Gallopoulos, E. and Saad, Y. (1989). On the parallel solution of parabolic equations. In Proc. ACM SIGARCH-89, 17--28.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... process, we are faced with the computation of the quantities e GammaD(t Gammat 0 ) z ( t ) where is the index which corresponds to the accepted iterate z ( Efficient techniques for doing this based on Krylov techniques are described in Sidje [21] and Gallopoulos and Saad [7]. As will be explained later for some iterative schemes (block Gauss Jacobi for example) the matrix D will have such a form that the computation of e Dt and e GammaDt will be trivial, if Q has a block tridiagonal structure. In the case that C and D commute (equivalently, Q and D commute) ....

E. Gallopoulos and Y. Saad, On the parallel solution of parabolic equations. In Proc. ACM SIGARCH-89, 17-28.

....to be of more value to the practitioner now. The reader interested in large scale parallelism for IVPs should see [1, 3, 4, 17, 18, 29, 30, 31, 33, 37, 38, 41, 43, 44, 46, 47, 48, 51, 54, 60] and the references therein. For the applications of these methods to partial differential equations, see [13, 14, 35, 36]. General Discussion The desire for parallel IVP solvers arises from the need to solve many important problems more rapidly than is currently possible. This may be because the solution is needed in real time, as is the case for flight simulators or control systems, or it may be because ....

E. Gallopoulos and Y. Saad, "On the parallel solution of parabolic equations", in Proc. 1989 ACM Int'l Conf. on Supercomputing, Herakleion, Greece, pp. 17-- 28, 1989.

....Aw (16) w(0) b (17) by any technique and save the vectors w(t i ) i = 1; 2; m at the points t i . Alternatively, one may compute w(t i 1 ) from w(t i ) by w(t i 1 ) e A Deltat i w(t i ) where Deltat i = t i 1 Gamma t i , in a number of efficient ways as has been recently suggested in [4]. The problem considered in [4] is to approximate the product of the exponential of a matrix times a vector. We will describe one such approach in some detail in Section 4. Thus, the class of algorithms based on numerical quadrature would begin by choosing a quadrature rule and performing the ....

....and save the vectors w(t i ) i = 1; 2; m at the points t i . Alternatively, one may compute w(t i 1 ) from w(t i ) by w(t i 1 ) e A Deltat i w(t i ) where Deltat i = t i 1 Gamma t i , in a number of efficient ways as has been recently suggested in [4] The problem considered in [4] is to approximate the product of the exponential of a matrix times a vector. We will describe one such approach in some detail in Section 4. Thus, the class of algorithms based on numerical quadrature would begin by choosing a quadrature rule and performing the following procedure. Algorithm 1 ....

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E. Gallopoulos and Y. Saad. On the parallel solution of parabolic equations. In R. De Groot, editor, Proceedings of the International Conference on Supercomputing 1989, Heraklion, Crete, June 5-9, 1989. ACM press, 1989.

....difficult to do because of the fact that e A will in general be a dense matrix even when A is very sparse. However, it is often not the exponential of the matrix that is sought but its product with some vector v. The question of approximating e A v for any given vector v was considered in [13] where polynomial and rational approximations to the exponential were used. Here we summarize only the method proposed in [13] that is based on polynomial approximation to e A v: e A v p m Gamma1 (A)v (14) where p m Gamma1 is a polynomial of degree m Gamma 1. Thus, the above approximation ....

....is often not the exponential of the matrix that is sought but its product with some vector v. The question of approximating e A v for any given vector v was considered in [13] where polynomial and rational approximations to the exponential were used. Here we summarize only the method proposed in [13] that is based on polynomial approximation to e A v: e A v p m Gamma1 (A)v (14) where p m Gamma1 is a polynomial of degree m Gamma 1. Thus, the above approximation is an element of the Krylov subspace (1) and it is convenient to express it in the orthonormal basis Vm = v 1 ; v 2 ; v 3 ; ....

[Article contains additional citation context not shown here]

E. Gallopoulos and Y. Saad. On the parallel solution of parabolic equations. In R. De Groot, editor, Proceedings of the International Conference on Supercomputing 1989, Heraklion, Crete, June 5-9, 1989. ACM press, 1989.

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Gallopoulos, E. and Saad, Y. (1989). On the parallel solution of parabolic equations. In Proc. ACM SIGARCH-89, 17--28.

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