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Performance Analysis and Optimization Report

This document outlines the performance analysis on the Jacobi test for the 2D Poisson equation solver, using default parameters specified in test.csv. The analysis includes initial benchmarking, profiling with gprof, and subsequent optimization strategies applied to enhance performance.

An initial benchmarking memory bandwidth with STREAM provides an insight into the memory bandwidth capabilities of the hardware utilized for the performance analysis:

Function Best Rate MB/s Avg time Min time Max time
Copy 23498.3 0.007372 0.006809 0.009033
Scale 23286.3 0.007183 0.006871 0.007961
Add 27881.5 0.009217 0.008608 0.011332
Triad 28027.4 0.008923 0.008563 0.009630

Profiling (no optimization)

Profiling the application with gprof highlighted several hotspots:

% Time Cumulative Seconds Self Seconds Calls s/call s/call Total Function
11.68% 0.35 0.35 374,740,052 0.00 0.00 std::vector::at
11.51% 0.70 0.35 549,685,071 0.00 0.00 std::vector::_M_range_check
9.21% 0.98 0.28 549,695,069 0.00 0.00 std::vector::size
8.72% 1.25 0.27 549,685,071 0.00 0.00 std::vector::at
6.25% 1.44 0.19 374,740,052 0.00 0.00 std::vector::_M_range_check
6.25% 1.63 0.19 1 0.19 3.01 testJacobi
5.10% 1.78 0.15 599,710,067 0.00 0.00 std::vector::size
4.28% 1.92 0.13 274,800,040 0.00 0.00 Mesh::getNode
3.95% 2.04 0.12 549,685,071 0.00 0.00 std::vector::operator[]
3.95% 2.15 0.12 1 0.12 0.86 Boundaries::setInnerNodes
3.62% 2.27 0.11 2 0.06 0.18 std::vector::_M_fill_initialize
3.29% 2.37 0.10 374,740,052 0.00 0.00 std::vector::operator[]
3.29% 2.46 0.10 1 0.10 1.15 jacobi
2.47% 2.54 0.07 1 0.07 0.35 Mesh::swapGrids
2.30% 2.61 0.07 10,000 0.00 0.00 Node* std::__uninitialized_copy
2.14% 2.67 0.07 174,945,017 0.00 0.00 Mesh::numCols

Total Runtime: 3.52s

  • Frequent calls to std::vector::at(unsigned long).
  • Frequent calls to Mesh::numCols() and Mesh::numCols() that leads to excessive use of std::size().
  • Poorly implemented Mesh::swapGrids().

To profile your application, execute:

./src/PDEsolver --test
gprof ./src/PDEsolver gmon.out > analysis.txt

The profiling output is available in analysis.txt within the build directory.

Optimization Strategies

Optimizing Frequent std::vector::size() Calls

  • Issue: Repeated calls to std::vector::size() within loops.

  • Solution: Cached the size of vectors before entering loops to avoid repeated function calls.

    After optimization:

    • Total Runtime: 2.24s
    • Runtime optimization: 36%

Reducing std::vector::at() Calls

  • Issue: A significant amount of time was spent on bounds-checked access via std::vector::at().

  • Solution: Replaced std::vector::at() with operator[] for direct access, after ensuring index bounds are checked ahead of time.

    After optimization:

    • Total Runtime: 1.28s
    • Runtime optimization: 27%

Optimizing Mesh::swapGrids()

  • Issue: The function swaps the grid entries one by one, inefficient use of std::swap.

  • Solution: Efficient use of std::swap.

    After oprimization:

    • Total Runtime: 1.12s
    • Runtime optimization: 5%

Profiling after optimization

% Time Cumulative Seconds Self Seconds Calls s/call s/call Total Function
19.57% 0.23s 0.23s 274,800,040 0.00s 0.00s Mesh::getNode
13.91% 0.39s 0.16s 324,780,050 0.00s 0.00s std::vector::operator[]
11.74% 0.52s 0.14s 324,780,044 0.00s 0.00s std::vector::operator[]
10.00% 0.64s 0.12s 50,000,000 0.00s 0.00s std::_Construct
8.70% 0.73s 0.10s 1 0.10s 0.35s Boundaries::setInnerNodes
6.09% 0.81s 0.07s 10,000 0.00s 0.00s std::__uninitialized_copy
5.65% 0.87s 0.07s 2 0.03s 0.17s std::vector::vector
4.78% 0.93s 0.06s 25,000,000 0.00s 0.00s Mesh::setNode
4.35% 0.97s 0.05s 50,010,000 0.00s 0.00s __gnu_cxx::operator!=
4.35% 1.02s 0.05s 1 0.05s 0.29s jacobi
2.61% 1.05s 0.03s 1 0.03s 0.07s printToCSV
2.61% 1.08s 0.03s 1 0.03s 0.38s Mesh::Mesh
1.74% 1.10s 0.02s 100,020,004 0.00s 0.00s __gnu_cxx::__normal_iterator::base
1.74% 1.12s 0.02s 50,000,000 0.00s 0.00s __gnu_cxx::__normal_iterator::operator++

  • Total Runtime: 1.12s

  • Total Runtime Optimization: 68%

Compiler optimizations

  • Optimization Level: -O2
  • Impact on Runtime: Reduced runtime from 1.12 seconds to 1.02 seconds.
  • Runtime Optimization: 3%

The -O2 optimization level enables loop optimizations, function inlining, and other improvements. While the -O2 optimization did lead to a runtime improvement, it was not as substantial as expected. To achieve more significant improvements, a more cache friendly way to get the data from the grid or a more efficient way to trigger loop unrolling by the compiler can be implemented.