大型稀疏矩阵计算的现代方法介绍
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前面有2篇博文：矩阵迭代， 稀疏矩阵相关。初学的认识。最近又来咀嚼回头草了，仍然参考两本书：
Iterative Methods for linear sparse systems
Lectcutres on computational numerical analysis of PDE. J.M.McDonough, Kentucky.
第一本是hpc课，老师主推荐；第二本的作者，写了几个lectures都非常好，e.g.
－ 经典迭代方法
* Jacobi, Gauss-Seidel, SOR serials :
- to solve Au=b, decompose A = D - L - U
D: dialogue part of A, L : lower part of A; U : upper part of A
u^(n+1) = G u^n + k
- for convergence test, contrast fixed-point iteration form, and through contraction mapping principle( 压缩映射定理). requiring spectral of system less than 1 .
namely rho(G) < 1
Jacobi G_ja = D;
G-S G_gs = D - L;
SOR G_sor = (D- wL)/w, w is relaxation paramter
SSOR G_ssor = (D - wL)D^(-1)(D - wF)/ (w(2-w))
* ADI
decompose A = H + V + S
H: x-direction term; V: y/z-direction term; S: 0-order term
(rI + H1) u* = (rI - V1) u^n + b;
(rI + V1) u^(n+1) = (rI - H1)u* + b;
* ILU
in this case, the A is sparse banded but not compact, so find a nearly sparse matrix M, and easily do LU decomposition
M = A + E , E is the error matrix
总结： Jacobi, GS, SOR, ADI, ILU 等传统迭代方法，收敛速度慢(?)。一般用作其他现代迭代算法的预处理算子(preconditioners)， 而较少场合单独使用。
预处理算子的作用，使矩阵A的条件数（最大特征值非负）小于1， 越小越好。那么最优预处理算子是将矩阵A化为单位矩阵。
二，迭代加速算法
经典迭代算法的收敛慢的原因，在于每个迭代步，仅抵消了部分自由度上的相对误差。而迭代加速算法的基本思想就是，在每个迭代步上，都抵消所有自由度上的（相对）误差，即第n近似解的误差。加速算法实际上是寻找近似解空间的投影算法 。 包括cg, gmres, ksp
＊ CG, KSP
见前两篇博文。
*Multi-Grid
1 using relaxation-type methods as smoothing operators on fine grids;
2 recursive application of coarse-grid corrections
3 nested iteration
--> defect equation
Accurately solving defect equation needs same amount of computing as the original equation. while if e_h is smooth enough we can solve it on a coarser grid, and still obtain a useful improvement in u_h, namely:
then,
there are two transfer, first from fine-to-coarse grid
and second from coarse-to-fine grid _h = P_{2h}^h e_{2h} " />
In fixed-point iteration form, we can see the convergence can be satisfied when
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If the system have lots of DoF ( >10e6), high-level grid will be helpful, and also full MG with nested iterations.
* Domain Decomposition
1 initially guess boundary values on Boundary B13, solving PDE in subdomain S23 = ( S2 U S3)
2 use the computed results to provide boundary values on Boundary B23, then solving PDE in subdomain S13 = (S1 U S3), and yielding a better boundary values fro B13
3 convergence test or go to next loop
总结： MG, DD, 都解决了传统迭代方法和ksp方法的一个严重问题，当系统自由度增大，收敛速度下降。当然，MG, DD也使用了更多case信息。这使它们相比前面的方法复杂一点。dd 的另一个优势，在于和mpi域分解的 并行思路不谋而合，非常promising.