P. N. Brown and A. C. Hindmarsh, Matrix-free methods for stiff systems of ODE's, SIAM J. Numer. Anal., 23 (1986), pp. 610--638.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....aimed at reducing the amount of computations. Instead of solving exactly a linear system at each simplified Newton step, we apply an iterative method to a corresponding preconditioned linear system. The use of iterative methods for the numerical solution of stiff ODE s were already considered in [3, 7, 12], with an emphasis on preconditioning in [4] Such inexact Newton methods are generally considered to be among the most efficient ways to solve nonlinear system of equations [11, 25] We construct a preconditioner requiring s independent decompositions of matrices of dimension n, i.e. whose ....

P. N. Brown and A. C. Hindmarsh, Matrix-free methods for stiff systems of ODE's, SIAM J. Numer. Anal., 23 (1986), pp. 610--638.

....on explicit matrix representations of the Jacobian operator. The advent of Krylov iterative methods (see, e.g. 65] for a survey) inside of inexact Newton iterations in a matrix free context can be traced to the ODE oriented papers of Gear Saad [30] Chan Jackson [18] and Brown Hindmarsh [9] and the PDE oriented work of Brown Saad [10] The term Newton Krylov seems first to have been applied to such problems in [10] The GMRES [67] method was firmly established in CFD following the work of Wigton, Yu, Young [91] and Johann, Hughes, Shakib [38, 70] Venkatakrishnan ....

P. N. Brown and A. C. Hindmarsh. Matrix-free methods for stiff systems of ODE's. SIAM J. Numerical Analysis, 23:610--638, 1986.

....types, assuming the matrix exports the required interface. A second level of flexibility is available in that the user may specialize the mtl: mult(A,x,y) function for a custom matrix type, and bypass the MTL generic algorithms. An example use of this would be to perform matrix free computations [9, 8]. Another possibility would be in using a distributed matrix with parallel versions of the linear algebra operations. 7.2 Ease of Implementation The most significant benefit of layering the ITL on top of the MTL interface is the ease of implementation. The ITL algorithms can be expressed in a ....

P. N. Brown and A. C. Hindmarsh. Matrix-free methods for stiff systems of ODE's. SIAM J. Numer. Anal., 23(3):610--638, June 1986.

....use of sparse direct methods for solving (6) in the previous section; we consider iterative schemes here. Recently, most attention has been focused on preconditioned Krylov subspace methods. For a discussion of this class of iterative linear equation solvers in the context of stiff ODE codes, see [6, 7, 14, 20, 31] and the references therein. In the introductory comments of VODPK [4] Brown, Byrne and Hindmarsh state: To achieve an efficient solution, the preconditioned Krylov methods in VODPK generally require a thoughtful choice of preconditioners. If the ODE system produces linear systems that are ....

....iteration is used in effect only to stabilize the explicit method that computes the initial guess y (0) n . However this leads to an important question: What is an appropriate stopping criterion for the iterative solution of (6) Although this question has been addressed by several authors (see [6, 7, 14, 20, 31] and the references therein) I do not believe it has been satisfactorily resolved. 9 As noted in x4, Krylov subspace methods are particularly effective when the system of ODEs has only a few stiff components. In this context, preconditioning may not be needed and a matrix free implementation ....

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P. N. Brown and A. C. Hindmarsh, Matrix-Free Methods for Stiff Systems of ODE's, SIAM J. Numer. Anal., 23, 1986, 610--638.

....linear residual is small, i.e. when kF 0 ## c #s F ## c #k## r kF## c #k (2.5) for some small # r . 2.5) is called the inexact Newton condition. The parameter # r is called the forcing term. In applications to temporal integration, one can use absolute residuals and gain some efficiency [3]. We can express convergence results for both approaches in terms of the termination condition kF 0 ## c #s F ## c #k## a # r kF## c #k: 2.6) If the initial iterate is near the solution, the nonlinearity is sufficiently smooth, and (2.6) holds for some # a and 0 # # r # 1 then the error in ....

....convergence results for both approaches in terms of the termination condition kF 0 ## c #s F ## c #k## a # r kF## c #k: 2.6) If the initial iterate is near the solution, the nonlinearity is sufficiently smooth, and (2. 6) holds for some # a and 0 # # r # 1 then the error in # satisfies, [3, 11, 19], ke k = O#ke c k 2 # r ke c k # a #: 2.7) Clearly, if # c is near # # and # r is sufficiently small the iteration will converge rapidly. However, solving the equation for the Newton step, 2.4) to very high accuracy may be wasteful, particularly if the initial iterate is far from ....

P. N. BROWN AND A. C. HINDMARSH, Matrix-free methods for stiff systems of ODE's,SIAMJ.Numer.Anal., 23 (1986), pp. 610--638.

....the computational load. Instead of solving exactly the linear systems of the simplified Newton iterations, we solve approximately and iteratively a preconditioned version of those linear systems. The use of linear iterative methods for the solution of implicit integration methods was considered in [3, 8, 15], with an emphasis on preconditioning in [4] Here, we use a preconditioner requiring at most s independent decompositions of matrices of 13 dimension n. Hence, the decomposition cost for a parallel implementation is equivalent to the cost for the implicit Euler method. A detailed presentation ....

P. N. Brown and A. C. Hindmarsh, Matrix-free methods for stiff systems of ODE's, SIAM J. Numer. Anal., 23 (1986), pp. 610--638.

....linear residual is small, i.e. when kF 0 ( c )s F ( c )k j r kF ( c )k (2.5) for some small j r . 2.5) is called the inexact Newton condition. The parameter j r is called the forcing term. In applications to temporal integration, one can use absolute residuals and gain some efficiency [3]. We can express convergence results for both approaches in terms of the termination condition kF 0 ( c )s F ( c )k j a j r kF ( c )k: 2.6) If the initial iterate is near the solution, the nonlinearity is sufficiently smooth, and (2.6) holds for some j a and 0 j r 1 then the error ....

....convergence results for both approaches in terms of the termination condition kF 0 ( c )s F ( c )k j a j r kF ( c )k: 2.6) If the initial iterate is near the solution, the nonlinearity is sufficiently smooth, and (2. 6) holds for some j a and 0 j r 1 then the error in satisfies, [3, 11, 19], ke k = O(ke c k 2 j r ke c k j a ) 2.7) Clearly, if c is near and j r is sufficiently small the iteration will converge rapidly. However, solving the equation for the Newton step, 2.4) to very high accuracy may be wasteful, particularly if the initial iterate is far from the ....

P. N. BROWN AND A. C. HINDMARSH, Matrix-free methods for stiff systems of ODE's, SIAM J. Numer. Anal., 23 (1986), pp. 610--638.

....of the nonlinear problem is based on both kffix n k1 and kF (x n )k 2 . In results presented here we require that they both be less than 10 Gamma5 . The Newton Krylov implementation used here replaces the Jacobian vector products in the GMRES algorithm with finite difference projections [4]. Consequently, terms of the form JP Gamma1 v are computed from, JP Gamma1 v F (x fflP Gamma1 v) Gamma F (x) ffl : 2) Here v is a GMRES vector and ffl is a small perturbation constant. The advantage of using this implementation is that it approximates the action of the Jacobian ....

Brown, P.N., Hindmarsh, A.C. : Matrix-free methods for stiff systems of ODE's. SIAM J. Numer. Anal. 23 (1986),

....available when integrating an ODE system. For example, the solution at the last timestep, or the solution derived from a less accurate solution method, such as an explicit or multistep solver, are both good possibilities. We have implemented the parallel ODE solver with matrix free methods [18] [4], where matrices are not assembled as collections of numbers, but are instead passed around as functions. The kernel of matrix free methods may be stated quite simply: The fundamental concept is a linear transformation, not the matrix that represents it. Iterative methods access the matrix ....

....is approximated by ignoring its off diagonal elements. To use this, the definition of a matrix must be expanded, so that there are three methods: the action of the matrix, the action of the transpose, and the action of the inverse of the diagonal. Although preconditioners have been suggested [4] that are purely matrix free, such as the Incomplete Orthogonalization Method [17] it is still generally true that more effective preconditioners require more complex software interfaces. 2.3.5 Archiving the Solution To compute the error bound on the approximate solution of the ODE, we solve a ....

Brown, P. N. and Hindmarsh, A. C., Matrix-free methods for stiff systems of ODE's, SIAM J. Numer. Anal., 23 (1986), 610-638.

....aimed at reducing the amount of computations. Instead of solving exactly a linear system at each simplified Newton step, we apply an iterative method to a corresponding preconditioned linear system. The use of iterative methods for the numerical solution of stiff ODE s was already considered in [3,7,12], with an emphasis on preconditioning in [4] Inexact Newton methods are generally considered to be among the most efficient ways to solve nonlinear system of equations [11,25] We construct here a preconditioner requiring s independent decompositions of matrices of dimension n, i.e. whose ....

P. N. Brown and A. C. Hindmarsh. Matrix-free methods for stiff systems of ODE's. SIAM J. Numer. Anal., 23:610--638, 1986.

....spend most of their computing time for the evaluation of the Jacobian and in the solution of the linear equations. Recently, Krylov methods have been used extensively in implicit methods for stiff ordinary differential equations to accelerate the linear algebra in these integration schemes, e.g. [2, 3, 4, 15]. An alternative approach for large stiff equations are methods containing matrix exponentials as exact solutions of linearized equations. An ODE integrator based on Krylov approximations for such matrix functions multiplied by a vector is discussed in [7] We will study the use of Krylov ....

P.N. Brown and A.C. Hindmarsh, Matrix-free methods for stiff systems of ODE's, SIAM J. Numer. Anal. 23 (1986), 610--638.

....is to use Newton s method or a variant such as the chord method. In this paper we consider another variant called an inexact Newton method [6] 5] In an inexact Newton method, the linear system for the Newton step is solved using an iterative method, such as those described above. Previous work [2], 3] 4] has shown how to implement this method into an ODE DAE solver, but some questions remain on how to terminate the linear iteration effectively. If the termination criteria is too tight, then the method may be inefficient, and if the termination criteria is too loose, the method may no ....

....criteria is too loose, the method may no longer be accurate. To select an appropriate termination criterion, we present a theorem in x2 and, in x3, an adaptive algorithm and some numerical results. 2. Inexact Newton Methods. In this section, we present an extension of the convergence results in [2] and [3] on the convergence of Newton s method when an iterative method is used to solve the linear system for the step. Our convergence result shows the relation between the effect of the linear solver s convergence criterion and the reduction in the nonlinear error in a way that can be used ....

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Peter N. Brown and Alan C. Hindmarsh. Matrix-free methods for stiff systems of ODE's. SIAM Journal of Numerical Analysis, 23(3):610 -- 638, 1986.

....ffl Solve (7) with WGMRES ffl Update x m 1 For the WGMRES algorithm applied to solving (7) the required operator function product can be computed using the formulas in Section 2. 3, with the splitting J F (x m (t) M(t) Gamma N (t) It is also possible to use a Jacobian free approach [8], but the nature of the linearization in the operator Newton algorithm makes that approach somewhat unreliable [24] Within each nonlinear (operator Newton) iteration, the initial residual for the WGMRES algorithm must be computed. Denote the initial guess for x m 1 in the WGMRES part of the ....

P. N. Brown and A. C. Hindmarsh, Matrix-free methods for stiff systems of ODE's, SIAM J. Numer. Anal., 23 (1986), pp. 610--638.

....assuming the matrix exports the required interface. A second level of flexibility is available in that the user may specialize the matvec: mult(A,x,y) function for a custom matrix type, and bypass the MTL generic algorithms. An example use of this would be to perform matrix free computations [3, 4]. Another possibility would be in using a distributed matrix with parallel versions of the linear algebra operations. 7.2 Ease of Implementation The most significant benefit of layering the ITL on top of the MTL interface is the ease of implementation. The ITL algorithms can be expressed in a ....

Peter N. Brown and Alan C. Hindmarsh. Matrix-free methods for stiff systems of ODE's. SIAM J. Numer. Anal., 23(3):610--638, June 1986.

....65L05, 65F10. 1. Introduction. In recent years, there have been several investigations of the use of iterative linear equation solvers in codes for the numerical solution of large systems of stiff initial value problems (IVPs) for ordinary differential equations (ODEs) See, for example, [5, 6, 8, 11]. Such work has established clearly the potential effectiveness of the combination of these methods, but there are many open questions concerning both the choice of iterative method and the way in which it interacts with strategies used in the ODE solver. Frequently, the iterative methods ....

....l 1 n = y l n Delta l n This iteration converges at least linearly if kd l n k j l kF (y l n )k; 4) where 0 j l j 1. In the case where an iterative linear equation solver is used to find y l n , the residual d l n is the final residual of the linear iteration. Brown and Hindmarsh [5] consider iterative linear equation solvers based on Arnoldi s method for use in the ODE code LSODE [12] In their approach, the matrixvector product required in the iterative method is approximated by a finite difference based on f . This approach avoids explicitly forming the iteration matrix W ....

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P. Brown and A. Hindmarsh, Matrix-free methods for stiff systems of ODE's, SIAM J. Numer. Anal., 23 (1986), pp. 610--638.

....of Lemma 2.1 to hold. Then there are ffl 3 0 and j such that if j n j, kx 0 Gamma x k ffl 3 , and ffi 0 0, either inf n ffi n = 0 or x n x and ffi n ffi max . Inexact Newton methods, in particular Newton Krylov solvers, have been applied to ODE DAE solvers in [1] 5] [4], 6] 7] and [9] The context here is different in that the nonlinear residual F (x) does not reflect the error in the transient solution but in the steady state solution. The analysis of the initial phase changes through (2.10) We must now estimate ffi n V Gamma1 F (x n ) s n : Set r ....

P. N. BROWN AND A. C. HINDMARSH, Matrix-free methods for stiff systems of ODE's, SIAM J. Numer. Anal., 23 (1986), pp. 610--638.

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P. N. Brown and A. C. Hindmarsh, Matrix-free methods for stiff systems of ODE's, SIAM J. Numer. Anal., 23 (1986), pp. 610--638.

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