FIELD OF THE INVENTION
[0001] The present invention generally relates to parallel ILU factorization for distributed sparse linear systems. More particularly, the present invention relates to a method for ordering the nodes underlying the equations in distributed sparse linear systems before solving the system(s) using a parallel ILU factorization preconditioner.
BACKGROUND OF THE INVENTION
[0002] Many types of physical processes, including fluid flow in a petroleum reservoir, are governed by partial differential equations. These partial differential equations, which can be very complex, are often solved using finite difference, finite volume, or finite element methods. All of these methods divide the physical model into units called gridblocks, cells, or elements. In each of these physical units the solution is given by one or more solution variables or unknowns. Associated with each physical unit is a set of equations governing the behavior of these unknowns, with the number of equations being equal to the number of unknowns. These equations also contain unknowns from neighboring physical units.
[0003] Thus, there is a structure to the equations, with the equations for a given physical unit containing unknowns from that physical unit and from its neighbors. This is most conveniently depicted using a combination of nodes and connections, where a node is depicted by a small circle and a connection is depicted by a line between two nodes. The equations at a node contain the unknowns at that node and at the neighboring nodes to which it is connected.
[0004] The equations at all nodes are assembled into a single matrix equation. Often the critical task in obtaining the desired solution to the partial differential equations is solving this matrix equation. One of the most effective ways to do this is through the use of incomplete LU factorization or ILU, in which the original matrix is approximately decomposed to the product of two matrices L and U. The matrices L and U are lower triangular and upper triangular and have similar nonzero structures as the lower and upper parts of the original matrix, respectively. With this decomposition, the solution is obtained iteratively by forward and backward substitutions.
[0005] There is an ongoing need to obtain better solution accuracy. One way to do this is to divide the physical model into smaller physical units, or in other words to use more nodes, perhaps millions of them. Of course, the time needed to perform the computations increases as this is done. One way to avoid this time increase is to perform the computations in parallel on multiple processors.
[0006] There are two types of parallel computers, those using shared memory and those using distributed memory. Shared memory computers use only a handful of processors, which limits the potential reduction in run time. Distributed memory computers using tens of processors are common, while some exist that use thousands of processors. It is desired to use distributed memory parallel processing.
[0007] When using distributed memory, the computations are parallelized by dividing the physical model into domains, with the number of domains being equal to the number of processors to be used simultaneously. Each domain is assigned to a particular processor, which performs the computations associated with that domain. Each domain contains a specified set of nodes, and each node is placed in a domain.
[0008] The entire modeling process involves many computations, nearly all of which are performed node by node. Some of these computations at a node require only information local to the node. Information is local to a node when it is contained completely within the same domain as the node. These computations are sometimes called embarrassingly parallel, since they require no special treatment to perform in parallel. Other computations require information at the node and its neighbors. If the node is on the boundary of its domain with another domain, one or more of its neighbors will reside in the other domain. To perform computations that require neighbor information at boundary nodes, information about these neighbors must be obtained from the domains in which these neighbors reside. If the information needed is known in advance, it can be obtained easily by "message passing," and the computations are easily parallelized. It is important that the information be known in advance because message passing takes time. In particular, there is a large, compared to normal computational times, latency; in other words, it takes a finite time for the first element of a message to reach its recipient. If the information is known in advance, the message containing it can be sent before it is needed in the other process. In this manner, it can arrive at the other process before it is needed.
[0009] Unfortunately, in factorization computations, the information needed is not known in advance. Instead, it is generated during the factorization. The computations are "inherently sequential." The general flow of the computations is as follows:
- 1. Update the current node's equations based on computations performed at its neighbors that have already been factored.
- 2. Factor the resulting modified equations at the current node.
- 3. Provide information about the current node's factorization to its neighbors that have not yet been factored.
"Neighbors" need not be immediate neighbors. They can be several nodes away.
[0010] The sequential nature of these calculations is not a problem if there is one domain. The sequential nature of these calculations is a problem if there is more than one domain. Information must be sent from one process to another. If this information is not known until immediately before it is needed at the other process, there will be a delay while the message containing it is sent. These delays can be avoided if the computations are ordered such that any information to be sent to another process is known well before it is needed at the process.
[0011] To consider this further, assume two domains. Each domain has interior nodes that communicate only with nodes in the same domain and boundary nodes that communicate with nodes in both domains. The processing could be performed in the following order:
- 1. Process interior nodes in domain 1.
- 2. Process boundary nodes in domain 1.
- 3. Send boundary node information from domain 1 to domain 2.
- 4. Process boundary nodes in domain 2.
- 5. Process interior nodes in domain 2.
[0012] If this order is used, domain 2 cannot begin its processing until domain 1 is completely finished. There is no parallelization of the computations at all.
[0013] A better processing order is as follows:
- 1. In parallel, process interior nodes in domain 1 and interior nodes in domain 2.
- 2. Process boundary nodes in domain 1.
- 3. Send boundary node information from domain 1 to domain 2.
- 4. Process boundary nodes in domain 2.
[0014] Using this order, the calculations on interior nodes are performed in parallel. This is significant since there are more interior nodes than boundary nodes. However, the boundary nodes are still processed sequentially. Typically 20%-40% of total nodes are boundary nodes.
[0015] There are quite a few parallel algorithms for ILU factorization. In the book "
Iterative Methods for Sparse Linear Systems (second edition)" written by Yousef Saad, Society for Industrial and Applied Mathematics, 2003 ("Saad"), two algorithms are introduced. One of them is the multi-elimination algorithm described on p.p. 295-392, which takes advantage of independent sets existing in the sparse linear system. However, this approach may be unsuitable for a distributed data structure. A second algorithm described on p.p. 396-399 factors interior nodes simultaneously on each processor then processes the boundary nodes in some order. The drawback of this second algorithm is that some processors remain idle while waiting for data coming from other processors. A third algorithm described in Parallel Threshold-based ILU Factorization by
George Karypis and Vipin Kumar, 1998, Technical Report #96-061, is similar to the second algorithm, except that it colors the boundary nodes and then factors the nodes of each color. A few colors, however, may be required. As more colors are required, more messages must be passed between processors, usually impairing the overall performance of the solver.
[0017] There is therefore, a need for an improved parallel ILU factorization algorithm that is suitable for distributed sparse linear systems and reduces processing time.
SUMMARY OF INVENTION
[0018] The present invention meets the above needs and overcomes one or more deficiencies in the prior art by providing systems and methods for parallel ILU factorization in distributed sparse linear systems, which utilize a unique method for ordering nodes underlying the equations in the systems(s) and reducing processing time.
[0019] According to a first aspect of the present invention, there is provided a computer-implemented method, according to claim 1.
[0020] In a second aspect of the present invention, there is provided a program carrier device for carrying computer executable instructions according to claim 8.
[0021] Additional aspects, advantages and embodiments of the invention will become apparent to those skilled in the art from the following description of the various embodiments and related drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
[0022] The present invention is described below, by way of example only, with references to the accompanying drawings in which like elements are referenced with like reference numerals, and in which:
FIG. 1 is a block diagram illustrating a system for implementing the present invention.
FIG. 2A is a flow diagram illustrating one embodiment of a method for implementing the present invention.
FIG. 2B is a continuation of the method illustrated in FIG. 2A.
FIG. 2C is a continuation of the method illustrated in FIG. 2B.
FIG. 2D is a continuation of the method illustrated in FIG. 2C.
FIG. 2E is a continuation of the method illustrated in FIG. 2D.
FIG. 2F is a continuation of the method illustrated in FIG. 2E.
FIG. 2G is a continuation of the method illustrated in FIG. 2F.
FIG. 2H is a continuation of the method illustrated in FIG. 2B.
FIG. 3 is a continuation of the method illustrated in FIG. 2E.
FIG. 4 is a continuation of the method illustrated in FIG. 3.
FIG. 5 illustrates an example of domain decomposition.
FIGS. 6A and 6B illustrate examples of coloring regular boundary nodes.
FIGS. 7A, 7B and 7C illustrate examples of coloring super boundary nodes.
FIG. 8 illustrates an example of incomplete factorization.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0023] The subject matter of the present invention is described with specificity, however, the description itself is not intended to limit the scope of the invention. The subject matter thus, might also be embodied in other ways, to include different steps or combinations of steps similar to the ones described herein, in conjunction with other present or future technologies. Moreover, although the term "step" may be used herein to describe different elements of methods employed, the term should not be interpreted as implying any particular order among or between various steps herein disclosed unless otherwise expressly limited by the description to a particular order.
System Description
[0024] The present invention may be implemented through a computer-executable program of instructions, such as program modules, generally referred to as software applications or application programs executed by a computer. The software may include, for example, routines, programs, objects, components, and data structures that perform particular tasks or implement particular abstract data types. The software forms an interface to allow a computer to react according to a source of input. NEXUS
_{™}, which is a commercial software application marketed by Landmark Graphics Corporation, may be used as an interface application to implement the present invention. The software may also cooperate with other code segments to initiate a variety of tasks in response to data received in conjunction with the source of the received data. The software may be stored and/or carried on any variety of memory media such as CD-ROM, magnetic disk, bubble memory and semiconductor memory (e.g., various types of RAM or ROM). Furthermore, the software and its results may be transmitted over a variety of carrier media such as optical fiber, metallic wire, free space and/or through any of a variety of networks such as the Internet.
[0025] Moreover, those skilled in the art will appreciate that the invention may be practiced with a variety of computer-system configurations, including hand-held devices, multiprocessor systems, microprocessor-based or programmable-consumer electronics, minicomputers, mainframe computers, and the like. Any number of computer-systems and computer networks are acceptable for use with the present invention. The invention may be practiced in distributed-computing environments where tasks are performed by remote-processing devices that are linked through a communications network. In a distributed-computing environment, program modules may be located in both local and remote computer-storage media including memory storage devices. The present invention may therefore, be implemented in connection with various hardware, software or a combination thereof, in a computer system or other processing system.
[0026] Referring now to
FIG. 1, a block diagram of a system for implementing the present invention on a computer is illustrated. The system includes a computing unit, sometimes referred to as a computing system, which contains memory, application programs, a client interface, and a processing unit. The computing unit is only one example of a suitable computing environment and is not intended to suggest any limitation as to the scope of use or functionality of the invention.
[0027] The memory primarily stores the application programs, which may also be described as program modules containing computer-executable instructions, executed by the computing unit for implementing the methods described herein and illustrated in
FIGS. 2A-4. The memory therefore, includes an ILU Factorization Module, which enables the methods illustrated and described in reference to
FIGS 2A-4, and NEXUS™.
[0028] Although the computing unit is shown as having a generalized memory, the computing unit typically includes a variety of computer readable media. By way of example, and not limitation, computer readable media may comprise computer storage media and communication media. The computing system memory may include computer storage media in the form of volatile and/or nonvolatile memory such as a read only memory (ROM) and random access memory (RAM). A basic input/output system (BIOS), containing the basic routines that help to transfer information between elements within the computing unit, such as during start-up, is typically stored in ROM. The RAM typically contains data and/or program modules that are immediately accessible to and/or presently being operated on by the processing unit. By way of example, and not limitation, the computing unit includes an operating system, application programs, other program modules, and program data.
[0029] The components shown in the memory may also be included in other removable/nonremovable, volatile/nonvolatile computer storage media. For example only, a hard disk drive may read from or write to nonremovable, nonvolatile magnetic media, a magnetic disk drive may read from or write to a removable, non-volatile magnetic disk, and an optical disk drive may read from or write to a removable, nonvolatile optical disk such as a CD ROM or other optical media. Other removable/non-removable, volatile/non-volatile computer storage media that can be used in the exemplary operating environment may include, but are not limited to, magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the like. The drives and their associated computer storage media discussed above therefore, store and/or carry computer readable instructions, data structures, program modules and other data for the computing unit.
[0030] A client may enter commands and information into the computing unit through the client interface, which may be input devices such as a keyboard and pointing device, commonly referred to as a mouse, trackball or touch pad. Input devices may include a microphone, joystick, satellite dish, scanner, or the like.
[0031] These and other input devices are often connected to the processing unit through the client interface that is coupled to a system bus, but may be connected by other interface and bus structures, such as a parallel port or a universal serial bus (USB). A monitor or other type of display device may be connected to the system bus via an interface, such as a video interface. In addition to the monitor, computers may also include other peripheral output devices such as speakers and printer, which may be connected through an output peripheral interface.
[0032] Although many other internal components of the computing unit are not shown, those of ordinary skill in the art will appreciate that such components and their interconnection are well known.
Method Description
[0033] Referring now to
FIG. 5, an example of domain decomposition is illustrated for purposes of describing how parallel processing, sometimes referred to herein as parallelism, may be utilized by the present invention for solving distributed sparse linear systems. In
FIG. 5, a two-dimensional gridded rectangular model
500 is decomposed into two domains (Ω1 and Ω2), which are separated by a broken line
506. This example is based on a finite difference or finite volume calculation using the five-point stencil
502, which comprises solid lines and dots. The resulting coefficient matrix has the same connectivity as the model
500 shown in
FIG. 5. The present invention may be applied to other more complex examples however, this example is used here because of its simplicity. The linear equations corresponding to domains Ω1 and Ω2 are loaded to processor P1 and P2, respectively. The parallelism takes advantage of the fact that all of the interior nodes in one domain are not connected to any nodes in the other domain, which allows both processors to process their local data simultaneously. The boundary nodes
504, shown as open circles, should be accessible for both processors, but the parallelism still can be realized among them by using a color-based ordering.
[0034] For example, to obtain a better ordering, regular boundary nodes may be colored as illustrated in
FIGS. 6A and
6B. In
FIGS. 6A and
6B, each domain is separated by a partitioning interface
602 and
604, respectively. The lighter broken lines represent the connections between nodes in the same domain. Each solid line therefore, represents a cross-domain connection. Although the term "color" is referred to herein to distinguish boundary nodes from interior nodes and to describe a process for ordering the nodes, other types of coding may be used, instead, to achieve the same objectives.
[0035] The present invention utilizes the following rules, which were applied to the nodes in
FIGS. 6A and
6B:
- 1. I (interior) nodes can connect only to nodes in the same domain;
- 2. C1 nodes must not connect to C1 nodes in a different domain;
- 3. C2 nodes must not connect to C2 nodes in a different domain; and
- 4. C3 nodes can connect to nodes of any color in any domain.
[0036] The C3 nodes act as a buffer between the C1 nodes in one domain and in a different domain and between the C2 nodes in one domain and in a different domain. If the node connectivity is as shown in
Fig. 6A, this buffer is not needed and there are no C3 nodes. If there are diagonal connections as shown in
Fig. 6B, the buffer is needed and is provided by the two C3 nodes shown.
[0037] Now, the nodes may be computed (processed) in the following order:
- 1. In parallel, send domain 1 C3 information to domain 2 and send domain 2 C3 information to domain 1.
- 2. In parallel, process interior nodes in domain 1 followed by C1 boundary nodes in domain 1, and interior nodes in domain 2 followed by C1 boundary nodes in domain 2.
- 3. In parallel, send domain 1 C1 boundary node information to domain 2, send domain 2 C1 boundary node information to domain 1, and process C3 nodes.
- 4. In parallel, process C2 boundary nodes in domain 1 and C2 boundary nodes in domain 2.
Typically, around 1%-5% of the total nodes are C3 nodes.
[0038] This invention divides all of the nodes into two categories, i.e. the interior nodes and the boundary nodes. The interior nodes are processed across all the processors simultaneously, and the computation for the interior nodes is very intensive in CPU time because there are usually more interior nodes than boundary nodes. Conventional methods do not use this local computation to cover any communication between processors, which is a inefficient. A new coloring algorithm is designed to utilize this local computation in interior nodes to overlap cross-processor communication and at the same time limit the number of colors for the boundary nodes to three. The basic principles for the coloring are that any adjacent nodes which belong to two different domains cannot have the same color except the accessory color C3 and each of the major colors (C1 and C2) must occur on all the processors. In most cases with regular stencils and reasonable domain decomposition, two colors are adequate for the coloring. The accessory color C3 is needed to switch the major colors cross-boundary to realize the second part of the basic coloring principles. In this invention, the boundary nodes with the third color (C3), if any, are made accessible for both sides of this boundary by message passing at the very beginning so that the communication can be overlapped with the local computation in factorization of the interior nodes.
[0039] With C3 nodes accessible for both sides of the partitioning, the factorization and the forward and backward substitution procedures are much same as for the cases with only two boundary colors, i.e. only one message passing is required for each of these procedures and the message passing is readily overlapped with local computation in boundary nodes. With that, it is possible to design a scalable parallel ILU preconditioner.
[0040] The following description applies the foregoing rules and processes the nodes in three major stages: i) coloring; ii) factoring; and iii) solving. The coloring stage determines an ordering of the nodes, which in turn determines the order in which the equations will be factored and the solution will be performed. Once the coloring is determined, the factorization and solution are routine, although perhaps complex.
COLORING
[0041] One embodiment of a method for implementing the coloring stage is described in reference to
FIGS. 2A-2H. In addition,
FIGS. 7A-7C are referenced to illustrate examples of coloring super-boundary nodes.
[0042] In
FIGS. 7A-7C, the boundary nodes connected to multiple external domains are defined as super-boundary nodes. All other boundary nodes are called regular-boundary nodes. The connections in
FIGS. 6A, 6B and
7A-7C represent the non-zero entries in a coefficient matrix, which are not necessarily the same as the grid connectivity because the connectivity in a coefficient matrix depends on what kind of stencil is used for the discretization of the partial differential equations. In these figures, only the cross-domain connections, which are represented by the solid lines, are considered for the coloring procedure. The broken lines represent a partitioning interface, which separates each domain.
[0043] Referring now to
FIG. 2A, the method
200A is the beginning of the coloring stage.
[0044] In step
202, find all connections that will be used to determine the node colors, which are cross-domain connections. Only these connections are used in determining node colors.
[0045] In step
204, the interior nodes, which have no connections across domain boundaries, are identified.
[0046] In step
206, the boundary nodes, which have connections across domain boundaries, are identified.
[0047] In step
208, the super-boundary nodes, which are those that connect to at least two domains other than the domain in which they are located, are identified.
[0048] In step
210, a connectivity matrix of the super-boundary nodes is constructed.
[0049] In step
212, the greedy multi-coloring algorithm, which is well known in the art, is applied to color the super-boundary nodes. The basic principle of this coloring is that no two connected nodes have the same color. The multi-coloring algorithm is run several times using different seed nodes in order to find the minimum number of colors.
[0050] In step
214, determine if the number of colors is greater than three.
[0051] In step
216, all colors greater than 3 are made color 3 (C3). With that, some adjacent super-boundary nodes may have the same color and this color must be C3, as shown in
FIG. 7B. For reasonable domain decompositions, there will be a very limited number of such color 3-color 3 (C3-C3) connections.
[0052] Referring now to
FIG. 2B, the method
200B is a continuation of the method 200A for implementing the coloring stage.
[0053] In step
216, s is equal to the number of partitioning interfaces.
[0054] In step
218, for each partitioning interface p(s), construct a connectivity matrix with only connections that cross the p(s).
[0055] In step
220, all nodes that have been previously colored with color 1 (C1), color 2 (C2), and color 3 (C3) are found.
[0056] In step
222, if C1 or C2 exist, method
200B continues to step
284 in
FIG. 2H. If C1 or C2 do not exist, step
224 picks the side with more uncolored nodes as the starting side, and puts the first uncolored node on this side in the 1-queue.
[0057] In step
226, C1 is assigned to C
_{l} as the starting color on the starting side, and C2 is assigned to C
_{r}.
[0058] Referring now to
FIG. 2C, the method
200C is a continuation of the method
200B for implementing the coloring stage.
[0059] In step
228, x is equal to the number of nodes in 1-queue.
[0060] In step
228a, for node i(x) adjacent nodes j are determined.
[0061] In step
228b, y is equal to the number of adjacent nodes j of i.
[0062] In step
230, determine if node j(y) is colored. If j(y) is colored, step 232 determines if j(y) is colored C
_{l}. If j(y) is not colored, then j is colored to be C
_{r} and added to the r-queue in step
232a.
[0063] In step
232, if j(y) is colored C
_{l}, recolor j(y) to be C3 in step
232b, then continue to step
234. If j(y) is not colored C
_{l} the method continues to step
234.
[0064] In step
234, determine if y is greater than one. If y is not greater than one, all adjacent nodes j(y) of i have been visited and the method
200C continues to step
236. If y is greater than one in step
234, some adjacent nodes j(y) of i have not been visited and the method
200C continue0s to step
234a.
[0065] In step
234a, y is equal to y minus 1. The method returns to step
230 from step
234a.
[0066] In step
236, node i(x) is removed from the 1-queue.
[0067] In step
238, determine if x is greater than one. If x is greater than one, then 1-queue is not empty and step
238a sets x equal to x minus 1. If x is not greater than one, then 1-queue is empty and the method
200C continues to
FIG. 2D.
[0068] Referring now to
FIG. 2D, the method
200D is a continuation of the method
200C for implementing the coloring stage.
[0069] In step
240, m equals the number of nodes in r-queue.
[0070] In step
240a, for node i(m) adjacent nodes j are determined.
[0071] In step
240b, n is equal to the number of adjacent nodes j of i.
[0072] In step
242, determine if node j(n) is colored. If j(n) is colored, step
244 determines if j(n) is colored C
_{r}. If j(n) is not colored, then j(n) is colored to be C
_{l} and added to the 1-queue in step
244b.
[0073] In step
244, if j(n) is colored C
_{r}, recolor j(n) C3 in step
244a, then continue to step
246. If j(n) is not colored C
_{r} the method continues to step
246.
[0074] In step
246, determine if n is greater than one. If n is not greater than one, all adjacent nodes j(n) of i(m) have been visited and the method
200D continues to step
248. If n is greater than one in step
246, some adjacent nodes j(n) of i(m) have not been visited and the method
200D continues to step
246a.
[0075] In step
246a, n is equal to n minus 1. The method returns to step
242 from step
246a.
[0076] In step
248, node i(m) is removed from the r-queue.
[0077] In step
250, determine if m is greater than one. If m is greater than one, then r-queue is not empty and step
250a sets m equal to m minus 1. If m is not greater than one, then r-queue is empty and the method
200D continues to
FIG. 2E.
[0078] Referring now to
FIG. 2E, the method
200E is a continuation of the method
200D for implementing the coloring stage.
[0079] In step
252, determine if the number of colored nodes reaches half of the total number of nodes on either side. If the number of colored nodes is equal to half the total number of nodes on either side, then the method proceeds to
FIG. 2F to exchange colors C
_{l} and C
_{r}. If the number of colored nodes is not equal to half the total number of nodes on either side, then the method proceeds to step
254.
[0080] In step
254, determine if all nodes are colored. If all nodes are colored, the method proceeds to step
254a. If some nodes are not colored, the method proceeds to step
254c.
[0081] In step
254a, determine if s is greater than one. If s is not greater than one, all interfaces have been visited and the method continues to step
302 in
FIG. 3. If s is greater than one, step
254b sets s equal to s minus 1 and the method returns to step
218 in
FIG. 2B.
[0082] In step
254c, determine if 1-queue is empty. If 1-queue is empty, step
256 finds the first uncolored node on the starting side, colors it C
_{l}, places it in the 1-queue, and continues to step
228 in
FIG. 2C. If 1-queue is not empty, the method continues to step
228 in
FIG. 2C.
[0083] Referring now to
FIG. 2F, the method
200F is a continuation of the method
200E for implementing the coloring stage. During the coloring process, when the number of nodes colored on either side reaches half of the total number of nodes on that side, colors C
_{l} and C
_{r} must be exchanged. Before the colors can be exchanged, the method
200F must first determine whether any nodes have been colored with color 3 for this switch.
[0084] In step
256a, t equals the number of nodes in 1-queue
[0085] In step
258, determine if t is greater than one. If t is not greater than one, 1-queue is empty and the method continues to step
276 in
FIG. 2G. If t is greater than one, 1-queue is not empty; and step
260 determines adjacent nodes j for node i(t) in 1-queue.
[0086] In step
262, determine if all adjacent nodes j of i(t) are colored C
_{r}. If all adjacent nodes j of i(t) are colored C
_{r}, step
274 removes node i(t) from the 1-queue. If some adjacent nodes j of i(t) are not colored C
_{r}, step
264 sets z equal to the number of adjacent nodes j of i(t).
[0087] In step
266, determine if adjacent node j(z) is colored. If j(z) is colored, step
266a determines if j(z) is colored C
_{l}. If j(z) is not colored, then step
270 places j(z) in the r-queue without coloring.
[0088] In step
266a, if node j(z) is colored C
_{l}, step
268 recolors i(t) to be C3, and the method continues to step
274. If node j is not colored C
_{l}, the method
200F continues to step
272.
[0089] In step
270, j(z) is placed in the r-queue without coloring.
[0090] In step
272, determine if z is greater than one. If z is greater than one, then some adjacent nodes have not been visited and step
272a sets z equal to z minus 1. If z is not greater than one, then all adjacent nodes j(z) have been visited and the method
200D continues to step
274.
[0091] In step
274, node i(t) is removed from the 1-queue.
[0092] In step
274a, t is equal to t minus 1, and the method returns to step
258.
[0093] Referring now to
FIG. 2G, the method
200G is a continuation of the method
200F for implementing the coloring stage.
[0094] In step
274b, u equals the number of nodes in r-queue.
[0095] In step
276, determine if r-queue is empty. If r-queue is empty, then colors C
_{l} and C
_{r} are exchanged in step
276a and the method
200G returns to step
254 in FIG.
2E. If r-queue is not empty, in step
278, adjacent nodes j are determined for node i(u) in r-queue.
[0096] In step
280, determine if all adjacent nodes j are colored C
_{l}. If some adjacent nodes j are not colored C
_{l}, then step
280a colors node i(u) to be C3. If all adjacent nodes j are colored C
_{l}, then step
280b colors node i(u) to be C
_{r}.
[0097] In step
282, node i(u) is removed from the r-queue.
[0098] In step
282a, u is equal to u-1, and the method returns to step
276.
[0099] Referring now to
FIG. 2H, the method
200H is a continuation of the method
200B for implementing the coloring stage.
[0100] In
step 284, nodes with colors C1 and C2 on both sides of the interface are counted.
[0101] In step
286, the color with the most nodes on either side of the interface is selected as the starting color, denoted by C
_{l}.
[0102] In step
288, the interface side with the most C
_{l} nodes is selected as the starting side.
[0103] In step
290, the C
_{l} nodes on the starting side are stored in a queue named 1-queue.
[0104] In step
292, the color different from C
_{l} is assigned to C
_{r}.
[0105] In step
294, determine if C
_{r} exists among nodes that have cross-domain connections to nodes in 1-queue. If C
_{r} exists among nodes that have cross-domain connections to nodes in 1-queue, those C
_{r} nodes are stored in a queue named r-queue in step
296. If C
_{r} does not exist among nodes that have cross-domain connections to nodes in 1-queue, the method continues with step
228 in method
200C.
FACTORING
[0106] Referring now to
FIG. 3, the method
300 is a continuation of the method
200E for implementing the factoring stage.
[0107] Method
300 illustrates the incomplete factorization according to the present invention. The following steps are performed on each processor in parallel.
[0108] In step
302, if C3 nodes exist, their equations are made accessible to both sides of the boundary by message passing. Step
302 initiates the non-blocking message sends and receives.
[0109] In step
304, all local interior nodes are approximately factored by ILU factorization. There are several well known variants of ILU factorization, which are available and are described in Saad. This step is CPU time intensive and should allow time for the messages sent in step
302 to arrive at their destinations.
[0110] In step
306, C1 boundary nodes are factored.
[0111] In step
308, non-blocking sends and receives for upper factorization coefficients related to C1 nodes, and interior coefficients related to C3 nodes are issued.
[0112] In step
310, C2 nodes are updated using information from local interior nodes.
[0113] In step
312, when the messages of step
302 have arrived, update C3 nodes using information from local interior and C1 nodes.
[0114] In step
314, C3 and C2 nodes are factored when the messages of step
308 have arrived.
SOLVING
[0115] Referring now to
FIG. 4, method
400 is a continuation of the method
300 for implementing the solving stage through forward and backward substitution, which are techniques well known in the art.
[0116] In step
402, if C3 nodes exist, initiate message passing of their right hand side.
[0117] In step
404, the forward solution for interior nodes is performed.
[0118] In step
406, the forward solution for C1 nodes is performed.
[0119] In step
408, non-blocking sends and receives of the forward solution at C1 nodes and interior nodes connected to C3 nodes are issued.
[0120] In step
410, C2 and C3 nodes are updated based on local interior and C1 solutions.
[0121] In step
412, C2 and C3 nodes are updated due to remote interior and C1 solutions when the messages of step
408 have arrived.
[0122] In step
414, the forward solution for C3 and C2 nodes is performed.
[0123] In step
416, the backward solution for C2 nodes is performed.
[0124] In step
418, the non-blocking sends and receives of C2 node solutions are issued.
[0125] In step
420, C3 and interior nodes are updated due to local C2 solution.
[0126] In step
422, C3, C1, and interior nodes are updated due to remote C2 solutions when the messages of step
418 have arrived.
[0127] In step
424, the backward solution for C3 nodes is performed.
[0128] In step
426, the backward solution for C1 nodes is performed.
[0129] In step
428, the backward solution for interior nodes is performed.
[0130] Referring now to
FIG. 8, an example of incomplete factoring and solving is illustrated for a re-ordered matrix for two processors. I and C represent the interior nodes and boundary nodes, respectively. Both procedures of factorization and solution can be explained readily in this figure. In the factorization and the forward substitution, the processors (P1/P2) treat interior nodes first and then C1 boundary nodes, and finally on C2/C3 boundary nodes. In the backward substitution, both processors solve C2 boundary nodes first then C3, C1 and I nodes. Message passing happens after C1 boundary nodes are treated in the factorization and the forward substitution or after C2 boundary nodes are treated in the backward substitution. Before the factorization and the forward substitution are taken, information about C3 boundary nodes, if any exist, is exchanged for both processors.
[0131] While the present invention has been described in connection with presently preferred embodiments, it will be understood by those skilled in the art that it is not intended to limit the invention to those embodiments.
[0132] It is therefore, contemplated that various alternative embodiments and modifications may be made to the disclosed embodiments without departing from the scope of the invention defined by the appended claims.
1. Computerimplementiertes Verfahren zum Simulieren eines Fluidflusses in einem Petroleumreservoir, wobei der Fluidfluss durch partielle Differentialgleichungen geregelt wird, wobei das Simulieren aufweist, dass eine Lösung der partiellen Differentialgleichungen durch das Lösen einer zugehörigen Gruppe linearer Gleichungen auf parallelen Prozessoren mit verteiltem Speicher erhalten wird,
wobei die linearen Gleichungen Knoten und Verbindungen zugeordnet sind, wobei die Gleichungen an einem Knoten die Unbekannten an dem Knoten und an benachbarten Knoten, mit denen die Knoten verbunden sind, aufweisen, wobei jeder Knoten einer physikalischen Einheit entspricht, wobei die Gruppe an physikalischen Einheiten in Bereiche aufgeteilt wird, die durch Aufteilungsgrenzflächen getrennt sind, wobei alle Knoten eines Bereichs einem bestimmten Prozessor zugewiesen sind, der dazu ausgestaltet ist, die linearen Gleichungen, die den Knoten in diesem Bereich zugeordnet sind, zu lösen,
wobei Knoten, die keine Verbindung haben, welche eine Aufteilungsgrenzfläche schneidet, als innere Knoten bezeichnet werden, und
Knoten, die eine Verbindung haben, die eine Aufteilungsgrenzfläche schneidet, als Grenzknoten bezeichnet werden,
wobei jeder Bereich eine Gruppe an Grenzknoten und eine Gruppe an inneren Knoten aufweist,
das Verfahren mit dem Ordnen der mehreren Knoten, die den linearen Gleichungen zugrundeliegen, das Ordnen mit:
dem Zuweisen von genau einem Code von nicht mehr als drei Codes zu jedem Grenzknoten, wobei die drei Codes aus einem ersten Code, einem zweiten Code und einem dritten Code bestehen, wobei die ersten, zweiten und dritten Codes verschieden sind, durch:
dem Zuweisen des ersten Codes zu jedem Grenzknoten, der einen ersten Grenzknoten (C1) darstellt, wobei jede erste Grenzknotenverbindung nicht eine Aufteilungsgrenzfläche so queren kann, dass sie zwei erste Grenzknoten (C1) verbindet,
dem Zuweisen des zweiten Codes zu jedem Grenzknoten, der einen zweiten Grenzknoten (C2) darstellt, wobei jede zweite Grenzknotenverbindung nicht eine Aufteilungsgrenzfläche so queren kann, dass sie sie zwei zweite Grenzknoten (C2) verbindet, und
dem Zuweisen des dritten Codes zu jedem Grenzknoten, der einen dritten Grenzknoten (C3) darstellt, wobei jede dritte Grenzknotenverbindung nicht eine Aufteilungsgrenzfläche so queren kann, dass sie mit einem inneren Knoten verbindet, und wobei jeder dritte Grenzknoten (C3) sich mit Knoten von jedwedem Code in jedwedem Bereich verbunden sein kann und
wobei Knoten sowohl des ersten als auch des zweiten Codes auf allen Prozessoren vorkommen müssen, das Verfahren ferner mit dem Lösen der zugrundeliegenden linearen Gleichungen, beginnend mit den inneren Knoten und in einer Reihenfolge, die durch die drei Codes bestimmt ist,
und wobei die Grenzknoten mit dem dritten Code zugänglich gemacht werden durch das Weitergeben einer Nachricht vor dem Lösen der zugrundeliegenden linearen Gleichungen in einer Reihenfolge, die durch die drei Codes bestimmt ist, so dass das Weitergeben der Nachricht mit dem lokalen Bearbeiten der inneren Knoten überlappen kann.
2. Verfahren nach Anspruch 1, wobei jede dritte Grenzknotenverbindung zwei dritte Grenzknoten (C3), einen dritten Grenzknoten (C3) und einen zweiten Grenzknoten (C2) oder einen dritten Grenzknoten (C3) und einen ersten Grenzknoten (C1) verbinden kann.
3. Verfahren nach Anspruch 1, ferner mit:
dem Bearbeiten jedes inneren Knotens in einem ersten Bereich gefolgt von jedem ersten Grenzknoten (C1) in dem ersten Bereich,
dem Bearbeiten jedes inneren Knotens in einem zweiten Bereich gefolgt von jedem ersten Grenzknoten (C1) in dem zweiten Bereich, und
dem Durchführen jedes Bearbeitungsschritts parallel zueinander.
4. Verfahren nach Anspruch 3, ferner mit:
dem Übertragen von Informationen, die jedem ersten Grenzknoten (C1) in dem Bereich zugeordnet sind, zu dem anderen Bereich,
dem Übertragen von Informationen, die jedem ersten Grenzknoten (C1) in dem anderen Bereich zugeordnet sind, zu dem Bereich, und
dem Durchführen jedes Übertragungsschritts parallel zueinander.
5. Verfahren nach Anspruch 4, ferner mit:
dem Bearbeiten jedes zweiten Grenzknotens (C2) in dem Bereich,
dem Bearbeiten jedes zweiten Grenzknotens (C2) in dem anderen Bereich, und
dem Durchführen jedes Bearbeitungsschritts parallel zueinander.
6. Verfahren nach Anspruch 1, wobei jeder Code eine verschiedene Farbe ist.
7. Verfahren nach Anspruch 1, wobei jede innere Knotenverbindung nur Knoten innerhalb eines einzigen Bereichs verbindet.
8. Datenträgereinrichtung zum Tragen von Instruktionen, die auf einem Computer ausführbar sind, um einen Computer dazu zu bringen, das Verfahren nach einem der Ansprüche 1 bis 7 durchzuführen.