Case Study: Poisson Equation

• Poisson1 (source):
  problem:

  • find approximative solution of the Poisson equation
    
  • on a rectangular grid for given given sources f and boundary values
  remarks:

  • grid approximates continuum, solution by Jacobi iteration:
    
  • source example: "electronic lens"
  • implementation uses two alternating U matrices (avoids additional copies)
  • there are much better algorithms (SOR, multigrid)!
• Poisson2 (source):
  parallelisation:

  • PARALLEL around it loop, DO around sweep loop
  synchronisation:

  • barriers between sweeps are necessary
  • automatic by END DO
  data locality:

  • variables in sweep (incl. sweep itself) private by default
  • it private, undefined after END PARALLEL
    (LASTPRIVATE only possible with DO)
    solution: print in SINGLE block inside parallel block
  • error: private (multiple writes)
  problem: each thread computes only partial error

  • error has different value in each thread
  • loop exit condition doesn't work, deadlock possible (try n=20)
• Poisson3 (source):
  global exit criterion necessary
  summation of error to (shared) total_error

  • can't use REDUCTION at PARALLEL, since summation is done only at the end of the PARALLEL block.
  • can't use REDUCTION at DO in sweep, since the return value is - like all local variables - thread private.
  initialisation of total_error in SINGLE
  summation inside CRITICAL:

  • only one thread at a time inside critical block
  • threads run through it serially
  • must be short (performance problem)
  BARRIER before test of total_error
• Poisson4 (source):
  idea: sweep returns total error, main program is simple
  problem: all variables in sweep are local
  solution: variable in extra module (or COMMON block) are shared
  summation of partial errors as before, but hidden in sweep