2D Excitable Media Simulations on an MPI Cluster

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2D Excitable Media Simulations on an MPI Cluster

• CMPT 495 / 585
• Group 3
• Sajid Gulzar
• Peter Proudfoot
• Andrew Gershman

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Initial Condition

• Initial Condition Initial condition for the
  stimulation were produced by using two kinds of
  rectangular patches.
• Excited
• Refractory
• uv1

V1
U1
-
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Snapshot for Classic IC
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Code for the ClassisIC()
void classicIC() int i,j, Ndat-gtN, Mdat-gtM
Array udat-gtu, vdat-gtv   //single classic
spiral (charge 1) for(i0iltN/2i)
for(j0jlt2M/3j) (u)(i,j)1.0
for(i0iltN/2i) for(j0jltM/3j)
(v)(i,j)1.0
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Random Patches


Shift 10 Side 30
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Snapshot for Random Patch
Snapshot with one patch and after 10 sec
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Snapshot for Ten Random Patches
Snapshot with ten patches and after 10 sec
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Snapshot for Ten Random Patches
  Snapshot with ten patch and after 1min
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Snapshot for forty Random Patches
  Snapshot with forty random patches and after 4
min
 
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Code for the randomPatches()
void randomPatches() int i,j, Ndat-gtN,
Mdat-gtM Array udat-gtu, vdat-gtv
srand(23) // seeds random number generator with
system time int side 30 int shift 10
for (int pp 0pplt20pp) int myI rand()
((N-side)-shift) int myJ rand()
((M-side)-shift) for(imyIiltmyIsidei)
for(jmyJjltmyJsidej) (u)(i,j)1.0
for(imyIshiftiltmyIsideshifti)
for(jmyJshiftjltmyJsideshiftj)
(v)(i,j)1.0
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Torus
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Boundary Conditions
Periodic (Torus)
No-Flux (Rectangle)
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Logic

• Rectangle
• 4 edges
• No-Flux waves terminate at edges

• Torus
• No edges
• Periodic waves continue to other side of screen
  
• Left to right, right to left
• Top to bottom, bottom to top

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Implementation (Right and Left)
// right and left Definitions, no-flux boundry
conditions if(lrank!1) leftltorlrank-1
else leftltorcsize if(lrank!csize)
rightltorlrank1 else rightltor1

Step 1 Redefine right and left
// right and left definations, Torus boundry
conditions if(lrank!1) leftltorlrank-1
else leftltorlrank3 if(lrank!csize)
rightltorlrank1 else
rightltorlrank-3
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Implementation (Right and Left)

• Step 2 Always send, always receive

//BOUNDRY CONDITIONS (Torus) //RIGHT SEND
MPI_Send((void)uu-gtGetColumnPtr(L_min_2),
N_plus_1, MPIFPTYPE, right,
11DiscrTime, MPICOMM_WORLD) //LEFT
RECEIVE MPI_Recv((void)uu-gtGetColumnPtr(
0), N_plus_1, MPIFPTYPE, left,
11DiscrTime, MPICOMM_WORLD, status)
//LEFT SEND MPI_Send((void)uu-gtGetColum
nPtr(1), N1, MPIFPTYPE, left,
33DiscrTime, MPICOMM_WORLD) //RIGHT
RECEIVE MPI_Recv((void)uu-gtGetColumnPtr(
L_min_1), N_plus_1, MPIFPTYPE, right,
33DiscrTime, MPICOMM_WORLD,
status)
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Implementation (Top and Bottom)
Step 1 Set u and v of (0,j) (N-1,j), for all j
Step 2 Set u and v of (1,j) (N,j), for all j
(0 , j)
(1 , j)
(N-1 , j)
(N, j)
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Implementation (Top and Bottom)

• //TOPBOTTOM (Torus)
• for(j0jltL_min_1j)
• (uu)(0,j) (u)(N-1,j)
• (v)(0,j) (v)(N-1,j)
• (uu)(N,j) (u)(1,j)
• (v)(N,j) (v)(1,j)

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Results
Single spiral / torus after 30 secs.
20 random patches / torus after 1 min.
10 random patches / torus after 3 mins.
40 random patches / torus after 3 mins.
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Probe Analysis
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Goal

• Insert a probe into the media, and find the
  beat-to-beat time intervals (the time intervals
  between consecutive wave fronts).
• Extract probe data on test cases involving a
  single spiral, multiple random patches, as well
  as measuring observed complex dynamics on a torus.

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General Approach

• The beat-to-beat time intervals between wave
  fronts were taken by maintaining a counter.
• Each time a wave passes, the current value of the
  counter is outputted (in model time units ?
  counterdt), and reset to 0 for then next
  interval timing.
• An integer value for keeping track the of the
  current wave to hit the probe was also
  maintained, for added data clarity in output.
• A boolean variable and conditional if statements
  were used to control excess output, and maintain
  data integrity.
• Screen output was made possible by simply using
  the C cout statement By default, the output
  data was directed from the slave holding the
  probe, to the master node.

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Implementation

• int currWave 1 // number of
  waves that have passed the probe
• int counter 0 // keeps
  track of the beat-to-beat time intervals
• bool hitByWave false // set if hit by
  wave used for output control
• // probe sensing work is done here
• if(lrank 3)
  
  ? probe sits in slave 3
• if((u)(L/2,M/2) lt 0.2 (uu)(L/2,M/2)
  lt 0.2) ? waiting for wave front
• increment counter
• hitByWave false
• counter
• if(((u)(L/2,M/2) gt 0.8)
  ((uu)(L/2,M/2) lt 0.2) !hitByWave)
  ? wave front hit probe
• output beat-to-beat
  time for
• cout ltlt "Wave " ltlt currWave ltlt " " ltlt
  counterdt ltlt endl current wave
  reset time interval
• counter 0 counter
  increment wave count
• currWave set
  boolean control variable
• hitByWave true

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Single Spiral / No Flux
Beat-to-Beat Time Interval
Wave
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10 Random Patches / No Flux
Beat-to-Beat Time Interval
Wave
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40 Random Patches / Torus
Beat-to-Beat Time Interval
Wave
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Difficulties

• The approach to measuring the beat-to-beat
  intervals went through roughly three different
  refinements, where the end of each refinement
  brought to light an approach better than the
  previous.
• Initially, probe data was gathered in a straight
  forward manner. If the current value of u (u)
  was detected to be above 0.8, the current
  discrete time step would be outputted to the
  screen. In this iteration, I also decided to
  include a counter to keep track of the current
  wave to hit the probe.
• In the second refinement, I revamped the
  conditional if statement to respond to a window
  (threshold) of change in the current value of u.
  The window was set to check for a wave front by
  detecting a value of (u) between 0.7 and 0.73.
  Unfortunately, this method lacks accuracy, and
  simply doesnt provide sound output.
• In the last refinement, wave front detection via
  a u-window was replaced by a coded interval
  counter, and boolean control variable. Applying
  these variables in cooperation with modification
  of the conditional if statements, finally enabled
  me to accurately record and output the
  beat-to-beat intervals between wave fronts.

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Speedup Graph
? Parallelism yields increasing returns in
speedup, though diminishing returns sets in past
5 CPUs

• Runs were not performed in RI-105, thus there
  was some factor of slowdown
• due to high communication overhead (network
  communication).

? A speedup factor of approximately 2 ½ was
recognized
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Scientific Findings

• Observation of the behavior associated with the
  various combinations of scenarios, including runs
  involving no-flux boundary conditions, torus
  topology, randomly placed patches, and probe
  data, might lead one to believe several things
• On a domain of size NM, placement of a single
  spiral (using the classic initial conditions ?
  classicIC()) has a wave period which is fairly
  regular a scatter plot of probe data in this
  scenario yields a regression which is fairly
  linear.
• Random placement of several patches in a no-flux
  boundary setting made for fairly interesting
  spiral interaction, though lacked consistency
  with triplet development. This scenario also
  produced a set of spiral interactions, which took
  longer to stabilize, than if the same setup was
  used on a torus topology.
• Using a torus topology by far created the most
  interesting and varied results Triplets always
  developed within the first minute of a run as
  well. Initial probe data shows irregular
  beat-to-beat intervals, though as the structure
  begins to stabilize, the intervals become quite
  regular. Structures stabilized sooner than
  structures in a no-flux boundary setting.