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Journal articles on the topic 'Implicit iteration'

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Published: 3 June 2025

Last updated: 6 July 2025

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1

Chugh, Renu, Preety Malik, and Vivek Kumar. "On a New Faster Implicit Fixed Point Iterative Scheme in Convex Metric Spaces." Journal of Function Spaces 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/905834.

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The purpose of this paper is to consider a new implicit iteration and study its strong convergence, stability, and data dependence. It is proved through numerical examples that newly introduced iteration has better convergence rate than well known implicit Mann iteration as well as implicit Ishikawa iteration and implicit iterations converge faster as compared to corresponding explicit iterations. Applications of implicit iterations to RNN (Recurrent Neural Networks) analysis are also presented.

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2

Joodi, Omar Mohammed Abbas, and Zena Hussein Maibed. "On analytical convergence of multi iterative procedure for finite family of generalized contractive." Journal of Interdisciplinary Mathematics 26, no. 7 (2023): 1635–46. http://dx.doi.org/10.47974/jim-1654.

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This paper introduces new iterations type three steps as multi_ implicit Noor iteration, multi_explicit Noor iteration and multi_Picard S-iteration, and defines a generalized quasi-like contractive mapping in convex metric space. Also, the convergence and stability of it are studied. On the other hand, the multi_ implicit Noor iteration is faster than the multi_explicit Noor iteration and multi_Picard S-iteration

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3

Wang, Cheng, and Zhi Ming Wang. "The Equivalence of Mann and Implicit Mann Iterations for Uniformly Pseudocontractive Mappings in Uniformly Smooth Banach Spaces." Applied Mechanics and Materials 50-51 (February 2011): 718–22. http://dx.doi.org/10.4028/www.scientific.net/amm.50-51.718.

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In this paper, suppose is an arbitrary uniformly smooth real Banach space, and is a nonempty closed convex subset of . Let be a generalized Lipschitzian and uniformly pseudocontractive self-map with . Suppose that , are defined by Mann iteration and implicit Mann iteration respectively, with the iterative parameter satisfying certain conditions. Then the above two iterations that converge strongly to fixed point of are equivalent.

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4

Rhoades, B. E., and Ştefan M. Şoltuz. "The convergence of an implicit mean value iteration." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–7. http://dx.doi.org/10.1155/ijmms/2006/68369.

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5

Sablin, M. "An iterative method for solving difference problems of gas dynamics in the mixed Euler-Lagrangian variables." Journal of Physics: Conference Series 2099, no. 1 (2021): 012013. http://dx.doi.org/10.1088/1742-6596/2099/1/012013.

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Abstract The method proposed is intended to solve implicit conservative operator difference schemes for a grid initial-boundary value problems on a simplex grid for a system of equations of gas dynamics in the mixed Euler-Lagrangian variables. To find a solution to such a scheme at a time step, it is represented as a single equation for a nonlinear function of two arguments from space – the direct product of the grid spaces of gas-dynamic quantities. To solve such an equation, a combination of the generalized Gauss-Seidel iterative method (external iterations) and an implicit two-layer iteration scheme (internal iterations at each external iteration) is used. The feature of the method is that, the equation, which is solved by internal iterations, is obtained from the equation of the difference scheme using symmetrization – such a non-degenerate linear transformation that the function in this equation has a self-adjoint positive Frechet derivative.

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6

YEE, H. C., and P. K. SWEBY. "GLOBAL ASYMPTOTIC BEHAVIOR OF ITERATIVE IMPLICIT SCHEMES." International Journal of Bifurcation and Chaos 04, no. 06 (1994): 1579–611. http://dx.doi.org/10.1142/s0218127494001210.

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The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing three models of 2×2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed using the theory of dynamical systems. The iterative procedures include simple iteration and full and modified Newton iterations. The results are compared with standard Runge-Kutta explicit methods, a noniterative implicit procedure, and the Newton method of solving the steady part of the ODEs. Studies showed that aside from exhibiting spurious asymptotes, all of the four implicit LMMs can change the type and stability of the steady states of the differential equations (DEs). They also exhibit a drastic distortion but less shrinkage of the basin of attraction of the true solution than standard nonLMM explicit methods. The simple iteration procedure exhibits behavior which is similar to standard nonLMM explicit methods except that spurious steady-state numerical solutions cannot occur. The numerical basins of attraction of the noniterative implicit procedure mimic more closely the basins of attraction of the DEs and are more efficient than the three iterative implicit procedures for the four implicit LMMs. Contrary to popular belief, the initial data using the Newton method of solving the steady part of the DEs may not have to be close to the exact steady state for convergence. These results can be used as an explanation for possible causes and cures of slow convergence and nonconvergence of steady-state numerical solutions when using an implicit LMM time-dependent approach in computational fluid dynamics.

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7

Yildirim, Isa, and Mujahid Abbas. "Convergence Rate of Implicit Iteration Process and a Data Dependence Result." European Journal of Pure and Applied Mathematics 11, no. 1 (2018): 189. http://dx.doi.org/10.29020/nybg.ejpam.v11i1.2911.

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The aim of this paper is to introduce an implicit S-iteration processand study its convergence in the framework of W-hyperbolic spaces. We showthat the implicit S-iteration process has higher rate of convergence than implicit Mann type iteration and implicit Ishikawa-type iteration processes. We present a numerical example to support the analytic result proved herein. Finally, we prove a data dependence result for a contractive type mapping using implicit S-iteration process.

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8

Agwu, Imo Kalu, Umar Ishtiaq, Naeem Saleem, Donatus Ikechi Igbokwe, and Fahd Jarad. "Equivalence of novel IH-implicit fixed point algorithms for a general class of contractive maps." AIMS Mathematics 8, no. 1 (2022): 841–72. http://dx.doi.org/10.3934/math.2023041.

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<abstract><p>In this paper, a novel implicit IH-multistep fixed point algorithm and convergence result for a general class of contractive maps is introduced without any imposition of the "sum conditions" on the countably finite family of the iteration parameters. Furthermore, it is shown that the convergence of the proposed iteration scheme is equivalent to some other implicit IH-type iterative schemes (e.g., implicit IH-Noor, implicit IH-Ishikawa and implicit IH-Mann) for the same class of maps. Also, some numerical examples are given to illustrate that the equivalence is true. Our results complement, improve and unify several equivalent results recently announced in literature.</p></abstract>

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9

Balicki, Linus. "Low-Rank Alternating Direction Implicit Iteration in pyMOR." GAMM Archive for Students 2, no. 1 (2020): 1–13. http://dx.doi.org/10.14464/gammas.v2i1.420.

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The low-rank alternating direction implicit (LR-ADI) iteration is an effective method for solving large-scale Lyapunov equations. In the software library pyMOR, solutions to Lyapunov equations play an important role when reducing a model using the balanced truncation method. In this article we introduce the LR-ADI iteration as well as pyMOR, while focusing on its features which are relevant for integrating the iteration into the library. We compare the run time of the iteration's pure pyMOR implementation with those achieved by external libraries available within the pyMOR framework.

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10

Li, Teng-fei, and Heng-you Lan. "On New Picard-Mann Iterative Approximations with Mixed Errors for Implicit Midpoint Rule and Applications." Journal of Function Spaces 2019 (March 3, 2019): 1–13. http://dx.doi.org/10.1155/2019/4042965.

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In order to solve (partial) differential equations, implicit midpoint rules are often employed as a powerful numerical method. The purpose of this paper is to introduce and study a class of new Picard-Mann iteration processes with mixed errors for the implicit midpoint rules, which is different from existing methods in the literature, and to analyze the convergence and stability of the proposed method. Further, some numerical examples and applications to optimal control problems with elliptic boundary value constraints are considered via the new Picard-Mann iterative approximations, which shows that the new Picard-Mann iteration process with mixed errors for the implicit midpoint rule of nonexpansive mappings is brand new and more effective than other related iterative processes.

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11

Li, Hong Jun, and Yong Fu Su. "A Note on "Implicit Mann Fixed Point Iterations for Pseudo-Contractive Mappings"." Advanced Materials Research 393-395 (November 2011): 543–45. http://dx.doi.org/10.4028/www.scientific.net/amr.393-395.543.

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Ljubomir Ciric, Arif Rafiq, Nenad Cakic, Jeong Sheok Umed [ Implicit Mann fixed point iterations for pseudo-contractive mappings, Applied Mathematics Letters 22 (2009) 581-584] introduced and investigated a modified Mann implicit iteration process for continuous hemi-contractive map. They proved the relatively convergence theorem. However, the content of mann theorem is fuzzy. In this paper, we will give some comments . Let be a Banach space and be a nonempty subset of . A mapping is called hemi-contractive (see [1]) if and In [1], the authors introduced and investigated a modified Mann implicit iteration process for continuous hemi-contractive map. They proved the following convergence theorem.

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12

Ma, Huaifa, Jikai Zhou, and Guoping Liang. "Implicit Damping Iterative Algorithm to Solve Elastoplastic Static and Dynamic Equations." Journal of Applied Mathematics 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/486171.

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This paper presents an implicit damping iterative algorithm to simultaneously solve equilibrium equations, yield function, and plastic flow equations, without requiring an explicit expression of elastoplastic stiffness matrices and local iteration for “return mapping” stresses to the yield surface. In addition, a damping factor is introduced to improve the stiffness matrix conformation in the nonlinear iterative process. The incremental iterative scheme and whole amount iterative scheme are derived to solve the dynamical and static and dynamical elastoplastic problems. To validate the proposed algorithms, computation procedures are designed and the numerical tests are implemented. The computational results verify the correctness and reliability of the proposed implicit iteration algorithms.

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13

Zidan, Ahmed, Ahmed Soliman, Tamer Nabil, and Mohamed Barakat. "An investigation of new quicker implicit iterations in hyperbolic spaces." Thermal Science 24, Suppl. 1 (2020): 199–207. http://dx.doi.org/10.2298/tsci20199z.

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The present paper investigates the convergence of some new implicit iterations of coupled fixed point for non-linear contractive like functions on W-hyperbolic metric spaces. Moreover, we provide a theoretical comparison of our new iterations to illustrate the fastest iteration the coupled fixed point.

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14

Zidan, Ahmed, Ahmed Soliman, Tamer Nabil, and Mohamed Barakat. "An investigation of new quicker implicit iterations in hyperbolic spaces." Thermal Science 24, Suppl. 1 (2020): 199–207. http://dx.doi.org/10.2298/tsci20s1199z.

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The present paper investigates the convergence of some new implicit iterations of coupled fixed point for non-linear contractive like functions on W-hyperbolic metric spaces. Moreover, we provide a theoretical comparison of our new iterations to illustrate the fastest iteration the coupled fixed point.

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15

Ceng, Lu-Chuan, and Meijuan Shang. "Generalized Mann Viscosity Implicit Rules for Solving Systems of Variational Inequalities with Constraints of Variational Inclusions and Fixed Point Problems." Mathematics 7, no. 10 (2019): 933. http://dx.doi.org/10.3390/math7100933.

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In this work, let X be Banach space with a uniformly convex and q-uniformly smooth structure, where 1 < q ≤ 2 . We introduce and consider a generalized Mann-like viscosity implicit rule for treating a general optimization system of variational inequalities, a variational inclusion and a common fixed point problem of a countable family of nonexpansive mappings in X. The generalized Mann-like viscosity implicit rule investigated in this work is based on the Korpelevich’s extragradient technique, the implicit viscosity iterative method and the Mann’s iteration method. We show that the iterative sequences governed by our generalized Mann-like viscosity implicit rule converges strongly to a solution of the general optimization system.

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16

Liu, Zhi-bin, Yi-shen Chen, Xue-song Li, and Yi-bin Xiao. "Implicit Ishikawa Approximation Methods for Nonexpansive Semigroups in CAT(0) Spaces." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/503198.

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This paper is devoted to the convergence of the implicit Ishikawa iteration processes for approximating a common fixed point of nonexpansive semigroup in CAT(0) spaces. We obtain theΔ-convergence results of the implicit Ishikawa iteration sequences for a family of nonexpansive mappings in CAT(0) spaces. Under certain and different conditions, we also get the strong convergence theorems of implicit Ishikawa iteration sequences for nonexpansive semigroups in the CAT(0) spaces. The results presented in this paper extend and generalize some previous results.

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17

YİLDİRİM, İsa, and Muhammed Emin BATUHAN. "Random fixed point results for generalized asymptotically nonexpansive random operators." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 72, no. 3 (2023): 570–86. http://dx.doi.org/10.31801/cfsuasmas.1211661.

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In this paper, we define an implicit random iterative process with errors for three finite families of generalized asymptotically nonexpansive random operators. We also prove some convergence theorems using this iteration method in separable Banach spaces.

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18

Sheth, Soham M., and Rami M. Younis. "Localized Linear Systems in Sequential Implicit Simulation of Two-Phase Flow and Transport." SPE Journal 22, no. 05 (2017): 1542–69. http://dx.doi.org/10.2118/173320-pa.

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Summary Implicit-reservoir-simulation models offer improved robustness compared with semi-implicit or explicit alternatives. The implicit treatment gives rise to a large nonlinear algebraic system of equations that must be solved at each timestep. Newton-like iterative methods are often used to solve these nonlinear systems. At each nonlinear iteration, large and sparse linear systems must be solved to obtain the Newton update vector. It is observed that these computed Newton updates are often sparse, even though the sum of the Newton updates over a converged iteration may not be. Sparsity in the Newton update suggests the presence of a spatially localized propagation of corrections along the nonlinear iteration sequence. Substantial computational savings may be realized by restricting the linear-solution process to obtain only the nonzero update elements. This requires an a priori identification of the set of nonzero update elements. To preserve the convergence behavior of the original Newton-like process, it is necessary to avoid missing any nonzero element in the identification procedure. This ensures that the localized and full linear computations result in the same solution. As a first step toward the development of such a localization method for general fully implicit simulation, the focus is on sequential implicit methods for general two-phase flow. Theoretically conservative, a priori estimates of the anticipated Newton-update sparsity pattern are derived. The key to the derivation of these estimates is in forming and solving simplified forms of infinite-dimensional Newton iteration for the semidiscrete residual equations. Upon projection onto the discrete mesh, the analytical estimates produce a conservative indication on the update's sparsity pattern. The algorithm is applied to several large-scale computational examples, and more than a 10-fold reduction in simulation time is attained. The results of the localized and full simulations are identical, as is the nonlinear convergence behavior.

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19

Wachspress, E. L. "Three-variable alternating-direction-implicit iteration." Computers & Mathematics with Applications 27, no. 3 (1994): 1–7. http://dx.doi.org/10.1016/0898-1221(94)90040-x.

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20

Şoltuz, Ştefan M. "Errors estimation for implicit Mann iteration." Journal of Numerical Analysis and Approximation Theory 35, no. 1 (2006): 117–18. http://dx.doi.org/10.33993/jnaat351-1019.

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21

Akewe, H., and A. A. Mogbademu. "Equivalence results for implicit Jungck–Kirk type iterations." Acta et Commentationes Universitatis Tartuensis de Mathematica 22, no. 1 (2018): 75–89. http://dx.doi.org/10.12697/acutm.2018.22.08.

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We show that the implicit Jungck–Kirk-multistep, implicit Jungck–Kirk–Noor, implicit Jungck–Kirk–Ishikawa, and implicit Jungck–Kirk–Mann iteration schemes are equivalently used to approximate the common fixed points of a pair of weakly compatible generalized contractive-like operators defined on normed linear spaces. Our results contribute to the existing results on the equivalence of fixed point iteration schemes by extending them to pairs of maps. An example to show the applicability of the main results is included.

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22

Sarfaraz, M., K. S. Nisar, and M. K. Ahmad. "Solving nonlinear implicit variational inclusion problems using S-iteration via relaxed resolvent operator." Journal of Interdisciplinary Mathematics 26, no. 1 (2023): 17–32. http://dx.doi.org/10.47974/jim-1644.

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The goal of the draft is to launch a fresh iteration scheme for nonlinear implicit variational inclusion problems with the help of strongly ()I−-monotone operator. This scheme is known as relaxed S-iteration scheme. Here, we use relaxed resolvent operator into S-iteration and create a new operator and with the help of that operator we will establish the strong convergence result for the nonlinear implicit variational inclusion problems.

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23

Chen, Yingxia, Yuqi Li, Tingting Wang, Yan Chen, and Faming Fang. "DPDU-Net: Double Prior Deep Unrolling Network for Pansharpening." Remote Sensing 16, no. 12 (2024): 2141. http://dx.doi.org/10.3390/rs16122141.

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The objective of the pansharpening task is to integrate multispectral (MS) images with low spatial resolution (LR) and to integrate panchromatic (PAN) images with high spatial resolution (HR) to generate HRMS images. Recently, deep learning-based pansharpening methods have been widely studied. However, traditional deep learning methods lack transparency while deep unrolling methods have limited performance when using one implicit prior for HRMS images. To address this issue, we incorporate one implicit prior with a semi-implicit prior and propose a double prior deep unrolling network (DPDU-Net) for pansharpening. Specifically, we first formulate the objective function based on observation models of PAN and LRMS images and two priors of an HRMS image. In addition to the implicit prior in the image domain, we enforce the sparsity of the HRMS image in a certain multi-scale implicit space; thereby, the feature map can obtain better sparse representation ability. We optimize the proposed objective function via alternating iteration. Then, the iterative process is unrolled into an elaborate network, with each iteration corresponding to a stage of the network. We conduct both reduced-resolution and full-resolution experiments on two satellite datasets. Both visual comparisons and metric-based evaluations consistently demonstrate the superiority of the proposed DPDU-Net.

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24

Zhang, Yanmei, Xia Cui, and Guangwei Yuan. "Nonlinear iteration acceleration solution for equilibrium radiation diffusion equation." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 5 (2020): 1465–90. http://dx.doi.org/10.1051/m2an/2019095.

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This paper discusses accelerating iterative methods for solving the fully implicit (FI) scheme of equilibrium radiation diffusion problem. Together with the FI Picard factorization (PF) iteration method, three new nonlinear iterative methods, namely, the FI Picard-Newton factorization (PNF), FI Picard-Newton (PN) and derivative free Picard-Newton factorization (DFPNF) iteration methods are studied, in which the resulting linear equations can preserve the parabolic feature of the original PDE. By using the induction reasoning technique to deal with the strong nonlinearity of the problem, rigorous theoretical analysis is performed on the fundamental properties of the four iteration methods. It shows that they all have first-order time and second-order space convergence, and moreover, can preserve the positivity of solutions. It is also proved that the iterative sequences of the PF iteration method and the three Newton-type iteration methods converge to the solution of the FI scheme with a linear and a quadratic speed respectively. Numerical tests are presented to confirm the theoretical results and highlight the high performance of these Newton acceleration methods.

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Hou, Fei, Chiyu Wang, Wencheng Wang, Hong Qin, Chen Qian, and Ying He. "Iterative poisson surface reconstruction (iPSR) for unoriented points." ACM Transactions on Graphics 41, no. 4 (2022): 1–13. http://dx.doi.org/10.1145/3528223.3530096.

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Poisson surface reconstruction (PSR) remains a popular technique for reconstructing watertight surfaces from 3D point samples thanks to its efficiency, simplicity, and robustness. Yet, the existing PSR method and subsequent variants work only for oriented points. This paper intends to validate that an improved PSR, called iPSR, can completely eliminate the requirement of point normals and proceed in an iterative manner. In each iteration, iPSR takes as input point samples with normals directly computed from the surface obtained in the preceding iteration, and then generates a new surface with better quality. Extensive quantitative evaluation confirms that the new iPSR algorithm converges in 5--30 iterations even with randomly initialized normals. If initialized with a simple visibility based heuristic, iPSR can further reduce the number of iterations. We conduct comprehensive comparisons with PSR and other powerful implicit-function based methods. Finally, we confirm iPSR's effectiveness and scalability on the AIM@SHAPE dataset and challenging (indoor and outdoor) scenes. Code and data for this paper are at https://github.com/houfei0801/ipsr.

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Zhang, Laiping, Ming Li, Wei Liu, and Xin He. "An Implicit Algorithm for High-Order DG/FV Schemes for Compressible Flows on 2D Arbitrary Grids." Communications in Computational Physics 17, no. 1 (2014): 287–316. http://dx.doi.org/10.4208/cicp.091113.280714a.

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AbstractA Newton/LU-SGS (lower-upper symmetric Gauss-Seidel) iteration implicit method was developed to solve two-dimensional Euler and Navier-Stokes equations by the DG/FV hybrid schemes on arbitrary grids. The Newton iteration was employed to solve the nonlinear system, while the linear system was solved with LU-SGS iteration. The effect of several parameters in the implicit scheme, such as the CFL number, the Newton sub-iteration steps, and the update frequency of Jacobian matrix, was investigated to evaluate the performance of convergence history. Several typical test cases were simulated, and compared with the traditional explicit Runge-Kutta (RK) scheme. Firstly the Couette flow was tested to validate the order of accuracy of the present DG/FV hybrid schemes. Then a subsonic inviscid flow over a bump in a channel was simulated and the effect of parameters was investigated also. Finally, the implicit algorithm was applied to simulate a subsonic inviscid flow over a circular cylinder and the viscous flow in a square cavity. The numerical results demonstrated that the present implicit scheme can accelerate the convergence history efficiently. Choosing proper parameters would improve the efficiency of the implicit scheme. Moreover, in the same framework, the DG/FV hybrid schemes are more efficient than the same order DG schemes.

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Tian, Ming, and Xin Jin. "Implicit Iterative Scheme for a Countable Family of Nonexpansive Mappings in 2-Uniformly Smooth Banach Spaces." Abstract and Applied Analysis 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/264910.

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Implicit Mann process and Halpern-type iteration have been extensively studied by many others. In this paper, in order to find a common fixed point of a countable family of nonexpansive mappings in the framework of Banach spaces, we propose a new implicit iterative algorithm related to a strongly accretive and Lipschitzian continuous operatorF:xn=αnγV(xn)+βnxn-1+((1-βn)I-αnμF)Tnxnand get strong convergence under some mild assumptions. Our results improve and extend the corresponding conclusions announced by many others.

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Maddulapalli, K., S. Azarm, and A. Boyars. "Interactive Product Design Selection With an Implicit Value Function." Journal of Mechanical Design 127, no. 3 (2005): 367–77. http://dx.doi.org/10.1115/1.1829727.

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We present a new method to aid a decision maker (DM) in selecting the “most preferred” from a set of design alternatives. The method is deterministic and assumes that the DM’s preferences reflect an implicit value function that is quasi-concave. The method is interactive, with the DM stating preferences in the form of attribute tradeoffs at a series of trial designs, each a specific design under consideration. The method is iterative and uses the gradient of the value function obtained from the preferences of the DM to eliminate lower value designs at each trial design. We present an approach for finding a new trial design at each iteration. We provide an example, the design selection for a cordless electric drill, to demonstrate the method. We provide results showing that (within the limit of our experimentation) our method needs only a few iterations to find the most preferred design alternative. Finally we extend our deterministic selection method to account for uncertainty in the attributes when the probability distributions governing the uncertainty are known.

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Du, Chunjuan, and Tongxin Yan. "SOR-based alternately linearized implicit iteration method for nonsymmetric algebraic Riccati equations." AIMS Mathematics 8, no. 9 (2023): 19876–91. http://dx.doi.org/10.3934/math.20231013.

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<abstract><p>In this paper, we propose a class of successive over relaxation-based alternately linearized implicit iteration method for computing the minimal nonnegative solution of nonsymmetric algebraic Riccati equations. Under certain conditions, we prove the convergence of the iterative method. Finally, numerical examples are given to show the iterative method is efficient.</p></abstract>

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Akinbo, G., and O. O. Fabelurin. "Convergence of Implicit Noor Iteration in Convex b-Metric Space." International Journal of Mathematical Sciences and Optimization: Theory and Applications 11, no. 1 (2025): 89–95. https://doi.org/10.5281/zenodo.15175819.

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 The convergence to a fixed point and stability of the implicit Noor iteration in a convex b-metric  space is established in this work. The class of mappings considered here is an extension of a  class of weak contractions which has been used by several authors to obtain quite interesting  results on the existence of unique fixed points as well as convergence and stability of iterative  schemes in the literature.

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Xu, Hong-Kun, and Ramesh G. Ori. "AN IMPLICIT ITERATION PROCESS FOR NONEXPANSIVE MAPPINGS." Numerical Functional Analysis and Optimization 22, no. 5-6 (2001): 767–73. http://dx.doi.org/10.1081/nfa-100105317.

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32

Atanacković‐Vukmanović, O., L. Crivellari та E. Simonneau. "A Forth‐and‐Back Implicit Λ‐Iteration". Astrophysical Journal 487, № 2 (1997): 735–46. http://dx.doi.org/10.1086/304626.

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Thong, Duong Viet. "An implicit iteration process for nonexpansive semigroups." Nonlinear Analysis: Theory, Methods & Applications 74, no. 17 (2011): 6116–20. http://dx.doi.org/10.1016/j.na.2011.05.090.

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Houwen, P. J. "Parallel iteration schemes for implicit ODEIVP methods." Rendiconti del Seminario Matematico e Fisico di Milano 63, no. 1 (1993): 151–70. http://dx.doi.org/10.1007/bf02925099.

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35

Krajíček, Jan. "Implicit proofs." Journal of Symbolic Logic 69, no. 2 (2004): 387–97. http://dx.doi.org/10.2178/jsl/1082418532.

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Abstract.We describe a general method how to construct from a prepositional proof system P a possibly much stronger proof system iP. The system iP operates with exponentially long P-proofs described “implicitly” by polynomial size circuits.As an example we prove that proof system iEF, implicit EF, corresponds to bounded arithmetic theory and hence, in particular, polynomially simulates the quantified prepositional calculus G and the -consequences of proved with one use of exponentiation. Furthermore, the soundness of iEF is not provable in . An iteration of the construction yields a proof system corresponding to T2 + Exp and, in principle, to much stronger theories.

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Sabinin, Vladimir. "ON INCOMPLETE FACTORIZATION IMPLICIT TECHNIQUE FOR 2D ELLIPTIC FD EQUATIONS." Mathematical Modelling and Analysis 25, no. 1 (2020): 37–52. http://dx.doi.org/10.3846/mma.2020.8485.

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A new variant of Incomplete Factorization Implicit (IFI) iterative technique for 2D elliptic finite-difference (FD) equations is suggested which is differed by applying the matrix tridiagonal algorithm. Its iteration parameter is shown be linked with the one for Alternating Direction Implicit method. An effective set of values for the parameter is suggested. A procedure for enhancing the set of iteration parameters for IFI is proposed. The technique is applied to a 5-point FD scheme, and to a 9-point FD scheme. It is suggested applying the solver for 5-point scheme to solving boundary-value problems for the 9-point scheme, too. The results of numerical experiment with Dirichlet and Neumann boundary-value problems for Poisson equation in a rectangle, and in a quasi-circle are presented. Mixed boundary-value problems in square are considered, too. The effectiveness of IFI is high, and weakly depends on the type of boundary conditions.

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Dai, Jingguo, Yeqing Yi, and Chengzhi Liu. "Fast surface reconstruction algorithm with adaptive step size." PLOS ONE 20, no. 1 (2025): e0314756. https://doi.org/10.1371/journal.pone.0314756.

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In (Dai et al. 2023), the authors proposed a fast algorithm for surface reconstruction that converges rapidly from point cloud data by alternating Anderson extrapolation with implicit progressive iterative approximation (I-PIA). This algorithm employs a fixed step size during iterations to enhance convergence. To further improve the computational efficiency, an adaptive step size adjustment strategy for surface reconstruction algorithm is investigated. During each iteration, the step size is adaptively chosen based on the current residual—larger residuals may necessitate larger steps, while smaller ones might permit smaller steps. Numerical experiments indicate that, for equivalent reconstruction errors, the adaptive step size algorithm demands substantially fewer iterations and less computation time than the fixed step size approach. These improvements robustly enhance computational performance in surface reconstruction, offering valuable insights for further research and applications.

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Cheng, Y., X. Yang, and I. А. Matveev. "A fast normal splitting preconditioner for attractive coupled nonlinear Schroedinger equations with fractional Laplacian." Teoriâ i sistemy upravleniâ, no. 4 (December 24, 2024): 3–32. https://doi.org/10.31857/s0002338824040014.

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A linearly implicit conservative difference scheme is applied to discretize the attractive coupled nonlinear Schroedinger equations with fractional Laplacian. In this case complex symmetric linear systems appear, with indefinite and Toeplitz-plus-diagonal system matrices. Standard fast methods of direct solution or iteration using a preconditioner are not applicable for such systems. A novel iterative method is proposed, based on the normal splitting of the equivalent real block form of linear systems. Unconditional convergence is proved and the quasi-optimal iteration parameter is deducted. The preconditioner for this method is obtained naturally; it is constructed and efficiently implemented using the fast Fourier transform. Theoretical analysis shows that the eigenvalues of the preconditioned system matrix are closely clustered. Numerical experiments demonstrate new preconditioner significantly speeds up the convergence rate of iterative Krylov subspace methods. In particular, the convergence behavior of the corresponding preconditioned generalized minimum residual method is independent of the mesh size and almost insensitive to the fractional order. Moreover, the linearly implicit conservative difference scheme in this case preserves mass and energy with a given accuracy.

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Igbokwe, DI, and UE Udofia. "Approximation of common fixed points of a finite family of Ø - demicontractive mappings by an implicit iteration method." Global Journal of Mathematical Sciences 12, no. 1 (2015): 31–38. http://dx.doi.org/10.4314/gjmas.v12i1.15.

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We prove that the Implicit Iteration process of Xu and Ori (2001) converges strongly to the commonfixed pointsof a finite family of Ø - demicontractive mappings in real Hilbertand Banachspaces. Our results extend the results of Osilike (2004a) from strictl pseudocontractive maps to the much more general Ø - demicontractive maps;complement and generalize several others in the literature.KEY WORDS AND PHRASES: Ø - Demicontractive Maps, Implicit Iteration Process, Fixed Points,Strong Convergence.

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Chen, Jiahua, Yangui Zhou, Hexiang He, and Yongyao Li. "Adaptive Construction of Freeform Surface to Integrable Ray Mapping Using Implicit Fixed-Point Iteration." Photonics 12, no. 2 (2025): 134. https://doi.org/10.3390/photonics12020134.

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Constructing a freeform surface that accurately satisfies both integrable condition and Snell’s law under a given invariant source–target map is challenging for freeform design. Here, we propose a fixed-point iteration method to address this problem. This process involves solving a set of balanced gradient equations in the form of fixed-point iterations that are derived from equivalent integrability conditions and Snell’s law. By using the convergence theorem of fixed-point iteration, a unique solution for the balanced gradient equations exists, which is determined by the natural geometric properties of the freeform surface and is independent of the mapping. The gradient operators on the left-hand side of the equations are converted into a differential matrix form via a finite difference scheme. In one iteration, differential operations are forward-performed on the right-hand side of the equations, and the system of linear equations is solved on the left-hand side of the equation. The constructed freeform surfaces work well in both the paraxial and nonparaxial regions, and convergence in the nonparaxial region is faster than that in the paraxial region. The robustness and high efficiency of the proposed method are demonstrated with several design examples.

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Razon, Alan Marquez, Yizhou Chen, Han Yushan, Steven Gagniere, Michael Tupek, and Joseph Teran. "A Linear and Angular Momentum Conserving Hybrid Particle/Grid Iteration for Volumetric Elastic Contact." Proceedings of the ACM on Computer Graphics and Interactive Techniques 6, no. 3 (2023): 1–25. http://dx.doi.org/10.1145/3606924.

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We present a momentum conserving hybrid particle/grid iteration for resolving volumetric elastic collision. Our hybrid method uses implicit time stepping with a Lagrangian finite element discretization of the volumetric elastic material together with impulse-based collision-correcting momentum updates designed to exactly conserve linear and angular momentum. We use a two-step process for collisions: first we use a novel grid-based approach that leverages the favorable collision resolution properties of Particle-In-Cell (PIC) techniques, then we finalize with a classical collision impulse strategy utilizing continuous collision detection. Our PIC approach uses Affine-Particle-In-Cell momentum transfers as collision preventing impulses together with novel perfectly momentum conserving boundary resampling and downsampling operators that prevent artifacts in portions of the boundary where the grid resolution is of disparate resolution. We combine this with a momentum conserving augury iteration to remove numerical cohesion and model sliding friction. Our collision strategy has the same continuous collision detection as traditional approaches, however our hybrid particle/grid iteration drastically reduces the number of iterations required. Lastly, we develop a novel symmetric positive semi-definite Rayleigh damping model that increases the convexity of the nonlinear systems associated with implicit time stepping. We demonstrate the robustness and efficiency of our approach in a number of collision intensive examples.

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Diamantakis, Michail, and Linus Magnusson. "Sensitivity of the ECMWF Model to Semi-Lagrangian Departure Point Iterations." Monthly Weather Review 144, no. 9 (2016): 3233–50. http://dx.doi.org/10.1175/mwr-d-15-0432.1.

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Accurate estimation of the position of the departure points (d.p.) is crucial for the accuracy of a semi-Lagrangian NWP model. This calculation is often performed applying an implicit discretization to a kinematic equation solved by a fixed-point iteration scheme. A small number of iterations is typically used, assuming that this is sufficient for convergence. This assumption, derived from a past theoretical analysis, is revisited here. Analyzing the convergence of a generic d.p. iteration scheme and testing the ECMWF Integrated Forecast System (IFS) model, it is demonstrated that 2–3 iterations may not be sufficient for convergence to satisfactory accuracy in a modern high-resolution global model. Large forecast improvements can be seen by increasing the number of iterations. The extratropical geopotential error decreases and the simulated vertical structure of tropical cyclones improves. These findings prompted the implementation of an algorithm in which stopping criteria based on estimated convergence rates are used to “dynamically” stop d.p. iterations when an error tolerance criterion is satisfied. This is applied consistently to the nonlinear forecast, tangent linear, and adjoint models used by the ECMWF data assimilation system (4DVAR). Although the additional benefit of dynamic iteration is small, its testing reinforces the conclusion that a larger number of iterations is needed in regions of strong winds and shear. Furthermore, experiments suggest that dynamic iteration may prevent occasional 4DVAR failures in cases of strong stratospheric cross-polar flow in which the tangent linear model becomes unstable.

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Amina, Sabir, and Abudusaimaiti Mairemunisa. "Stepwise Alternating Direction Implicit Method of the Three Dimensional Convective-Diffusion Equation." Annals of Mathematics and Physics 7, no. 3 (2024): 248–76. http://dx.doi.org/10.17352/amp.000131.

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A stepwise alternating direction implicit method of the three dimensional convective-diffusion equation is considered in this paper. We constructed an implicit difference scheme and analyzes it's truncation error, convergence and stabilities. The theoretical and numerical analysis shows that the implicit difference scheme is unconditional stable. Then the Greedy Algorithm is proposed to solve the numerical solution on x,y and z axis separately by using implicit difference scheme and the numerical solution is convergent theoretically, however with no physical meaning. The Stepwise Alternating Direction Implicit Method (SADIM) is proposed, which uses the implicit difference scheme in this paper. Using Sauls scheme to pretreat the initial-boundary condition before iterating, thus eliminate the numerical oscillation caused by discontinuous initial boundary conditions. This SADIM is at least six ordered convergent, and is one of high ordered numerical methods for three dimensional problem. Our implicit difference scheme is more ideal than the standard Galerkin centered on finite difference scheme, quicker than SOR iteration method. The convergence of our implicit scheme is better than finite element method, characteristic line method, and mesh-less method. Our method eliminates the numerical oscillation caused by the convection dominant, resists the dispersion effectively and addresses dissipation caused by diffusion dominant.The implicit difference scheme has good theoretical and practical value.

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Ma, Chao, and Zhen Zhou Lu. "Iterative Algorithm for Structure Reliability Analysis Based on Support Vector Classification Method." Key Engineering Materials 353-358 (September 2007): 1009–12. http://dx.doi.org/10.4028/www.scientific.net/kem.353-358.1009.

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For reliability analysis of structure with implicit limit state function, an iterative algorithm is presented on the basis of support vector classification machine. In the present method, the support vector classification machine is employed to construct surrogate of the implicit limit state function. By use of the proposed rational iteration and sampling procedure, the constructed support vector classification machine can converge to the actual limit state function at the important region, which contributes to the failure probability significantly. Then the precision of the reliability analysis is improved. The implementation of the presented method is given in detail, and the feasibility and the efficiency are demonstrated by the illustrations.

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Rafiq, Arif, and Byung-Soo Lee. "ON ERROR ESTIMATES OF AN IMPLICIT ITERATION SCHEME." East Asian mathematical journal 31, no. 3 (2015): 379–82. http://dx.doi.org/10.7858/eamj.2015.030.

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Plubtieng, Somyot, and Rattanaporn Punpaeng. "Implicit iteration process of nonexpansive non-self-mappings." International Journal of Mathematics and Mathematical Sciences 2005, no. 19 (2005): 3103–10. http://dx.doi.org/10.1155/ijmms.2005.3103.

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SupposeCis a nonempty closed convex subset of real Hilbert spaceH. LetT:C→Hbe a nonexpansive non-self-mapping andPis the nearest point projection ofHontoC. In this paper, we study the convergence of the sequences{xn},{yn},{zn}satisfyingxn=(1−αn)u+αnT[(1−βn)xn+βnTxn],yn=(1−αn)u+αnPT[(1−βn)yn+βnPTyn], andzn=P[(1−αn)u+αnTP[(1−βn)zn+βnTzn]], where{αn}⊆(0,1),0≤βn≤β<1andαn→1asn→∞. Our results extend and improve the recent ones announced by Xu and Yin, and many others.

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van der Houwen, P. J., and J. J. B. de Swart. "Triangularly Implicit Iteration Methods for ODE-IVP Solvers." SIAM Journal on Scientific Computing 18, no. 1 (1997): 41–55. http://dx.doi.org/10.1137/s1064827595287456.

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Han, Linchang, Liming Yang, Zhihui Li, Jie Wu, Yinjie Du, and Xiang Shen. "Unlocking the Key to Accelerating Convergence in the Discrete Velocity Method for Flows in the Near Continuous/Continuous Flow Regimes." Entropy 25, no. 12 (2023): 1609. http://dx.doi.org/10.3390/e25121609.

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How to improve the computational efficiency of flow field simulations around irregular objects in near-continuum and continuum flow regimes has always been a challenge in the aerospace re-entry process. The discrete velocity method (DVM) is a commonly used algorithm for the discretized solutions of the Boltzmann-BGK model equation. However, the discretization of both physical and molecular velocity spaces in DVM can result in significant computational costs. This paper focuses on unlocking the key to accelerate the convergence in DVM calculations, thereby reducing the computational burden. Three versions of DVM are investigated: the semi-implicit DVM (DVM-I), fully implicit DVM (DVM-II), and fully implicit DVM with an inner iteration of the macroscopic governing equation (DVM-III). In order to achieve full implicit discretization of the collision term in the Boltzmann-BGK equation, it is necessary to solve the corresponding macroscopic governing equation in DVM-II and DVM-III. In DVM-III, an inner iterative process of the macroscopic governing equation is employed between two adjacent DVM steps, enabling a more accurate prediction of the equilibrium state for the full implicit discretization of the collision term. Fortunately, the computational cost of solving the macroscopic governing equation is significantly lower than that of the Boltzmann-BGK equation. This is primarily due to the smaller number of conservative variables in the macroscopic governing equation compared to the discrete velocity distribution functions in the Boltzmann-BGK equation. Our findings demonstrate that the fully implicit discretization of the collision term in the Boltzmann-BGK equation can accelerate DVM calculations by one order of magnitude in continuum and near-continuum flow regimes. Furthermore, the introduction of the inner iteration of the macroscopic governing equation provides an additional 1–2 orders of magnitude acceleration. Such advancements hold promise in providing a computational approach for simulating flows around irregular objects in near-space environments.

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Postolache, Mihai, Ashish Nandal, and Renu Chugh. "Strong Convergence of a New Generalized Viscosity Implicit Rule and Some Applications in Hilbert Space." Mathematics 7, no. 9 (2019): 773. http://dx.doi.org/10.3390/math7090773.

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In this paper, based on the very recent work by Nandal et al. (Nandal, A.; Chugh, R.; Postolache, M. Iteration process for fixed point problems and zeros of maximal monotone operators. Symmetry 2019, 11, 655.), we propose a new generalized viscosity implicit rule for finding a common element of the fixed point sets of a finite family of nonexpansive mappings and the sets of zeros of maximal monotone operators. Utilizing the main result, we first propose and investigate a new general system of generalized equilibrium problems, which includes several equilibrium and variational inequality problems as special cases, and then we derive an implicit iterative method to solve constrained multiple-set split convex feasibility problem. We further combine forward–backward splitting method and generalized viscosity implicit rule for solving monotone inclusion problem. Moreover, we apply the main result to solve convex minimization problem.

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Karatson, Janos, and Tamas Kurics. "A PRECONDITIONED ITERATIVE SOLUTION SCHEME FOR NONLINEAR PARABOLIC SYSTEMS ARISING IN AIR POLLUTION MODELING." Mathematical Modelling and Analysis 18, no. 5 (2013): 641–53. http://dx.doi.org/10.3846/13926292.2013.868841.

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A preconditioned iterative solution method is presented for nonlinear parabolic transport systems. The ingredients are implicit Euler discretization in time and finite element discretization in space, then an outer-inner (outer damped inexact Newton method with inner preconditioned conjugate gradient) iteration, further, as a main part, preconditioning via an l-tuple of independent elliptic operators. Numerical results show that the suggested method works properly for a test problem in air pollution modeling.

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