VPMR C++ Implementation

Call For Help

• more performant parallel SVD algorithm: eigen only provides sequential SVD
• alternative integration: currently only Gauss-Legendre quadrature is available

What Is This?

This is a C++ implementation of the VPMR algorithm to compute the approximation of arbitrary smooth kernel.

Check the reference paper 10.1007/s10915-022-01999-1 and the original MATLAB implementation for more details.

Dependency

1. gmp for multiple precision arithmetic
2. mpfr for multiple-precision floating-point computations
3. mpreal mpreal type C++ wrapper, included
4. BigInt BigInt arbitrary large integer for combinatorial number, included
5. Eigen for matrix decomposition, included
6. tbb for parallel computing
7. exprtk for expression parsing, included
8. exprtk-custom-types for mpreal support, included

How To

Python Package

[!WARNING] The Python module needs external libraries to be installed.

On RPM-based Linux distributions (using dnf), if you are:

1. compiling the application from source (or wheels are not available), sudo dnf install -y gcc-c++ tbb-devel mpfr-devel gmp-devel
2. using the packaged binary (wheels are available), sudo dnf install -y gmp mpfr tbb

Then install the package with pip.

pip install pyvpmr

Compile

[!WARNING] The application relies on eigen and exprtk, which depend on very heavy usage of templates. The compilation would take minutes and around 2 GB memory. You need to install libraries gmp, mpfr and tbb before compiling.

Windows

Use the following instructions based on MSYS2, or follow the Linux instructions below with WSL.

# install necessary packages
pacman -S git mingw-w64-x86_64-cmake mingw-w64-x86_64-tbb mingw-w64-x86_64-gcc mingw-w64-x86_64-ninja mingw-w64-x86_64-gmp mingw-w64-x86_64-mpfr
# clone the repository
git clone --depth 1 https://github.com/TLCFEM/vpmr.git
# initialise submodules
cd vpmr
git submodule update --init --recursive
# apply patch to enable parallel evaluation of some loops in SVD
cd eigen && git apply --ignore-space-change --ignore-whitespace ../patch_size.patch && cd ..
# configure and compile
cmake -G Ninja -DCMAKE_BUILD_TYPE=Release .
ninja

Linux

The following is based on Fedora.

sudo dnf install gcc g++ gfortran cmake git -y
sudo dnf install tbb-devel mpfr-devel gmp-devel -y
git clone --depth 1 https://github.com/TLCFEM/vpmr.git
cd vpmr
git submodule update --init --recursive
cd eigen && git apply --ignore-space-change --ignore-whitespace ../patch_size.patch && cd ..
cmake -DCMAKE_BUILD_TYPE=Release .
make

Usage

All available options are:

Usage: vpmr [options]

Options:

   -n <int>     number of terms (default: 10)
   -d <int>     number of digits (default: 512)
   -q <int>     quadrature order (default: 500)
   -m <int>     digit multiplier (default: 6)
   -nc <int>    controls the maximum exponent (default: 4)
   -e <float>   tolerance (default: 1E-8)
   -k <string>  file name of kernel function (default: exp(-t^2/4))
   -s           print singular values
   -w           print weights
   -h           print this help message

Example

The default kernel is exp(-t^2/4). One can run the application with the following command:

./vpmr -n 30

The output is:

Using the following parameters:
        nc = 4.
         n = 30.
     order = 500.
 precision = 336.
 tolerance = 1.0000e-08.
    kernel = exp(-t*t/4).

[1/6] Computing weights... [60/60]
[2/6] Solving Lyapunov equation...
[3/6] Solving SVD...
[4/6] Transforming (P=+9)...
[5/6] Solving eigen decomposition...
[6/6] Done.

M = 
+1.1745193571738943e+01-1.4261645574068720e-100j
-5.5143304351134397e+00+5.7204056791636839e+00j
-5.5143304351134397e+00-5.7204056791636839e+00j
-1.6161617424833762e-02+2.3459542440459513e+00j
-1.6161617424833762e-02-2.3459542440459513e+00j
+1.6338578576177487e-01+1.9308431539218418e-01j
+1.6338578576177487e-01-1.9308431539218418e-01j
-5.4905134221689715e-03+2.2104939243740062e-03j
-5.4905134221689715e-03-2.2104939243740062e-03j
S = 
+1.8757961592204051e+00-0.0000000000000000e+00j
+1.8700580506914817e+00+6.2013413918954552e-01j
+1.8700580506914817e+00-6.2013413918954552e-01j
+1.8521958553280000e+00-1.2601975249082220e+00j
+1.8521958553280000e+00+1.2601975249082220e+00j
+1.8197653300065935e+00+1.9494562062795735e+00j
+1.8197653300065935e+00-1.9494562062795735e+00j
+1.7655956664692953e+00-2.7555720406099038e+00j
+1.7655956664692953e+00+2.7555720406099038e+00j

Running time: 3 s.

Arbitrary Kernel

For arbitrary kernel, it is necessary to provide the kernel function in a text file. The file should contain the kernel expressed as a function of variable t.

The exprtk is used to parse the expression and compute the value. The provided kernel function must be valid and supported by exprtk.

For example, to compute the approximation of exp(-t^2/10), one can create a file kernel.txt with the following content:

exp(-t*t/10)

In the following, the kernel function is echoed to a file and then used as an input to the application.

echo "exp(-t*t/10)" > kernel.txt
 ./vpmr -n 60 -k kernel.txt -e 1e-12

Visualisation

The plotter folder contains a Python script to plot the result.

It is necessary to have click, matplotlib and numpy available in, for example, the virtual environment.

The usage is similar:

Usage: plotter.py [OPTIONS]

Options:
  -n INTEGER   number of terms
  -q INTEGER   quadrature order
  -d INTEGER   number of precision digits
  -nc INTEGER  controls the maximum exponents
  -e FLOAT     tolerance
  -o TEXT      save plot to file
  -k TEXT      file name of kernel function
  --exe TEXT   path to vpmr executable
  --help       Show this message and exit.

[!Note] Remember to change the kernel function so that the plot is meaningful.

# change this kernel before plotting
def kernel(x):
    return np.exp(-x * x / 4)

Binary

The binary requires available gmp, mpfr and tbb libraries.

❯ ldd vpmr
     linux-vdso.so.1 (0x00007ffcf3121000)
     libgmp.so.10 => /lib64/libgmp.so.10 (0x00007f72087e8000)
     libmpfr.so.6 => /lib64/libmpfr.so.6 (0x00007f7208736000)
     libtbb.so.2 => /lib64/libtbb.so.2 (0x00007f72086f2000)
     libstdc++.so.6 => /lib64/libstdc++.so.6 (0x00007f7208400000)
     libm.so.6 => /lib64/libm.so.6 (0x00007f7208320000)
     libgcc_s.so.1 => /lib64/libgcc_s.so.1 (0x00007f72086d0000)
     libc.so.6 => /lib64/libc.so.6 (0x00007f7208143000)
     /lib64/ld-linux-x86-64.so.2 (0x00007f72088a1000)

The distributed appimage is portable.