NumPy vs. MIR using multigrid

TLDR
Comparison of the implementations of a multigrid method in Python and in D. Pictures are here.

Acknowledgements
We would like to thank Ilya Yaroshenko for the pull request with the improvements of the D implementation. Furthermore, we would like to thank Jan Hönig for the supervision.

Feedback is welcome
If you have suggestions for further improvements, feel free to open an issue or a pull request.

Content

NumPy vs. MIR using multigrid

Motivation

Python is a well known and often used programming language. Its C-based package NumPy allows efficient computation for a wide variety of problems. Although Python is popular and commonly used it is worth to look if there are other languages and if they perform similar or even better than the established Python + NumPy combination.

In this sense we want to compare it with the D programming language and its library MIR and find out to which extend they are comparable and how the differences in the performance are.

To do so we choose a more complex application from HPC and implement a multigrid solver in both languages. The measurement is done by solving the Poisson equation in 2D with our solvers. The animation below shows the result of these calculations.

Visualization of the results of the Multigrid algorithm after each cycle (lower left) and the L2-Norm of the residual (lower right).

Related Work

There are already some comparisons between D and other competitors, like this blog entry from Dr. Chibisi Chima-Okereke which deals with the comparison of D, Chapel and Julia. It aims at kernel matrix operations like dot products, exponents, Cauchy, Gaussian, Power and some more.

In MIR Benchmark, D is compared to Python and Julia with respect to simple numerical operations like dot product, multiplication and sorting. Similar to our approach, MIR and NumPy is used in those implementations.

Both works compare the performance of individual matrix or vector operations. We compare a more complex application, and not just individual functions, by implementing a multigrid solver in D and Python using MIR and NumPy.

Methods

Poisson Equation

The Poisson Equation is -Δu = f and is used in various fields to describe processes like fluid dynamics or heat distribution. To solve it numerically, the finte-difference method is usually applied for discretization. The discrete version on rectangular 2D-Grid looks like this:

(∇²u)_i,j = ¹⁄_(h²) (u_i+1,j + u_{i - 1, j} + u_{i, j+1} + u_{i, j-1} - 4 * u_{i, j} ) = f_i,j

Where h is the distance between the grid points. (see here)

Red-Black Gauss Seidel

The Gauss-Seidel method is a common iterative technique to solve systems of linear equations.

For Ax = b the element wise formula is this:

x^(k+1)_i = ¹⁄_{(a_i,i)} (b_i - Σ _i<j a _i,j x_i,j^(k+1) - Σ _i>j a _i,j x_i,j^(k))

The naive implementation is not good to parallelize since the computation is forced to be sequential by construction. This issue is tackled by grouping the grid points into two independent groups. The corresponding method is called Red-Black Gauss-Seidel. Therefore, the inside of the grid is divided into so-called red and black dots like a chessboard. It first calculates updates where the sum of indices is even (red), because these are independent. This step can be done in parallel. Afterwards the same is done for the cells where the sum of indices is odd (black). (see here)

Multigrid

A Multigrid method is an iterative solver for systems of equations in the form of Ax = b. Where A is N × M matrix and x and b are Vectors with N entries. The main idea of multigrid is to solve a relaxed version of the problem with less variables instead of solving the problem directly. The solution for Ax = b is approximated by using the residual r = b - Ax. This residual describes the distance of the current x to the targeted solution and is used to calculate the error e. The error e is the solution to the system of equations Ae = r. It is solved in a restricted version which means that the problem has been transformed to a lower resolution. The current approximation x is then corrected by adding the correction error e. Since e has a lower resolution, it has to be interpolated to a higher resolution first. The solving of Ae = r can be done recursively until the costs for solving are negligible and can be done directly due to the smaller problem size.

The basic scheme of a multigrid cycle looks like the following:

Pre-Smoothing – reducing initial errors calculating a few iterations of the Gauss–Seidel method.
Residual Computation – computing residual error.
Restriction – downsampling the residual to a lower resolution.
Compute error – solve (recursively) the problem Ae = r on the restricted residual.
Prolongation – interpolating the correction e back to the previous resolution.
Correction – Adding prolongated correction error onto the solution approximation x.
Post-Smoothing – reducing further errors using a few iterations of the Gauss–Seidel method.

Various cycle types can be defined by looping the computation of the correction error μ times. Well known cycle types are V-Cycle (μ = 1) and W-Cycle (μ = 2). Performing multiple multigrid cycles will reduce the error of the solution approximation. (see here)

Implementation

Python Multigrid

The Python multigrid implementation is based on an abstract class Cycle. It contains the basic logic of a multigrid cycle and how the correction shall be computed.

    def _compute_correction(self, r, l, h):
        e = np.zeros_like(r)
        for _ in range(self.mu):
            e = self.do_cycle(r, e, l, h)
        return e

    def do_cycle(self, F, U, l, h=None):
        if h is None:
            h = 1 / U.shape[0]

        if l <= 1 or U.shape[0] <= 1:
            return self._solve(F, U, h)

        U = self._presmooth(F=F, U=U, h=h)

        r = self._compute_residual(F=F, U=U, h=h)

        r = self.restriction(r)

        e = self._compute_correction(r, l - 1, 2 * h)

        e = prolongation(e, U.shape)

        # correction
        U = U + e

        return self._postsmooth(F=F, U=U, h=h)

The class PoissonCycle is a specialization of this abstract Cycle. Here, the class specific methods like pre- and post-smoothing are implemented. Both smoothing implementations and also the solver are using the Red-Black Gauss-Seidel algorithm.

D Multigrid

The implementation in D is very similar to the implementation in Python. It is simply a translated version from Python to D.

    @nogc void compute_correction(Slice!(T*, Dim) e, Slice!(const(T)*, Dim) r, uint d, T current_h)
    {
        e[] = 0;
        foreach (_; 0 .. mu)
        {
            do_cycle(r, e, d, current_h);
        }
    }

    /++ adds the correction vector to the U +/
    @nogc void add_correction(Slice!(T*, Dim) U, Slice!(const(T)*, Dim) e)
    {
        U[] += e;
    }

    @nogc void do_cycle(Slice!(const(T)*, Dim) F, Slice!(T*, Dim) U, uint d, T current_h)
    {
        if (d + 1 >= l || U.length <= 1)
        {
            solve(F, U, current_h);
            return;
        }

        presmooth(F, U, current_h);

        auto r = compute_residual(F, U, current_h);

        auto res = temp[d][0];
        restriction(res, r);

        auto cor = temp[d][1];
        compute_correction(cor, res, d + 1, current_h * 2);

        auto e = R(U.shape);
        prolongation(e, cor);

        add_correction(U, e);

        postsmooth(F, U, current_h);
    }

Differences in Red-Black Gauss–Seidel

The implementations of the multigrid only differ essentially in syntactical matters. The main difference is in the used solver and smoother. More precisely, the difference is the Gauss-Seidel method.

In Python, we use Numba to speed up the sweep method in the Red-Black Gauss-Seidel algorithm. It basically performs the update step by using the NumPy array slices. The efficiency differences in using Numba or not are considered in the Python-Benchmark.

In order to estimate the fastest approach in D, we consider three variations of the Red-Black Gauss-Seidel sweep:

Slice: Python like. Uses D Slices and Strides for grouping (Red-Black).
Naive: one for-loop for each dimension. Matrix-Access via multi-dimensional Array.
Field: one for-loop. Matrix is flattened. Access via flattened index.
NdSlice: D like. Uses just MIR functionalities.

The first one is the approach to implement the Gauss-Seidel in a way, that it "looks" syntactical like the Python implementation. But since the indexing operator of the MIR slices does not support striding, it is needed to do with a extra function call of strided. The second, the "naive" version is an implementation as it can be found in an textbook. The third one is the most C/C++ like version with accessing the underling D-array of the MIR slice directly. The last one is implemented by Ilya Yaroshenko and gets along with surprisingly few code lines.

We do not compare these different variations in Python, because this would mean to use high level Python for-loops which are not competitive for this application at least without further optimizations. A sweep implementation with loops can be accelerated by using Numba such that it runs faster than the implementation based on the Numpy array operations. However, it turns out that the performance increase in our use case is not too high, so we do not consider a loop version in our benchmarks. Especially because a loop version would have been significantly slower without Numba.

Measurements

Hardware/Software Setup

Software:
- Python
  - Python 3.7.3
  - NumPy 1.19.3
  - Numba 0.51.2
  - Intel Python Distribution 2020.4.912
    - NumPy 1.18.5
    - Numba 0.51.2
- D
  - LDC 1.24 pre-built package
  - mir-algorithm 3.10.11
  - mir-random 2.2.14
Hardware:

Model Name CPU min CPU max L1d cache L1i cache L2 cache L3 cache

Intel(R) Core(TM) i7-9700 CPU @ 3.00GHz 800 MHz 4700 MHz 32K 32K 256K 12288K
Kernel: Linux cip1e3 4.19.144-1-cip-amd64 x86_64 (gcc version 8.3.0 (Debian 8.3.0-6))

RAM: 64GB | Speed measured with STREAM:

-------------------------------------------------------------
STREAM version $Revision: 5.10 $
-------------------------------------------------------------
This system uses 8 bytes per array element.
-------------------------------------------------------------
Array size = 10000000 (elements), Offset = 0 (elements)
Memory per array = 76.3 MiB (= 0.1 GiB).
Total memory required = 228.9 MiB (= 0.2 GiB).
Each kernel will be executed 10 times.
The *best* time for each kernel (excluding the first iteration)
will be used to compute the reported bandwidth.
-------------------------------------------------------------
Number of Threads requested = 8
Number of Threads counted = 8
-------------------------------------------------------------
Your clock granularity/precision appears to be 1 microseconds.
Each test below will take on the order of 4697 microseconds.
  (= 4697 clock ticks)
Increase the size of the arrays if this shows that
you are not getting at least 20 clock ticks per test.
-------------------------------------------------------------
WARNING -- The above is only a rough guideline.
For best results, please be sure you know the
precision of your system timer.
-------------------------------------------------------------
Function    Best Rate MB/s  Avg time     Min time     Max time
Copy:           22945.6     0.006989     0.006973     0.007011
Scale:          22490.3     0.007176     0.007114     0.007224
Add:            25086.2     0.009590     0.009567     0.009694
Triad:          25104.3     0.009577     0.009560     0.009598
-------------------------------------------------------------
Solution Validates: avg error less than 1.000000e-13 on all three arrays
-------------------------------------------------------------

What was measured?

As performance measures we use the execution time and the number of floating-point operations (FLOP) per second (FLOP/s).

The example problem we choose is the Poisson-Equation with the two dimensional function f(x,y) = sin(2πx)cos(2πy). The analytical solution is given by u(x,y) = sin(2πx)cos(2πy) / (8π²). Therefore, it is very simple to check our results. The boundaries are managed with the Dirichlet boundary condition, that means the boundaries of the Matrices are not updated. The iteration starts initially from zero. Except from the boundary cells, there already initialized with the correct solution. This animation visualizes the results after each multigrid cycle. In the lower left corner it shows the number of the visualized cycle, as well as in the lower right corner the L2-Norm of the residual after each cycle.

As benchmarks for the multigrid implementations (D Benchmark, Python Benchmark, Benchmarks Combined) we solve problems in size of 16, 32, 48, 64, 128, 192, .. 1216, 1280, 1408, 1536, ..., 2432, 2560, 2816, ..., 3840, 4096. Each problem is solved with a Multigrid V-cycle with 2 pre- and postsmoothing steps. As stop criteria we use an epsilon of 1e-6 multiplied with the squared problem size. This makes the stop criteria independent of the problem size. For each permutation of the setup option a run is done 3 times. And in evaluation the median of them is considered.

We also compare the performance of the solvers in the different versions in a Solver Benchmark. Since the multigrid algorithm uses the solver only on relative small problems, we also use problems up to a size of 1280 × 1280. We generate 20 problems from size 16 × 16 to 320 × 320 by increasing the problem size by 16 in each step. From problem size 384 × 384 to 1280 × 1280 we increase the step size to 64. The number of Gauss-Seidel iterations is fixed to 5000 for each problem size.

In order to make sure that each benchmark deals with the same problems, these are generated beforehand and written into .npy files. These files are later imported to the benchmark programs.

To see if parallelization causes some differences in performance, we measure the Python code with 1 and with 8 threads. We do not actively parallelize the code and only set the number of allowed threads through the environment variables (see here). In addition, we also distinguish measurements between with or without optimizations using Numba. As mentioned above, the sweep method in the Gauss-Seidel algorithm as well as the intergrid operations restriction and prolongation are accelerated with the Numba jit decorator. During development we also used the jit decorator with the argument parallel=True. Nevertheless, this shows no significant effect in our implementation, so we not use that argument for our benchmarks. We also experiment with the Intel Python Distribution to speed up our implementation. The Intel Python Distribution is a combination of many Python-packages like Numba or NumPy that is optimized for Intel CPUs. These packages are accelerated with the Intel MKL. When not using the Intel Python Distribution, NumPy is accelerated with the OpenBlas libary.

In the D-Benchmark we differentiate the measurements between the sweep implementations slice, naive, field and ndslice in the Gauss-Seidel method.

How was measured?

To measure the execution time we use the perf_counter() from the Python time package and in the D implementation the Stopwatch from the D standard library is used.

To count the floating-point operations that occur while execution we use the Linux tool perf, which is build into the Linux kernel. It allows to gather a enormous variety of performance counters, if they are implemented by the CPU. The CPU we use offers the performance counters scalar_single, scalar_double, 128b_packed_double, 128b_packed_single, 256b_packed_double, 256b_packed_single for different floating-point operations. Perf offers the Metric Group GFLOPS for these which counts all this hardware events.

Before starting the actual benchmark, there is the need for a startup phase were the original problem is loaded and a small warm-up problem is solved. This is especially crucial for the Python implementation when it is accelerated with numba, since in this initialization phase the JIT-Compiler of numba is doing his work. So we want to avoid that perf counts the FLOP/s that occur while this phase. To achieve this we use the delay option for perf, which delays the start of the measurement, and also delay our programs accordingly. The delay for the program is meant to be a bit longer than the actual startup phase. So the program needs to sleep after the warm-up until the delay is over. It is meant that perf starts to measure while the benchmark program is waiting till its delay is over. This is not problematic, because while waiting there should be no floating-point operation that would spoil our results. The actual execution time is measured separately on program side.

This procedure is sufficient for our kind and complexity of project, but for more advanced projects it might be more suitable to use tools like PAPI or likwid, which allow a more fine grain measurement. But it would be necessary to provide an interface, especially for D, so that it can be used in the benchmarks.

Results and Discussion

The vertical doted lines in the follwing pictures indicate the size of the L1, L2 and L3 cache. The place of the vertical line is calculated with √(Cachesize/8) since we are calculating with 64-bit values and the arrays are N × N big.

The memory-bandwidth (horizontal dotted line) in the figures below is calculated with the value of the Triad-Strem benchmark as it is mentioned here. This value is multiplied by 4/3, since the Stream benchmark is not aware of the write allocated (see here on slide 20) and needs to be corrected. To get the FLOP/s this value is then divided by 16, since for every 32 MB that is written there are 2 floating point operations. (see here on slide 21)

In the following the results are shown in graphs. In the graph legend the variations are named as follows:

D: D with MIR (<sweep type>), while <sweep type> can be field, naive, slice or ndslice.
Python: <environment> <# threads> <numba type>, where <environment> can be intel for the Intel Python Disribution or openblas for the default Python environment with OpenBlas backend. The <# threads> is the number of allowed threads used by Numpy and Numba which is set to 1 or 8. Finally, the <numba type> indicates if Numba was used or not. Here, numba and nonumba are possible values.

Table Multigrid-Cycles

The following table contains some details about how many multigrid cycles and levels are performed for the according problem sizes:

Problem size	16	32	48	64	128	192	256	320	384	448	512	576	640	704	768	832	896	960	1024	1088	1152	1216	1280	1408	1536	1664	1792	1920	2048	2176	2304	2432	2560	2816	3072	3328	3584	3840	4096
Multigird cycles	4	7	8	8	9	10	10	10	11	11	11	11	11	11	11	11	11	11	11	12	11	12	12	12	12	12	12	12	12	12	12	12	12	12	12	12	12	12	12
# levels	3	4	4	5	6	6	7	7	7	7	8	8	8	8	8	8	8	8	9	9	9	9	9	9	9	9	9	9	10	10	10	10	10	10	10	10	10	10	11

This table is representative for all benchmarks. Since every variation of or Multigrid implementation does the same calculation the number of cycles and levels are always the same. The number of levels are simply calculated with ⌊log2(N)⌋ - 1.

Solver Benchmark

The Y-axis of the graphs below have a logarithmic scale to make these more concisely.

Flop/s	Time

Here is already apparent that the D version with using fields and ndslices are the fastest ones. While the Python implementation using the Intel Distribution without Numba is the slowest one. Furthermore, there is no difference in the single- and the multithreaded runs visible. This could be an effect of the relatively small array size, so multithreading would not be worthwhile.

Here it is noticeable that for all implementations - except Python without Numba - the FLOP/s increase sharply before the L1 cache limit. Between the L1 and L2 cache lines the graphs stop rising that much. Up to the L3 cache line and beyond, the graphs start to decrease slightly and level out. In contrast, Python implementations without Numba show no response to all cache limits. This may be a consequence of the missing optimization of Numba.

All setups that are using Numba take almost the same time for the corresponding problem sizes. So the FLOP/s graphs for these setups look verry similar and close (see detailed view of FLOP/s and Time). Without Numba, the runs using the Intel environment are below the OpenBlas environment FLOP/s on smaller problem sizes. They soar and overtake OpenBlas between problem size 320 and 384, although the Intel's execution time is even longer at this point. This effect can only be caused by an abrupt increase of the FLOP by Intel. Until problem size 576, the distance of Intel and OpenBlas is remarkably, then it shrinks (see detailed view of FLOP/s and Time). Therefore, using the Intel environment without Numba is not appropriate. However, both runs without Numba perform very poorly. For big problems, the Python runs can be grouped by applying and not applying the Numba jit.

D Benchmark

Flop/s	Time

In the left figure we see the FLOP/s achieved during the benchmarks with the different D implementations. The ndslice version performs best, then follows close the field and naive version. The slice version achieves the lowest FLOP/s, since it is the most time consuming version, as it can be seen in the right figure.

The execution time of the slice implementation has a higher slope for greater problems than the ndslice, field and naive implementations that remain closely. This results in a greater gap of FLOP/s for greater problem sizes between the slice the other three implementations.

Furthermore, the execution times for small problem sizes are in the range of several milliseconds. As a result, even small changes in the execution time have a great impact to the corresponding FLOP/s value.

Compared to the Solver Benchmark, the FLOP/s of the different sweep versions for sizes smaller than 1280 are closer together. This is mainly an effect of the multigrid architecture since all these versions need to perform the same overhead like the intergrid operations.

Python Benchmark

Flop/s	Time

We split up the figures in different groups, the upper two pictures show the curves for the benchmarks that are accelerated with Numba, the lower ones are without Numba. What stands out in all benchmarks just like in the Solver Benchmark, is that there is no big difference visible between single and multithreaded versions. This might be an effect of the small problem sizes in the lowest multigrid-level provided to the solver. When Numba is used, there is no big difference between the Intel Python distribution, that uses the Intel MKL and the "plain" Python version accelerated with Openblas. In the runs where Numba is not used, the Intel version is slower than the Openblas version, but runs more FLOP/s for problem sizes above 800. One aspect that possibly plays into is the relatively old NumPy version that is used in the Intel Python distribution. The smaller steps in the time curve are caused by the increased multigrid-level for the corresponding problem size (see table). These jumps in the required execution time also influence the ups and downs of the FLOP/s values accordingly.

Benchmarks combined

For the sake of clarity the Y-axis of the combined views below have a logarithmic scale, but the single graphs remain with a linear Y-axis.

Flop/s	Time

As already seen in the Solver Benchmark, the multigrid implementation in D outperforms the Python implementations. Even the slowest D version (slice) is faster than the fastest Python version. This may be due to the optimization level of the D compiler, but also to the fact that compiled programs tend to be faster than interpreted ones.

More or less all implementations follow a similar pattern in behaviour of execution time and FLOP/s as it can be seen in the figures with the single graphs. For smaller problem sizes we can observe a sharp increase in FLOP/s until they reach a peak round about problem size 500. For bigger problem sizes the FLOP/s slightly drop and finally level out.

Summary

From a performance perspective the MIR implementations are superior to the NumPy implementation. The big difference is especially visible in these figures. For the biggest multigird problem, the D versions take around 3 to 4 seconds, while the Python versions take from 15 to 20 seconds. Probably this is mainly caused by the overhead of the Python interpreter and might be reduced by more optimization efforts.

From a programming perspective it was a bit easier to use NumPy than MIR. This is partially caused that we are somehow biased with the experience we already had in the use with Python and NumPy. In contrast, we only got to know D and MIR during this project. Furthermore, the resources, especially the available documentation, for NumPy is more exhaustive and helpful then the one for MIR. However, the D-community in the D-Forum is very helpful and we got quick replies to our questions.

At the current point in time NumPy provides much more functionalities and utilities than MIR, mainly due to its longer existence and big community. But during our project, we did not miss essential features in MIR, since the main data structure of MIR, the slices, provides similar functionalities as the NumPy arrays. The main difference, from a programmers point of view, lies in the indexing operator and how striding is handled.

For convenience, there is also the library numir, which provides some NumPy-like helper functions. This allows a similar use of MIR compared to NumPy. From the rework we have learned that MIR provides a lot of convenient functionalities that allow to write short and pragmatic code like the ndslice sweep implementation.

For those who are not afraid of statically typed programming languages and want to leave a lot of optimization to a compiler, D in combination with MIR seems to be good choice for HPC applications.

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D		D
Python		Python
graphs		graphs
problems		problems
scripts		scripts
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README.md		README.md

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