JPH04365171A - Parallel numerical arithmetic system - Google Patents

Abstract

PURPOSE:To execute arithmetic operation at high speed by using MIMD type parallel computers by parallelizing a positive solution part by dividing a vector, and parallelizing a negative solution part by pipeline processing. CONSTITUTION:In a positive solution to update the vector U from the vectors U, W shown in a figure, the new element 13 of the vector U is updated by referring to the elements 10, 11, 12. At that time, by arranging the elements in a processor PE2 as shown in the figure, the processor PE2 can calculate the positive solution part by only receiving the elements 10, 14 from the neighboring processors E1, E3. In the processing to obtain Vn,k from the vector Un,k by forward elimination, and further, to obtain Wn,k by back-substitution, by obtaining Vn,k, Wn,k by using plural processors in a pipeline-like way, the negative solution part can be calculated.

Description

[Detailed description of the invention]

0001

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a parallel numerical calculation method using a plurality of processors.

0002

[Prior Art] Numerical operations consist of an explicit method and an implicit method, and there are N independent arithmetic expressions with i as an index, and the relational expression of the explicit method part to obtain yi from xi is yi = f(xi The relational expression of the implicit part from ui to wi is shown as ui
Consider a numerical calculation for a problem that includes an operation expressed as =Ei wi where Ei is a sparse matrix.

[0003] In this case, it is difficult to vectorize the implicit method portion, and conventionally, speed has been improved by vector processing only the explicit method portion, which requires a large amount of calculation, using a vector computer. However, even if the speed of the explicit part is improved, the implicit part, which cannot process vectors, becomes a bottleneck in improving the speed of the vector computer, and there is a problem in that no significant speed improvement can be expected.

[0004] Therefore, there was an idea to improve the speed by parallelizing the implicit part of the calculation by taking advantage of the fact that the implicit part of the calculation consists of N independent arithmetic expressions.

[0005]

[Problem to be Solved by the Invention] The following speed improvement method can be considered simply. 1. The explicit part is processed by a vector computer, and the implicit part is processed by N parallel processors. 2. Both the explicit method part and the implicit method part are processed by N processors.

However, the first item requires an expensive vector computer and also requires data redistribution between the vector computer and the parallel processor, and communication processing overhead may impede speed improvement. . Also, the second
In this section, unless N is sufficiently large, it is difficult to speed up the explicit method part, which requires a large amount of processing. Generally, N is about 10 to several 10, which is not so large that it is difficult to achieve a large speed improvement.

[0007]

[Means for Solving the Problems] The parallel numerical calculation method according to the present invention is a numerical calculation consisting of an explicit method and an implicit method.
x) is expressed as an operation to derive y from x in the relational expression,
The implicit part uses N vectors ui, wi and N independent matrices Ai to derive wi from ui using the relational expression ui = Ai wi using P processors. In this method, the explicit method part divides the x vector into P consecutive small vectors xk and
Parallelization is performed by solving xk with
LU decomposition, ui = Li vi from ui to v
Forward elimination to calculate i and v with vi = Ui wi
This results in backward substitution to calculate wi from i, and in accordance with the division of the x vector, the ui, vi, wi vector is divided into ui, k, vi, k, wi, k, and in forward elimination, the k-th processor PEk Then, the intermediate solution vi received from ui,k and Li and PEk-1
, k-1 and perform partial forward elimination to obtain an intermediate solution vi,
After creating k and sending it to PEk+1, PEk is
Compute vi in parallel by pipeline processing that partially performs another forward cancellation using another vector uj,k and the intermediate solution vj,k-1 received from PEk-1,
In the backward substitution, the k-th processor PEk calculates vi,k, Ui, and PEk+1 calculated in the forward elimination process.
Partial backward substitution is performed using the partial solution wi,k+1 received from PEk-1 to create a partial solution wi,k.
After sending PEk to another vector vj,k
The implicit method part is parallelized by calculating wi in parallel by pipeline processing that partially performs another backward substitution using the intermediate solution wj,k+1 received from PEk+1 and PEk+1.

[0008]

[Operation] In the present invention, parallelization can be performed by utilizing the high degree of parallelism that exists as many as the number of elements in the vector x. Therefore, there is no limit on the degree of parallelism as mentioned in the second term above, and by using a parallel machine consisting of a large number of processors, it is possible to improve the speed of the explicit method part by exceeding that of a vector computer.

In the present invention, the implicit method part can also be parallelized using the degree of parallelism of N. At this time, since the shape of data division is the same between the implicit method and the explicit method, there is no need for data redistribution as described in the first section. Communication is performed using explicit and implicit methods, and the amount of communication is small because only data exchange at the boundary between adjacent processors is required.Furthermore, since each processor can communicate simultaneously, communication overhead can be extremely reduced.

As described above, according to the present invention, parallelization of the implicit method part, which has been a problem with vector computers, has been achieved, and numerical calculations can be performed at a speed exceeding that of vector computers. Further, since the coefficient matrix and data can be distributed and arranged using all processors, numerical operations on large-scale data are possible.

[0011]

DESCRIPTION OF THE PREFERRED EMBODIMENTS The parallel numerical calculation system of the present invention will be explained with reference to FIGS. 1 and 2, taking the solution of equations (1) and (2) as an example.

(Equation)

[Math 1]

[Formula 2] However, ak,l,m (r) ,bk,l,m (r
) , ck, l, m (r) are coefficients determined depending on k, l, m and r. Furthermore, the operator []m,n is defined by equation (3).

[Math 3]

(Numerical calculation method) In the r direction, discretization is performed using the central difference shown in equation 4.

##EQU00004## Then, the time integration of equation (1) can be performed using an explicit method.

On the other hand, equation (2) can be solved from time-integrated U, but this calculation is an implicit method that involves matrix inversion. At this time, equation (2) is
Considering that each n is independent, it is expressed by N matrix equations.

[0015] In the matrix operation, the discretized Um, n, φl, n are un (elements are ui, m, n), wn
(Elements are wi, l, n) Matrix representation (5)
It becomes a ceremony. At this time, if un wn is sorted so that m or l elements are continuous, matrix A becomes a block tridiagonal matrix with block size m shown in FIG. However, the range of i is from 1 to I, and the range of m, l is from 1 to M.
, n range from 1 to N.

Since An does not change over time, the intermediate vector vn is decomposed by LU decomposition of equation (5) as An = Ln Un
is introduced and decomposed into the forward elimination equation (6) and the backward substitution equation (7). un = An wn
(5) un = Ln vn
(6) un = Un wn
(7)

(Flow of numerical calculations) Time integration is performed by repeating the following processing. 1. Integration of one time step of un by explicit method 2.
Solving the intermediate solution vn by forward elimination from un 3. v
Solving wn from n by backward substitution

[0018](
Parallelization) As an embodiment of the present invention, P processors are used to parallelize the numerical operations. After that, the processor is PEk
, (1<k<P). Divide un , wn into P consecutive subvectors, and set PEk as k-th subvector un,k , wn,k , where the elements are ui,m,n , wi,l,n and imin(k)
Place items such that <i<imax(k).

(Parallelization of explicit method part) The explicit method part is
i, m, n , wi, l, n for i (imi
This can be solved by referring to n(k)-1<i<imax(k)+1).

FIG. 1 shows how data is referenced when the explicit method part is parallelized. To update element 13, neighboring elements 10, 11, 12 are referenced.

[0021] Processing flow of the parallelized explicit method part
The processing of the explicit method part is as follows: 1. PEk, where k<=P-1 is ui, m, n
, where i=imax(k) is transferred to PEk+1. 2. PEk, where 2<=k, ui,m,n,
However, i=imin(k) is transferred to PEk-1. 3. PEk 0<=k<=P independently calculates un,k. In FIG. 1, vector elements 10 and 14 necessary for calculation of PE2 are transferred by the processing of the first and second terms.

Parallelization of the implicit method part The implicit method part is parallelized by performing pipeline processing between PEk and PEk-1, as shown in FIG. However, for simplicity, FIG. 2 shows a case where P=3 and n=2.

Processing flow of parallelized implicit method part
The flow of processing for each n component is shown below with reference to FIG. However, for simplicity in Figure 2, P=3, n=
Case 2 is shown below. 1. (Hereafter, the forward erasing part) 2. PE1 calculates vn,o from un,o. 3. PE1 is vi, m, n (where i=imax(
1)) Send the element to PE2. 4. PE2 is vi, m, n (where i=imax
(1)) Receive the element and calculate vn,2. 5. PE2 is vi, m, n (where i=imax(
2)) Send the element to PE3. 6. …7. PEP is vi, m, n (where i=imax
(P-1)) Receive the element and calculate vn,P. 8. (Backward substitution elimination part) 9. PEP calculates wn,P from vn,P. 10. PEP is wi, m, n (where i=imin
(P)) Send the element to PEP-1. 11. PEP-1 is wi, m, n (where i=i
min(P)) element and calculate wn,P-1. 12. PEP-1 is wi, m, n (where i=im
in(P-1)) element to PEP-2. 13. ...14. PE1 is wi, m, n (where i=imi
n(2)) elements and calculate wn,1. Then, wn,1 is obtained. For PEk, forward elimination is realized in a pipeline manner as shown in FIG. 2 by starting the solution for a different n after solving for one n.

[0024]

According to the present invention, since the vector length of a problem is generally larger than N, the explicit solution part can be significantly speeded up by using a highly parallel machine with a large degree of parallelism in the explicit solution part. On the other hand, since the shape of data distribution to PEs is the same between the implicit method part and the explicit method part, there is no need for data redistribution and communication overhead is small.

Furthermore, since the coefficient matrix and data can be distributed and arranged using all the processors, the scale of the target problem can be expanded by increasing the number of processors. The parallel numerical calculation method of the present invention is a parallelization method suitable for MIMD type highly parallel machines.

[Brief explanation of drawings]

FIG. 1 is a conceptual diagram showing parallelization of an explicit method part.

FIG. 2 is a conceptual diagram showing parallelization of an implicit method part.

FIG. 3 is a diagram showing a coefficient matrix of an implicit solution part.

FIG. 4 is a diagram showing forward elimination in matrix representation.

FIG. 5 is a diagram showing backward substitution in matrix representation.

[Explanation of symbols]

10 Elements 11, 12 of wn, un that are needed in the calculation of PE2 and received from PE1 Element 1 of wn, un that is referenced
3 Elements of un to be updated 14 Elements 20, 21, 22 of wn, un that are required in the calculation of PE2 and received from PE3 Elements 23, 24, 25 of v1, k calculated by forward elimination from u1, k v2, calculated from u2,k by forward elimination, elements 26, 27, 28 of k, w1, calculated by backward substitution from v1,k, elements 29, 30, 31 of k, calculated by backward substitution from v2,k w2, element of k

Claims (1)

[Claims] Claim 1: Numerical calculation consisting of an explicit method and an implicit method, where the explicit method part uses vectors x and y to calculate y
It is expressed as an operation to derive y from x using the relational expression =f(x), and the implicit method part uses N vectors ui, wi and N independent matrices Ai to calculate ui =Ai w
This is a method of parallelizing the numerical operation shown as the process of deriving wi from ui in the relational expression of i using P processors, and the explicit method part consists of dividing the x vector into P consecutive small vectors xk. kth processor P
Parallelization is performed by solving xk with Ek, and the implicit solution part is the LU decomposition of matrix Ai, and ui with ui = Li vi
Forward elimination to calculate vi from vi = Ui wi
This results in backward substitution to calculate wi from vi , and x
According to the vector division, the ui, vi, wi vector is divided into ui,k, vi,k, wi,k, and in forward elimination, the k-th processor PEk divides the ui,k
and the intermediate solution v received from Li and PEk-1
i, k−1 and perform partial forward elimination to obtain an intermediate solution v
After creating i,k and sending it to PEk+1, PEk
calculates vi in parallel by a pipeline process that partially performs another forward elimination using another vector uj,K and an intermediate solution vj,k−1 received from PEk−1, and in backward substitution, the k-th vi,k, Ui and PE calculated by the forward elimination process using the processor PEk of
Using the partial solution wi,k+1 received from k+1, perform partial backward substitution to create a partial solution wi,k, and PE
After sending it to k-1, PEk is sent to another vector vJ
, k and the intermediate solution wj,k+ received from PEk+1
1. A parallel numerical calculation method characterized in that the implicit method part is parallelized by calculating wi in parallel by pipeline processing that partially performs another backward substitution using 1.