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2018年6月29日 (金) 05:50時点における最新版

Performance Testing of Solvers

In this document we present the performance evaluation of some numerical solvers for our Laplacian Smooth system.

I need to solve the following system of equations.

Difusion equation for mesh smoothing:

<math>\frac{\partial V}{\partial t}=\lambda L\left( V\right) </math>

Integrating the diffusion equation with a explicit Euler scheme:

<math>\left( I+\lambda dtL\right) V^{t+1}=V^{t}</math>

Where <math>V^{t}</math> is the actual position of vertex, and <math>V^{t+1}</math> is the vertex after smoothing.

Solving the sparse linear system

<math>Ax=b</math>

Where:

<math>A=\left( I+\lambda dtL\right) </math>

<math>x=V^{t+1}</math>

<math>b=V^{t}</math>


Setup

Hardware Specification

  • Processor: AMD Quad-Core 2.40 GHz
  • RAM: 8.0 GB
  • OS: Windows 7 Professional
  • Graphics controller: NVIDIA Quadro FX 570


Software Specification

  • CGAL Computational Geometry Algorithms Library
  • Graphite Research platform for computer graphics


Numeric Solvers Used

  • CG: Conjugate gradient method.
  • BICGSTAB: Biconjugate gradient stabilized method.
  • GMRES: Generalized minimal residual method.
  • SUPERLU: Sparse Direct Solver, LU decomposition with partial pivoting.
  • TAUCS_LLT: A library of sparse linear solvers with LLT factorization.
  • TAUCS_LU: A library of sparse linear solvers with LU decomposition.
  • TAUCS_LDLT: A library of sparse linear solvers with LDLT factorization.
  • CHOLMOD: Supernodal sparse Cholesky factorization.
  • FASTCG: Fast Conjugate gradient.
  • ARPACK: Solver for symmetric matrices.

Results

Numerical solver SuperLU is faster than the other methods.

The difference between the solvers is not representative.

Model Vertices Edges Faces CG BICGSTAB GMRES SUPERLU TAUCS_LLT TAUCS_LU TAUCS_LDLT CHOLMOD FASTCG ARPACK
cruz 24 44 22 0.05 0.05 0.04 0.04 0.05 0.04 0.05 0.05 0.05 0.50
Motionblender31 538 1608 1072 0.83 0.63 0.71 0.61 0.69 0.63 0.68 0.79 0.62 0.63
Ymodel 4770 14304 9536 19.60 16.44 16.93 16.06 16.65 17.61 16.88 17.95 18.23 16.56
mesh_0120 10002 30000 20000 33.43 27.76 29.91 28.54 29.72 31.43 29.53 30.80 30.51 30.97
neptune 28052 84168 56112 133.97 136.46 136.39 133.21 139.98 140.79 142.87 142.76 146.22 142.96
Armadillo 34594 103776 69184 194.48 174.88 175.80 169.92 179.88 174.37 181.70 183.49 183.56 185.99
Vertices Vs Seconds
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