We develop and analyze an algorithm for the generalized Darcy–Forchheimer equations in porous media, combining a two-level mixed finite element discretization with a backtracking correction step. The governing operator is G(u) =μ/ρK^-1u+β/ρ|u|^m-1u with exponent m ∈ (1,2] , which recovers the classical quadratic Forchheimer law at m = 2 and captures sub-quadratic inertial corrections for m < 2 . The algorithm proceeds in three steps: a nonlinear solve on a coarse mesh 𝒯_H , a linearized solve on a fine mesh 𝒯_h , and a backtracking correction on 𝒯_H that refines the coarse-grid solution using second-order information from the fine-grid residual. We establish well-posedness of all three discrete problems and derive a priori error estimates for the velocity in L^m(Ω ) and the pressure gradient in L^(m+1)/m(Ω ) . Under the mesh-size and regularization balancing conditions h = O(H^2) and ε = O(h^m/2(m-1)) , the corrected two-level approximation achieves the same first-order accuracy as the fine-grid solution. Explicit stability and convergence bounds are derived for both the velocity and pressure corrections, showing that the correction step introduces no degradation in the overall convergence rate. Numerical experiments for m = 2 , 3/2 , and 6/5 confirm the theoretical rates, extend the two-level analysis to the deep sub-quadratic Darcy–Forchheimer regime for the first time, and demonstrate that the backtracking correction reduces errors consistently at negligible additional computational cost.
We propose a scale-normalized Gaussian–wavelet algorithm for zero localization in oscillatory functions. Unlike pointwise sign-change or residual-based strategies, the proposed approach identifies candidate zeros as minimizers of an integral multiscale functional that couples local amplitude and derivative-sensitive variation. Under standard smoothness assumptions, we prove a local consistency result: near an isolated simple zero, minimizers converge to the true zero as the scale parameter tends to zero. Under an additional structural assumption, we obtain a conditional second-order localization estimate. Numerical experiments on representative oscillatory functions, including the Riemann Ξ -function and residual functions arising from Sturm–Liouville eigenvalue problems, indicate consistent localization behavior in the tested noisy sampling regimes. These results support the proposed method as a practical preprocessing stage for basin identification and initialization of classical root-finding methods when reliable initialization is not available.
The approximation of complex-valued functions is of fundamental importance as it provides a framework for amplitude and phase-dependent phenomena. In this paper, we propose a family of complex rational interpolation operators to approximate analytic as well as non-analytic functions along with real-life application. In this direction, the first operator is constructed using Chebyshev polynomials of the first kind, namely classical complex rational interpolation operators for approximating complex-valued continuous functions. We establish approximation results for these operators utilizing the notion of the modulus of continuity. To approximate not-necessarily analytic but integrable function, we define Kantorovich-type complex rational interpolation operators and establish their boundedness and convergence. Furthermore, in order to approximate functions preserving higher derivatives, we introduce Hermite-type complex rational interpolation operators and study their approximation capabilities using higher order of modulus of continuity. To validate the theoretical results, we provide numerical simulations illustrating approximation ability of proposed family of complex rational interpolation operators. Moreover, we present the utility of our construction in denoising of signals corrupted by Gaussian noise. In addition, we demonstrate its real-life application in brain image reconstruction by applying these operators to real brain imaging data, particularly magnetic resonance imaging (MRI) dataset. The algorithms are provided at the end to formally explain the reconstruction procedure of the proposed family of complex rational interpolation operators.
Let X and Y be reflexive strictly convex Banach spaces, and let A: X→ Y be a bounded linear operator with a closed range ℛ(A) . In this paper, utilizing the geometric properties of Banach spaces in conjunction with duality mappings, under suitable quasi-additivity assumptions, we propose an efficient parametric iterative algorithm for computing the metric projection operators π _𝒩(A) and π _ℛ(A) , where 𝒩(A) denotes the kernel of the operator A . We conduct a sensitivity analysis of the parameter to assess its impact on convergence. The convergence properties of the proposed approach are thoroughly examined, yielding first and second order convergence rates. Additionally, we provide detailed error estimates to quantify the performance of the proposed methods. The theoretical results are validated through numerical examples that demonstrate the effectiveness of the proposed methods. Furthermore, we explore some applications of the iterative algorithm in relevant contexts, including the representation of the Moore–Penrose metric generalized inverses and the resolution of two types of approximation solution problems. To the best of our knowledge, this study may represent the first instance in the literature of an iterative algorithm specifically designed to compute nonlinear metric projections associated with Banach space operators. The approaches and findings presented in this paper provide novel insights into the computation of nonlinear metric projection operators.
The Ramshaw–Mesina hybrid regularization combines pressure penalty and artificial compression for the incompressible Navier–Stokes equations but requires careful parameter tuning. We develop a self-adaptive strategy that automatically adjusts both parameters at each time step from computable solution indicators. A key mechanism is a lower bound on the hybridization ratio: when it exceeds the initial ratio, it acts as a cross-coupling amplifier, propagating divergence-driven increases in the penalty parameter to the compression parameter simultaneously, and it automatically restores the Ramshaw–Mesina calibration scaling α ^2 = 𝒪(Δ t^-2) , β = 𝒪(Δ t^-1) under which the method is first-order accurate in time. We prove unconditional stability and error estimates with complete proofs, showing that time-varying parameters perturb neither the spatial nor the temporal convergence rates. Numerical experiments demonstrate that the adaptive method reduces divergence error by up to nearly two orders of magnitude relative to untuned parameters — for example, from 8.1× 10^-3 to 1.1× 10^-4 starting from the severely untuned pair (α ^2_0,β _0)=(10,1) — and achieves two to three orders of magnitude improvement over penalty-only or compression-only methods, confirming the landmark Ramshaw–Mesina result. The adaptation constants themselves require no tuning: a sweep over their values shows only mild sensitivity. Tests include convergence studies on structured and unstructured meshes, a Reynolds-number study up to Re=2000 , and lid-driven cavity benchmarks in two and three dimensions.
This paper focuses on solving dual quaternion Sylvester matrix equations and proposes two global Arnoldi-based iterative methods, namely the global dual quaternion full orthogonalization method and the global dual quaternion generalized minimum residual method. Both methods are constructed based on the theories of global orthogonal projection and oblique projection on the Krylov subspace of dual quaternion matrices. To support the implementation of the proposed methods, we develop the global dual quaternion Arnoldi process and the generalized dual quaternion Givens rotation method, and complete the convergence analysis of the proposed algorithms. Numerical examples verify the effectiveness of the proposed algorithms, and their application to color image encryption and decryption preliminarily illustrates their potential practical value.
In this paper, we propose a new method for solving the Stokes interface problem in a convex polyhedral domain Ω⊂ℝ^d (d=2, 3) with smooth, radially parameterizable interfaces of the form r = f(θ _1, θ _2) (or r = f(θ ) in 2D). The method employs a diffeomorphic geometric transformation that maps curved, smooth interfaces exactly onto flat geometries, thereby eliminating discontinuities in the viscosity coefficients across the interface. Two novel algorithms are provided to explicitly construct such transformations for three-dimensional radially parameterizable smooth interfaces. The proposed approach is simple, robust, and computationally efficient. Furthermore, we establish optimal a priori error estimates in the L^2 , H^1 -norms, and quasi-optimal error bound in the L^∞ -norm, demonstrating both the theoretical accuracy and practical effectiveness of the method. Numerical experiments confirm the accuracy and efficiency of the geometric approach in addressing complex interface problems.
Due to the complexity of the Hamilton–Jacobi–Bellman (HJB) equation, obtaining an exact analytical solution is often difficult or impossible. A new strategy is proposed in this paper to solve the partial differential HJB equation that appears in the nonlinear optimal control problem. The optimal homotopy asymptotic method (OHAM) is employed to compute the approximate solution of the HJB equation. Unlike previous techniques, such as the homotopy perturbation method (HPM), the proposed OHAM strategy offers a flexible systematic structure, where the convergence rate can be adjusted to achieve a faster response with appropriate accuracy, requiring minimal repetition and computational work. The rate of convergence and the convergence zone in this algorithm are adjustable by tuning the auxiliary constants. The proposed technique can be implemented to achieve optimal control policies for various nonlinear systems and for real-time control applications and practical implementations. The capacities of the presented structure provide a more efficient alternative to existing semi-analytical techniques for solving complex nonlinear HJB optimal control problems. We validate the effectiveness and simplicity of the presented scheme by providing several numerical examples.
In this paper, we investigate the convergence domains of an inverse-free third-order Ulm’s method under weaker continuity conditions for solving nonlinear equations. Recurrence relations are used to derive the convergence estimates and establish the existence and uniqueness theorem. The present approach provides sufficient conditions for cubic convergence, where the existing results involving the Lipschitz condition are not applicable. A comparative study of the basins of attractions of the family shows that a third-order Ulm’s method offers broader stability and accessibility. Theoretical results are validated through various numerical examples involving Hammerstein-type integral equations, a boundary value ordinary differential equation, and an elliptic partial differential equation with nonlinear boundary conditions. Further, we apply third-order inverse-free Ulm’s method to nonlinear equations to obtain the numerical solution and to demonstrate that the numerical solution remains within the predicted convergence domains.
GMRES is known to determine a least squares (LS) solution of A x= b where A ∈R^n × n without breakdown for arbitrary b∈R^n , and initial iterate x_0 ∈R^n if and only if A is range-symmetric, i.e. ℛ(A^T) = ℛ(A) holds, where A may be singular and b may not be in the range space ℛ(A) of A . On the other hand, when ℛ(A^T) ℛ(A) , including the case when the index of A is greater than or equal to 1, there exist x_0 and b such that GMRES breaks down without giving a LS solution. To solve this difficulty, we propose applying the Range Restricted GMRES (RRGMRES) to A C A^Tz= b , where C ∈R^n × n is symmetric positive definite (SPD). The proposed RRGMRES robustly determines a least squares solution x= CA^Tz of A x= b without breakdown for arbitrary (singular) matrix A ∈R^n × n and b, x_0 ∈R^n since A C A^T is symmetric even if ℛ(A^T) ℛ(A) . In particular, we prove that C ∈R^n × n for NR-SSOR right preconditioner is SPD and propose the NR-SSOR right preconditioned RRGMRES, which also works efficiently for rectangular (rank-deficient) least squares problems min_x∈R^n‖b- A x‖ _2 for A ∈R^m × n and arbitrary b∈R^m . We also propose applying NR-SSOR right preconditioning to Range Restricted MINRES (RRMINRES). Numerical experiments demonstrate the validity of the proposed methods.
This work discusses the application of the moving least squares (MLS) method for the numerical solution of two-dimensional nonlinear fractional integro-differential equations (2D-NFIDEs) with multi-term kernels on both regular and irregular domains. The time derivative and the Riemann-Liouville (R–L) fractional integral are approximated using the Euler backward method and the product trapezoidal integration rule, respectively. The theoretical analysis of the final scheme is carried out to establish its stability condition and to determine the convergence rates in both time and space. Furthermore, we solve several examples to demonstrate the convergence and accuracy of the proposed strategy when moving from regular to complex geometries. Additionally, we compare our method with an existing approach from the literature, showing that our strategy yields superior results.
In this paper, we extend fourth-order alternating direction implicit (ADI) methods to solve high-dimensional wave equations with nonseparable coefficients in each spatial direction. By combining a compact finite difference approximation with the linear θ -method, we derive several ADI algorithms with fourth-order accuracy in space and second-order accuracy in time. For the intermediate solutions generated by the ADI splitting, the required boundary values are derived from the original boundary data and one-sided ghost-point extrapolations that preserve the overall accuracy, so that the compact solvers can be implemented without losing the spatial accuracy. Furthermore, the proposed algorithms are capable of handling nonseparable coefficients depending simultaneously on spatial and temporal variables. A stability analysis is also provided for a representative variable-coefficient problem, which provides theoretical insight into the stability of the proposed schemes. To match the spatial accuracy, Richardson extrapolation is employed to enhance the temporal accuracy. Three-dimensional algorithms are also constructed when coefficients are independent of time. Extensive numerical examples with different values of the parameter θ are carried out to validate the accuracy.
This paper addresses uncertain multiobjective optimization problems by reformulating them into a deterministic form using objective-wise worst-case robust counterparts. To efficiently solve the resulting robust problem, we propose a BFGS-based quasi-Newton descent method combined with Wolfe-type line search strategies. The Wolfe-type inexact line search is employed to compute suitable step sizes, while a modified BFGS update guarantees the positive definiteness of the Hessian approximation at each iteration. We establish the global convergence of the proposed method and show that, under strong convexity assumptions, it achieves an R-linear convergence rate to a Pareto optimal solution. Moreover, for locally strongly convex objective functions with Lipschitz continuous Hessians, the algorithm exhibits Q-superlinear convergence. Numerical experiments compare the proposed method with the Armijo-type quasi-Newton method and the Newton method using performance profiles, demonstrating the efficiency and robustness of the proposed approach.
The aim of this work is twofold: first, to develop a memory-based Kurchatov conformable iterative method to solve nonlinear equations; second, to provide a convergence sphere for global visualization of real dynamics. We establish the R-order of convergence for the proposed method and demonstrate how the convergence sphere maps initial guesses onto a closed surface, allowing for a compact analysis of global stability and basin connectivity. Moreover, this study investigates the root convergence trajectory, the dynamic behavior of the method, numerical results, and the concept of Approximated Order of Convergence (ACOC), all of which support and strengthen the theoretical results.
This paper proposes a theoretical framework to address the reduced biquaternion equality-constrained total least squares (RBTLSE) problem. The objective is to find an approximate solution to the system AX ≈ B , subject to linear constraints CX = D , while explicitly accounting for errors in both the coefficient matrix A and the observation matrix B. We establish conditions under which real, complex, and reduced biquaternion solutions exist and develop corresponding solution methods using real and complex representations of reduced biquaternion matrices. To assess the sensitivity of the solutions to data perturbations, we establish upper bounds for the relative forward errors. These results ensure the computational reliability of the proposed framework. Extensive numerical experiments validate the theoretical findings and demonstrate that the RBTLSE approach significantly outperforms the conventional reduced biquaternion equality-constrained least squares (RBLSE) method, particularly in noisy environments affecting both sides of the system. Furthermore, the proposed framework is applied to color image encryption and decryption to demonstrate its practical effectiveness.
In this manuscript, we develop an efficient numerical scheme for solving time distributed-order fractional integro-differential equations. The proposed method utilizes orthogonal generating polynomials (OGPs) and Chebyshev-polynomial-based operational matrices. First, we construct OGPs associated with a distributed-order weight function defined in the Caputo sense. Using these polynomials, we derive operational matrices corresponding to both the integral and distributed-order fractional derivative terms, incorporating the Legendre–Gauss quadrature rule for numerical integration. The original complex problem is then transformed into a system of linear algebraic equations using the standard tau method, combined with uniformly distributed random number as collocation points to enforce the initial and boundary conditions. The resulting approximate solutions demonstrate the accuracy and efficiency of the proposed approach. Furthermore, we establish error bounds, conduct a detailed convergence analysis, and verify the numerical stability of the scheme. Several illustrative examples are provided to highlight the reliability and computational performance of the method. For comparison, results obtained using the Chebyshev polynomial based operational matrix method are also presented.
This paper considers the distributed solution of large-scale sparse overdetermined linear systems and proposes an event-triggered adaptive accelerated parallel randomized Kaczmarz (ET-AAPRK) algorithm. The method is designed to reduce the communication bottleneck in parallel solvers and to improve the projection efficiency of conventional parallel randomized Kaczmarz methods with fixed relaxation parameters. Within a parallel randomized projection framework, ET-AAPRK combines greedy sampling with an adaptive relaxation strategy. The relaxation parameter is computed online by approximating residual energy minimization over a small set of probe rows, which improves the quality of local projections and reduces numerical oscillations. To avoid unnecessary synchronization, an event-triggered communication rule based on the residual variation rate is introduced. Global communication is performed only when the effectiveness of local projections becomes limited. After synchronization, a residual-driven weighted aggregation strategy is used to combine local iterates. Theoretical analysis shows that, for consistent overdetermined systems with full-column-rank coefficient matrices, ET-AAPRK achieves stable error reduction and local linear convergence in expectation. Numerical experiments were carried out on the Shanhe Supercomputer for four sparse overdetermined systems using 1 to 32 processes. Compared with the parallel randomized Kaczmarz projection (PRKP) algorithm and the parallel conjugate gradient (PCG) method, ET-AAPRK achieves the shortest runtime on the tested problems, with average speedups of 4.15 × and 156.06 × , respectively. The results indicate that the proposed event-triggered and adaptive projection strategies can reduce communication overhead and improve parallel efficiency for the tested large-scale sparse systems.
This paper introduces interpolatory quadrature formulas for definite integrals associated with a measure supported on an arc of the unit circle. Their nodes are the nodes of Szegő quadratures corresponding to a Geronimus measure supported on that arc and their weights are asymptotically positive where the original measure is positive.
In this paper, we introduce a new class of complex functions incorporating an exponential factor and generate its Mandelbrot- and Julia-like sets using the K^* iteration with s-convexity. An escape criterion is established for the combination of the function and iteration scheme and is then implemented in the escape-time algorithm for the visualization of the fractal sets. The dynamical behavior of the resulting fractals is analyzed through graphical and numerical experiments using the average escape time and the non-escaping area index. The results demonstrate that the proposed iteration framework significantly influences the geometry and escape dynamics of the generated sets, leading to distinct structural variations governed by the iteration parameters.
