Scientific Computing

Transcription

1 Lecture on Scientific Computing Dr. Kersten Schmidt Lecture 21 Technische Universität Berlin Institut für Mathematik Wintersemester 2014/2015

2 Syllabus Linear Regression, Fast Fourier transform Modelling by partial differential equations (PDEs) Maxwell, Helmholtz, Poisson, Linear elasticity, Navier-Stokes equation boundary value problem, eigenvalue problem boundary conditions (Dirichlet, Neumann, Robin) handling of infinite domains (wave-guide, homogeneous exterior: DtN, PML) boundary integral equations Computer aided-design (CAD) Mesh generators Space discretisation of PDEs Finite difference method Finite element method Discontinuous Galerkin finite element method Solvers Linear Solvers (direct, iterative), preconditioner Nonlinear Solvers (Newton-Raphson iteration) Eigenvalue Solvers Parallelisation Computer hardware (SIMD, MIMD: shared/distributed memory) Programming in parallel: OpenMP, MPI VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/2015 2

3 Generic distributed memory computer Clusters with partly distributed memory and shared memory A process is an instance of a program that is executing more or less autonomously on a physical processor. VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/2015 3

4 Message passing Communication on parallel computers with distributed memory (multicomputers) is most commonly done by message passing. Processes coordinate their activities by explicitely sending and receiving messages. Assume that (as in MPI) processes are statically allocated. That is, the number of processors is set at the beginning of the program execution, and no further processes are created during execution. There is usually one process executing on one processor. Each process is assigned a unique integer rank in the range 0, 1,..., p 1, where p is the number of processes. VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/2015 4

5 The Message Passing Interface: MPI Like OpenMP for shared memory programming, MPI is an application programmer interface to message passing. MPI extends programming languages (like C/C++ or Fortran) with a library of functions for point-to-point and collective communication and additional functions for managing the processes at the computation and for querying their status. MPI has become a de facto standard for message passing on multicomputers. Standardization by the MPI forum ( Implementations: MPICH: Open MPI: On clusters with many processors, VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/2015 5

6 First MPI demo program in C: mpi1st.c #include <stdio.h> #include <mpi.h> int main(int argc, char* argv[]) { } int rank; /* rank of process */ int p; /* number of processes */ MPI_Init(&argc, &argv); /* Start up MPI */ MPI_Comm_rank(MPI_COMM_WORLD, &rank); /* Find out proc rank */ MPI_Comm_size(MPI_COMM_WORLD, &p); /* Find out number of processes */ printf("proc %d from %d is ready.\n", rank, p); MPI_Finalize(); /* Shut down MPI */ Calling % mpicc mpi1st.c -o mpi1st % mpirun -np 4 mpi1st Proc 1 from 4 is ready. Proc 2 from 4 is ready. Proc 3 from 4 is ready. Proc 0 from 4 is ready. VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/2015 6

7 Simple send and receive commands A typical usage of sending and receiving is given by the following example, where process 0 sends a single float x to process 1. Process 0 executes MPI_Send(&x, 1, MPI_FLOAT, 1, 0, MPI_COMM_WORLD); while process 1 executes MPI_Recv(&x, 1, MPI_FLOAT, 0, 0, MPI_COMM_WORLD, &status); Process 0 and process 1 execute different statements. However, the single-program multi-data (SPMD) programming model permits individual processes executing different statements by means of conditional branches. float x = 0; if (rank == 0) { x = 1; /* e.g. read from an input file */ MPI_Send(&x, 1, MPI_FLOAT, 1, 0, MPI_COMM_WORLD); } else if (rank == 1) MPI_Recv(&x, 1, MPI_FLOAT, 0, 0, MPI_COMM_WORLD, &status); VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/2015 7

8 Simple send and receive commands int MPI_Send(void* buffer /* in */, int count /* in */, MPI_Datatype datatype /* in */ int destination /* in */, int tag /* in */, MPI_Comm communicator /* in */) int MPI_Recv(void* buffer /* out */, int count /* in */, MPI_Datatype datatype /* in */ int destination /* in */, int tag /* in */, MPI_Comm communicator /* in */, MPI_Status* status /* out */) The communicators of MPI_Send and MPI_Recv have to match. The communicator indicates a collection of processes that can send messages to each other. The predefined communicator MPI_COMM_WORLD denotes the set of all processes that participate at the computation. Communicators are an important tool when writing library routines. Defining a communicator that is known only to these routines, messages issued by these routines cannot mix up with messages sent in other parts of the program (even if tags are identical). VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/2015 8

9 Simple send and receive commands The tag in the above example is 0. The tag of the send message must match the expected tag of the receive call. The tag is (can be) used to avoid confusion if several messages are communicated between sender and receiver, e.g., in an iteration. Receiver can wildcard. To receive from any source: use MPI_ANY_SOURCE. To receive with any tag: use MPI_ANY_TAG. The status of MPI_recv returns information on the data that was actually received. status is a (pointer to a) C structure with (at least) three members. status -> MPI_SOURCE status -> MPI_TAG If, e.g., tag or source have been set to be a wildcard, then status->mpi_source and status->mpi_tag return the actual values of these parameters. VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/2015 9

10 Another demo program in C: greetings.c #include <stdio.h> #include <mpi.h> int main(int argc, char* argv[]) { int rank; /* rank of process */ int p; /* number of processes */ char message[100]; /* storage of the message */ MPI_Status status; /* return status for receiver */ MPI_Init(&argc, &argv); MPI_Comm_rank(MPI_COMM_WORLD, &rank); MPI_Comm_size(MPI_COMM_WORLD, &p); if (rank == 0) { sprintf(message, "Greetings from process %d!", rank); MPI_Send(message, strlen(message) + 1, MPI_CHAR, 0, 0, MPI_COMM_WOLRD); } else for (int source = 1; source < p; ++source) { MPI_Recv(message, 100, MPI_CHAR, source, 0, MPI_COMM_WOLRD, &status); printf("%s\n", message); } MPI_Finalize(); } /* main */ VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

11 Communication modes What happens if a send or receive command is issued? Standard communication : MPI_Send, MPI_Recv Sending and receiving is asynchron, blocking communication. MPI_Recv can be called before the associated MPI_Send is called. MPI_Recv blocks until the message has been received completely, blocks forever if message is not sent. MPI_Send can also be called before the associated MPI_Recv is called. MPI_Send blocks until the message is copied out of memory (wherever). The message might be copied directly into the matching receive buffer, or it might be copied into a temporary system buffer. MPI offers the choice of several communication modes that allow one to control the choice of the communication protocol. VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

12 Communication modes What happens if a send or receive command is issued? Synchronous communication (avoid buffering) : MPI_Ssend, MPI_Srecv Sender waits until receiver is ready. Then the system copies from memory to memory. Both processors block forever if messages are not awaited or sent: deadlock, e.g. VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

13 Communication modes Non-blocking communication : MPI_Isend, MPI_Irecv Messages are copied into a system buffer on the sender or receiver side (or both). MPI_Irecv returns immediately, even if nothing has been received yet. int MPI_Irecv(... MPI_Comm communicator /* in */, MPI_Request* request /* out */) Additional output parameter request to check whether a message actually has been received (i.e. has been copied in memory) int MPI_Test(MPI_Request* request /* in */, int* flag /* out */, MPI_Status* status /* out */) Instead of waiting of receiving data from other processors some other calculations can be performed in meantime. VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

14 Communication modes Non-blocking communication : MPI_Isend, MPI_Irecv Messages are copied into a system buffer on the sender or receiver side (or both). MPI_Irecv returns immediately, even if nothing has been received yet. int MPI_Irecv(... MPI_Comm communicator /* in */, MPI_Request* request /* out */) Additional output parameter request. If, finally, one wants to wait until receiving the message, i.e., switching it to a blocking one, use int MPI_Wait(MPI_Request* request /* in */, MPI_Status* status /* out */) VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

15 Collective modes Let s assume that process 0 reads some input data that it needs to make available to all other processes in the group. We know how process 0 could proceed: for (dest = 1; dest < p; ++dest) MPI_Send(data, size, MPI_INT, dest, tag, MPI_COMM_WOLRD); In this approach p 1 messages are sent, all with the same sender. We know that there are more elegant (and in general more efficient) algorithms to do the above: a tree-structured algorithm (remember the hypercube) VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

16 Broadcast implements the tree-structured algorithm int MPI_Bcast(void* message /* in/out */, int count /* in */, MPI_Datatype datatype /* in */, int root /* in */, MPI_Comm communicator /* in */) Message data is sent from the source process (root) to all other processes. So, data is input data in the source process and output data otherwise. Remark: There is no tag, as broadcasts have been used historically for synchronization. VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

17 Reduction: as OpenMP, MPI provides a reduction function (that uses a tree-structured algorithm) int MPI_Reduce(void* operand /* in */, void* result /* out */, int count /* in */, MPI_Datatype datatype /* in */, MPI_Op operator /* in */, int root /* in */, MPI_Comm communicator /* in */) combines the operands and stores the result in *result in process root. Both operand and result refer to count memory locations with data type datatype. MPI_Reduce must be called by all processes in the communicator comm, and count, datatype, operator, and root must be the same in each invocation. VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

18 Operations for MPI_Reduce Operation name Meaning MPI_MAX MPI_MIN MPI_SUM MPI_PROD MPI_LAND MPI_BAND MPI_LOR MPI_BOR MPI_LXOR MPI_BXOR MPI_MAXLOC MPI_MINLOC Maximum Minimum Sum Product Logical and Bitwise and Logical or Bitwise or Logical exclusive or Bitwise exclusive or Maximum and location of maximum Minimum and location of minimum VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

19 Example: dot product Serial version float Serial_dot( float x[] /* in */, floag y[] /* in */, int n /* in */) { } float sum = 0.0; for (i = 0; i < n; ++i) sum = sum + x[i] * y[i]; return sum; Parallel version float sum = 0.0; float local_sum = Serial_dot(local_x, local_y, local_n); MPI_Reduce(&local_sum, &sum, 1, MPI_FLOAT, MPI_SUM, 0, MPI_COMM_WORLD); Use MPI_Allreduce instead of MPI_Reduce and all processes get the result (combination of MPI_Reduce and MPI_Bcast). There is no parameter root for MPI_Allreduce. VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

20 Matrix-vector multiplication Let s consider what integrients we need to do a matrix-vector multiplication, y = A x. For simplicity we assume that A is a n n square matrix. Then, n 1 y k = a ki x i, 0 k < n. i=0 In OpenMP we could parallize this by #pragma omp parallel for(k = 0; k < n; ++k) { y[k] = 0.0; for(i = 0; i < n; ++i) { y[k] = y[k] + a[k,i] * x[i]; } VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

21 As the outer-most loop is parallized and we access the matrix row-wise, we can visualise the matrix-vector product as follows. This is not quite correct as all processes access all of y. VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

22 How can we do this in a distributed memory environment? Let us assume that the matrix A and the vectors x, y are distributed in the block-wise fashion as displayed on the previous page. Let x k R m, y k R m, A k R m n, m = n p, 0 k < p, be portions of x, y, and A, respectively, stored in the process k (usually on processor k). Then, y k = A k x. Thus, each element of the vector y is the result of the inner product of a row of A with the vector x. In order to form the inner product of each row of A with x we either have to gather all of x onto each process or we have to scatter each (block-)row of A across the processes. In our previous OpenMP code the former has been done. If we had parallelized the inner loop then the latter approach had been taken. VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

23 Gather (sammeln) vector, take parts from all processors and send to one VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

24 In MPI gathering vector x on process 0 can be done by the call /* Space allocated in calling program */ float local_x[]; /* Local storage for x */ float global_x[]; /* Storage for all x */ MPI_Gather(local_x, n/p, MPI_FLOAT, global_x, n/p, MPI_FLOAT, 0, MPI_COMM_WORLD); The syntax is MPI_Gather(void* send_data /* in */, int send_count /* in */, MPI_Datatype send_type /* in */, void* recv_data /* out */, int recv_count /* in */ MPI_Datatype recv_type /* in */, int dest /* in */, MPI_Comm comm /* in */) MPI_Gather: Collecting pieces of a distributed vector on a single processor. MPI_Allgather: Collecting pieces of a distributed vector on all processors. VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

25 The alternative to gathering vector x is to scatter matrix A. VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

26 The syntax of MPI_Scatter is int MPI_Scatter(void* send_data /* in */, int send_count /* in */, MPI_Datatype send_type /* in */, void* recv_data /* out */, int recv_count /* in */ MPI_Datatype recv_type /* in */, int origin /* in */, MPI_Comm comm /* in */) MPI_Scatter splits the data referenced by send_data on the process with rank origin in p segments, each of which consists of send_count elements of type send_type. The first segment is sent to process 0, the second to process 1, etc. VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

27 If A is stored block-wise, we have Here, x k R m, y k R m, A k R n m VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/

28 Formally, we have to do the following 1. Compute (locally) y k = A k x x, 0 k < p 2. Reduce y = p 1 k=0 y k with root process 0 (for example), 3. Scatter y on the p processes. The last steps can be combined by the call MPI_Reduce_Scatter VL Scientific Computing WS 2014/2015, Dr. K. Schmidt, 02/03/