Parallelization with MPI and OpenMPI
Contents
15. Parallelization with MPI and OpenMPI¶
15.1. How much is parallelizable¶
If any non-parallel code slips into the application, the parallel performance is limited.
In many simulations, however, the fraction of non-parallelizable work is \(10^{-6}\) or less due to large arrays or objects that are perfectly parallelizable.
15.2. Today’s situation of parallel computing¶
Distributed memory is the dominant hardware configuration. There is a large diversity in these machines, from MPP (massively parallel processing) systems to clusters of off-the-shelf PCs, which are very cost-effective.
Message-passing is a mature programming paradigm and widely accepted. It often provides an efficient match to the hardware. It is primarily used for the distributed memory systems, but can also be used on shared memory systems.
Modern nodes have nowadays several cores, which makes it interesting to use both shared memory (the given node) and distributed memory (several nodes with communication). This leads often to codes which use both MPI and OpenMP.
Our lectures will focus on both MPI and OpenMP.
15.3. Overhead present in parallel computing¶
Uneven load balance: not all the processors can perform useful work at all time.
Overhead of synchronization
Overhead of communication
Extra computation due to parallelization
Due to the above overhead and that certain parts of a sequential algorithm cannot be parallelized we may not achieve an optimal parallelization.
15.4. Parallelizing a sequential algorithm¶
Identify the part(s) of a sequential algorithm that can be executed in parallel. This is the difficult part,
Distribute the global work and data among \(P\) processors.
15.5. Strategies¶
Develop codes locally, run with some few processes and test your codes. Do benchmarking, timing and so forth on local nodes, for example your laptop or PC.
When you are convinced that your codes run correctly, you can start your production runs on available supercomputers.
15.6. How do I run MPI on a PC/Laptop? MPI¶
To install MPI is rather easy on hardware running unix/linux as operating systems, follow simply the instructions from the OpenMPI website. See also subsequent slides. When you have made sure you have installed MPI on your PC/laptop,
Compile with mpicxx/mpic++ or mpif90
# Compile and link
mpic++ -O3 -o nameofprog.x nameofprog.cpp
# run code with for example 8 processes using mpirun/mpiexec
mpiexec -n 8 ./nameofprog.x
15.7. Can I do it on my own PC/laptop? OpenMP installation¶
If you wish to install MPI and OpenMP on your laptop/PC, we recommend the following:
For OpenMP, the compile option -fopenmp is included automatically in recent versions of the C++ compiler and Fortran compilers. For users of different Linux distributions, simply use the available C++ or Fortran compilers and add the above compiler instructions, see also code examples below.
For OS X users however, install libomp
brew install libomp
and compile and link as
c++ -o <name executable> <name program.cpp> -lomp
15.8. Installing MPI¶
For linux/ubuntu users, you need to install two packages (alternatively use the synaptic package manager)
sudo apt-get install libopenmpi-dev
sudo apt-get install openmpi-bin
For OS X users, install brew (after having installed xcode and gcc, needed for the gfortran compiler of openmpi) and then install with brew
brew install openmpi
When running an executable (code.x), run as
mpirun -n 10 ./code.x
where we indicate that we want the number of processes to be 10.
15.9. Installing MPI and using Qt¶
With openmpi installed, when using Qt, add to your .pro file the instructions here
You may need to tell Qt where openmpi is stored.
15.10. What is Message Passing Interface (MPI)?¶
MPI is a library, not a language. It specifies the names, calling sequences and results of functions or subroutines to be called from C/C++ or Fortran programs, and the classes and methods that make up the MPI C++ library. The programs that users write in Fortran, C or C++ are compiled with ordinary compilers and linked with the MPI library.
MPI programs should be able to run on all possible machines and run all MPI implementetations without change.
An MPI computation is a collection of processes communicating with messages.
15.11. Going Parallel with MPI¶
Task parallelism: the work of a global problem can be divided into a number of independent tasks, which rarely need to synchronize. Monte Carlo simulations or numerical integration are examples of this.
MPI is a message-passing library where all the routines have corresponding C/C++-binding
MPI_Command_name
and Fortran-binding (routine names are in uppercase, but can also be in lower case)
MPI_COMMAND_NAME
15.12. MPI is a library¶
MPI is a library specification for the message passing interface, proposed as a standard.
independent of hardware;
not a language or compiler specification;
not a specific implementation or product.
A message passing standard for portability and ease-of-use. Designed for high performance.
Insert communication and synchronization functions where necessary.
15.13. Bindings to MPI routines¶
MPI is a message-passing library where all the routines have corresponding C/C++-binding
MPI_Command_name
and Fortran-binding (routine names are in uppercase, but can also be in lower case)
MPI_COMMAND_NAME
The discussion in these slides focuses on the C++ binding.
15.14. Communicator¶
A group of MPI processes with a name (context).
Any process is identified by its rank. The rank is only meaningful within a particular communicator.
By default the communicator contains all the MPI processes.
MPI_COMM_WORLD
Mechanism to identify subset of processes.
Promotes modular design of parallel libraries.
15.15. Some of the most important MPI functions¶
\(MPI\_Init\) - initiate an MPI computation
\(MPI\_Finalize\) - terminate the MPI computation and clean up
\(MPI\_Comm\_size\) - how many processes participate in a given MPI communicator?
\(MPI\_Comm\_rank\) - which one am I? (A number between 0 and size-1.)
\(MPI\_Send\) - send a message to a particular process within an MPI communicator
\(MPI\_Recv\) - receive a message from a particular process within an MPI communicator
\(MPI\_reduce\) or \(MPI\_Allreduce\), send and receive messages
15.16. The first MPI C/C++ program¶
Let every process write “Hello world” (oh not this program again!!) on the standard output.
using namespace std;
#include <mpi.h>
#include <iostream>
int main (int nargs, char* args[])
{
int numprocs, my_rank;
// MPI initializations
MPI_Init (&nargs, &args);
MPI_Comm_size (MPI_COMM_WORLD, &numprocs);
MPI_Comm_rank (MPI_COMM_WORLD, &my_rank);
cout << "Hello world, I have rank " << my_rank << " out of "
<< numprocs << endl;
// End MPI
MPI_Finalize ();
15.17. The Fortran program¶
PROGRAM hello
INCLUDE "mpif.h"
INTEGER:: size, my_rank, ierr
CALL MPI_INIT(ierr)
CALL MPI_COMM_SIZE(MPI_COMM_WORLD, size, ierr)
CALL MPI_COMM_RANK(MPI_COMM_WORLD, my_rank, ierr)
WRITE(*,*)"Hello world, I've rank ",my_rank," out of ",size
CALL MPI_FINALIZE(ierr)
END PROGRAM hello
15.18. Note 1¶
The output to screen is not ordered since all processes are trying to write to screen simultaneously.
It is the operating system which opts for an ordering.
If we wish to have an organized output, starting from the first process, we may rewrite our program as in the next example.
15.19. Ordered output with MPIBarrier¶
int main (int nargs, char* args[])
{
int numprocs, my_rank, i;
MPI_Init (&nargs, &args);
MPI_Comm_size (MPI_COMM_WORLD, &numprocs);
MPI_Comm_rank (MPI_COMM_WORLD, &my_rank);
for (i = 0; i < numprocs; i++) {}
MPI_Barrier (MPI_COMM_WORLD);
if (i == my_rank) {
cout << "Hello world, I have rank " << my_rank <<
" out of " << numprocs << endl;}
MPI_Finalize ();
15.20. Note 2¶
Here we have used the \(MPI\_Barrier\) function to ensure that that every process has completed its set of instructions in a particular order.
A barrier is a special collective operation that does not allow the processes to continue until all processes in the communicator (here \(MPI\_COMM\_WORLD\)) have called \(MPI\_Barrier\).
The barriers make sure that all processes have reached the same point in the code. Many of the collective operations like \(MPI\_ALLREDUCE\) to be discussed later, have the same property; that is, no process can exit the operation until all processes have started.
However, this is slightly more time-consuming since the processes synchronize between themselves as many times as there are processes. In the next Hello world example we use the send and receive functions in order to a have a synchronized action.
15.21. Ordered output¶
.....
int numprocs, my_rank, flag;
MPI_Status status;
MPI_Init (&nargs, &args);
MPI_Comm_size (MPI_COMM_WORLD, &numprocs);
MPI_Comm_rank (MPI_COMM_WORLD, &my_rank);
if (my_rank > 0)
MPI_Recv (&flag, 1, MPI_INT, my_rank-1, 100,
MPI_COMM_WORLD, &status);
cout << "Hello world, I have rank " << my_rank << " out of "
<< numprocs << endl;
if (my_rank < numprocs-1)
MPI_Send (&my_rank, 1, MPI_INT, my_rank+1,
100, MPI_COMM_WORLD);
MPI_Finalize ();
15.22. Note 3¶
The basic sending of messages is given by the function \(MPI\_SEND\), which in C/C++ is defined as
int MPI_Send(void *buf, int count,
MPI_Datatype datatype,
int dest, int tag, MPI_Comm comm)}
This single command allows the passing of any kind of variable, even a large array, to any group of tasks. The variable buf is the variable we wish to send while count is the number of variables we are passing. If we are passing only a single value, this should be 1.
If we transfer an array, it is the overall size of the array. For example, if we want to send a 10 by 10 array, count would be \(10\times 10=100\) since we are actually passing 100 values.
15.23. Note 4¶
Once you have sent a message, you must receive it on another task. The function \(MPI\_RECV\) is similar to the send call.
int MPI_Recv( void *buf, int count, MPI_Datatype datatype,
int source,
int tag, MPI_Comm comm, MPI_Status *status )
The arguments that are different from those in MPI_SEND are buf which is the name of the variable where you will be storing the received data, source which replaces the destination in the send command. This is the return ID of the sender.
Finally, we have used \(MPI\_Status\_status\),
where one can check if the receive was completed.
The output of this code is the same as the previous example, but now process 0 sends a message to process 1, which forwards it further to process 2, and so forth.
15.24. Numerical integration in parallel¶
Integrating \(\pi\).
The code example computes \(\pi\) using the trapezoidal rules.
The trapezoidal rule
Click on this link for the full program.
15.25. Dissection of trapezoidal rule with \(MPI\_reduce\)¶
// Trapezoidal rule and numerical integration usign MPI
using namespace std;
#include <mpi.h>
#include <iostream>
// Here we define various functions called by the main program
double int_function(double );
double trapezoidal_rule(double , double , int , double (*)(double));
// Main function begins here
int main (int nargs, char* args[])
{
int n, local_n, numprocs, my_rank;
double a, b, h, local_a, local_b, total_sum, local_sum;
double time_start, time_end, total_time;
15.26. Dissection of trapezoidal rule¶
// MPI initializations
MPI_Init (&nargs, &args);
MPI_Comm_size (MPI_COMM_WORLD, &numprocs);
MPI_Comm_rank (MPI_COMM_WORLD, &my_rank);
time_start = MPI_Wtime();
// Fixed values for a, b and n
a = 0.0 ; b = 1.0; n = 1000;
h = (b-a)/n; // h is the same for all processes
local_n = n/numprocs;
// make sure n > numprocs, else integer division gives zero
// Length of each process' interval of
// integration = local_n*h.
local_a = a + my_rank*local_n*h;
local_b = local_a + local_n*h;
15.27. Integrating with MPI¶
total_sum = 0.0;
local_sum = trapezoidal_rule(local_a, local_b, local_n,
&int_function);
MPI_Reduce(&local_sum, &total_sum, 1, MPI_DOUBLE,
MPI_SUM, 0, MPI_COMM_WORLD);
time_end = MPI_Wtime();
total_time = time_end-time_start;
if ( my_rank == 0) {
cout << "Trapezoidal rule = " << total_sum << endl;
cout << "Time = " << total_time
<< " on number of processors: " << numprocs << endl;
}
// End MPI
MPI_Finalize ();
return 0;
} // end of main program
15.28. How do I use \(MPI\_reduce\)?¶
Here we have used
MPI_reduce( void *senddata, void* resultdata, int count,
MPI_Datatype datatype, MPI_Op, int root, MPI_Comm comm)
The two variables \(senddata\) and \(resultdata\) are obvious, besides the fact that one sends the address of the variable or the first element of an array. If they are arrays they need to have the same size. The variable \(count\) represents the total dimensionality, 1 in case of just one variable, while \(MPI\_Datatype\) defines the type of variable which is sent and received.
The new feature is \(MPI\_Op\). It defines the type of operation we want to do.
15.29. More on \(MPI\_Reduce\)¶
In our case, since we are summing the rectangle contributions from every process we define \(MPI\_Op = MPI\_SUM\). If we have an array or matrix we can search for the largest og smallest element by sending either \(MPI\_MAX\) or \(MPI\_MIN\). If we want the location as well (which array element) we simply transfer \(MPI\_MAXLOC\) or \(MPI\_MINOC\). If we want the product we write \(MPI\_PROD\).
\(MPI\_Allreduce\) is defined as
MPI_Allreduce( void *senddata, void* resultdata, int count,
MPI_Datatype datatype, MPI_Op, MPI_Comm comm)
15.30. Dissection of trapezoidal rule¶
We use \(MPI\_reduce\) to collect data from each process. Note also the use of the function \(MPI\_Wtime\).
// this function defines the function to integrate
double int_function(double x)
{
double value = 4./(1.+x*x);
return value;
} // end of function to evaluate
15.31. Dissection of trapezoidal rule¶
// this function defines the trapezoidal rule
double trapezoidal_rule(double a, double b, int n,
double (*func)(double))
{
double trapez_sum;
double fa, fb, x, step;
int j;
step=(b-a)/((double) n);
fa=(*func)(a)/2. ;
fb=(*func)(b)/2. ;
trapez_sum=0.;
for (j=1; j <= n-1; j++){
x=j*step+a;
trapez_sum+=(*func)(x);
}
trapez_sum=(trapez_sum+fb+fa)*step;
return trapez_sum;
} // end trapezoidal_rule
15.32. The quantum dot program for two electrons¶
// Variational Monte Carlo for atoms with importance sampling, slater det
// Test case for 2-electron quantum dot, no classes using Mersenne-Twister RNG
#include "mpi.h"
#include <cmath>
#include <random>
#include <string>
#include <iostream>
#include <fstream>
#include <iomanip>
#include "vectormatrixclass.h"
using namespace std;
// output file as global variable
ofstream ofile;
// the step length and its squared inverse for the second derivative
// Here we define global variables used in various functions
// These can be changed by using classes
int Dimension = 2;
int NumberParticles = 2; // we fix also the number of electrons to be 2
// declaration of functions
// The Mc sampling for the variational Monte Carlo
void MonteCarloSampling(int, double &, double &, Vector &);
// The variational wave function
double WaveFunction(Matrix &, Vector &);
// The local energy
double LocalEnergy(Matrix &, Vector &);
// The quantum force
void QuantumForce(Matrix &, Matrix &, Vector &);
// inline function for single-particle wave function
inline double SPwavefunction(double r, double alpha) {
return exp(-alpha*r*0.5);
}
// inline function for derivative of single-particle wave function
inline double DerivativeSPwavefunction(double r, double alpha) {
return -r*alpha;
}
// function for absolute value of relative distance
double RelativeDistance(Matrix &r, int i, int j) {
double r_ij = 0;
for (int k = 0; k < Dimension; k++) {
r_ij += (r(i,k)-r(j,k))*(r(i,k)-r(j,k));
}
return sqrt(r_ij);
}
// inline function for derivative of Jastrow factor
inline double JastrowDerivative(Matrix &r, double beta, int i, int j, int k){
return (r(i,k)-r(j,k))/(RelativeDistance(r, i, j)*pow(1.0+beta*RelativeDistance(r, i, j),2));
}
// function for square of position of single particle
double singleparticle_pos2(Matrix &r, int i) {
double r_single_particle = 0;
for (int j = 0; j < Dimension; j++) {
r_single_particle += r(i,j)*r(i,j);
}
return r_single_particle;
}
void lnsrch(int n, Vector &xold, double fold, Vector &g, Vector &p, Vector &x,
double *f, double stpmax, int *check, double (*func)(Vector &p));
void dfpmin(Vector &p, int n, double gtol, int *iter, double *fret,
double(*func)(Vector &p), void (*dfunc)(Vector &p, Vector &g));
static double sqrarg;
#define SQR(a) ((sqrarg=(a)) == 0.0 ? 0.0 : sqrarg*sqrarg)
static double maxarg1,maxarg2;
#define FMAX(a,b) (maxarg1=(a),maxarg2=(b),(maxarg1) > (maxarg2) ?\
(maxarg1) : (maxarg2))
// Begin of main program
int main(int argc, char* argv[])
{
// MPI initializations
int NumberProcesses, MyRank, NumberMCsamples;
MPI_Init (&argc, &argv);
MPI_Comm_size (MPI_COMM_WORLD, &NumberProcesses);
MPI_Comm_rank (MPI_COMM_WORLD, &MyRank);
double StartTime = MPI_Wtime();
if (MyRank == 0 && argc <= 1) {
cout << "Bad Usage: " << argv[0] <<
" Read also output file on same line and number of Monte Carlo cycles" << endl;
}
// Read filename and number of Monte Carlo cycles from the command line
if (MyRank == 0 && argc > 2) {
string filename = argv[1]; // first command line argument after name of program
NumberMCsamples = atoi(argv[2]);
string fileout = filename;
string argument = to_string(NumberMCsamples);
// Final filename as filename+NumberMCsamples
fileout.append(argument);
ofile.open(fileout);
}
// broadcast the number of Monte Carlo samples
MPI_Bcast (&NumberMCsamples, 1, MPI_INT, 0, MPI_COMM_WORLD);
// Two variational parameters only
Vector VariationalParameters(2);
int TotalNumberMCsamples = NumberMCsamples*NumberProcesses;
// Loop over variational parameters
for (double alpha = 0.5; alpha <= 1.5; alpha +=0.1){
for (double beta = 0.1; beta <= 0.5; beta +=0.05){
VariationalParameters(0) = alpha; // value of alpha
VariationalParameters(1) = beta; // value of beta
// Do the mc sampling and accumulate data with MPI_Reduce
double TotalEnergy, TotalEnergySquared, LocalProcessEnergy, LocalProcessEnergy2;
LocalProcessEnergy = LocalProcessEnergy2 = 0.0;
MonteCarloSampling(NumberMCsamples, LocalProcessEnergy, LocalProcessEnergy2, VariationalParameters);
// Collect data in total averages
MPI_Reduce(&LocalProcessEnergy, &TotalEnergy, 1, MPI_DOUBLE, MPI_SUM, 0, MPI_COMM_WORLD);
MPI_Reduce(&LocalProcessEnergy2, &TotalEnergySquared, 1, MPI_DOUBLE, MPI_SUM, 0, MPI_COMM_WORLD);
// Print out results in case of Master node, set to MyRank = 0
if ( MyRank == 0) {
double Energy = TotalEnergy/( (double)NumberProcesses);
double Variance = TotalEnergySquared/( (double)NumberProcesses)-Energy*Energy;
double StandardDeviation = sqrt(Variance/((double)TotalNumberMCsamples)); // over optimistic error
ofile << setiosflags(ios::showpoint | ios::uppercase);
ofile << setw(15) << setprecision(8) << VariationalParameters(0);
ofile << setw(15) << setprecision(8) << VariationalParameters(1);
ofile << setw(15) << setprecision(8) << Energy;
ofile << setw(15) << setprecision(8) << Variance;
ofile << setw(15) << setprecision(8) << StandardDeviation << endl;
}
}
}
double EndTime = MPI_Wtime();
double TotalTime = EndTime-StartTime;
if ( MyRank == 0 ) cout << "Time = " << TotalTime << " on number of processors: " << NumberProcesses << endl;
if (MyRank == 0) ofile.close(); // close output file
// End MPI
MPI_Finalize ();
return 0;
} // end of main function
// Monte Carlo sampling with the Metropolis algorithm
void MonteCarloSampling(int NumberMCsamples, double &cumulative_e, double &cumulative_e2, Vector &VariationalParameters)
{
// Initialize the seed and call the Mersienne algo
std::random_device rd;
std::mt19937_64 gen(rd());
// Set up the uniform distribution for x \in [[0, 1]
std::uniform_real_distribution<double> UniformNumberGenerator(0.0,1.0);
std::normal_distribution<double> Normaldistribution(0.0,1.0);
// diffusion constant from Schroedinger equation
double D = 0.5;
double timestep = 0.05; // we fix the time step for the gaussian deviate
// allocate matrices which contain the position of the particles
Matrix OldPosition( NumberParticles, Dimension), NewPosition( NumberParticles, Dimension);
Matrix OldQuantumForce(NumberParticles, Dimension), NewQuantumForce(NumberParticles, Dimension);
double Energy = 0.0; double EnergySquared = 0.0; double DeltaE = 0.0;
// initial trial positions
for (int i = 0; i < NumberParticles; i++) {
for (int j = 0; j < Dimension; j++) {
OldPosition(i,j) = Normaldistribution(gen)*sqrt(timestep);
}
}
double OldWaveFunction = WaveFunction(OldPosition, VariationalParameters);
QuantumForce(OldPosition, OldQuantumForce, VariationalParameters);
// loop over monte carlo cycles
for (int cycles = 1; cycles <= NumberMCsamples; cycles++){
// new position
for (int i = 0; i < NumberParticles; i++) {
for (int j = 0; j < Dimension; j++) {
// gaussian deviate to compute new positions using a given timestep
NewPosition(i,j) = OldPosition(i,j) + Normaldistribution(gen)*sqrt(timestep)+OldQuantumForce(i,j)*timestep*D;
// NewPosition(i,j) = OldPosition(i,j) + gaussian_deviate(&idum)*sqrt(timestep)+OldQuantumForce(i,j)*timestep*D;
}
// for the other particles we need to set the position to the old position since
// we move only one particle at the time
for (int k = 0; k < NumberParticles; k++) {
if ( k != i) {
for (int j = 0; j < Dimension; j++) {
NewPosition(k,j) = OldPosition(k,j);
}
}
}
double NewWaveFunction = WaveFunction(NewPosition, VariationalParameters);
QuantumForce(NewPosition, NewQuantumForce, VariationalParameters);
// we compute the log of the ratio of the greens functions to be used in the
// Metropolis-Hastings algorithm
double GreensFunction = 0.0;
for (int j = 0; j < Dimension; j++) {
GreensFunction += 0.5*(OldQuantumForce(i,j)+NewQuantumForce(i,j))*
(D*timestep*0.5*(OldQuantumForce(i,j)-NewQuantumForce(i,j))-NewPosition(i,j)+OldPosition(i,j));
}
GreensFunction = exp(GreensFunction);
// The Metropolis test is performed by moving one particle at the time
if(UniformNumberGenerator(gen) <= GreensFunction*NewWaveFunction*NewWaveFunction/OldWaveFunction/OldWaveFunction ) {
for (int j = 0; j < Dimension; j++) {
OldPosition(i,j) = NewPosition(i,j);
OldQuantumForce(i,j) = NewQuantumForce(i,j);
}
OldWaveFunction = NewWaveFunction;
}
} // end of loop over particles
// compute local energy
double DeltaE = LocalEnergy(OldPosition, VariationalParameters);
// update energies
Energy += DeltaE;
EnergySquared += DeltaE*DeltaE;
} // end of loop over MC trials
// update the energy average and its squared
cumulative_e = Energy/NumberMCsamples;
cumulative_e2 = EnergySquared/NumberMCsamples;
} // end MonteCarloSampling function
// Function to compute the squared wave function and the quantum force
double WaveFunction(Matrix &r, Vector &VariationalParameters)
{
double wf = 0.0;
// full Slater determinant for two particles, replace with Slater det for more particles
wf = SPwavefunction(singleparticle_pos2(r, 0), VariationalParameters(0))*SPwavefunction(singleparticle_pos2(r, 1),VariationalParameters(0));
// contribution from Jastrow factor
for (int i = 0; i < NumberParticles-1; i++) {
for (int j = i+1; j < NumberParticles; j++) {
wf *= exp(RelativeDistance(r, i, j)/((1.0+VariationalParameters(1)*RelativeDistance(r, i, j))));
}
}
return wf;
}
// Function to calculate the local energy without numerical derivation of kinetic energy
double LocalEnergy(Matrix &r, Vector &VariationalParameters)
{
// compute the kinetic and potential energy from the single-particle part
// for a many-electron system this has to be replaced by a Slater determinant
// The absolute value of the interparticle length
Matrix length( NumberParticles, NumberParticles);
// Set up interparticle distance
for (int i = 0; i < NumberParticles-1; i++) {
for(int j = i+1; j < NumberParticles; j++){
length(i,j) = RelativeDistance(r, i, j);
length(j,i) = length(i,j);
}
}
double KineticEnergy = 0.0;
// Set up kinetic energy from Slater and Jastrow terms
for (int i = 0; i < NumberParticles; i++) {
for (int k = 0; k < Dimension; k++) {
double sum1 = 0.0;
for(int j = 0; j < NumberParticles; j++){
if ( j != i) {
sum1 += JastrowDerivative(r, VariationalParameters(1), i, j, k);
}
}
KineticEnergy += (sum1+DerivativeSPwavefunction(r(i,k),VariationalParameters(0)))*(sum1+DerivativeSPwavefunction(r(i,k),VariationalParameters(0)));
}
}
KineticEnergy += -2*VariationalParameters(0)*NumberParticles;
for (int i = 0; i < NumberParticles-1; i++) {
for (int j = i+1; j < NumberParticles; j++) {
KineticEnergy += 2.0/(pow(1.0 + VariationalParameters(1)*length(i,j),2))*(1.0/length(i,j)-2*VariationalParameters(1)/(1+VariationalParameters(1)*length(i,j)) );
}
}
KineticEnergy *= -0.5;
// Set up potential energy, external potential + eventual electron-electron repulsion
double PotentialEnergy = 0;
for (int i = 0; i < NumberParticles; i++) {
double DistanceSquared = singleparticle_pos2(r, i);
PotentialEnergy += 0.5*DistanceSquared; // sp energy HO part, note it has the oscillator frequency set to 1!
}
// Add the electron-electron repulsion
for (int i = 0; i < NumberParticles-1; i++) {
for (int j = i+1; j < NumberParticles; j++) {
PotentialEnergy += 1.0/length(i,j);
}
}
double LocalE = KineticEnergy+PotentialEnergy;
return LocalE;
}
// Compute the analytical expression for the quantum force
void QuantumForce(Matrix &r, Matrix &qforce, Vector &VariationalParameters)
{
// compute the first derivative
for (int i = 0; i < NumberParticles; i++) {
for (int k = 0; k < Dimension; k++) {
// single-particle part, replace with Slater det for larger systems
double sppart = DerivativeSPwavefunction(r(i,k),VariationalParameters(0));
// Jastrow factor contribution
double Jsum = 0.0;
for (int j = 0; j < NumberParticles; j++) {
if ( j != i) {
Jsum += JastrowDerivative(r, VariationalParameters(1), i, j, k);
}
}
qforce(i,k) = 2.0*(Jsum+sppart);
}
}
} // end of QuantumForce function
#define ITMAX 200
#define EPS 3.0e-8
#define TOLX (4*EPS)
#define STPMX 100.0
void dfpmin(Vector &p, int n, double gtol, int *iter, double *fret,
double(*func)(Vector &p), void (*dfunc)(Vector &p, Vector &g))
{
int check,i,its,j;
double den,fac,fad,fae,fp,stpmax,sum=0.0,sumdg,sumxi,temp,test;
Vector dg(n), g(n), hdg(n), pnew(n), xi(n);
Matrix hessian(n,n);
fp=(*func)(p);
(*dfunc)(p,g);
for (i = 0;i < n;i++) {
for (j = 0; j< n;j++) hessian(i,j)=0.0;
hessian(i,i)=1.0;
xi(i) = -g(i);
sum += p(i)*p(i);
}
stpmax=STPMX*FMAX(sqrt(sum),(double)n);
for (its=1;its<=ITMAX;its++) {
*iter=its;
lnsrch(n,p,fp,g,xi,pnew,fret,stpmax,&check,func);
fp = *fret;
for (i = 0; i< n;i++) {
xi(i)=pnew(i)-p(i);
p(i)=pnew(i);
}
test=0.0;
for (i = 0;i< n;i++) {
temp=fabs(xi(i))/FMAX(fabs(p(i)),1.0);
if (temp > test) test=temp;
}
if (test < TOLX) {
return;
}
for (i=0;i<n;i++) dg(i)=g(i);
(*dfunc)(p,g);
test=0.0;
den=FMAX(*fret,1.0);
for (i=0;i<n;i++) {
temp=fabs(g(i))*FMAX(fabs(p(i)),1.0)/den;
if (temp > test) test=temp;
}
if (test < gtol) {
return;
}
for (i=0;i<n;i++) dg(i)=g(i)-dg(i);
for (i=0;i<n;i++) {
hdg(i)=0.0;
for (j=0;j<n;j++) hdg(i) += hessian(i,j)*dg(j);
}
fac=fae=sumdg=sumxi=0.0;
for (i=0;i<n;i++) {
fac += dg(i)*xi(i);
fae += dg(i)*hdg(i);
sumdg += SQR(dg(i));
sumxi += SQR(xi(i));
}
if (fac*fac > EPS*sumdg*sumxi) {
fac=1.0/fac;
fad=1.0/fae;
for (i=0;i<n;i++) dg(i)=fac*xi(i)-fad*hdg(i);
for (i=0;i<n;i++) {
for (j=0;j<n;j++) {
hessian(i,j) += fac*xi(i)*xi(j)
-fad*hdg(i)*hdg(j)+fae*dg(i)*dg(j);
}
}
}
for (i=0;i<n;i++) {
xi(i)=0.0;
for (j=0;j<n;j++) xi(i) -= hessian(i,j)*g(j);
}
}
cout << "too many iterations in dfpmin" << endl;
}
#undef ITMAX
#undef EPS
#undef TOLX
#undef STPMX
#define ALF 1.0e-4
#define TOLX 1.0e-7
void lnsrch(int n, Vector &xold, double fold, Vector &g, Vector &p, Vector &x,
double *f, double stpmax, int *check, double (*func)(Vector &p))
{
int i;
double a,alam,alam2,alamin,b,disc,f2,fold2,rhs1,rhs2,slope,sum,temp,
test,tmplam;
*check=0;
for (sum=0.0,i=0;i<n;i++) sum += p(i)*p(i);
sum=sqrt(sum);
if (sum > stpmax)
for (i=0;i<n;i++) p(i) *= stpmax/sum;
for (slope=0.0,i=0;i<n;i++)
slope += g(i)*p(i);
test=0.0;
for (i=0;i<n;i++) {
temp=fabs(p(i))/FMAX(fabs(xold(i)),1.0);
if (temp > test) test=temp;
}
alamin=TOLX/test;
alam=1.0;
for (;;) {
for (i=0;i<n;i++) x(i)=xold(i)+alam*p(i);
*f=(*func)(x);
if (alam < alamin) {
for (i=0;i<n;i++) x(i)=xold(i);
*check=1;
return;
} else if (*f <= fold+ALF*alam*slope) return;
else {
if (alam == 1.0)
tmplam = -slope/(2.0*(*f-fold-slope));
else {
rhs1 = *f-fold-alam*slope;
rhs2=f2-fold2-alam2*slope;
a=(rhs1/(alam*alam)-rhs2/(alam2*alam2))/(alam-alam2);
b=(-alam2*rhs1/(alam*alam)+alam*rhs2/(alam2*alam2))/(alam-alam2);
if (a == 0.0) tmplam = -slope/(2.0*b);
else {
disc=b*b-3.0*a*slope;
if (disc<0.0) cout << "Roundoff problem in lnsrch." << endl;
else tmplam=(-b+sqrt(disc))/(3.0*a);
}
if (tmplam>0.5*alam)
tmplam=0.5*alam;
}
}
alam2=alam;
f2 = *f;
fold2=fold;
alam=FMAX(tmplam,0.1*alam);
}
}
#undef ALF
#undef TOLX
15.33. What is OpenMP¶
OpenMP provides high-level thread programming
Multiple cooperating threads are allowed to run simultaneously
Threads are created and destroyed dynamically in a fork-join pattern
An OpenMP program consists of a number of parallel regions
Between two parallel regions there is only one master thread
In the beginning of a parallel region, a team of new threads is spawned
The newly spawned threads work simultaneously with the master thread
At the end of a parallel region, the new threads are destroyed
Many good tutorials online and excellent textbook
Many tutorials online like OpenMP official site
15.34. Getting started, things to remember¶
Remember the header file
#include <omp.h>
Insert compiler directives in C++ syntax as
#pragma omp...
Compile with for example c++ -fopenmp code.cpp
Execute
Remember to assign the environment variable OMP NUM THREADS
It specifies the total number of threads inside a parallel region, if not otherwise overwritten
15.35. OpenMP syntax¶
Mostly directives
#pragma omp construct [ clause ...]
Some functions and types
#include <omp.h>
Most apply to a block of code
Specifically, a structured block
Enter at top, exit at bottom only, exit(), abort() permitted
15.36. Different OpenMP styles of parallelism¶
OpenMP supports several different ways to specify thread parallelism
General parallel regions: All threads execute the code, roughly as if you made a routine of that region and created a thread to run that code
Parallel loops: Special case for loops, simplifies data parallel code
Task parallelism, new in OpenMP 3
Several ways to manage thread coordination, including Master regions and Locks
Memory model for shared data
15.37. General code structure¶
#include <omp.h>
main ()
{
int var1, var2, var3;
/* serial code */
/* ... */
/* start of a parallel region */
#pragma omp parallel private(var1, var2) shared(var3)
{
/* ... */
}
/* more serial code */
/* ... */
/* another parallel region */
#pragma omp parallel
{
/* ... */
}
}
15.38. Parallel region¶
A parallel region is a block of code that is executed by a team of threads
The following compiler directive creates a parallel region
#pragma omp parallel { ... }
Clauses can be added at the end of the directive
Most often used clauses:
default(shared) or default(none)
public(list of variables)
private(list of variables)
15.39. Hello world, not again, please!¶
#include <omp.h>
#include <cstdio>
int main (int argc, char *argv[])
{
int th_id, nthreads;
#pragma omp parallel private(th_id) shared(nthreads)
{
th_id = omp_get_thread_num();
printf("Hello World from thread %d\n", th_id);
#pragma omp barrier
if ( th_id == 0 ) {
nthreads = omp_get_num_threads();
printf("There are %d threads\n",nthreads);
}
}
return 0;
}
15.40. Hello world, yet another variant¶
#include <cstdio>
#include <omp.h>
int main(int argc, char *argv[])
{
omp_set_num_threads(4);
#pragma omp parallel
{
int id = omp_get_thread_num();
int nproc = omp_get_num_threads();
cout << "Hello world with id number and processes " << id << nproc << endl;
}
return 0;
}
Variables declared outside of the parallel region are shared by all threads If a variable like id is declared outside of the
#pragma omp parallel,
it would have been shared by various the threads, possibly causing erroneous output
Why? What would go wrong? Why do we add possibly?
15.41. Important OpenMP library routines¶
int omp get num threads (), returns the number of threads inside a parallel region
int omp get thread num (), returns the a thread for each thread inside a parallel region
void omp set num threads (int), sets the number of threads to be used
void omp set nested (int), turns nested parallelism on/off
15.42. Private variables¶
Private clause can be used to make thread- private versions of such variables:
#pragma omp parallel private(id)
{
int id = omp_get_thread_num();
cout << "My thread num" << id << endl;
}
What is their value on entry? Exit?
OpenMP provides ways to control that
Can use default(none) to require the sharing of each variable to be described
15.43. Master region¶
It is often useful to have only one thread execute some of the code in a parallel region. I/O statements are a common example
#pragma omp parallel
{
#pragma omp master
{
int id = omp_get_thread_num();
cout << "My thread num" << id << endl;
}
}
15.44. Parallel for loop¶
Inside a parallel region, the following compiler directive can be used to parallelize a for-loop:
#pragma omp for
Clauses can be added, such as
schedule(static, chunk size)
schedule(dynamic, chunk size)
schedule(guided, chunk size) (non-deterministic allocation)
schedule(runtime)
private(list of variables)
reduction(operator:variable)
nowait
15.45. Parallel computations and loops¶
OpenMP provides an easy way to parallelize a loop
#pragma omp parallel for
for (i=0; i<n; i++) c[i] = a[i];
OpenMP handles index variable (no need to declare in for loop or make private)
Which thread does which values? Several options.
15.46. Scheduling of loop computations¶
We can let the OpenMP runtime decide. The decision is about how the loop iterates are scheduled and OpenMP defines three choices of loop scheduling:
Static: Predefined at compile time. Lowest overhead, predictable
Dynamic: Selection made at runtime
Guided: Special case of dynamic; attempts to reduce overhead
15.47. Example code for loop scheduling¶
#include <omp.h>
#define CHUNKSIZE 100
#define N 1000
int main (int argc, char *argv[])
{
int i, chunk;
float a[N], b[N], c[N];
for (i=0; i < N; i++) a[i] = b[i] = i * 1.0;
chunk = CHUNKSIZE;
#pragma omp parallel shared(a,b,c,chunk) private(i)
{
#pragma omp for schedule(dynamic,chunk)
for (i=0; i < N; i++) c[i] = a[i] + b[i];
} /* end of parallel region */
}
15.48. Example code for loop scheduling, guided instead of dynamic¶
#include <omp.h>
#define CHUNKSIZE 100
#define N 1000
int main (int argc, char *argv[])
{
int i, chunk;
float a[N], b[N], c[N];
for (i=0; i < N; i++) a[i] = b[i] = i * 1.0;
chunk = CHUNKSIZE;
#pragma omp parallel shared(a,b,c,chunk) private(i)
{
#pragma omp for schedule(guided,chunk)
for (i=0; i < N; i++) c[i] = a[i] + b[i];
} /* end of parallel region */
}
15.49. More on Parallel for loop¶
The number of loop iterations cannot be non-deterministic; break, return, exit, goto not allowed inside the for-loop
The loop index is private to each thread
A reduction variable is special
During the for-loop there is a local private copy in each thread
At the end of the for-loop, all the local copies are combined together by the reduction operation
Unless the nowait clause is used, an implicit barrier synchronization will be added at the end by the compiler
// #pragma omp parallel and #pragma omp for
can be combined into
#pragma omp parallel for
15.50. What can happen with this loop?¶
What happens with code like this
#pragma omp parallel for
for (i=0; i<n; i++) sum += a[i]*a[i];
All threads can access the sum variable, but the addition is not atomic! It is important to avoid race between threads. So-called reductions in OpenMP are thus important for performance and for obtaining correct results. OpenMP lets us indicate that a variable is used for a reduction with a particular operator. The above code becomes
sum = 0.0;
#pragma omp parallel for reduction(+:sum)
for (i=0; i<n; i++) sum += a[i]*a[i];
15.51. Inner product¶
1
< < < ! ! M A T H _ B L O C K
int i;
double sum = 0.;
/* allocating and initializing arrays */
/* ... */
#pragma omp parallel for default(shared) private(i) reduction(+:sum)
for (i=0; i<N; i++) sum += a[i]*b[i];
}
15.52. Different threads do different tasks¶
Different threads do different tasks independently, each section is executed by one thread.
#pragma omp parallel
{
#pragma omp sections
{
#pragma omp section
funcA ();
#pragma omp section
funcB ();
#pragma omp section
funcC ();
}
}
15.53. Single execution¶
#pragma omp single { ... }
The code is executed by one thread only, no guarantee which thread
Can introduce an implicit barrier at the end
#pragma omp master { ... }
Code executed by the master thread, guaranteed and no implicit barrier at the end.
15.54. Coordination and synchronization¶
#pragma omp barrier
Synchronization, must be encountered by all threads in a team (or none)
#pragma omp ordered { a block of codes }
is another form of synchronization (in sequential order). The form
#pragma omp critical { a block of codes }
and
#pragma omp atomic { single assignment statement }
is more efficient than
#pragma omp critical
15.55. Data scope¶
OpenMP data scope attribute clauses:
shared
private
firstprivate
lastprivate
reduction
What are the purposes of these attributes
define how and which variables are transferred to a parallel region (and back)
define which variables are visible to all threads in a parallel region, and which variables are privately allocated to each thread
15.56. Some remarks¶
When entering a parallel region, the private clause ensures each thread having its own new variable instances. The new variables are assumed to be uninitialized.
A shared variable exists in only one memory location and all threads can read and write to that address. It is the programmer’s responsibility to ensure that multiple threads properly access a shared variable.
The firstprivate clause combines the behavior of the private clause with automatic initialization.
The lastprivate clause combines the behavior of the private clause with a copy back (from the last loop iteration or section) to the original variable outside the parallel region.
15.57. Parallelizing nested for-loops¶
Serial code
for (i=0; i<100; i++)
for (j=0; j<100; j++)
a[i][j] = b[i][j] + c[i][j];
}
}
Parallelization
#pragma omp parallel for private(j)
for (i=0; i<100; i++)
for (j=0; j<100; j++)
a[i][j] = b[i][j] + c[i][j];
}
}
Why not parallelize the inner loop? to save overhead of repeated thread forks-joins
Why must j be private? To avoid race condition among the threads
15.58. Nested parallelism¶
When a thread in a parallel region encounters another parallel construct, it may create a new team of threads and become the master of the new team.
#pragma omp parallel num_threads(4)
{
/* .... */
#pragma omp parallel num_threads(2)
{
//
}
}
15.59. Parallel tasks¶
#pragma omp task
#pragma omp parallel shared(p_vec) private(i)
{
#pragma omp single
{
for (i=0; i<N; i++) {
double r = random_number();
if (p_vec[i] > r) {
#pragma omp task
do_work (p_vec[i]);
15.60. Common mistakes¶
Race condition
int nthreads;
#pragma omp parallel shared(nthreads)
{
nthreads = omp_get_num_threads();
}
Deadlock
#pragma omp parallel
{
...
#pragma omp critical
{
...
#pragma omp barrier
}
}
15.61. Not all computations are simple¶
Not all computations are simple loops where the data can be evenly divided among threads without any dependencies between threads
An example is finding the location and value of the largest element in an array
for (i=0; i<n; i++) {
if (x[i] > maxval) {
maxval = x[i];
maxloc = i;
}
}
15.62. Not all computations are simple, competing threads¶
All threads are potentially accessing and changing the same values, maxloc and maxval.
OpenMP provides several ways to coordinate access to shared values
#pragma omp atomic
Only one thread at a time can execute the following statement (not block). We can use the critical option
#pragma omp critical
Only one thread at a time can execute the following block
Atomic may be faster than critical but depends on hardware
15.63. How to find the max value using OpenMP¶
Write down the simplest algorithm and look carefully for race conditions. How would you handle them? The first step would be to parallelize as
#pragma omp parallel for
for (i=0; i<n; i++) {
if (x[i] > maxval) {
maxval = x[i];
maxloc = i;
}
}
15.64. Then deal with the race conditions¶
Write down the simplest algorithm and look carefully for race conditions. How would you handle them? The first step would be to parallelize as
#pragma omp parallel for
for (i=0; i<n; i++) {
#pragma omp critical
{
if (x[i] > maxval) {
maxval = x[i];
maxloc = i;
}
}
}
Exercise: write a code which implements this and give an estimate on performance. Perform several runs, with a serial code only with and without vectorization and compare the serial code with the one that uses OpenMP. Run on different archictectures if you can.
15.65. What can slow down OpenMP performance?¶
Give it a thought!
15.66. What can slow down OpenMP performance?¶
Performance poor because we insisted on keeping track of the maxval and location during the execution of the loop.
We do not care about the value during the execution of the loop, just the value at the end.
This is a common source of performance issues, namely the description of the method used to compute a value imposes additional, unnecessary requirements or properties
Idea: Have each thread find the maxloc in its own data, then combine and use temporary arrays indexed by thread number to hold the values found by each thread
15.67. Find the max location for each thread¶
int maxloc[MAX_THREADS], mloc;
double maxval[MAX_THREADS], mval;
#pragma omp parallel shared(maxval,maxloc)
{
int id = omp_get_thread_num();
maxval[id] = -1.0e30;
#pragma omp for
for (int i=0; i<n; i++) {
if (x[i] > maxval[id]) {
maxloc[id] = i;
maxval[id] = x[i];
}
}
}
15.68. Combine the values from each thread¶
#pragma omp flush (maxloc,maxval)
#pragma omp master
{
int nt = omp_get_num_threads();
mloc = maxloc[0];
mval = maxval[0];
for (int i=1; i<nt; i++) {
if (maxval[i] > mval) {
mval = maxval[i];
mloc = maxloc[i];
}
}
}
Note that we let the master process perform the last operation.
15.69. Matrix-matrix multiplication¶
This code computes the norm of a vector using OpenMp
// OpenMP program to compute vector norm by adding two other vectors
#include <cstdlib>
#include <iostream>
#include <cmath>
#include <iomanip>
#include <omp.h>
# include <ctime>
using namespace std; // note use of namespace
int main (int argc, char* argv[])
{
// read in dimension of vector
int n = atoi(argv[1]);
double *a, *b, *c;
int i;
int thread_num;
double wtime, Norm2, s, angle;
cout << " Perform addition of two vectors and compute the norm-2." << endl;
omp_set_num_threads(4);
thread_num = omp_get_max_threads ();
cout << " The number of processors available = " << omp_get_num_procs () << endl ;
cout << " The number of threads available = " << thread_num << endl;
cout << " The matrix order n = " << n << endl;
s = 1.0/sqrt( (double) n);
wtime = omp_get_wtime ( );
// Allocate space for the vectors to be used
a = new double [n]; b = new double [n]; c = new double [n];
// Define parallel region
# pragma omp parallel for default(shared) private (angle, i) reduction(+:Norm2)
// Set up values for vectors a and b
for (i = 0; i < n; i++){
angle = 2.0*M_PI*i/ (( double ) n);
a[i] = s*(sin(angle) + cos(angle));
b[i] = s*sin(2.0*angle);
c[i] = 0.0;
}
// Then perform the vector addition
for (i = 0; i < n; i++){
c[i] += a[i]+b[i];
}
// Compute now the norm-2
Norm2 = 0.0;
for (i = 0; i < n; i++){
Norm2 += c[i]*c[i];
}
// end parallel region
wtime = omp_get_wtime ( ) - wtime;
cout << setiosflags(ios::showpoint | ios::uppercase);
cout << setprecision(10) << setw(20) << "Time used for norm-2 computation=" << wtime << endl;
cout << " Norm-2 = " << Norm2 << endl;
// Free up space
delete[] a;
delete[] b;
delete[] c;
return 0;
}
15.70. Matrix-matrix multiplication¶
This the matrix-matrix multiplication code with plain c++ memory allocation using OpenMP
// Matrix-matrix multiplication and Frobenius norm of a matrix with OpenMP
#include <cstdlib>
#include <iostream>
#include <cmath>
#include <iomanip>
#include <omp.h>
# include <ctime>
using namespace std; // note use of namespace
int main (int argc, char* argv[])
{
// read in dimension of square matrix
int n = atoi(argv[1]);
double **A, **B, **C;
int i, j, k;
int thread_num;
double wtime, Fsum, s, angle;
cout << " Compute matrix product C = A * B and Frobenius norm." << endl;
omp_set_num_threads(4);
thread_num = omp_get_max_threads ();
cout << " The number of processors available = " << omp_get_num_procs () << endl ;
cout << " The number of threads available = " << thread_num << endl;
cout << " The matrix order n = " << n << endl;
s = 1.0/sqrt( (double) n);
wtime = omp_get_wtime ( );
// Allocate space for the two matrices
A = new double*[n]; B = new double*[n]; C = new double*[n];
for (i = 0; i < n; i++){
A[i] = new double[n];
B[i] = new double[n];
C[i] = new double[n];
}
// Define parallel region
# pragma omp parallel for default(shared) private (angle, i, j, k) reduction(+:Fsum)
// Set up values for matrix A and B and zero matrix C
for (i = 0; i < n; i++){
for (j = 0; j < n; j++) {
angle = 2.0*M_PI*i*j/ (( double ) n);
A[i][j] = s * ( sin ( angle ) + cos ( angle ) );
B[j][i] = A[i][j];
}
}
// Then perform the matrix-matrix multiplication
for (i = 0; i < n; i++){
for (j = 0; j < n; j++) {
C[i][j] = 0.0;
for (k = 0; k < n; k++) {
C[i][j] += A[i][k]*B[k][j];
}
}
}
// Compute now the Frobenius norm
Fsum = 0.0;
for (i = 0; i < n; i++){
for (j = 0; j < n; j++) {
Fsum += C[i][j]*C[i][j];
}
}
Fsum = sqrt(Fsum);
// end parallel region and letting only one thread perform I/O
wtime = omp_get_wtime ( ) - wtime;
cout << setiosflags(ios::showpoint | ios::uppercase);
cout << setprecision(10) << setw(20) << "Time used for matrix-matrix multiplication=" << wtime << endl;
cout << " Frobenius norm = " << Fsum << endl;
// Free up space
for (int i = 0; i < n; i++){
delete[] A[i];
delete[] B[i];
delete[] C[i];
}
delete[] A;
delete[] B;
delete[] C;
return 0;
}