Parallel Computation in Simulating Diffusion and Deformation in Human Brain - Parallel Computing for Bioinformatics and Computational Biology - page 129

matrix A in some way. Preconditioning is an iterative method for solving the linear
system, Eq. (5.10), whichmeans applying this method instead to the equivalent system
P − 1 Ax P − 1 b . The preconditioned system is expected to become easier to solve than

the original problem, that is, the matrix P − 1 A is better conditioned than A or P − 1 A and
has a more favorable eigenvalue distribution than A . For an efficient preconditioning,
P should be in some sense close to A and its construction should be inexpensive.
Numerical experiments are performed in Ref. [22] to investigate the efficiency and
effectiveness of several standard ILU preconditioners in solving the linear systems
arising at each Newton iteration, which are conducted on a Sun Blade 100 workstation
with a single 500 MHz UltraSPARC-IIe processor and 128 MB memory. The overall
time cost runs in several hundred seconds, including the preconditioner setup time and
iteration time. The ILU preconditioners based on the static pattern scheme, ILU( k ),
are found to be inefficient for solving the current problem. The best choice among all
the preconditioners that are investigated is the dynamic-pattern-scheme-based ILUT,
with optimum choices of its two thresholding parameters p and τ . The composite
preconditioner in which ILU(0) uses the ILUT data pattern shows better performance
than ILU(0), but no better than ILUT. The numerical tests in Ref. [22] illustrate that,
regardless of the thresholding strategies applied in the ILUpreconditioning, the choice
of the corresponding parameters has direct and distinct influences on the accuracy
and the construction cost of the preconditioner, the convergence rate of the iterative
solution, and the total computational efforts. Generally, the more the entries kept
in the factorization, the higher the quality of the ILU preconditioner will be, which
makes the iterative solution more robust but with the price of more construction time,
and higher per iteration cost.
Figure 5.4 shows the simulated concentration distribution profiles of the anisotropic
diffusion in the human brain from Ref. [22]. The simulated result on the 3D Cartesian

=

Figure 5.4 The concentration distribution profiles of the anisotropic diffusion in the human
brain [22]. Left: the profile on the cutting plane of z = 8. Right: the profile on the cutting plane
of y = 75.