Abstract: This paper explores an iterative approach to solve linear thermo-poroelasticity problems, with its application as a high-fidelity discretization utilizing finite elements during the training of projection-based reduced order models. One of the main challenges in addressing coupled multi-physics problems is the complexity and computational expenses involved. In this study, we introduce a decoupled iterative solution approach, integrated with reduced order modeling, aimed at augmenting the efficiency of the computational algorithm. The iterative technique we employ builds upon the established fixed-stress splitting scheme that has been extensively investigated for Biot's poroelasticity. By leveraging solutions derived from this coupled iterative scheme, the reduced order model employs an additional Galerkin projection onto a reduced basis space formed by a small number of modes obtained through proper orthogonal decomposition. The effectiveness of the proposed algorithm is demonstrated through numerical experiments, showcasing its computational prowess.

Abstract: This paper presents an iterative planning framework for multi-agent systems with hybrid state spaces. The framework uses transition systems to mathematically represent planning tasks and employs multiple solvers to iteratively improve the plan until computational resources are exhausted. When integrating different solvers for iterative planning, we establish theoretical guarantees for recursive feasibility. The proposed framework enables continual improvement of solutions to reduce sub-optimality, efficiently using allocated computational resources. The proposed method is validated by applying it to an energy-aware UAV-UGV cooperative task site assignment problem. The results demonstrate continual solution improvement while preserving real-time implementation ability compared to algorithms proposed in the literature. Note to Practitioners —This paper presents an iterative planning solution for cooperative planning problems in multi-agent systems, which integrates multiple solvers to create an optimization framework. The proposed planning framework has been theoretically validated and applied in an energy-aware cooperative planning scenario for multi-vehicle task site assignments. The proposed framework can be applied to plan for any generalized task site assignment using multiple solvers iteratively.

Abstract: Global Nash equilibrium is an optimal solution for each player in a graphical game. This paper proposes an iterative adaptive dynamic programming-based algorithm to solve the global Nash equilibrium solution for optimal containment control problem with robustness analysis to the iterative error. The containment control problem is transferred into the graphical game formulation. Sufficient conditions are given to decouple the Hamilton-Jacobi equations, which guarantee the solvability of the global Nash equilibrium solution. The iterative algorithm is designed to obtain the solution without any knowledge of system dynamics. Conditions of iterative error for global stability are given with rigorous proof. Compared with existing works, the design procedures of control gain and coupling strength are separated, which avoids trivial cases in the design procedure. The robustness analysis exactly quantifies the effect of the iterative error caused by various sources in engineering practice. The theoretical results are validated by two numerical examples with marginally stable and unstable dynamics of the leader.

Abstract: In this paper we develop an iterative method to perform single and multiple scattering analyses of both convex and non-convex scatterers at very low computational cost and time. The computational method proposed will resolve the current inaccuracy of the On Surface Radiation Conditions (OSRC) in estimating the scattered field from non-convex obstacles and allows a fast and reliable single- and multiple-scattering analyses in reduced spatial dimensions. We demonstrate the reliability and efficiency of the proposed method in solving two- and three-dimensional time-harmonic acoustic scattering problems. Non-Uniform Rational B-Splines were used to represent the obstacle boundaries providing a seamless connection to geometrical representation. The proposed Iterative On Surface Radiation Conditions (ITOSRC) method tremendously reduces the meshing and analysis time and paves the way for shape and topology optimization of future devices relying on wave prpagation phenomena.

Abstract: We modify the successive overrelaxation (SOR) method and accelerated overrelaxation (AOR) method for solving linear equations systems. The optimal value of the acceleration parameter is determined, using the maximal reduction method of the residual vector’s length, or equivalently an orthogonality condition. Rather than the constant value, the MAOR method endows a step-by-step varying acceleration parameter to possess the property of absolute convergence and the orthogonality of consecutive residual vector. In SOR, the relaxation parameter is also optimized by using the orthogonality condition. Numerical examples ensure that the MSOR and MAOR iterative schemes converge faster than the original SOR and AOR iterative schemes. They are easily implemented with low computational cost, and without needing of a detailed spectral analysis to determine the optimal values of parameters has a great advantage.

Abstract: This study presents some new iterative algorithms based on the gradient method to solve general constrained systems of conjugate transpose matrix equations for both real and complex matrices. In addition, we analyze the convergence properties of these methods and provide numerical techniques to determine the solutions. The effectiveness of the proposed iterative methods is demonstrated through various numerical examples employed in this study.

Abstract: We formulate for the first time the economic dispatch problem among prosumers in an integrated electrical and gas distribution system (IEGDS) as a game equilibrium problem. Specifically, by approximating the nonlinear gas-flow equations either with a mixed-integer second-order cone (MISOC) or a piecewise affine (PWA) model and by assuming that electricity and gas prices depend linearly on the total consumption, we obtain a potential mixed-integer game. To compute an approximate generalized Nash equilibrium (GNE), we propose an iterative two-stage method that exploits a problem convexification and the gas-flow models. We quantify the quality of the computed solution and perform a numerical study to evaluate the performance of our method.

Abstract: In this paper, we present a technique that improves the applicability of the result obtained by Cordero et al. in 2024 for solving nonlinear equations. Cordero et al. assumed the involved operator to be differentiable at least five times to extend a two-step p-order method to order p+3. We obtained the convergence order of Cordero et al.’s method by assuming only up to the third-order derivative of the operator. Our analysis is in a more general commutative Banach algebra setting and provides a radius of the convergence ball. Finally, we validate our theoretical findings with several numerical examples. Also, the concept of basin of attraction is discussed with examples.

Abstract: The symmetric successive overrelaxation (SSOR) and symmetric accelerated overrelaxation (SAOR) are conventional iterative methods for solving linear equations. In this paper, novel approaches are presented by combining a splitting–linearizing method with SSOR and SAOR for solving a system of nonlinear equations. The nonlinear terms are decomposed at two sides through a splitting parameter, which are linearized around the values at the previous step, obtaining a linear equation system at each iteration step. The optimal values of parameters are determined to minimize the reciprocal of the maximal projection, which are sought in preferred ranges using the golden section search algorithm. Numerical tests assess the performance of the developed methods, namely, the optimal splitting symmetric successive over-relaxation (OSSSOR), and the optimal splitting symmetric accelerated over-relaxation (OSSAOR). The chief advantages of the proposed methods are that they do not need to compute the inverse matrix at each iteration step, and the computed orders of convergence by OSSSOR and OSSAOR are between 1.5 and 5.61; they, without needing the inner iterations loop, converge very fast with saving CPU time to find the true solution with a high accuracy.

Abstract: The behavior of connected and autonomous vehicles (CAVs) in traffic environments is very complex. Giving efficient cooperative driving strategies in traffic intersection scenarios is still a challenge, especially at roundabout intersections. In this article, we propose a decision making and motion planning method for roundabouts by an iterative learning-based optimization. Time-optimal safe driving strategies are given in a hybrid architecture. Based on the crossing mode modeling, a data-efficient learning-based iterative optimization (DELIO) method for offline trajectory generation is proposed. We build the optimization problem by introducing stage and terminal costs and describing the static and dynamic constraints using support vector machine (SVM) method. Iterative safety constraints are constructed to store historical data for closed-loop efficient data-driven learning. The algorithm finally converges to a time-optimal collision-free trajectory only after several iterations. In order to rationally apply the optimal trajectories, a multivehicle cooperative decision making and motion planning method based on adaptive Monte Carlo tree search (AMCTS) is proposed. The algorithm eliminates intergroup conflicts and quickly converges to the decision sequence of the optimal grouping. We validate our method in a typical roundabout intersection scenario. Results show that our method enables vehicles to pass through the roundabout intersection efficiently and safely, which significantly improves the traffic efficiency.

Abstract: In this paper, we present a three-step sixth-order iterative schemes to estimate the solutions of a nonlinear systems of equations, for predicting the shear strength of a reinforced concrete beam. This procedure is designed by means of a weight function technique. The values for the parameters of this system were randomly selected inside the prescribed ranges by technical standards for structural concrete; moreover, some of this parameters were fixed taking into consideration the solvability region of the adopted steel constitutive model. The efficiency of the new class is compared with other current schemes, with very good results.

Abstract: Passive position determination of the radiating source is widely applied in wireless sensor networks (WSN). Compared with traditional two-step methods, the direct position determination (DPD) methods estimate the position in a single step and exhibit conspicuous performance superiority at a low signal-to-noise ratio (SNR). Unfortunately, the computational load for the exhaustive maximum likelihood (ML) DPD search is unacceptable for real-time processing. Although the conventional gradient-based iterative DPD method has lower computational complexity, it easily converges to the local maxima. In this paper, for a stationary source, we aim to tackle the problem of DPD using Doppler frequency shifts in a computationally attractive way. The waveform of the transmitted signal is a priori knowledge, and the transmitted frequency offset exists as the nuisance parameter. Based on the expectation maximization (EM) concept, we propose an approximately globally-optimal iterative ML DPD algorithm. The proposed method regards the source-receiver vector as the unobserved latent variable, and then by exploiting the Laplace approximation technique and polar coordinate system, the original multi-dimensional grid search is transformed into multiple one-dimensional estimations for polar angle. To guarantee global optimality for the estimation of transmitted frequency offsets, a method different from its counterpart in the iterative cycle is utilized when the local maxima are achieved. Moreover, by correctly simplifying the estimation cost function in the expectation step, both the search of the polar angle and the design of the iteration parameter is avoided, which contributes to prominent computational savings. Simulation results demonstrate that the proposed modified EM-based DPD estimator (MEM-DPD) has high probability of converging to the global maximum. Meanwhile, the estimation performance is very close to the Cramér-Rao lower bound (CRLB) with minor iterations.

Abstract: In this paper, based on the extrapolated Landweber‐type operators, we present new strongly convergent block‐iterative schemes for solving the split common fixed point problem (SCFPP) with demiclosed strongly quasi‐nonexpansive operators on Hilbert spaces. The strong convergence is proved without the additional assumptions such as the boundedly regular condition and the closedness property of the range of the transformation operator, assumed recently in the literature for the problem. A necessary and sufficient condition that ensures that a th iterate is a solution is given. An application of our results to solve the multiple‐sets split convex feasibility problem (MSSCFP) is showed with computational experiments for illustration.

Abstract: This letter proposes a quaternion projection gradient ascent (QPGA) iterative algorithm based on generalized $\mathbb {HR}$ calculus for computing the principal eigenvalues and its eigenvectors of quaternion Hermitian matrices. We also prove the convergence of the QPGA algorithm, demonstrating that the estimated sequence of principal eigenvalues is monotonically increasing. Numerical experiments demonstrate the superiority of the proposed iterative method over traditional algebraic methods in terms of accuracy and speed, as well as the application of principal eigenvalues and their eigenvectors obtained by the QPGA algorithm in denoising with quaternion principal component analysis and quaternion least mean square (QLMS) algorithms in filtering fetal electrocardiograms. Overall, the fast quaternion eigenvalue solving method provides a novel and effective technical tool for quaternion signal processing.

Abstract: In this article, we introduce a novel three-step iterative algorithm with memory for finding the roots of nonlinear equations. The convergence order of an established eighth-order iterative method is elevated by transforming it into a with-memory variant. The improvement in the convergence order is achieved by introducing two self-accelerating parameters, calculated using the Hermite interpolating polynomial. As a result, the R-order of convergence for the proposed bi-parametric with-memory iterative algorithm is enhanced from 8 to 10.5208. Notably, this enhancement in the convergence order is accomplished without the need for extra function evaluations. Moreover, the efficiency index of the newly proposed with-memory iterative algorithm improves from 1.5157 to 1.6011. Extensive numerical testing across various problems confirms the usefulness and superior performance of the presented algorithm relative to some well-known existing algorithms.

Abstract: The objective of this paper is to suggest and construct a general four step semi-implicit iterative scheme involving midpoint rule. We establish and analyse the convergence of the suggested scheme to reckon the fixed point of an almost contraction mapping. Also, we prove the stability of the suggested iterative scheme. Finally, suggested method and our main results are applied to examine a general variational inequality and a nonlinear integral equation.

Abstract: A noniterative and stable microwave non-resonant method is devised for accurate permittivity $\varepsilon _{r}$ extraction of solid samples without the need for information of reference plane and sample thickness $L $ . The method first evaluates reference plane transformation distances $L_{01}$ and $L_{02}$ and then calculates L using the location of first peaks (but not their values) of the magnitudes of forward and backward reflection scattering (S-) parameters in the time domain. Then, this information is used for noniterative and stable determination of $\varepsilon _{r}$ in the frequency domain. Accuracy and uncertainty analyses are performed to examine the performance of the method. Permittivities of acrylonitrile butadiene styrene and polyethylene dielectric samples were measured by coaxial line (30 kHz–26.5 GHz) and free-space (8.2–12.4 GHz) measurements.

Abstract: Multiple connected regions bounded by circles are crucial from the point of view of analyzing physical problems and reducing the amount of computation. However, finding a conformal mapping function that maps a multiple connected region to circular domain is challenging. Koebe’s iterative method provides a theoretically feasible path for the conformal mapping of a multiple connected region to a circular region. In this study, a numerical implementation of Koebe’s iterative method is accomplished using the charge simulation method, and an algorithm for conformal circular mapping of unbounded multiple connected regions is proposed. Through numerical experiments, this paper successfully verifies the effectiveness of the proposed algorithm.

Abstract: : This paper presents an innovative Iterative Learning Control (ILC) strategy for Linear Time-Varying (LTV) systems subject to uncertainties. In a real-world environment, implementing ILC causes the uncertainties to vary concerning both time and iteration. To address this challenge, we introduce a metric to quantify the impact of the uncertainties on the tracking error’s variation. First, an equivalent 2D Roesser model is established for the uncertain ILC system. It has uncertain parameters and is subject to an external disturbance caused by the time-varying model uncertainties of the original system. Then, a Linear Matrix Inequality (LMI) condition is proposed to design the ILC law to provide an upper metric bound. The strategy aims to lower this bound, thereby reducing the impact of uncertainties on the system. Finally, preliminary numerical simulation verifies the effectiveness and robustness of the proposed strategy.