Fixed-point Iteration Research Articles

Gaussian-inverse gamma mixture distribution–based extended Kalman observer for dynamic positioning control system of offshore platform

Under different working conditions and control objectives, fuzzy model predictive control is used to optimize the control effect by adjusting the weight to achieve stable and accurate control for the dynamic positioning control of offshore platforms. Also, aiming at the situation where the measurement noise of dynamic positioning control may be time-varying and non-Gaussian and the statistics of the noise are not exact, this paper proposes a variational Bayesian extended Kalman filter observer. In order to model non-Gaussian noise more reasonably, the joint posterior distribution of non-Gaussian noise with state and time-varying covariance is described as the product of Gaussian and independent inverse gamma distributions. Then, the variational Bayesian (VB) method was used to simultaneously estimate the system state, the intermediate variable values of the new distribution, and the noise parameters through fixed-point iteration. The fuzzy algorithm is applied to the model predictive control, and the weight parameters of the model predictive control are adaptively changed to better adapt to the changing working conditions in the control process. The simulation results show that the variational Bayesian extended Kalman filter observer has better performance than the existing algorithms when dealing with the time-varying non-Gaussian observation noise, whose statistics are not exact. The variational Bayesian extended Kalman filter observer can significantly improve the accuracy of the control when it is used in fuzzy model predictive control.

A fixed point evolution algorithm based on expanded Aitken rapid iteration method for global numeric optimization

Evolution algorithms based on mathematical models whose reproduction operators are derived from mathematical models are a promising branch of metaheuristic algorithms. Aitken rapid iteration method, as a fixed point iteration technique for solving nonlinear equations, performs a procedure of progressive display of a root and generates an iterative sequence that exhibits a convergent trend. Inspired by the idea that an iterative sequence gradually converges to the optimal point during the progressive display procedure of a fixed point of an equation, a fixed point evolution algorithm based on the expanded Aitken rapid iteration method (FPEea) is proposed. To develop FPEea, an expanded Aitken rapid model is first constructed. Then, three polynomials which are derived from the expanded Aitken rapid model are used as the reproduction operator of FPEea to produce offspring. The performance of FPEea is investigated on CEC2019 and CEC2020 benchmark function sets, as well as four engineering design problems. Experimental results show that FPEea is an effective and competitive algorithm compared with several classical evolution algorithms and state-of-the-art algorithms.

Development of multi-physics calculation method in lead cooled fast reactor code system MOSASAUR

A full-core three-dimensional multi-physics calculation method was researched based on the Lead cooled fast reactor (LFR) code system MOSASAUR, which is focused on both core steady-state and transient analyses of LFR. In the previous version of MOSASAUR, cross-sections generation module and core steady-state simulation module have been developed. In this research, the stiffness confinement method is employed to solve the time-dependent multi-group neutron transport equation to expand the capabilities of core transient analyses. And the thermal hydraulic calculation method is developed as thermal feedback module for core steady-state and transient analyses. Based on the fixed-point iteration scheme, the neutron module and thermal hydraulic module are coupled at each time step. The LMW benchmark and the problem based on ORAL 19-rod bundle are employed to verify the accuracy of transient calculation and thermal hydraulic module respectively. Finally, the multi-physics calculation method is utilized to simulate the transient process of the MicroURANUS core. The simulation results demonstrate the capability of the constructed multi-physics calculation framework to accurately simulate the transient behaviors of LMRs. Numerical results showed the good accuracy of the newly-developed multi-physics calculation module with MOSASAUR.

Stability Analysis Study of Time-Fractional Nonlinear Modified Kawahara Equation Based on the Homotopy Perturbation Sadik Transform

In this manuscript, we survey a numerical algorithm based on the combination of the homotopy perturbation method and the Sadik transform for solving the time-fractional nonlinear modified shallow water waves (called Kawahara equation) within the frame of the Caputo–Prabhakar (CP) operator. The nonlinear terms are handled with the assistance of the homotopy polynomials. The stability analysis of the implemented method is studied by using S-stable mapping and the Banach contraction principle. Also, we use the fixed-point method to determine the existence and uniqueness of solutions in the given suggested model. Finally, some numerical simulations are illustrated to display the accuracy and efficiency of the present numerical method. Moreover, numerical behaviors are captured to validate the reliability and efficiency of the scheme.

A projected fixed point method for a class of vertical tensor complementarity problems

A projected fixed point method for a class of vertical tensor complementarity problems

A Game-Theoretic Framework for Generic Second-Order Traffic Flow Models Using Mean Field Games and Adversarial Inverse Reinforcement Learning

A traffic system can be interpreted as a multiagent system, wherein vehicles choose the most efficient driving approaches guided by interconnected goals or strategies. This paper aims to develop a family of mean field games (MFG) for generic second-order traffic flow models (GSOM), in which cars control individual velocity to optimize their objective functions. GSOMs do not generally assume that cars optimize self-interested objectives, so such a game-theoretic reinterpretation offers insights into the agents’ underlying behaviors. In general, an MFG allows one to model individuals on a microscopic level as rational utility-optimizing agents while translating rich microscopic behaviors to macroscopic models. Building on the MFG framework, we devise a new class of second-order traffic flow MFGs (i.e., GSOM-MFG), which control cars’ acceleration to ensure smooth velocity change. A fixed-point algorithm with fictitious play technique is developed to solve GSOM-MFG numerically. In numerical examples, different traffic patterns are presented under different cost functions. For real-world validation, we further use an inverse reinforcement learning approach (IRL) to uncover the underlying cost function on the next-generation simulation (NGSIM) data set. We formulate the problem of inferring cost functions as a min-max game and use an apprenticeship learning algorithm to solve for cost function coefficients. The results show that our proposed GSOM-MFG is a generic framework that can accommodate various cost functions. The Aw Rascle and Zhang (ARZ) and Light-Whitham-Richards (LWR) fundamental diagrams in traffic flow models belong to our GSOM-MFG when costs are specified. History: This paper has been accepted for the Transportation Science Special Issue on ISTTT25 Conference. Funding: X. Di is supported by the National Science Foundation [CAREER Award CMMI-1943998]. E. Iacomini is partially supported by the Italian Research Center on High-Performance Computing, Big Data and Quantum Computing (ICSC) funded by MUR Missione 4-Next Generation EU (NGEU) [Spoke 1 “FutureHPC & BigData”]. C. Segala and M. Herty thank the Deutsche Forschungsgemeinschaft (DFG) for financial support [Grants 320021702/GRK2326, 333849990/IRTG-2379, B04, B05, and B06 of 442047500/SFB1481, HE5386/18-1,19-2,22-1,23-1,25-1, ERS SFDdM035; Germany’s Excellence Strategy EXC-2023 Internet of Production 390621612; and Excellence Strategy of the Federal Government and the Länder]. Support through the EU DATAHYKING is also acknowledged. This work was also funded by the DFG [TRR 154, Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks, Projects C03 and C05, Project No. 239904186]. Moreover, E. Iacomini and C. Segala are members of the Indam GNCS (Italian National Group of Scientific Calculus).

A stable localized weak strong form radial basis function method for modelling variably saturated groundwater flow induced by pumping and injection

The unsaturated zone profoundly affects groundwater (GW) flow induced by pumping and injection due to the capillary forces. However, current radial basis function (RBF) numerical models for GW pumping and injection mostly ignore the unsaturated zone. To bridge this gap, we developed a new three-dimensional weak strong form RBF model in this study, called CCHE3D-GW-RBF. Flow processes were modelled by the mixed-form Richards equation which was iteratively solved by the modified Picard iteration. Soil-water characteristic curves were represented by the widely applicable formulas, the van Genuchten (1980) model. Differential operators were approximated by the localized Gaussian RBF, and the weighted singular value decomposition method was used to construct stable bases. The injection/pumping wells and the flux boundaries were handled by the weak formulation using Meshless Local Petrov Galerkin method, and the strong-form equation using the collocation RBF method was enforced on the other points. Good agreement was found between the simulation results from our numerical model and the well-accepted solutions in all three verification cases, demonstrating the accuracy and applicability of this model. In addition, a smaller RBF shape parameter was found to produce a more accurate modelling resulting, indicating the necessity of implementing stable bases for RBF models.

A novel multi-frame image super-resolution model based on regularized nonlinear diffusion with Caputo time fractional derivative

In this work, we introduce an innovative fractional nonlinear parabolic model using a time-fractional order derivative, specifically employing the Caputo sense for fractional differentiation. This model aims to enhance traditional super-resolution models, particularly in the context of multi-frame image super-resolution. Additionally, we incorporate a regularized Perona–Malik diffusion mechanism to control the speed and direction of diffusion at each image location. We begin our study by exploring the theoretical solvability of our proposed model. Firstly, we employ the Faedo–Galerkin approach to establish the existence and uniqueness of a weak solution for an auxiliary fractional super-resolution model. Subsequently, we use the Schauder fixed point method to demonstrate the existence and uniqueness of a weak solution for our model. To validate the effectiveness of our model in the multi-frame super-resolution (SR) context, we conduct numerical experiments on images featuring diverse characteristics, including corners and edges, while applying various warping, decimation, and blurring matrices to the low-resolution (LR) images. We start the evaluation by introducing an adaptive discrete scheme tailored to the proposed model. To prove the robustness of our approach, we subject our images to varying levels of noise. Additionally, we perform simulations on real data (videos). The obtained high-resolution (HR) results demonstrate notable efficiency and robustness against noise, outperforming competitive models both visually and quantitatively.

Deriving with Derivatives: Optimizing Incremental Fixpoints for Higher-Order Flow Analysis

At the heart of efficient program analysis implementations are incremental solutions to fixpoint problems. These solutions can be interpreted as the derivative of the underlying analysis function. Methods that describe how to systematically derive higher-order analyses from program semantics, such as Abstracting Abstract Machines, don’t shed light on how to efficiently implement those analyses. In this paper, we explore complementary techniques to optimize the derivative computation towards deriving efficient implementations. In particular, we use static specializations (by partial evaluation and rewriting) and dynamic specializations (in the form of tracking dependencies during the fixpoint), yielding efficient incremental fixpoints. We present how these optimizations apply to an example analysis of continuation-passing-style λ-calculus, and describe how they pair particularly well with tunable and optimized workset-based fixpoint methods. We demonstrate the efficacy of this approach on a flow analysis for the Standard ML language, yielding an average speed-up of 56x over an existing fixpoint method for higher-order analyses from the literature.

Implicit Low-Rank Riemannian Schemes for the Time Integration of Stiff Partial Differential Equations

We propose two implicit numerical schemes for the low-rank time integration of stiff nonlinear partial differential equations. Our approach uses the preconditioned Riemannian trust-region method of Absil, Baker, and Gallivan, 2007. We demonstrate the efficiency of our method for solving the Allen–Cahn and the Fisher–KPP equations on the manifold of fixed-rank matrices. Our approach allows us to avoid the restriction on the time step typical of methods that use the fixed-point iteration to solve the inner nonlinear equations. Finally, we demonstrate the efficiency of the preconditioner on the same variational problems presented in Sutti and Vandereycken, 2021.

Mean-field backward stochastic differential equations with mean reflection and nonlinear resistance

The present paper is devoted to the study of the well-posedness of mean field BSDEs with mean reflection and nonlinear resistance. By the contraction mapping argument, we first prove that the mean-field BSDE with mean reflection and nonlinear resistance admits a unique deterministic flat local solution on a small time interval. Moreover, we build the global solution by introducing a two-step approach, which is a combination of stitching method and fixed point method. We further provide an application to the super-hedging problem with risk constraint.

On the qualitative behaviors of Volterra-Fredholm integro differential equations with multiple time-varying delays

This article considers a Volterra-Fredholm integro-differential equation including multiple time-varying delays. The aim of this article is to study the uniqueness of solution, the Ulam–Hyers–Rassias stability and the Ulam–Hyers stability of the Volterra-Fredholm integro-differential equation including multiple time-varying delays. We prove four new results in connection with the uniqueness of solution, the Ulam–Hyers–Rassias stability and the Ulam–Hyers stability of the considered Volterra-Fredholm integro-differential equation, respectively. The new results of this article involve sufficient conditions. The techniques of the proofs depend on the fixed point method according to the definitions of a suitable metric, operators and the related calculations. In particular case of the considered Volterra-Fredholm integro-differential equation, two illustrative examples are presented to verify the applications of the results. This article also involves some new complementary outcomes in connection with qualitative theory of Volterra-Fredholm integro-differential equations with delays.

Uniform and optimal error estimates of a nested Picard integrator for the nonlinear Schrödinger equation with wave operator

AbstractWe propose a second‐order nested Picard iterative integrator sine pseudospectral (NPI‐SP) method for the nonlinear Schrödinger equation with wave operator (NLSW) involving a parameter and carry out rigorous error estimates. Actually, the equation propagates wave with wavelength in time, while the amplitude of the leading oscillation is for well‐prepared initial data, and for ill‐prepared initial data, respectively. Based on the exponential integrator and nested Picard iteration, the uniformly accurate (w.r.t. ) NPI‐SP scheme is proposed with the optimal uniform error bounds at in time and spectral accuracy in space for both well‐prepared and ill‐prepared data in ‐norm. This result significantly improves the error bounds of the finite difference methods and exponential wave integrator for the NLSW. Error estimates are rigorously carried out and numerical examples are provided to confirm the theoretical analysis.

Modified Inertial Krasnosel'skii-Mann Type Method for Solving Fixed PointProblems in Real Uniformly Convex Banach Spaces

We present an altered version of the inertial Krasnosel'skii-Mann algorithm and demonstrate convergence outcomes for mappings that are asymptotically nonexpansive within real, uniformly convex Banach spaces. To achieve our results, we skillfully construct the inequality in equation \eqref{6} and apply it accordingly. Our findings support and broadly generalize a number of significant findings from the literature. We demonstrate, as an application, the generation of maximal monotone operators' zeros via fixed point methods in Hilbert spaces. Additionally, we solve convex minimization issues using our fixed-point techniques.

Hydrodynamic Porosity: A New Perspective on Flow through Porous Media, Part I

Pore-scale flow velocity is an essential parameter in determining transport through porous media, but it is often miscalculated. Researchers use a static porosity value to relate volumetric or superficial velocities to pore-scale flow velocities. We know this modeling assumption to be an oversimplification. The variable fraction of porosity conducive to flow, what we define as hydrodynamic porosity, θmobile, exhibits a quantifiable dependence on the Reynolds number (i.e., pore-scale flow velocity) in the Laminar flow regime. This fact remains largely unacknowledged in the literature. In this work, we quantify the dependence of θmobile on the Reynolds number via numerical flow simulation at the pore scale for rectangular pores of various aspect ratios, i.e., for highly idealized dead-end pore spaces. We demonstrate that, for the chosen cavity geometries, θmobile decreases by as much as 42% over the Laminar flow regime. Moreover, θmobile exhibits an exponential dependence on the Reynolds number, Re = R. The fit quality is effectively perfect, with a coefficient of determination (R2) of approximately 1 for each set of simulation data. Finally, we show that this exponential dependence can be easily fitted for pore-scale flow velocity through use of only a few Picard iterations, even with an initial guess that is 10 orders of magnitude off. Not only is this relationship a more accurate definition of pore-scale flow velocity, but it is also a necessary modeling improvement that can be easily implemented. In the companion paper (Part 2), we build upon the findings reported here and demonstrate their applicability to media with other pore geometries: rectangular and non-rectangular cavities (circular and triangular).

Fractional Jacobi–Picard iteration method using Gauss–Seidel technique for solving a system of nonlinear fractional differential equations

The main objective of this study is to introduce an improvement of Picard’s method, a technique commonly used to effectively solve a set of nonlinear fractional differential equations based on Caputo’s fractional derivative. Using the Picard’s method to solve fractional differential equations is straightforward. However, dealing with the integral in each Picard’s iteration becomes tough or even impossible for nonlinear problems. Thus, we propose an iterative strategy called the fractional Jacobi–Picard iteration method, which combines Picard’s iteration method with the shifted Jacobi polynomial. The computation of the fractional integrals of the shifted Jacobi polynomials is easily achieved at each step by utilizing properties of the fractional integral and shifted Jacobi polynomial. Furthermore, this approach not only transforms the system of equations into a reversible form but also solves it using the Gauss–Seidel technique. The convergence analysis of the method has been carefully performed. We performed detailed numerical simulations to show how well our method performs compared to other methods. Our results demonstrate the effectiveness and accuracy of our approach, especially in handling problems with non-smooth solutions.

Acetylated lignin sulfonate as a biodegradable coating for controlled-release urea fertilizer: A novel acetylation method and diffusion coefficient analysis

Plants require essential nutrients to grow, which soil alone cannot provide. Chemical fertilizers like urea supply the necessary nutrients, including nitrogen. They quickly dissolve in water and can contaminate it with nitrate and nitrite, which can cause diseases. Slow-release fertilizers are a better option to reduce environmental risks. Researchers are exploring cheap and biodegradable alternatives, such as lignin. A critical discussion in the coated urea fertilizer is modeling the nitrogen diffusion process in the coating, which predicts the system's behavior. This article uses lignin sulfonate to coat urea fertilizer, which should first be acetylated with decanoyl chloride. One of the critical parameters is the diffusion coefficient (D). D is determined using the mass transfer flux and the completion time of the effective substance, and with its help, the graph of the total mass transferred from the membrane in a specific time (Mt) is determined. D equals 6.298813 × 10−8 cm2/s using the time lag method. Also, with the fixed-point convergence method, 5.8849 × 10−8 cm2/s was obtained, which has about 0.80 % error with the D obtained by the time lag method. The Mt obtained from the analytical method and the experimental data coincides with a minimal error, which indicates high accuracy.

The cell-centered positivity-preserving finite volume scheme for 3D convection–diffusion equation on distorted meshes

PurposeThis paper aims to construct positivity-preserving finite volume schemes for the three-dimensional convection–diffusion equation that are applicable to arbitrary polyhedral grids.Design/methodology/approachThe cell vertices are used to define the auxiliary unknowns, and the primary unknowns are defined at cell centers. The diffusion flux is discretized by the classical nonlinear two-point flux approximation. To ensure the fully discrete scheme has positivity-preserving property, an improved discretization method for the convection flux was presented. Besides, a new positivity-preserving vertex interpolation method is derived from the linear reconstruction in the discretization of convection flux. Moreover, the Picard iteration method may have slow convergence in solving the nonlinear system. Thus, the Anderson acceleration of Picard iteration method is used to solve the nonlinear system. A condition number monitor of matrix is employed in the Anderson acceleration method to achieve better robustness.FindingsThe new scheme is applicable to arbitrary polyhedral grids and has a second-order accuracy. The results of numerical experiments also confirm the positivity-preserving of the discretization scheme.Originality/value1. This article presents a new positivity-preserving finite volume scheme for the 3D convection–diffusion equation. 2. The new discretization scheme of convection flux is constructed. 3. A new second-order interpolation algorithm is given to eliminate the auxiliary unknowns in flux expressions. 4. An improved Anderson acceleration method is applied to accelerate the convergence of Picard iterations. 5. This scheme can solve the convection–diffusion equation on the distorted meshes with second-order accuracy.

Numerical solution of a hydrodynamic model with cavitation using finite difference method at arbitrary meshes

In this paper, we investigate the implementation of the finite difference method on arbitrary meshes in conjunction with a fixed-point algorithm for the lubrication problem of a journal bearing with cavitation, considering the Elrod-Adams model. We establish numerical properties of the generalized finite difference scheme and provide several illustrative examples.

An efficient PGD solver for structural dynamics applications

We propose in this paper a Proper Generalized Decomposition (PGD) solver for reduced-order modeling of linear elastodynamic problems. It primarily focuses on enhancing the computational efficiency of a previously introduced PGD solver based on the Hamiltonian formalism. The novelty of this work lies in the implementation of a solver that is halfway between Modal Decomposition and the conventional PGD framework, so as to accelerate the fixed-point iteration algorithm. Additional procedures such that Aitken’s delta-squared process and mode-orthogonalization are incorporated to ensure convergence and stability of the algorithm. Numerical results regarding the ROM accuracy, time complexity, and scalability are provided to demonstrate the performance of the new solver when applied to dynamic simulation of a three-dimensional structure.