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last change: 04.03.2008 1) Test case The Full Multigrid(FMG) solver uses Gauss-Seidel(GS) as pre- and post-smoother (FMG-GS). The result is compared with another solver already existing in NeuroFEM package, namely Conjugate Gradient with Algebraic Multigrid as preconditioner (AMG-CG). The solvers are tested on models are built from MRI measurement data and which is in a cube discretized into 94 by 94 by 94 grid points, with probes placed with an interval of 2mm. In the cube there is a ball with four layers with different conductivities. The AMG-CG solver uses an unstructured grid containing only non-zero conductivity. The FMG solver requires an structured grid, which makes it possible to exploit the ParExPDE library, so the grid read from NeuroFEM is embedded into a whole cube and the grid points around the ball are given zero conductivity as they stand for the air. In order to have comparable runtime, the convergence tolerance of FMG-GS is set to 1e-8, while the tolerance for AMG-CG is 1e-9. 2) Results
FMG-GS details: Pre-smoothing steps: 2 Post-smoothing steps: 3 Number of cycles on the finest level: 2 Number of corrections on each level: 10/30/45/55/250 3 Discussion The FMG-GS solver has advantages in less memory consumption and faster setup time. This is because it uses structured grid which can be dealt with by NeuroFEM very efficiently. The solutions of simulation on electric potential compared with analytical solutions shows that in the same runtime the FMG-GS can not converge, and the time for FMG-GS to converge takes about 108 seconds. The reason for this is that the conductivity jumps between different layers have large influence on the solving procedure. AMG-CG solver treats with the conductivity jumps better than FMG-GS because it uses CG iteration to accelerate the convergence. As a matter of fact, pure AMG also converges very slowly for conductivity jumps. So in order to make full use of the advantages of FMG-GS solver and achieve faster convergence, we believe that making FMG-GS as the preconditioner of CG iterations is the best solver for this kind of problems. | ||||||||||||||||