Vector 4 computational reordering and cache friendliness

Computation-reordering and cache friendly computation

In this example we show how reordering the data can significantly improve the computation speed. In order to do this we will re-work the molecular dynamic example.

Calculate forces

This function is the same as the molecular dynamic example with few changes:

• The function now take as argument CellList_hilb instead of CellList

   
  template<typename CellList> void calc_forces(vector_dist<3,double, aggregate<double[3],double[3]> > & vd, CellList & NN, double sigma12, double sigma6)
  {
   

• We get an iterator from the Cell list instead that from the vector

   
      // Get an iterator over particles
      auto it2 = NN.getIterator();
   

See also

Calculate forces

Calculate energy

In this function is the same as the molecular dynamic example with few changes:

• The function now take as argument CellList_hilb instead of CellList

   
  template<typename CellList> void calc_forces(vector_dist<3,double, aggregate<double[3],double[3]> > & vd, CellList & NN, double sigma12, double sigma6)
  {
   

• We get an iterator from the Cell list instead that from the vector

   
      // Get an iterator over particles
      auto it2 = NN.getIterator();
   

  The difference in doing this is that now we iterate on particles in a smarter way. We will explain more in detail later in the example

See also

Calculate energy

Initialization

The initialization is the same as the molecular dynamic example. The differences are in the parameters. We will use a bigger system, with more particles. The delta time for integration is chosen in order to keep the system stable.

See also

Initialization

 
    double dt = 0.0001;
    double r_cut = 0.03;
    double sigma = r_cut/3.0;
    double sigma12 = pow(sigma,12);
    double sigma6 = pow(sigma,6);
 
    openfpm::vector<double> x;
    openfpm::vector<openfpm::vector<double>> y;
 
    openfpm_init(&argc,&argv);
    Vcluster<> & v_cl = create_vcluster();
 
    // we will use it do place particles on a 40x40x40 Grid like
    size_t sz[3] = {40,40,40};
 
    // domain
    Box<3,double> box({0.0,0.0,0.0},{1.0,1.0,1.0});
 
    // Boundary conditions
    size_t bc[3]={PERIODIC,PERIODIC,PERIODIC};
 
    // ghost, big enough to contain the interaction radius
    Ghost<3,double> ghost(r_cut);
 
    vector_dist<3,double, aggregate<double[3],double[3]> > vd(0,box,bc,ghost);
 

Particles on a grid like position

Here we place the particles on a grid like manner

See also

Particles on a grid like position

 
    auto it = vd.getGridIterator(sz);
 
    while (it.isNext())
    {
        vd.add();
 
        auto key = it.get();
 
        vd.getLastPos()[0] = key.get(0) * it.getSpacing(0);
        vd.getLastPos()[1] = key.get(1) * it.getSpacing(1);
        vd.getLastPos()[2] = key.get(2) * it.getSpacing(2);
 
        vd.template getLastProp<velocity>()[0] = 0.0;
        vd.template getLastProp<velocity>()[1] = 0.0;
        vd.template getLastProp<velocity>()[2] = 0.0;
 
        vd.template getLastProp<force>()[0] = 0.0;
        vd.template getLastProp<force>()[1] = 0.0;
        vd.template getLastProp<force>()[2] = 0.0;
 
        ++it;
    }
 

Molecular dynamic steps

Here we do 30000 MD steps using verlet integrator the cycle is the same as the molecular dynamic example. with the following changes.

Cell lists

Instead of getting the normal cell list we get an hilbert curve cell-list. Such cell list has a function called getIterator used inside the function calc_forces and calc_energy that iterate across all the particles but in a smart-way. In practice given an r-cut a cell-list is constructed with the provided spacing. Suppose to have a cell-list \( m \times n \), an hilbert curve \( 2^k \times 2^k \) is contructed with \( k = ceil(log_2(max(m,n))) \). Cell-lists are explored according to this Hilbert curve, If a cell does not exist is simply skipped.

+------+------+------+------+     Example of Hilbert curve running on a 3 x 3 Cell
|      |      |      |      |     An hilbert curve of k = ceil(log_2(3)) = 4
|  X+---->X   |  X +---> X  |
|  ^   |  +   |  ^   |   +  |
***|******|******|****---|--+      *******
*  +   |  v   |  +   *   v  |      *     *
*  7   |  8+---->9   *   X  |      *     *  = Domain
*  ^   |      |      *   +  |      *     *
*--|-----------------*---|--+      *******
*  +   |      |      *   v  |
*  4<----+5   |   6<---+ X  |
*      |  ^   |   +  *      |
*---------|-------|--*------+
*      |  +   |   v  *      |
*  1+---->2   |   3+---> X  |
*      |      |      *      |
**********************------+

 this mean that we will iterate the following cells

 1,2,5,4,7,8,9,6,3

 Suppose now that the particles are ordered like described




Particles   id      Cell
             0         1
             1         7
             2         8
             3         1
             4         9
             5         9
             6         6
             7         7
             8         3
             9         2
            10         4
            11         3


The iterator of the cell-list will explore the particles in the following way

Cell     1  2 5 4  7  8  9  6 3
       |   | | | |   | |   | | |
        0,3,9,,10,1,7,2,4,5,6,8


 * 

We cannot explain here what is a cache, but in practice is a fast memory in the CPU able to store chunks of memory. The cache in general is much smaller than RAM, but the big advantage is its speed. Retrieve data from the cache is much faster than RAM. Unfortunately the factors that determine what is on cache and what is not are multiples: Type of cache, algorithm ... . Qualitatively all caches will tend to load chunks of data that you read multiple-time, or chunks of data that probably you will read based on pattern analysis. A small example is a linear memory copy where you read consecutively memory and you write on consecutive memory. Modern CPU recognize such pattern and decide to load on cache the consecutive memory before you actually require it.

Iterating the vector in the way described above has the advantage that when we do computation on particles and its neighborhood with the sequence described above it will happen that:

• If to process a particle A we read some neighborhood particles to process the next particle A+1 we will probably read most of the previous particles.

In order to show in practice what happen we first show the graph when we do not reorder

The measure has oscillation but we see an asymptotic behavior from 0.04 in the initial condition to 0.124 . Below we show what happen when we use iterator from the Cell list hilbert

In cases where particles does not move or move very slowly consider to use data-reordering, because it can give 8-10% speedup

See also

Data reordering, computation-reordering and cache friendly computation

Timers

In order to collect the time of the force calculation we insert two timers around the function calc_force. The overall performance is instead calculated with another timer around the time stepping

 
        timer time;
        if (i % 10 == 0)
            time.start();
 

 
        if (i % 10 == 0)
        {
            time.stop();
            x.add(i);
            y.add({time.getwct()});
        }
 

See also

Molecular dynamic steps

Plotting graphs

After we terminate the MD steps our vector x contains at which iteration we benchmark the force calculation time, while y contains the measured time at that time-step. We can produce a graph X Y

Note

The graph produced is an svg graph that can be view with a browser. From the browser we can also easily save the graph into pure svg format

 
    // Google charts options, it store the options to draw the X Y graph
    GCoptions options;
 
    // Title of the graph
    options.title = std::string("Force calculation time");
 
    // Y axis name
    options.yAxis = std::string("Time");
 
    // X axis name
    options.xAxis = std::string("iteration");
 
    // width of the line
    options.lineWidth = 1.0;
 
    // Object that draw the X Y graph
    GoogleChart cg;
 
    // Add the graph
    // The graph that it produce is in svg format that can be opened on browser
    cg.AddLinesGraph(x,y,options);
 
    // Write into html format
    cg.write("gc_plot2_out.html");
 

Finalize

At the very end of the program we have always to de-initialize the library

 
    openfpm_finalize();
 

Full code

 
#include "Vector/vector_dist.hpp"
#include "Decomposition/CartDecomposition.hpp"
#include "data_type/aggregate.hpp"
#include "Plot/GoogleChart.hpp"
#include "Plot/util.hpp"
#include "timer.hpp"
 
 
constexpr int velocity = 0;
constexpr int force = 1;
 
 
template<typename CellList> void calc_forces(vector_dist<3,double, aggregate<double[3],double[3]> > & vd, CellList & NN, double sigma12, double sigma6)
{
 
 
 
    // Uodate the cell-list
    vd.updateCellList(NN);
 
 
    // Get an iterator over particles
    auto it2 = NN.getIterator();
 
 
    // For each particle p ...
    while (it2.isNext())
    {
        // ... get the particle p
        auto p = it2.get();
 
        // Get the position xp of the particle
        Point<3,double> xp = vd.getPos(p);
 
        // Reset the force counter
        vd.template getProp<force>(p)[0] = 0.0;
        vd.template getProp<force>(p)[1] = 0.0;
        vd.template getProp<force>(p)[2] = 0.0;
 
        // Get an iterator over the neighborhood particles of p
        auto Np = NN.template getNNIterator<NO_CHECK>(NN.getCell(vd.getPos(p)));
 
        // For each neighborhood particle ...
        while (Np.isNext())
        {
            // ... q
            auto q = Np.get();
 
            // if (p == q) skip this particle
            if (q == p) {++Np; continue;};
 
            // Get the position of p
            Point<3,double> xq = vd.getPos(q);
 
            // Get the distance between p and q
            Point<3,double> r = xp - xq;
 
            // take the norm of this vector
            double rn = norm2(r);
 
            // Calculate the force, using pow is slower
            Point<3,double> f = 24.0*(2.0 *sigma12 / (rn*rn*rn*rn*rn*rn*rn) -  sigma6 / (rn*rn*rn*rn)) * r;
 
            // we sum the force produced by q on p
            vd.template getProp<force>(p)[0] += f.get(0);
            vd.template getProp<force>(p)[1] += f.get(1);
            vd.template getProp<force>(p)[2] += f.get(2);
 
            // Next neighborhood
            ++Np;
        }
 
        // Next particle
        ++it2;
    }
}
 
 
 
template<typename CellList> double calc_energy(vector_dist<3,double, aggregate<double[3],double[3]> > & vd, CellList & NN, double sigma12, double sigma6)
{
    double E = 0.0;
 
    // update cell-list
    vd.updateCellList(NN);
 
    // Get the iterator
    auto it2 = NN.getIterator();
 
    // For each particle ...
    while (it2.isNext())
    {
        // ... p
        auto p = it2.get();
 
        // Get the position of the particle p
        Point<3,double> xp = vd.getPos(p);
 
        // Reset the force
        vd.template getProp<force>(p)[0] = 0.0;
        vd.template getProp<force>(p)[1] = 0.0;
        vd.template getProp<force>(p)[2] = 0.0;
 
        // Get an iterator over the neighborhood of the particle p
        auto Np = NN.template getNNIterator<NO_CHECK>(NN.getCell(vd.getPos(p)));
 
        // For each neighborhood of the particle p
        while (Np.isNext())
        {
            // Neighborhood particle q
            auto q = Np.get();
 
            // if p == q skip this particle
            if (q == p) {++Np; continue;};
 
            // Get position of the particle q
            Point<3,double> xq = vd.getPos(q);
 
            // take the normalized direction
            double rn = norm2(xp - xq);
 
            // potential energy (using pow is slower)
            E += 4.0 * ( sigma12 / (rn*rn*rn*rn*rn*rn) - sigma6 / ( rn*rn*rn) );
 
            // Next neighborhood
            ++Np;
        }
 
        // Kinetic energy of the particle given by its actual speed
        E +=   (vd.template getProp<velocity>(p)[0]*vd.template getProp<velocity>(p)[0] +
                vd.template getProp<velocity>(p)[1]*vd.template getProp<velocity>(p)[1] +
                vd.template getProp<velocity>(p)[2]*vd.template getProp<velocity>(p)[2]) / 2;
 
        // Next Particle
        ++it2;
    }
 
    // Calculated energy
    return E;
}
 
 
 
int main(int argc, char* argv[])
{
 
    double dt = 0.0001;
    double r_cut = 0.03;
    double sigma = r_cut/3.0;
    double sigma12 = pow(sigma,12);
    double sigma6 = pow(sigma,6);
 
    openfpm::vector<double> x;
    openfpm::vector<openfpm::vector<double>> y;
 
    openfpm_init(&argc,&argv);
    Vcluster<> & v_cl = create_vcluster();
 
    // we will use it do place particles on a 40x40x40 Grid like
    size_t sz[3] = {40,40,40};
 
    // domain
    Box<3,double> box({0.0,0.0,0.0},{1.0,1.0,1.0});
 
    // Boundary conditions
    size_t bc[3]={PERIODIC,PERIODIC,PERIODIC};
 
    // ghost, big enough to contain the interaction radius
    Ghost<3,double> ghost(r_cut);
 
    vector_dist<3,double, aggregate<double[3],double[3]> > vd(0,box,bc,ghost);
 
 
 
    auto it = vd.getGridIterator(sz);
 
    while (it.isNext())
    {
        vd.add();
 
        auto key = it.get();
 
        vd.getLastPos()[0] = key.get(0) * it.getSpacing(0);
        vd.getLastPos()[1] = key.get(1) * it.getSpacing(1);
        vd.getLastPos()[2] = key.get(2) * it.getSpacing(2);
 
        vd.template getLastProp<velocity>()[0] = 0.0;
        vd.template getLastProp<velocity>()[1] = 0.0;
        vd.template getLastProp<velocity>()[2] = 0.0;
 
        vd.template getLastProp<force>()[0] = 0.0;
        vd.template getLastProp<force>()[1] = 0.0;
        vd.template getLastProp<force>()[2] = 0.0;
 
        ++it;
    }
 
 
 
    // Get the Cell list structure
    auto NN = vd.getCellList_hilb(r_cut);
 
    // calculate forces
    calc_forces(vd,NN,sigma12,sigma6);
    unsigned long int f = 0;
 
    timer time2;
    time2.start();
 
    #ifndef TEST_RUN
    size_t Nstep = 30000;
    #else
    size_t Nstep = 300;
    #endif
 
    // MD time stepping
    for (size_t i = 0; i < Nstep ; i++)
    {
        // Get the iterator
        auto it3 = vd.getDomainIterator();
 
        // integrate velicity and space based on the calculated forces (Step1)
        while (it3.isNext())
        {
            auto p = it3.get();
 
            // here we calculate v(tn + 0.5)
            vd.template getProp<velocity>(p)[0] += 0.5*dt*vd.template getProp<force>(p)[0];
            vd.template getProp<velocity>(p)[1] += 0.5*dt*vd.template getProp<force>(p)[1];
            vd.template getProp<velocity>(p)[2] += 0.5*dt*vd.template getProp<force>(p)[2];
 
            // here we calculate x(tn + 1)
            vd.getPos(p)[0] += vd.template getProp<velocity>(p)[0]*dt;
            vd.getPos(p)[1] += vd.template getProp<velocity>(p)[1]*dt;
            vd.getPos(p)[2] += vd.template getProp<velocity>(p)[2]*dt;
 
            ++it3;
        }
 
        // Because we mooved the particles in space we have to map them and re-sync the ghost
        vd.map();
        vd.template ghost_get<>();
 
        timer time;
        if (i % 10 == 0)
            time.start();
 
        // calculate forces or a(tn + 1) Step 2
        calc_forces(vd,NN,sigma12,sigma6);
 
        if (i % 10 == 0)
        {
            time.stop();
            x.add(i);
            y.add({time.getwct()});
        }
 
        // Integrate the velocity Step 3
        auto it4 = vd.getDomainIterator();
 
        while (it4.isNext())
        {
            auto p = it4.get();
 
            // here we calculate v(tn + 1)
            vd.template getProp<velocity>(p)[0] += 0.5*dt*vd.template getProp<force>(p)[0];
            vd.template getProp<velocity>(p)[1] += 0.5*dt*vd.template getProp<force>(p)[1];
            vd.template getProp<velocity>(p)[2] += 0.5*dt*vd.template getProp<force>(p)[2];
 
            ++it4;
        }
 
        // After every iteration collect some statistic about the confoguration
        if (i % 100 == 0)
        {   
            // We write the particle position for visualization (Without ghost)
            vd.deleteGhost();
            vd.write("particles_",f);
 
            // we resync the ghost
            vd.ghost_get<>();
 
            // We calculate the energy
            double energy = calc_energy(vd,NN,sigma12,sigma6);
            auto & vcl = create_vcluster();
            vcl.sum(energy);
            vcl.execute();
 
            // We also print on terminal the value of the energy
            // only one processor (master) write on terminal
            if (vcl.getProcessUnitID() == 0)
                std::cout << std::endl << "Energy: " << energy << std::endl;
 
            f++;
        }
    }
 
    time2.stop();
    std::cout << "Performance: " << time2.getwct() << std::endl;
 
 
 
    // Google charts options, it store the options to draw the X Y graph
    GCoptions options;
 
    // Title of the graph
    options.title = std::string("Force calculation time");
 
    // Y axis name
    options.yAxis = std::string("Time");
 
    // X axis name
    options.xAxis = std::string("iteration");
 
    // width of the line
    options.lineWidth = 1.0;
 
    // Object that draw the X Y graph
    GoogleChart cg;
 
    // Add the graph
    // The graph that it produce is in svg format that can be opened on browser
    cg.AddLinesGraph(x,y,options);
 
    // Write into html format
    cg.write("gc_plot2_out.html");
 
 
 
    openfpm_finalize();
 
 
}