly rainfall data between 2003 and 2021 of 37 rain gauge stations in the Pi River Basin were collected to classify rain patterns based on the universally acknowledged dynamic time warping (DTW) algorithm, and the classifications were treated as the benchmark result. After that, four other machine learning methods, including the Decision Tree (DT), Long- and Short-Term Memory (LSTM) neural network, Light Gradient Boosting Machine (LightGBM), and Support Vector Machine (SVM), were specifically selected to establish classification models and the model performances were compared. By adjusting the sampling size, the influence of different sizes on the classification was analyzed. Intercomparison results indicated that LightGBM achieved the highest accuracy and the fastest training speed, the accuracy and F₁ score were 98.95% and 98.58%, respectively, and the loss function and accuracy converged quickly after only 20 iterations. LSTM and SVM have satisfactory accuracy but relatively low training efficiency, and DT has fast classification speed but relatively low accuracy. With the increase in the sampling size, classification results became stable and more accurate. Besides the higher accuracy, the training efficiency of the four methods was also improved. Full article

In this paper, a new one-parameter class of fixed point iterative method is proposed to approximate the fixed points of contractive type mappings. The presence of an arbitrary parameter in the proposed family increases its interval of convergence. Further, we also propose new two-step and three-step fixed point iterative schemes. We also discuss the stability, strong convergence and fastness of the proposed methods. Furthermore, numerical experiments are performed to check the applicability of the new methods, and these have been compared with well-known similar existing methods in the literature. Full article

In this paper, Chebyshev polynomials—which are ultraspherical in the first and second kind and hence symmetric, while the third and fourth order are not ultraspherical and are hence non-symmetric—are used for the simulation of two-dimensional mass transfer equation arising during the convective air drying processes of food products subject to Robin and Neumann boundary conditions. These simulations are used to improve the quality of dried food products and for prediction of the moisture distributions. The equation is discretized in both temporal and special variables by using the second order finite difference scheme and spectral method based on Chebyshev polynomial with the help of fast Fourier transform on tensor product grid, respectively. A system of algebraic equations is obtained after applying the proposed numerical scheme, which is then solved by an appropriate iterative method. The error analysis of the proposed scheme is provided. Some numerical examples are presented to confirm the numerical efficiency and theoretical justification of the proposed scheme. Our numerical scheme has an exponential rate of convergence, which means that one can achieve a very accurate solution using a few collocation points, as opposed to the other available techniques which are very slow in terms of convergence and consume a lot of time. In order to further validate the accuracy of our numerical method, a comparison is made with the exact solution using different norms. Full article

This paper proposes the use of enhanced comprehensive learning particle swarm optimization (ECLPSO), combined with a Gaussian local search (GLS) technique, for the simultaneous optimal size and shape design of truss structures under applied forces and design constraints. The ECLPSO approach presents two novel enhancing techniques, namely perturbation-based exploitation and adaptive learning probability, in addition to its distinctive diversity of particles. This prevents the premature convergence of local optimal solutions. In essence, the perturbation enables the robust exploitation in the updating velocity of particles, whilst the learning probabilities are dynamically adjusted by ranking information on the personal best particles. Based on the results given by ECLPSO, the GLS technique takes data from the global best particle and personal best particles in the last iteration to generate samples from a Gaussian distribution to improve convergence precision. A combination of these techniques results in the fast convergence and likelihood to obtain the optimal solution. Applications of the combined GLS-ECLPSO method are illustrated through several successfully solved truss examples in two- and three-dimensional spaces. The robustness and accuracy of the proposed scheme are illustrated through comparisons with available benchmarks processed by other meta-heuristic algorithms. All examples show simultaneous optimal size and shape distributions of truss structures complying with limit state design specifications. Full article

For a railway or highway tunnel under high water pressure during operation, various factors such as the design of the drainage system, material aging, and pipeline blockage must be considered for the tunnels to work with the parallel adit to drain and control the external water pressure on the tunnel lining. A simplified steady-state seepage model in a semi-infinite multi-connected domain for the tunnel and parallel adit was established and was solved iteratively using the complex variable method and the Schwartz alternating method. After verifying the numerical simulation, parametric analysis, orthogonal tests, and multivariate nonlinear regression were also carried out. Results show that the simplified theoretical model and its semi-analytical algorithm have a fast convergence speed, and the obtained regression formula is simple, which is suitable for calculation and parameter analysis. A scheme that primarily relies on the parallel adit for drainage would make the external water pressure of the lining facing the parallel adit side less than that of the opposite side. Therefore, to reduce pressure uniformly and meet the requirements of surrounding rock stability, the horizontal net distance between the parallel adit and the tunnel should be no less than the tunnel diameter. Drainage volume of the parallel adit is linearly negatively correlated with tunnel water pressure on the lining and has the most significant effect on pressure reduction. The influence of the vertical distance between the parallel adit and the tunnel on water pressure is small. Full article

Aiming at the problem that the existing alternating direction method of multipliers (ADMM) cannot realize totally distributed computation, a totally distributed improved ADMM algorithm that combines logarithmic barrier function and virtual agent is proposed. We also investigate economic dispatch for microgrids considering demand response based on day-ahead real-time pricing (RTP), which forms a source-load-storage collaborative optimization scheme. First, three general distributed energy sources (DERs), renewable energy resources (RESs), conventional DERs and energy storage systems (ESSs), are considered in the method. Second, the goal of economic dispatch is to minimize the sum of three energy generation costs and implement the optimal power allocation of dispatchable DERs. Specifically, the approach not only inherits the fast computational speed of ADMM but also uses barrier function and virtual agent to handle inequality and equality, respectively. Moreover, the approach requires no coordination center and only the communication between current agent and adjacent agent to achieve totally distributed solution for every iteration, which can preserve information privacy well. Finally, a 30-node microgrid system is used for case analysis, and the simulation results demonstrate the feasibility and effectiveness of the proposed approach. It can be found that, the proposed approach converges to the optima when $p$ = 0.01, $v$ = 100, $t^0$ = 0.01 and $\mu$ = 2. Full article