M. Mascagni, (1990) High-Dimensional Numerical Integration and Massively Parallel Computing, Contemporary Mathematics, 115, pp. 53--73. 14  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of the quadrature rule pair to an interval requires a number of evaluations of the integrand which can be executed in parallel; this fine grain parallelism is well suited to SIMD machines and vector processors Genz [9] considers the approximation of multiple integrals on an ICL DAP and Mascagni [17] considers the same problem on a Connection Machine; Gladwell [13] considers the vectorisation of the NAG routine D01AKF on a CRAY 1. However this parallelism may be too fine grained (depending, of course, on the cost of integrand evaluation) for efficient implementation on a MIMD machine. It is ....

M. Mascagni, (1990) High-Dimensional Numerical Integration and Massively Parallel Computing, Contemporary Mathematics, 115, pp. 53--73. 14

....might apply probabilistic methods. We stress, incidentally, that in this approach parallelization is trivial, since the solution at every point x 2 Omega is evaluated as an average over Brownian trajectories which can be generated independently (and hence on several different processors) cf. [21], e.g. The direct application of such methods to the nonlinear evolution equation (44) is not straightforward, cf. 27] C) A single semilinear parabolic equation in C 0 ( Omega Gamma3 The following example shows an application in a Banach space which is not a Hilbert space (cf. Remark 3.4) ....

M. Mascagni, High dimensional numerical integration and massively parallel computing, Contemporary Mathematics 115 (1991), 53-73.

....For MIMD architectures, other sources of overhead, such as synchronisation and communication, may also be a problem for this approach, as we shall see in Chapter 5. Parallelism at this level is, however generally well suited to vector architectures (see [22] and array processors (see [18] and [34]) and variants of library routines (such as D01AUF in the NAG library) are designed to exploit this. For efficient parallel algorithms on MIMD machines, it is necessary to exploit coarser grained parallelism. To do this, Algorithm CP is modified by, at each stage, identifying a set of subintervals ....

Mascagni, M., (1990) High-dimensional numerical integration and massively parallel computing , Contemporary Mathematics, vol. 115, pp. 53--73.

....For MIMD architectures, other sources of overhead, such as synchronisation and communication, may also be a problem for this approach, as we shall see in Chapter 5. Parallelism at this level is, however generally well suited to vector architectures (see [22] and array processors (see [18] and [34]) and variants of library routines (such as D01AUF in the NAG library) are designed to exploit this. For efficient parallel algorithms on MIMD machines, it is necessary to exploit coarser grained parallelism. To do this, Algorithm CP is modified by, at each stage, identifying a set of subintervals ....

Mascagni, M., (1990) High-dimensional numerical integration and massively parallel computing , Contemporary Mathematics, vol. 115, pp. 53--73.

....equation) play a fundamental role in the construction of these Wiener measures. It turns out that this probabilistic theory for representing the solutions of linear elliptic and parabolic PDEs has many applications in analysis [2] 4] and as we will see below, in numerical computation as well [1] [3]. As a simple example of the application of these ideas to computation let us consider the Dirichlet BVP for the Laplace equation: Gamma Deltau(x) 0; x 2 Omega ; u(x) g(x) x 2 Omega (1) The probabilistic representation of equation 1, often called the Wiener integral representation, ....

....the starting point of a new forward random walk which first encountered the boundary at the given boundary location. In fact, it can be proven that scoring the boundary value at each point in the retraced path is probabilistically equivalent to the forward random walk algorithm discussed above [3]. In addition, retracing has the advantage that we obtain one sample per walker per step. Finally, it should be obvious that the notion of retracing is superfluous, as it is more efficient to start our walkers at the boundary. Thus by trying to avoid an extremely inefficient aspect of a particular ....

[Article contains additional citation context not shown here]

M. Mascagni, High dimensional numerical integration and massively parallel computing, Contemporary Mathematics, 115(1991), pg. 53--73.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC