galerkin / sparse-solver-benchmark

🚀 Sparse Positive Definite Linear System Solver Benchmark 📏

This is an informal benchmark for k-harmonic diffusion problems on triangle meshes found commonly in geometry processing.

Current upshot:

• CholMod is very good 🐐
• Pardiso holds up alright 🏆
• Eigen LLT/LDLT are OK but suffer for less sparse systems 🏅
• catamari is OK for medium but inaccurate for big systems (bonus: it's MPL2 🆓)
• Eigen LU¹ is significantly slower 🐌

¹These systems are SPD so LU is not a good choice, but provides a reference.

Clone

git clone --recursive https://github.com/alecjacobson/sparse-solver-benchmark

Build

mkdir build
cd build
cmake ../ -DCMAKE_BUILD_TYPE=Release
make

Run

./sparse_solver_benchmark [path to triangle mesh]

Example

Running

./sparse_solver_benchmark ../xyzrgb_dragon-720K.ply 

on my MacBook 2.3 GHz Quad-Core Intel Core i7 with 32 GB Ram will produce:

harmonic

Method	Factor	Solve	L∞ norm
Eigen::CholmodSupernodalLLT	0.86 secs	0.1 secs	1.13266e-10
Eigen::SimplicialLLT	1.5 secs	0.12 secs	1.19565e-10
Eigen::SimplicialLDLT	1.5 secs	0.14 secs	1.48007e-10
catamari::SparseLDL	1.7 secs	0.11 secs	1.27168e-10
Eigen::PardisoLLT	1.9 secs	0.59 secs	1.05662e-10
Eigen::SparseLU	4.8 secs	0.18 secs	6.85971e-11

biharmonic

Method	Factor	Solve	L∞ norm
Eigen::CholmodSupernodalLLT	1.7 secs	0.13 secs	8.33117e-05
Eigen::SimplicialLLT	12 secs	0.37 secs	5.58785e-05
Eigen::SimplicialLDLT	12 secs	0.41 secs	6.92762e-05
catamari::SparseLDL	13 secs	0.33 secs	0.0002359
Eigen::PardisoLLT	4 secs	0.69 secs	6.34374e-05
Eigen::SparseLU	31 secs	0.6 secs	4.10639e-05

triharmonic

Method	Factor	Solve	L∞ norm
Eigen::CholmodSupernodalLLT	3.3 secs	0.19 secs	42.2496
Eigen::SimplicialLLT	41 secs	0.8 secs	36.0525
Eigen::SimplicialLDLT	41 secs	0.89 secs	32.1019
catamari::SparseLDL	50 secs	0.78 secs	150.97
Eigen::PardisoLLT	6.9 secs	0.81 secs	96.7205
Eigen::SparseLU	1.3e+02 secs	1.2 secs	22.0579

Obviously YMMV, if you find something interesting let me know!.

What about this other solver XYZ?

Please submit a pull request with a wrapper for solver XYZ. The more the merrier.

What are the systems being solved?

This code will build a discretization of the ∆ᵏ operator and solve a system of the form:

∆ᵏ u + u = x

where x is the surface's embedding. In matrix form this is:

(Wᵏ + M) u = M x

where Wᵏ is defined recursively as:

W¹ = L
Wᵏ⁺¹ = Wᵏ M⁻¹ L

and L is the discrete Laplacian and M is the discrete mass matrix.

This is a form of smoothing (k=1 is implicit mean curvature flow "Implicit Fairing of Irregular Meshes" Desbrun et al. 1999, k≥2 is higher order, e.g., "Mixed Finite Elements for Variational Surface Modeling" Jacobson et al. 2010)

For k=1, the system is generally OK w.r.t. conditioning and the sparsity for a minifold mesh will be 7 non-zeros per row (on average).

For k=3, the system can get really badly scaled and starts to become more dense (~40 non-zeros per row).