Multigrid method for nonlinear poroelasticity equations

In this study, a nonlinear multigrid method is applied for solving the system of incompressible poroelasticity equations considering nonlinear hydraulic conductivity. For the unsteady problem, an additional artificial term is utilized to stabilize the solutions when the equations are discretized on collocated grids. We employ two nonlinear multigrid methods, i.e. the “full approximation scheme” and “Newton multigrid” for solving the corresponding system of equations arising after discretization. For the steady case, both homogeneous and heterogeneous cases are solved and two different smoothers are examined to search for an efficient multigrid method. Numerical results show a good convergence performance for all the strategies

Multigrid method for nonlinear poroelasticity equations

<p>In this study, a nonlinear multigrid method is applied for solving the system of incompressible poroelasticity equations considering nonlinear hydraulic conductivity. For the unsteady problem, an additional artificial term is utilized to stabilize the solutions when the equations are discretized on collocated grids. We employ two nonlinear multigrid methods, i.e. the “full approximation scheme” and “Newton multigrid” for solving the corresponding system of equations arising after discretization. For the steady case, both homogeneous and heterogeneous cases are solved and two different smoothers are examined to search for an efficient multigrid method. Numerical results show a good convergence performance for all the strategies.</p

Solution of the Least Squares Method problem of pairwise comparison matrices

The aim of the paper is to present a new global optimization method for determining all the optima of the Least Squares Method (LSM) problem of pairwise comparison matrices. Such matrices are used, e.g., in the Analytic Hierarchy Process (AHP). Unlike some other distance minimizing methods, LSM is usually hard to solve because of the corresponding nonlinear and non-convex objective function. It is found that the optimization problem can be reduced to solve a system of polynomial equations. Homotopy method is applied which is an efficient technique for solving nonlinear systems. The paper ends by two numerical example having multiple global and local minima

An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method

In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.Comment: 34 pages, 13 figures, Published in "Computer Physics Communications

Solving nonlinear problems by Ostrowski Chun type parametric families

In this paper, by using a generalization of Ostrowski' and Chun's methods two bi-parametric families of predictor-corrector iterative schemes, with order of convergence four for solving system of nonlinear equations, are presented. The predictor of the first family is Newton's method, and the predictor of the second one is Steffensen's scheme. One of them is extended to the multidimensional case. Some numerical tests are performed to compare proposed methods with existing ones and to confirm the theoretical results. We check the obtained results by solving the molecular interaction problem.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and FONDOCYT, Republica Dominicana.Cordero Barbero, A.; Maimo, J.; Torregrosa Sánchez, JR.; Vassileva, M. (2015). Solving nonlinear problems by Ostrowski Chun type parametric families. Journal of Mathematical Chemistry. 53(1):430-449. https://doi.org/10.1007/s10910-014-0432-zS430449531M.S. Petkovic̀, B. Neta, L.D. Petkovic̀, J. Dz̆unic̀, Multipoint Methods for Solving Nonlinear Equations (Academic, New York, 2013)M. Mahalakshmi, G. Hariharan, K. Kannan, The wavelet methods to linear and nonlinear reaction–diffusion model arising in mathematical chemistry. J. Math. Chem. 51(9), 2361–2385 (2013)P.G. Logrado, J.D.M. Vianna, Partitioning technique procedure revisited: Formalism and first application to atomic problems. J. Math. Chem. 22, 107–116 (1997)C.G. Jesudason, I. Numerical nonlinear analysis: differential methods and optimization applied to chemical reaction rate determination. J. Math. Chem. 49, 1384–1415 (2011)K. Maleknejad, M. Alizadeh, An efficient numerical scheme for solving hammerstein integral equation arisen in chemical phenomenon. Procedia Comput. Sci. 3, 361–364 (2011)R.C. Rach, J.S. Duan, A.M. Wazwaz, Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J. Math. Chem. 52, 255–267 (2014)J.F. Steffensen, Remarks on iteration. Skand. Aktuar Tidskr. 16, 64–72 (1933)J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970)H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration. J. ACM 21, 643–651 (1974)J.R. Sharma, R.K. Guha, R. Sharma, An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer. Algorithms 62, 307–323 (2013)J.R. Sharma, H. Arora, On efficient weighted-Newton methods for solving systems of nonlinear equations. Appl. Math. Comput. 222, 497–506 (2013)M. Abad, A. Cordero, J.R. Torregrosa, Fourth- and fifth-order methods for solving nonlinear systems of equations: an application to the Global positioning system. Abstr. Appl. Anal.(2013) Article ID:586708. doi: 10.1155/2013/586708F. Soleymani, T. Lotfi, P. Bakhtiari, A multi-step class of iterative methods for nonlinear systems. Optim. Lett. 8, 1001–1015 (2014)M.T. Darvishi, N. Darvishi, SOR-Steffensen-Newton method to solve systems of nonlinear equations. Appl. Math. 2(2), 21–27 (2012). doi: 10.5923/j.am.20120202.05F. Awawdeh, On new iterative method for solving systems of nonlinear equations. Numer. Algorithms 5(3), 395–409 (2010)D.K.R. Babajee, A. Cordero, F. Soleymani, J.R. Torregrosa, On a novel fourth-order algorithm for solving systems of nonlinear equations. J. Appl. Math. (2012) Article ID:165452. doi: 10.1155/2012/165452A. Cordero, J.R. Torregrosa, M.P. Vassileva, Pseudocomposition: a technique to design predictor–corrector methods for systems of nonlinear equations. Appl. Math. Comput. 218(23), 1496–1504 (2012)A. Cordero, J.R. Torregrosa, M.P. Vassileva, Increasing the order of convergence of iterative schemes for solving nonlinear systems. J. Comput. Appl. Math. 252, 86–94 (2013)A.M. Ostrowski, Solution of Equations and System of Equations (Academic, New York, 1966)C. Chun, Construction of Newton-like iterative methods for solving nonlinear equations. Numer. Math. 104, 297–315 (2006)R. King, A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal. 10, 876–879 (1973)A. Cordero, J.R. Torregrosa, Low-complexity root-finding iteration functions with no derivatives of any order of convergence. J. Comput. Appl. Math. (2014). doi: 10.1016/j.cam.2014.01.024A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, A modified Newton Jarratts composition. Numer. Algorithms 55, 87–99 (2010)P. Jarratt, Some fourth order multipoint methods for solving equations. Math. Comput. 20, 434–437 (1966)A. Cordero, J.R. Torregrosa, Variants of Newtons method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)Z. Liu, Q. Zheng, P. Zhao, A variant of Steffensens method of fourth-order convergence and its applications. Appl. Math. Comput. 216, 1978–1983 (2010)A. Cordero, J.R. Torregrosa, A class of Steffensen type methods with optimal order of convergence. Appl. Math. Comput. 217, 7653–7659 (2011)L.B. Rall, New York, Computational Solution of Nonlinear Operator Equations (Robert E. Krieger Publishing Company Inc, New York, 1969

Inexact Newton Methods Applied to Under-Determined Systems

Consider an under-determined system of nonlinear equations F(x)=0, F:R^m→R^n, where F is continuously differentiable and m \u3e n. This system appears in a variety of applications, including parameter-dependent systems, dynamical systems with periodic solutions, and nonlinear eigenvalue problems. Robust, efficient numerical methods are often required for the solution of this system. Newton\u27s method is an iterative scheme for solving the nonlinear system of equations F(x)=0, F:R^n→R^n. Simple to implement and theoretically sound, it is not, however, often practical in its pure form. Inexact Newton methods and globalized inexact Newton methods are computationally efficient variations of Newton\u27s method commonly used on large-scale problems. Frequently, these variations are more robust than Newton\u27s method. Trust region methods, thought of here as globalized exact Newton methods, are not as computationally efficient in the large-scale case, yet notably more robust than Newton\u27s method in practice. The normal flow method is a generalization of Newton\u27s method for solving the system F:R^m→R^n, m \u3e n. Easy to implement, this method has a simple and useful local convergence theory; however, in its pure form, it is not well suited for solving large-scale problems. This dissertation presents new methods that improve the efficiency and robustness of the normal flow method in the large-scale case. These are developed in direct analogy with inexact-Newton, globalized inexact-Newton, and trust-region methods, with particular consideration of the associated convergence theory. Included are selected problems of interest simulated in MATLAB

Memory in a new variant of King's family for solving nonlinear systems

[EN] In the recent literature, very few high-order Jacobian-free methods with memory for solving nonlinear systems appear. In this paper, we introduce a new variant of King's family with order four to solve nonlinear systems along with its convergence analysis. The proposed family requires two divided difference operators and to compute only one inverse of a matrix per iteration. Furthermore, we have extended the proposed scheme up to the sixth-order of convergence with two additional functional evaluations. In addition, these schemes are further extended to methods with memory. We illustrate their applicability by performing numerical experiments on a wide variety of practical problems, even big-sized. It is observed that these methods produce approximations of greater accuracy and are more efficient in practice, compared with the existing methods.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE).Kansal, M.; Cordero Barbero, A.; Bhalla, S.; Torregrosa Sánchez, JR. (2020). Memory in a new variant of King's family for solving nonlinear systems. Mathematics. 8(8):1-15. https://doi.org/10.3390/math8081251S11588Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2009). A modified Newton-Jarratt’s composition. Numerical Algorithms, 55(1), 87-99. doi:10.1007/s11075-009-9359-zCordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2011). Efficient high-order methods based on golden ratio for nonlinear systems. Applied Mathematics and Computation, 217(9), 4548-4556. doi:10.1016/j.amc.2010.11.006Babajee, D. K. R., Cordero, A., Soleymani, F., & Torregrosa, J. R. (2012). On a Novel Fourth-Order Algorithm for Solving Systems of Nonlinear Equations. Journal of Applied Mathematics, 2012, 1-12. doi:10.1155/2012/165452Zheng, Q., Zhao, P., & Huang, F. (2011). A family of fourth-order Steffensen-type methods with the applications on solving nonlinear ODEs. Applied Mathematics and Computation, 217(21), 8196-8203. doi:10.1016/j.amc.2011.01.095Sharma, J., & Arora, H. (2013). An efficient derivative free iterative method for solving systems of nonlinear equations. Applicable Analysis and Discrete Mathematics, 7(2), 390-403. doi:10.2298/aadm130725016sSharma, J. R., Arora, H., & Petković, M. S. (2014). An efficient derivative free family of fourth order methods for solving systems of nonlinear equations. Applied Mathematics and Computation, 235, 383-393. doi:10.1016/j.amc.2014.02.103Wang, X., Zhang, T., Qian, W., & Teng, M. (2015). Seventh-order derivative-free iterative method for solving nonlinear systems. Numerical Algorithms, 70(3), 545-558. doi:10.1007/s11075-015-9960-2Chicharro, F. I., Cordero, A., Garrido, N., & Torregrosa, J. R. (2020). On the improvement of the order of convergence of iterative methods for solving nonlinear systems by means of memory. Applied Mathematics Letters, 104, 106277. doi:10.1016/j.aml.2020.106277Petković, M. S., & Sharma, J. R. (2015). On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations. Numerical Algorithms, 71(2), 457-474. doi:10.1007/s11075-015-0003-9Narang, M., Bhatia, S., Alshomrani, A. S., & Kanwar, V. (2019). General efficient class of Steffensen type methods with memory for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 352, 23-39. doi:10.1016/j.cam.2018.10.048King, R. F. (1973). A Family of Fourth Order Methods for Nonlinear Equations. SIAM Journal on Numerical Analysis, 10(5), 876-879. doi:10.1137/0710072Hermite, M. C., & Borchardt, M. (1878). Sur la formule d’interpolation de Lagrange. Journal für die reine und angewandte Mathematik (Crelles Journal), 1878(84), 70-79. doi:10.1515/crelle-1878-18788405Petkovic, M., Dzunic, J., & Petkovic, L. (2011). A family of two-point methods with memory for solving nonlinear equations. Applicable Analysis and Discrete Mathematics, 5(2), 298-317. doi:10.2298/aadm110905021pCordero, A., & Torregrosa, J. R. (2007). Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. doi:10.1016/j.amc.2007.01.062Awawdeh, F. (2009). On new iterative method for solving systems of nonlinear equations. Numerical Algorithms, 54(3), 395-409. doi:10.1007/s11075-009-9342-8Noor, M. A., Waseem, M., & Noor, K. I. (2015). New iterative technique for solving a system of nonlinear equations. Applied Mathematics and Computation, 271, 446-466. doi:10.1016/j.amc.2015.08.125Pramanik, S. (2002). Kinematic Synthesis of a Six-Member Mechanism for Automotive Steering. Journal of Mechanical Design, 124(4), 642-645. doi:10.1115/1.150337

Momentum-Space Approach to Asymptotic Expansion for Stochastic Filtering

This paper develops an asymptotic expansion technique in momentum space for stochastic filtering. It is shown that Fourier transformation combined with a polynomial-function approximation of the nonlinear terms gives a closed recursive system of ordinary differential equations (ODEs) for the relevant conditional distribution. Thanks to the simplicity of the ODE system, higher order calculation can be performed easily. Furthermore, solving ODEs sequentially with small sub-periods with updated initial conditions makes it possible to implement a substepping method for asymptotic expansion in a numerically efficient way. This is found to improve the performance significantly where otherwise the approximation fails badly. The method is expected to provide a useful tool for more realistic financial modeling with unobserved parameters, and also for problems involving nonlinear measure-valued processes.Comment: revised version for publication in Ann Inst Stat Mat

Quantum Algorithm for Solving Quadratic Nonlinear System of Equations

High-dimensional nonlinear system of equations that appears in all kinds of fields is difficult to be solved on a classical computer, we present an efficient quantum algorithm for solving $n$-dimensional quadratic nonlinear system of equations. Our algorithm embeds the equations into a finite-dimensional system of linear equations with homotopy perturbation method and a linearization technique, then we solve the linear equations with quantum linear system solver and obtain a state which is $\epsilon$-close to the normalized exact solution of the original nonlinear equations with success probability $\Omega(1)$. The complexity of our algorithm is $O(\rm{poly}(\rm{log}(n/\epsilon)))$, which provides an exponential improvement over the optimal classical algorithm in dimension $n$.Comment: 9 pages; Modify the format error of tex source fil