Journal articles: 'Right Hand Side iteration'

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Freitag, Melina A., Patrick Kürschner, and Jennifer Pestana. "GMRES Convergence Bounds for Eigenvalue Problems." Computational Methods in Applied Mathematics 18, no. 2 (April 1, 2018): 203–22. http://dx.doi.org/10.1515/cmam-2017-0017.

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AbstractThe convergence of GMRES for solving linear systems can be influenced heavily by the structure of the right-hand side. Within the solution of eigenvalue problems via inverse iteration or subspace iteration, the right-hand side is generally related to an approximate invariant subspace of the linear system. We give detailed and new bounds on (block) GMRES that take the special behavior of the right-hand side into account and explain the initial sharp decrease of the GMRES residual. The bounds motivate the use of specific preconditioners for these eigenvalue problems, e.g., tuned and polynomial preconditioners, as we describe. The numerical results show that the new (block) GMRES bounds are much sharper than conventional bounds and that preconditioned subspace iteration with either a tuned or polynomial preconditioner should be used in practice.

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Humayoun, Muhammad, S. M. Aqil Burney, A. H. Sheikh, and Abdul Ghafoor. "Ritz Vectors-Based Deflation Preconditioner for Linear System with Multiple Right-Hand Sides." STATISTICS, COMPUTING AND INTERDISCIPLINARY RESEARCH 3, no. 2 (December 31, 2021): 155–68. http://dx.doi.org/10.52700/scir.v3i2.56.

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Computational mathematics have many tools to solve the large systems of equations which may be linear or nonlinear. Iterative methods are used to solve the nonsymmetric definite system of linear equations like Krylov methods. Linear systems with multiple right-hand sides find application in many areas of engineering and science. Considering generality and indefiniteness, Krylov subspace methods are frequently used for such problems. However, problem with system with multiple right-hand side vectors requires constructing subspace for every right-hand side. GMRES produces Ritz vectors, approximation to eigenvectors during iterations. These Ritz values, recycled vectors are used in Krylov solve while solving system with second and subsequent right-hand side vectors. This is applied as a deflation preconditioner to GMRES. The numerical results show that the computational time, residuals and number of iterations is reduced as compare to simple GMRES. Deflation technique with Ritz vectors is not expensive as compare to the GMRES and as well as exact eigenvectors deflations.

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Humayoun, Muhammad, S. M. Aqil Burney, A. H. Sheikh, and Abdul Ghafoor. "Ritz Vectors-Based Deflation Preconditioner for Linear System with Multiple Right-Hand Sides." STATISTICS, COMPUTING AND INTERDISCIPLINARY RESEARCH 3, no. 2 (December 31, 2021): 155–68. http://dx.doi.org/10.52700/scir.v3i2.56.

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Computational mathematics have many tools to solve the large systems of equations which may be linear or nonlinear. Iterative methods are used to solve the nonsymmetric definite system of linear equations like Krylov methods. Linear systems with multiple right-hand sides find application in many areas of engineering and science. Considering generality and indefiniteness, Krylov subspace methods are frequently used for such problems. However, problem with system with multiple right-hand side vectors requires constructing subspace for every right-hand side. GMRES produces Ritz vectors, approximation to eigenvectors during iterations. These Ritz values, recycled vectors are used in Krylov solve while solving system with second and subsequent right-hand side vectors. This is applied as a deflation preconditioner to GMRES. The numerical results show that the computational time, residuals and number of iterations is reduced as compare to simple GMRES. Deflation technique with Ritz vectors is not expensive as compare to the GMRES and as well as exact eigenvectors deflations.

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4

Humayoun, Muhammad, S. M. Aqil Burney, A. H. Sheikh, and Abdul Ghafoor. "Ritz Vectors-Based Deflation Preconditioner for Linear System with Multiple Right-Hand Sides." STATISTICS, COMPUTING AND INTERDISCIPLINARY RESEARCH 3, no. 2 (December 31, 2021): 155–68. http://dx.doi.org/10.52700/scir.v3i2.56.

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Computational mathematics have many tools to solve the large systems of equations which may be linear or nonlinear. Iterative methods are used to solve the nonsymmetric definite system of linear equations like Krylov methods. Linear systems with multiple right-hand sides find application in many areas of engineering and science. Considering generality and indefiniteness, Krylov subspace methods are frequently used for such problems. However, problem with system with multiple right-hand side vectors requires constructing subspace for every right-hand side. GMRES produces Ritz vectors, approximation to eigenvectors during iterations. These Ritz values, recycled vectors are used in Krylov solve while solving system with second and subsequent right-hand side vectors. This is applied as a deflation preconditioner to GMRES. The numerical results show that the computational time, residuals and number of iterations is reduced as compare to simple GMRES. Deflation technique with Ritz vectors is not expensive as compare to the GMRES and as well as exact eigenvectors deflations.

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5

Dhage, B. C., and S. Heikkilä. "On nonlinear boundary value problems with deviating arguments and discontinuous right hand side." Journal of Applied Mathematics and Stochastic Analysis 6, no. 1 (January 1, 1993): 83–91. http://dx.doi.org/10.1155/s1048953393000085.

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In this paper we shall study the existence of the extremal solutions of a nonlinear boundary value problem of a second order differential equation with general Dirichlet/Neumann form boundary conditions. The right hand side of the differential equation is assumed to contain a deviating argument, and it is allowed to possess discontinuities in all the variables. The proof is based on a generalized iteration method.

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Lampe, J., and H. Voss. "GLOBAL CONVERGENCE OF RTLSQEP: A SOLVER OF REGULARIZED TOTAL LEAST SQUARES PROBLEMS VIA QUADRATIC EIGENPROBLEMS." Mathematical Modelling and Analysis 13, no. 1 (March 31, 2008): 55–66. http://dx.doi.org/10.3846/1392-6292.2008.13.55-66.

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The total least squares (TLS) method is a successful approach for linear problems if both the matrix and the right hand side are contaminated by some noise. In a recent paper Sima, Van Huffel and Golub suggested an iterative method for solving regularized TLS problems, where in each iteration step a quadratic eigenproblem has to be solved. In this paper we prove its global convergence, and we present an efficient implementation using an iterative projection method with thick updates.

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Liu, Chein-Shan. "A Feasible Approach to Determine the Optimal Relaxation Parameters in Each Iteration for the SOR Method." Journal of Mathematics Research 13, no. 1 (January 7, 2021): 1. http://dx.doi.org/10.5539/jmr.v13n1p1.

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The paper presents a dynamic and feasible approach to the successive over-relaxation (SOR) method for solving large scale linear system through iteration. Based on the maximal orthogonal projection technique, the optimal relaxation parameter is obtained by minimizing a derived merit function in terms of right-hand side vector, the coefficient matrix and the previous step values of unknown variables. At each iterative step, we can quickly determine the optimal relaxation value in a preferred interval. When the theoretical optimal value is hard to be achieved, the new method provides an alternative choice of the relaxation parameter at each iteration. Numerical examples confirm that the dynamic optimal successive over-relaxation (DOSOR) method outperforms the classical SOR method.

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Liu, Chein-Shan. "A Feasible Approach to Determine the Optimal Relaxation \\[6pt]Parameters in Each Iteration for the SOR Method." Journal of Mathematics Research 13, no. 1 (January 7, 2021): 1. http://dx.doi.org/10.5539/jmr.v13n1p1.

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The paper presents a dynamic and feasible approach to the successive over-relaxation (SOR) method for solving large scale linear system through iteration. Based on the maximal orthogonal projection technique, the optimal relaxation parameter is obtained by minimizing a derived merit function in terms of right-hand side vector, the coefficient matrix and the previous step values of unknown variables. At each iterative step, we can quickly determine the optimal relaxation value in a preferred interval. When the theoretical optimal value is hard to be achieved, the new method provides an alternative choice of the relaxation parameter at each iteration. Numerical examples confirm that the dynamic optimal successive over-relaxation (DOSOR) method outperforms the classical SOR method.

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9

Laguta, V. V. "THE ITERATION METHOD OF THE FIRST BOUNDARY-VALUE PROBLEM SOLUTION WITH NONLINEAR RIGHT-HAND SIDE." Science and Transport Progress, no. 18 (October 25, 2007): 95–99. http://dx.doi.org/10.15802/stp2007/17452.

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The numerical method of the decision of the first boundery problem with the nonlinear ordinary differential equation of the second order is offered. The initial problem is reduced to the decision of the two integrated equations with use Green's function. The system is solved Picard's method.

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Aghamiry, Hossein S., Ali Gholami, and Stéphane Operto. "Accurate and efficient data-assimilated wavefield reconstruction in the time domain." GEOPHYSICS 85, no. 2 (January 30, 2020): A7—A12. http://dx.doi.org/10.1190/geo2019-0535.1.

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Wavefield reconstruction inversion (WRI) mitigates cycle skipping in full-waveform inversion by computing wavefields that do not exactly satisfy the wave equation to match data with inaccurate velocity models. We refer to these wavefields as data assimilated wavefields because they are obtained by combining the physics of wave propagation and the observations. Then, the velocity model is updated by minimizing the wave-equation errors, namely, the source residuals. Computing these data-assimilated wavefields in the time domain with explicit time stepping is challenging. This is because the right-hand side of the wave equation to be solved depends on the back-propagated residuals between the data and the unknown wavefields. To bypass this issue, a previously proposed approximation replaces these residuals by those between the data and the exact solution of the wave equation. This approximation is questionable during the early WRI iterations when the wavefields computed with and without data assimilation differ significantly. We have developed a simple backward-forward time-stepping recursion to refine the accuracy of the data-assimilated wavefields. Each iteration requires us to solve one backward and one forward problem, the former being used to update the right side of the latter. An application to the BP salt model indicates that a few iterations are enough to reconstruct data-assimilated wavefields accurately with a crude velocity model. Although this backward-forward recursion leads to increased computational overheads during one WRI iteration, it preserves its capability to extend the search space.

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11

DHAGE, BAPURAO C., JANHAVI B. DHAGE, and JAVID ALI. ""Monotone iteration method for general nonlinear two point boundary value problems with deviating arguments"." Carpathian Journal of Mathematics 38, no. 2 (February 28, 2022): 405–15. http://dx.doi.org/10.37193/cjm.2022.02.11.

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"In this paper we shall study the existence and approximation results for a nonlinear two point boundary value problem of a second order ordinary differential equation with general form of Dirichlet/Neumann type boundary conditions. The nonlinearity present on right hand side of the differential equation is assumed to be Caratho´eodory containing a deviating argument. The proofs of the main results are based on a monotone iteration method contained in the hybrid fixed point principles of Dhage (2014) in an ordered Banach space. Finally, some remarks concerning the merits of our monotone iteration method over other frequently used iteration methods in the theory of nonlinear differential equations are given in the conclusion."

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Su, LingDe, and V. I. Vasil’ev. "Identification of spacewise dependent right-hand side in two dimensional parabolic equation." Journal of Physics: Conference Series 2092, no. 1 (December 1, 2021): 012019. http://dx.doi.org/10.1088/1742-6596/2092/1/012019.

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Abstract In this paper numerical solution of the inverse problem of determining a spacewise dependent right-hand side function in two dimensional parabolic equation is considered. Usually, the right-hand side function dependent on spatial variable is obtained from measured data of the solution at the final time point. Many mathematical modeling problems in the field of physics and engineering will encounter the inverse problems to identify the right-hand terms. When studying an inverse problem of identifying the spacewise dependent right-hand function, iterative methods are often used. We propose a new conjugate gradient method based on the constructed self-adjoint operator of the equation for numerical solution of the function and numerical examples illustrate the efficiency and accuracy.

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Nguyen Thu, Thuy. "Parallel iteration of two-step Runge-Kutta methods." Journal of Science Natural Science 66, no. 1 (March 2021): 12–24. http://dx.doi.org/10.18173/2354-1059.2021-0002.

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In this paper, we introduce the Parallel iteration of two-step Runge-Kutta methods for solving non-stiff initial-value problems for systems of first-order differential equations (ODEs): y′(t) = f(t, y(t)), for use on parallel computers. Starting with an s−stage implicit two-step Runge-Kutta (TSRK) method of order p, we apply the highly parallel predictor-corrector iteration process in P (EC)mE mode. In this way, we obtain an explicit two-step Runge-Kutta method that has order p for all m, and that requires s(m+1) right-hand side evaluations per step of which each s evaluation can be computed parallelly. By a number of numerical experiments, we show the superiority of the parallel predictor-corrector methods proposed in this paper over both sequential and parallel methods available in the literature.

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Quang A, Dang, and Ngo Thi Kim Quy. "New fixed point approach for a fully nonlinear fourth order boundary value problem." Boletim da Sociedade Paranaense de Matemática 36, no. 4 (October 1, 2018): 209–23. http://dx.doi.org/10.5269/bspm.v36i4.33584.

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In this paper we propose a method for investigating the solvability and iterative solution of a nonlinear fully fourth order boundary value problem. Namely, by the reduction of the problem to an operator equation for the right-hand side function we establish the existence and uniqueness of a solution and the convergence of an iterative process. Our method completely differs from the methods of other authors and does not require the condition of boundedness or linear growth of the right-hand side function on infinity. Many examples, where exact solutions of the problems are known or not, demonstrate the effectiveness of the obtained theoretical results.

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van Driel, M., J. Kemper, and C. Boehm. "On the modelling of self-gravitation for full 3-D global seismic wave propagation." Geophysical Journal International 227, no. 1 (June 21, 2021): 632–43. http://dx.doi.org/10.1093/gji/ggab237.

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SUMMARY We present a new approach to the solution of the Poisson equation present in the coupled gravito-elastic equations of motion for global seismic wave propagation in time domain aiming at the inclusion of the full gravitational response into spectral element solvers. We leverage the Salvus meshing software to include the external domain using adaptive mesh refinement and high order shape mapping. Together with Neumann boundary conditions based on a multipole expansion of the right-hand side this minimizes the number of additional elements needed. Initial conditions for the iterative solution of the Poisson equation based on temporal extrapolation from previous time steps together with a polynomial multigrid method reduce the number of iterations needed for convergence. In summary, this approach reduces the extra cost for simulating full gravity to a similar order as the elastic forces. We demonstrate the efficacy of the proposed method using the displacement from an elastic global wave propagation simulation (decoupled from the Poisson equation) at $200\, \mbox{s}$ dominant period to compute a realistic right-hand side for the Poisson equation.

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Krejić, Nataša, Zorana Lužanin, and Sanja Rapajić. "Iterative method with modification of the right-hand side vector for nonlinear complementarity problems." International Journal of Computer Mathematics 83, no. 2 (February 2006): 193–201. http://dx.doi.org/10.1080/00207160500168508.

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Guo, Han, Jun Hu, Hanru Shao, and Zaiping Nie. "Hierarchical Matrices Method and Its Application in Electromagnetic Integral Equations." International Journal of Antennas and Propagation 2012 (2012): 1–9. http://dx.doi.org/10.1155/2012/756259.

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Hierarchical (H-) matrices method is a general mathematical framework providing a highly compact representation and efficient numerical arithmetic. When applied in integral-equation- (IE-) based computational electromagnetics,H-matrices can be regarded as a fast algorithm; therefore, both the CPU time and memory requirement are reduced significantly. Its kernel independent feature also makes it suitable for any kind of integral equation. To solveH-matrices system, Krylov iteration methods can be employed with appropriate preconditioners, and direct solvers based on the hierarchical structure ofH-matrices are also available along with high efficiency and accuracy, which is a unique advantage compared to other fast algorithms. In this paper, a novel sparse approximate inverse (SAI) preconditioner in multilevel fashion is proposed to accelerate the convergence rate of Krylov iterations for solvingH-matrices system in electromagnetic applications, and a group of parallel fast direct solvers are developed for dealing with multiple right-hand-side cases. Finally, numerical experiments are given to demonstrate the advantages of the proposed multilevel preconditioner compared to conventional “single level” preconditioners and the practicability of the fast direct solvers for arbitrary complex structures.

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A, Dang Quang, Hung Nguyen Quoc, and Quang Vu Vinh. "NUMERICAL METHOD FOR SOLVING THE DIRICHLET BOUNDARY VALUE PROBLEM FOR NONLINEAR TRIHARMONIC EQUATION." Journal of Computer Science and Cybernetics 38, no. 2 (June 23, 2022): 181–91. http://dx.doi.org/10.15625/1813-9663/38/2/16912.

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In this work, we consider the Dirichlet boundary value problem for nonlinear triharmonic equation. Due to the reduction of the problem to operator equation for the pair of the right hand side function and the unknown second normal derivative of the function to be sought, we design an iterative method at both continuous and discrete levels for numerical solution of the problem. Some examples demonstrate that the numerical method is of fourth order convergence. When the right hand side function does not depend on the unknown function and its derivatives, the numerical method gives more accurate results in comparison with the results obtained by the interior method of Gudi and Neilan.

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Heikkilä, S., M. Kumpulainen, and V. Lakshmikantham. "On solvability of mixed monotone operator equations with applications to mixed quasimonotone differential systems involving discontinuities." Journal of Applied Mathematics and Stochastic Analysis 5, no. 1 (January 1, 1992): 1–17. http://dx.doi.org/10.1155/s1048953392000017.

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In this paper we shall first study solvability of mixed monotone systems of operator equations in an ordered normed space by using a generalized iteration method. The obtained results are then applied to prove existence of coupled extremal quasisolutions of the systems of first and second order mixed quasimonotone differential equations with discontinuous right hand sides. Most of the results deal with systems in a Banach space ordered by a regular order cone.

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Iliev, O., R. Lazarov, and J. Willems. "Numerical Study of Two-grid Preconditioners for 1-d Elliptic Problems with Highly Oscillating Discontinuous Coefficients." Computational Methods in Applied Mathematics 7, no. 1 (2007): 48–67. http://dx.doi.org/10.2478/cmam-2007-0003.

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AbstractVarious advanced two-level iterative methods are studied numerically and compared with each other in conjunction with finite volume discretizations of symmetric 1-D elliptic problems with highly oscillatory discontinuous coefficients. Some of the methods considered rely on the homogenization approach for deriving the coarse grid operator. This approach is considered here as an alternative to the well-known Galerkin approach for deriving coarse grid operators. Different intergrid transfer operators are studied, primary consideration being given to the use of the so-called problemdependent prolongation. The two-grid methods considered are used as both solvers and preconditioners for the Conjugate Gradient method. The recent approaches, such as the hybrid domain decomposition method introduced by Vassilevski and the globallocal iterative procedure proposed by Durlofsky et al. are also discussed. A two-level method converging in one iteration in the case where the right-hand side is only a function of the coarse variable is introduced and discussed. Such a fast convergence for problems with discontinuous coefficients arbitrarily varying on the fine scale is achieved by a problem-dependent selection of the coarse grid combined with problem-dependent prolongation on a dual grid. The results of the numerical experiments are presented to illustrate the performance of the studied approaches.

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Karavaev, D. A., and Y. M. Laevsky. "Simulation of heat transfer with considering permafrost thawing in 3D media." Journal of Physics: Conference Series 2099, no. 1 (November 1, 2021): 012007. http://dx.doi.org/10.1088/1742-6596/2099/1/012007.

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Abstract An approach to mathematical modeling of heat transfer with a permafrost algorithm in 3D media based on the idea of localizing the phase transition area is considered. The paper presents a problem statement for a non-stationary heat transfer and a description of a numerical method based on a predictor-corrector scheme. For a better understanding of the proposed splitting method, the accuracy order of approximation considering inhomogeneous right-hand side was studied. The phase changes in the numerical implementation of permafrost thawing is considered in the temperature range and requires recalculation of coefficients values of the heat equation at each iteration step with respect to time. A brief description of the parallel algorithm based on a 3D decomposition method and the parallel sweep method is presented. A study of the parallel algorithm implementations using a high-performance computing system of the Siberian Supercomputer Center of the SB RAS was performed. The results of the permafrost algorithm on models with wellbores are also presented.

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Scheiber, Ernest. "On the numerical Picard iterations with collocations for the initial value problem." Journal of Numerical Analysis and Approximation Theory 48, no. 1 (September 8, 2019): 89–105. http://dx.doi.org/10.33993/jnaat481-1146.

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Some variants of the numerical Picard iterations method are presented to solve an IVP for an ordinary differential system. The term "numerical" emphasizes that a numerical solution is computed. The method consists in replacing the right hand side of the differential system by Lagrange interpolation polynomials followed by successive approximations. In the case when the number of interpolation point is fixed a convergence result is given. Finally some numerical experiments are reported.

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Cordero, Alicia, Ivan Girona, and Juan R. Torregrosa. "A Variant of Chebyshev’s Method with 3αth-Order of Convergence by Using Fractional Derivatives." Symmetry 11, no. 8 (August 6, 2019): 1017. http://dx.doi.org/10.3390/sym11081017.

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In this manuscript, we propose several iterative methods for solving nonlinear equations whose common origin is the classical Chebyshev’s method, using fractional derivatives in their iterative expressions. Due to the symmetric duality of left and right derivatives, we work with right-hand side Caputo and Riemann–Liouville fractional derivatives. To increase as much as possible the order of convergence of the iterative scheme, some improvements are made, resulting in one of them being of 3 α -th order. Some numerical examples are provided, along with an study of the dependence on initial estimations on several test problems. This results in a robust performance for values of α close to one and almost any initial estimation.

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Matysik, O. V., and V. F. Savchuk. "Method of iteration of solving invariant equations with an approximate operator in the case of an arbitrary choice of the regularization parameter." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 54, no. 4 (January 11, 2019): 408–16. http://dx.doi.org/10.29235/1561-2430-2018-54-4-408-416.

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In the introduction, the object of investigation is indicated – incorrect problems described by first-kind operator equations. The subject of the study is an explicit iterative method for solving first-kind equations. The aim of the paper is to prove the convergence of the proposed method of simple iterations with an alternating step alternately and to obtain error estimates in the original norm of a Hilbert space for the cases of self-conjugated and non self-conjugated problems. The a priori choice of the regularization parameter is studied for a source-like representable solution under the assumption that the operator and the right-hand side of the equation are given approximately. In the main part of the work, the achievement of the stated goal is expressed in four reduced and proved theorems. In Section 1, the first-kind equation is written down and a new explicit method of simple iteration with alternating steps is proposed to solve it. In Section 2, we consider the case of the selfconjugated problem and prove Theorem 1 on the convergence of the method and Theorem 2, in which an error estimate is obtained. To obtain an error estimate, an additional condition is required – the requirement of the source representability of the exact solution. In Section 3, the non-self-conjugated problem is solved, the convergence of the proposed method is proved, which in this case is written differently, and its error estimate is obtained in the case of an a priori choice of the regularization parameter. In sections 2 and 3, the error estimates obtained are optimized, that is, a value is found – the step number of the iteration, in which the error estimate is minimal. Since incorrect problems constantly arise in numerous applications of mathematics, the problem of studying them and constructing methods for their solution is topical. The obtained results can be used in theoretical studies of solution of first-kind operator equations, as well as applied ill-posed problems encountered in dynamics and kinetics, mathematical economics, geophysics, spectroscopy, systems for complete automatic processing and interpretation of experiments, plasma diagnostics, seismic and medicine.

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Waheed, Umair bin, and Tariq Alkhalifah. "A fast sweeping algorithm for accurate solution of the tilted transversely isotropic eikonal equation using factorization." GEOPHYSICS 82, no. 6 (November 1, 2017): WB1—WB8. http://dx.doi.org/10.1190/geo2016-0712.1.

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Traveltime computation is essential for many seismic data processing applications and velocity analysis tools. High-resolution seismic imaging requires eikonal solvers to account for anisotropy whenever it significantly affects the seismic wave kinematics. Moreover, computation of auxiliary quantities, such as amplitude and take-off angle, relies on highly accurate traveltime solutions. However, the finite-difference-based eikonal solution for a point-source initial condition has upwind source singularity at the source position because the wavefront curvature is large near the source point. Therefore, all finite-difference solvers, even the high-order ones, show inaccuracies because the errors due to source-singularity spread from the source point to the whole computational domain. We address the source-singularity problem for tilted transversely isotropic (TTI) eikonal solvers using factorization. We solve a sequence of factored tilted elliptically anisotropic (TEA) eikonal equations iteratively, each time by updating the right-hand-side function. At each iteration, we factor the unknown TEA traveltime into two factors. One of the factors is specified analytically, such that the other factor is smooth in the source neighborhood. Through this iterative procedure, we obtain an accurate solution to the TTI eikonal equation. Numerical tests show significant improvement in accuracy due to factorization. The idea can be easily extended to compute accurate traveltimes for models with lower anisotropic symmetries, such as orthorhombic, monoclinic, or even triclinic media.

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Agarwal, Ravi P., Snezhana Hristova, Donal O’Regan, and Ricardo Almeida. "Approximate Iterative Method for Initial Value Problem of Impulsive Fractional Differential Equations with Generalized Proportional Fractional Derivatives." Mathematics 9, no. 16 (August 19, 2021): 1979. http://dx.doi.org/10.3390/math9161979.

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The main aim of the paper is to present an algorithm to solve approximately initial value problems for a scalar non-linear fractional differential equation with generalized proportional fractional derivative on a finite interval. The main condition is connected with the one sided Lipschitz condition of the right hand side part of the given equation. An iterative scheme, based on appropriately defined mild lower and mild upper solutions, is provided. Two monotone sequences, increasing and decreasing ones, are constructed and their convergence to mild solutions of the given problem is established. In the case of uniqueness, both limits coincide with the unique solution of the given problem. The approximate method is based on the application of the method of lower and upper solutions combined with the monotone-iterative technique.

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GIRAUD, L., J. LANGOU, and G. SYLVAND. "ON THE PARALLEL SOLUTION OF LARGE INDUSTRIAL WAVE PROPAGATION PROBLEMS." Journal of Computational Acoustics 14, no. 01 (March 2006): 83–111. http://dx.doi.org/10.1142/s0218396x06002780.

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The use of Fast Multipole Methods (FMM) combined with embedded Krylov solvers preconditioned by a sparse approximate inverse is investigated for the solution of large linear systems arising in industrial acoustic and electromagnetic simulations. We use a boundary element integral equation method to solve the Helmholtz and the Maxwell equations in the frequency domain. The resulting linear systems are solved by iterative solvers using FMM to accelerate the matrix-vector products. The simulation code is developed in a distributed memory environment using message passing and it has out-of-core capabilities to handle very large calculations. When the calculation involves one incident wave, one linear system has to be solved. In this situation, embedded solvers can be combined with an approximate inverse preconditioner to design extremely robust algorithms. For radar cross section calculations, several linear systems have to be solved. They involve the same coefficient matrix but different right-hand sides. In this case, we propose a block variant of the single right-hand side scheme. The efficiency, robustness and parallel scalability of our approach are illustrated on a set of large academic and industrial test problems.

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Laitinen, E., A. Lapin, and S. Lapin. "On the Iterative Solution Methods for Finite-Dimensional Inclusions with Applications to Optimal Control Problems." Computational Methods in Applied Mathematics 10, no. 3 (2010): 283–301. http://dx.doi.org/10.2478/cmam-2010-0016.

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AbstractIterative methods for finite-dimensional inclusions which arise in applying a finite-element or a finite-difference method to approximate state-constrained optimal control problems have been investigated. Specifically, problems of control on the right- hand side of linear elliptic boundary value problems and observation in the entire domain have been considered. The convergence and the rate of convergence for the iterative algorithms based on the finding of the control function or Lagrange multipliers are proved.

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Raus, Toomas, and Uno Hämarik. "ON NUMERICAL REALIZATION OF QUASIOPTIMAL PARAMETER CHOICES IN (ITERATED) TIKHONOV AND LAVRENTIEV REGULARIZATION." Mathematical Modelling and Analysis 14, no. 1 (March 31, 2009): 99–108. http://dx.doi.org/10.3846/1392-6292.2009.14.99-108.

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We consider linear ill‐posed problems in Hilbert spaces with noisy right hand side and given noise level. For approximation of the solution the Tikhonov method or the iterated variant of this method may be used. In self‐adjoint problems the Lavrentiev method or its iterated variant are used. For a posteriori choice of the regularization parameter often quasioptimal rules are used which require computing of additionally iterated approximations. In this paper we propose for parameter choice alternative numerical schemes, using instead of additional iterations linear combinations of approximations with different parameters.

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Calvetti, D., and L. Reichel. "Application of a block modified Chebyshev algorithm to the iterative solution of symmetric linear systems with multiple right hand side vectors." Numerische Mathematik 68, no. 1 (June 1, 1994): 3–16. http://dx.doi.org/10.1007/s002110050045.

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Laitinen, E., and Alexander Lapin. "Solution of a Finite-Dimensional Problem With M-mappings and Diagonal Multivalued Operators." Computational Methods in Applied Mathematics 1, no. 3 (2001): 242–64. http://dx.doi.org/10.2478/cmam-2001-0017.

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AbstractThe existence of a solution of the finite-dimensional problem with continuous M-mappings and multivalued diagonal maximal monotone operators on an ordered interval, which is formed by the so-called subsolution, is proved. Under several additional assumptions on the operators the monotone dependence of a solution upon the right-hand side is investigated. This result implies, in particular, the uniqueness of a solution and serves as a basis for the analysis of the convergence for a multisplitting iterative method. As an illustrative example, the finite difference scheme approximating a model variational inequality is studied by using the general results.

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Vion, Alexandre, and Christophe Geuzaine. "Improved sweeping preconditioners for domain decomposition algorithms applied to time-harmonic Helmholtz and Maxwell problems." ESAIM: Proceedings and Surveys 61 (2018): 93–111. http://dx.doi.org/10.1051/proc/201861093.

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Sweeping-type algorithms have recently gained a lot of interest for the solution of highfrequency time-harmonic wave problems, in particular when used in combination with perfectly matched layers. However, an inherent problem with sweeping approaches is the sequential nature of the process, which makes them inadequate for efficient implementation on parallel computers. We propose several improvements to the double-sweep preconditioner originally presented in [18], which uses sweeping as a matrix-free preconditioner for a Schwarz domain decomposition method. Similarly, the improved preconditioners are based on approximations of the inverse of the Schwarz iteration operator: the general methodology is to apply well-known algebraic techniques to the operator seen as a matrix, which in turn is processed to obtain equivalent matrix-free routines that we use as preconditioners. A notable feature of the new variants is the introduction of partial sweeps that can be performed concurrently in order to make a better usage of the resources. As these modifications still leave some unexploited computational power, we also propose to combine them with right-hand side pipelining to further improve parallelism and achieve significant speed-ups. Examples are presented on high-frequency Helmholtz and Maxwell problems, in two and three dimensions, to demonstrate the properties of our improvements on parallel computer architectures.

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Calandra, Henri, Serge Gratton, Rafael Lago, Xavier Vasseur, and Luiz Mariano Carvalho. "A Modified Block Flexible GMRES Method with Deflation at Each Iteration for the Solution of Non-Hermitian Linear Systems with Multiple Right-Hand Sides." SIAM Journal on Scientific Computing 35, no. 5 (January 2013): S345—S367. http://dx.doi.org/10.1137/120883037.

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Alber, Y. I., A. G. Kartsatos, and E. Litsyn. "Iterative solution of unstable variational inequalities on approximately given sets." Abstract and Applied Analysis 1, no. 1 (1996): 45–64. http://dx.doi.org/10.1155/s1085337596000024.

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The convergence and the stability of the iterative regularization method for solving variational inequalities with bounded nonsmooth properly monotone (i.e., degenerate) operators in Banach spaces are studied. All the items of the inequality (i.e., the operatorA, the “right hand side”fand the set of constraintsΩ) are to be perturbed. The connection between the parameters of regularization and perturbations which guarantee strong convergence of approximate solutions is established. In contrast to previous publications by Bruck, Reich and the first author, we do not suppose here that the approximating sequence is a priori bounded. Therefore the present results are new even for operator equations in Hilbert and Banach spaces. Apparently, the iterative processes for problems with perturbed sets of constraints are being considered for the first time.

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Kim, Hyea Hyun, Atul S. Minhas, and Eung Je Woo. "An Iterative Method for Problems with Multiscale Conductivity." Computational and Mathematical Methods in Medicine 2012 (2012): 1–12. http://dx.doi.org/10.1155/2012/893040.

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A model with its conductivity varying highly across a very thin layer will be considered. It is related to a stable phantom model, which is invented to generate a certain apparent conductivity inside a region surrounded by a thin cylinder with holes. The thin cylinder is an insulator and both inside and outside the thin cylinderare filled with the same saline. The injected current can enter only through the holes adopted to the thin cylinder. The model has a high contrast of conductivity discontinuity across the thin cylinder and the thickness of the layer and the size of holes are very small compared to the domain of the model problem. Numerical methods for such a model require a very fine mesh near the thin layer to resolve the conductivity discontinuity. In this work, an efficient numerical method for such a model problem is proposed by employing a uniform mesh, which need not resolve the conductivity discontinuity. The discrete problem is then solved by an iterative method, where the solution is improved by solving a simple discrete problem with a uniform conductivity. At each iteration, the right-hand side is updated by integrating the previous iterate over the thin cylinder. This process results in a certain smoothing effect on microscopic structures and our discrete model can provide a more practical tool for simulating the apparent conductivity. The convergence of the iterative method is analyzed regarding the contrast in the conductivity and the relative thickness of the layer. In numerical experiments, solutions of our method are compared to reference solutions obtained from COMSOL, where very fine meshes are used to resolve the conductivity discontinuity in the model. Errors of the voltage inL2norm followO(h)asymptotically and the current density matches quitewell those from the reference solution for a sufficiently small mesh sizeh. The experimental results present a promising feature of our approach for simulating the apparent conductivity related to changes in microscopic cellular structures.

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Zubov, N. E., and V. N. Ryabchenko. "Solution of a Linear Nondegenerate Matrix Equation Based on the Zero Divisor." Herald of the Bauman Moscow State Technical University. Series Natural Sciences, no. 5 (98) (October 2021): 49–59. http://dx.doi.org/10.18698/1812-3368-2021-5-49-59.

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New formulas were obtained to solve the linear non-degenerate matrix equations based on zero divisors of numerical matrices. Two theorems were formulated, and a proof to one of them is provided. It is noted that the proof of the second theorem is similar to the proof of the first one. The proved theorem substantiates new formula in solving the equation equivalent in the sense of the solution uniqueness to the known formulas. Its fundamental difference lies in the following: any explicit matrix inversion or determinant calculation is missing; solution is "based" not on the left, but on the right side of the matrix equation; zero divisor method is used (it was never used in classical formulas for solving a matrix equation); zero divisor calculation is reduced to simple operations of permutating the vector elements on the right-hand side of the matrix equation. Examples are provided of applying the proposed method for solving a nondegenerate matrix equation to the numerical matrix equations. High accuracy of the proposed formulas for solving the matrix equations is demonstrated in comparison with standard solvers used in the MATLAB environment. Similar problems arise in the synthesis of fast and ultrafast iterative solvers of linear matrix equations, as well as in nonparametric identification of abnormal (emergency) modes in complex technical systems, for example, in the power systems

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Rossmann, M. G. "The molecular replacement method." Acta Crystallographica Section A Foundations of Crystallography 46, no. 2 (February 1, 1990): 73–82. http://dx.doi.org/10.1107/s0108767389009815.

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Molecular replacement can be used for obtaining approximate phasing of an unknown structure from a known related molecule and for phase improvement as well as extension in the presence of noncrystallographic symmetry. Emphasis is placed on the latter procedure. It is shown that the real-space method of iterative electron density averaging and Fourier back transformation corresponds to iterative phase substitution in the right-hand side of expressions to give a set of improved phases. Analysis of these expressions (the 'molecular replacement equations') provides insight into the limits of possible phase extension, and the implications for the use of calculated structure factors when there are no observed amplitudes. It is shown that the percentage of observed data and inaccuracy of the observed amplitudes available for phase extension are compensated by the extent of noncrystallographic redundancy and the fraction of crystal cell volume that may be flattened because it is outside the control of noncrystallographic symmetry.

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Fedorchenko, Ludmila, and Sergey Baranov. "Equivalent Transformations and Regularization in Context-Free Grammars." Cybernetics and Information Technologies 14, no. 4 (January 31, 2015): 29–44. http://dx.doi.org/10.1515/cait-2014-0003.

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Abstract Regularization of translational context-free grammar via equivalent transformations is a mandatory step in developing a reliable processor of a formal language defined by this grammar. In the 1970-ies, the multi-component oriented graphs with basic equivalent transformations were proposed to represent a formal grammar of ALGOL-68 in a compiler for IBM/360 compatibles. This paper describes a method of grammar regularization with the help of an algorithm of eliminating the left/right-hand side recursion of nonterminals which ultimately converts a context-free grammar into a regular one. The algorithm is based on special equivalent transformations of the grammar syntactic graph: elimination of recursions and insertion of iterations. When implemented in the system SynGT, it has demonstrated over 25% reduction of the memory size required to store the respective intermediate control tables, compared to the algorithm used in Flex/Bison parsers.

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Sumin, V. I., and M. I. Sumin. "Regularized classical optimality conditions in iterative form for convex optimization problems for distributed Volterra-type systems." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 31, no. 2 (June 2021): 265–84. http://dx.doi.org/10.35634/vm210208.

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We consider the regularization of the classical optimality conditions (COCs) — the Lagrange principle and the Pontryagin maximum principle — in a convex optimal control problem with functional constraints of equality and inequality type. The system to be controlled is given by a general linear functional-operator equation of the second kind in the space $L^m_2$, the main operator of the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is strongly convex. Obtaining regularized COCs in iterative form is based on the use of the iterative dual regularization method. The main purpose of the regularized Lagrange principle and the Pontryagin maximum principle obtained in the work in iterative form is stable generation of minimizing approximate solutions in the sense of J. Warga. Regularized COCs in iterative form are formulated as existence theorems in the original problem of minimizing approximate solutions. They “overcome” the ill-posedness properties of the COCs and are regularizing algorithms for solving optimization problems. As an illustrative example, we consider an optimal control problem associated with a hyperbolic system of first-order differential equations.

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Mamtiyev, Kamil, Tarana Aliyeva, and Ulviyya Rzayeva. "Analysis of one class of optimal control problems for distributed-parameter systems." Eastern-European Journal of Enterprise Technologies 5, no. 4 (113) (October 29, 2021): 26–33. http://dx.doi.org/10.15587/1729-4061.2021.241232.

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In the paper, the method of straight lines approximately solves one class of optimal control problems for systems, the behavior of which is described by a nonlinear equation of parabolic type and a set of ordinary differential equations. Control is carried out using distributed and lumped parameters. Distributed control is included in the partial differential equation, and lumped controls are contained both in the boundary conditions and in the right-hand side of the ordinary differential equation. The convergence of the solutions of the approximating boundary value problem to the solution of the original one is proved when the step of the grid of straight lines tends to zero, and on the basis of this fact, the convergence of the approximate solution of the approximating optimal problem with respect to the functional is established. A constructive scheme for constructing an optimal control by a minimizing sequence of controls is proposed. The control of the process in the approximate solution of a class of optimization problems is carried out on the basis of the Pontryagin maximum principle using the method of straight lines. For the numerical solution of the problem, a gradient projection scheme with a special choice of step is used, this gives a converging sequence in the control space. The numerical solution of one variational problem of the mentioned type related to a one-dimensional heat conduction equation with boundary conditions of the second kind is presented. An inequality-type constraint is imposed on the control function entering the right-hand side of the ordinary differential equation. The numerical results obtained on the basis of the compiled computer program are presented in the form of tables and figures. The described numerical method gives a sufficiently accurate solution in a short time and does not show a tendency to «dispersion». With an increase in the number of iterations, the value of the functional monotonically tends to zero

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Slobodyanenko, Alexandr A. "A projection method for solving discretezed inverse problems for antenna measurements." Proceedings of the Russian higher school Academy of sciences, no. 2 (June 30, 2022): 36–45. http://dx.doi.org/10.17212/1727-2769-2022-2-36-45.

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In the article, the problems of determining and monitoring antenna parameters, solved using measurements in the near field, are presented in a unified formulation corresponding to the gene-ralized inverse radiation problem in the form of an integral equation of the first kind. Within the framework of the numerical approach to the solution, the equation is discretized, and then the solution is sought approximately as a solution to the normal system of equations. It is shown that an approximate solution to the discretized problem that satisfies the normal system is also a solution to an equation with an orthogonal projector on its right-hand side. On the basis of this formulation, a new method of iterative projections is proposed. The efficiency of the method is demonstrated by the example of solving the problem of reconstructing the radiation pattern of a horn antenna from near-field data calculated for the electrodynamic model of the antenna in the HFSS program. The results of the proposed method are compared with the results of solving the problem by the method of decomposition in singular numbers.

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Hillier, Grant. "ON THE CONDITIONAL LIKELIHOOD RATIO TEST FOR SEVERAL PARAMETERS IN IV REGRESSION." Econometric Theory 25, no. 2 (April 2009): 305–35. http://dx.doi.org/10.1017/s0266466608090105.

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For the problem of testing the hypothesis that all m coefficients of the right-hand-side endogenous variables in an instrumental variables (IV) regression are zero, the likelihood ratio (LR) test can, if the reduced form covariance matrix is known, be rendered similar by a conditioning argument. To exploit this fact requires knowledge of the relevant conditional cumulative distribution function (c.d.f.) of the LR statistic, but the statistic is a function of the smallest characteristic root of an (m + 1)-square matrix and is therefore analytically difficult to deal with when m > 1. We show in this paper that an iterative conditioning argument used by Hillier (2009) and Andrews, Moreira, and Stock (2007 Journal of Econometrics 139, 116–132) to evaluate the c.d.f. in the case m = 1 can be generalized to the case of arbitrary m. This means that we can completely bypass the difficulty of dealing with the smallest characteristic root. Analytic results are obtained for the case m = 2, and a simple and efficient simulation approach to evaluating the c.d.f. is suggested for larger values of m.

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Boikov, I. V., Ya V. Zelina, and D. I. Vasyunin. "Approximate methods for solving amplitude-phase problem for discrete signals." Journal of Physics: Conference Series 2099, no. 1 (November 1, 2021): 012002. http://dx.doi.org/10.1088/1742-6596/2099/1/012002.

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Abstract Methods for solving amplitude and phase problems for one and two-dimensional discrete signals are proposed. Methods are based on using nonlinear singular integral equations. In the one-dimensional case amplitude and phase problems are modeled by corresponding linear and nonlinear singular integral equations. In the two-dimensional case amplitude and phase problems are modeled by corresponding linear and nonlinear bisingular integral equations. Several approaches are presented for modeling two-dimensional problems: 1) reduction of amplitude and phase problems to systems of linear and nonlinear singular integral equations; 2) using methods of the theory of functions of many complex variables, problems are reduced to linear and nonlinear bisingular integral equations. To solve the constructed nonlinear singular integral equations, methods of collocation and mechanical quadrature are used. These methods lead to systems of nonlinear algebraic equations, which are solved by the continuous method for solution of nonlinear operator equations. The choice of this method is due to the fact that it is stable against perturbations of coefficients in the right-hand side of the system of equations. In addition, the method is realizable even in cases where the Frechet and Gateaux derivatives degenerate at a finite number of steps in the iterative process. Some model examples have shown effectiveness of proposed methods and numerical algorithms.

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Shishkin, Gregorii I. "On Conditioning of a Schwarz Method for Singularly Perturbed Convection–diffusion Equations in the Case of Disturbances in the Data of the Boundary-value Problem." Computational Methods in Applied Mathematics 3, no. 3 (2003): 459–87. http://dx.doi.org/10.2478/cmam-2003-0030.

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AbstractIn this paper we discuss conditioning of a discrete Schwarz method on piecewise–uniform meshes with an example of a one-dimensional singularly perturbed boundary-value problem. We consider a Dirichlet problem for singularly perturbed ordinary differential equations with convection terms and a small perturbation parameter ε. To solve the problem numerically we use an ε-uniformly convergent (in the maximum norm) difference scheme on special piecewise–uniform meshes. For this base scheme we construct a decomposition scheme based on a Schwarz technique with overlapping subdomains, which converges ε-uniformly with respect to both the number of mesh points and the number of iterations. The step-size of such special meshes is extremely small in the neighborhood of the layer and changes sharply on its boundary, that (as was shown by A.A. Samarskii) can generally lead to a loss of well-conditioning of the above schemes. For the decomposition scheme we study the conditioning of the system (difference scheme) and the conditioning of the system matrix (difference operator), and also the influence of perturbations in the data of the boundary-value problem on disturbances of its numerical solutions. We derive estimates for the disturbances of the numerical solutions (in the maximum norm) depending on the subdomain in which the disturbance of the data appears. It is shown that the condition number of the difference operator associated with the Schwarz method, just as for the base scheme, is not ε-uniformly bounded. However, these difference schemes are well-conditioned ε-uniformly (with the ε-uniform estimate for the condition number being the same as for the schemes on uniform meshes for regular problems) when the right-hand side of the discrete equations is considered in a “natural” norm, i.e., in the maximum norm with a special weight multiplier. In the case of the boundary-value problem with perturbed data we give conditions under which the solution of the iterative scheme based on the overlapping Schwarz method is convergent ε-uniformly to the solution of this Dirichlet problem as the number of mesh points and the number of iterations increase.

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Simoncini, V., and E. Gallopoulos. "An Iterative Method for Nonsymmetric Systems with Multiple Right-Hand Sides." SIAM Journal on Scientific Computing 16, no. 4 (July 1995): 917–33. http://dx.doi.org/10.1137/0916053.

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Fischer, Paul F. "Projection techniques for iterative solution of with successive right-hand sides." Computer Methods in Applied Mechanics and Engineering 163, no. 1-4 (September 1998): 193–204. http://dx.doi.org/10.1016/s0045-7825(98)00012-7.

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Jbilou, K. "Smoothing iterative block methods for linear systems with multiple right-hand sides." Journal of Computational and Applied Mathematics 107, no. 1 (July 1999): 97–109. http://dx.doi.org/10.1016/s0377-0427(99)00083-7.

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Samnioti, Anna, Eirini Maria Kanakaki, Sofianos Panagiotis Fotias, and Vassilis Gaganis. "Rapid Hydrate Formation Conditions Prediction in Acid Gas Streams." Fluids 8, no. 8 (August 5, 2023): 226. http://dx.doi.org/10.3390/fluids8080226.

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Sour gas in hydrocarbon reservoirs contains significant amounts of H2S and smaller amounts of CO2. To minimize operational costs, meet air emission standards and increase oil recovery, operators revert to acid gas (re-)injection into the reservoir rather than treating H2S in Claus units. This process requires the pressurization of the acid gas, which, when combined with low-temperature conditions prevailing in subsurface pipelines, often leads to the formation of hydrates that can potentially block the fluid flow. Therefore, hydrates formation must be checked at each pipeline segment and for each timestep during a flow simulation, for any varying composition, pressure and temperature, leading to millions of calculations that become more intense when transience is considered. Such calculations are time-consuming as they incorporate the van der Walls–Platteeuw and Langmuir adsorption theory, combined with complex EoS models to account for the polarity of the fluid phases (water, inhibitors). The formation pressure is obtained by solving an iterative multiphase equilibrium problem, which takes a considerable amount of CPU time only to provide a binary answer (hydrates/no hydrates). To accelerate such calculations, a set of classifiers is developed to answer whether the prevailing conditions lie to the left (hydrates) or the right-hand (no hydrates) side of the P-T phase envelope. Results are provided in a fast, direct, non-iterative way, for any possible conditions. A set of hydrate formation “yes/no” points, generated offline using conventional approaches, are utilized for the classifier’s training. The model is applicable to any acid gas flow problem and for any prevailing conditions to eliminate the CPU time of multiphase equilibrium calculations.

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Калинина, А. Б., А. А. Корнев, and В. С. Назаров. "On the hybrid projection method to a stable manifold of a one-dimensional Burgers-type equation." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 2 (April 5, 2023): 170–81. http://dx.doi.org/10.26089/nummet.v24r213.

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В работе рассматривается уравнение типа Бюргерса с полиномиальной нелинейностью и нулевыми краевыми условиями. Для интересующего диапазона параметров тождественно нулевое решение задачи является локально неустойчивым, и в его окрестности существует устойчивое многообразие, имеющее конечную коразмерность. Для приближенного построения указанного многообразия предложен комбинированный итерационный алгоритм, начальное условие для которого строится аналитическим методом и имеет квадратичную точность. Численно показано, насколько существенно данная модификация позволяет уменьшить для типичных значений параметров вычислительную сложность проецирования на искомое многообразие по сравнению со стандартным линейным приближением. Полученные результаты допускают обобщение на многомерные диссипативные уравнения широкого класса и могут применяться при решении задач асимптотической стабилизации по начальным данным, краевым условиям и правой части. The paper considers a Burgers type equation with polynomial nonlinearity and zero boundary conditions. For the range of parameters of interest, the identically zero solution of the problem is locally unstable, and in its neighborhood there exists a stable manifold having finite codimension. For the approximate construction of this manifold a combined iterative algorithm the initial data for which is constructed by an analytical method and has quadratic accuracy is proposed. It is numerically shown how significant this modification is allows to reduce the computational complexity of projection on the desired manifold for typical parameter values compared to the standard linear approximation. The results obtained allow generalization to multidimensional dissipative equations of a wide class and can be used to solve problems of asymptotic stabilization based on initial data, boundary conditions and a right-hand side.

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Al Daas, Hussam, Laura Grigori, Pascal Hénon, and Philippe Ricoux. "Enlarged GMRES for solving linear systems with one or multiple right-hand sides." IMA Journal of Numerical Analysis 39, no. 4 (August 28, 2018): 1924–56. http://dx.doi.org/10.1093/imanum/dry054.

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Abstract We propose a variant of the generalized minimal residual (GMRES) method for solving linear systems of equations with one or multiple right-hand sides. Our method is based on the idea of the enlarged Krylov subspace to reduce communication. It can be interpreted as a block GMRES method. Hence, we are interested in detecting inexact breakdowns. We introduce a strategy to perform the test of detection. Furthermore, we propose a technique for deflating eigenvalues that has two benefits. The first advantage is to avoid the plateau of convergence after the end of a cycle in the restarted version. The second is to have very fast convergence when solving the same system with different right-hand sides, each given at a different time (useful in the context of a constrained pressure residual preconditioner). We test our method with these deflation techniques on academic test matrices arising from solving linear elasticity and convection–diffusion problems as well as matrices arising from two real-life applications, seismic imaging and simulations of reservoirs. With the same memory cost we obtain a saving of up to $50 \%$ in the number of iterations required to reach convergence with respect to the original method.

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Journal articles on the topic 'Right Hand Side iteration'

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