如何在Scipy中实施ILU先导者? [英] How to implement ILU precondioner in scipy?

问题描述

对于scipy.sparse.linalg中的迭代求解器，例如bicg，gmres等，可以选择为矩阵A添加前置条件.但是，文档不是我很清楚我应该作为先决条件.如果我使用ilu = sp.sparse.linalg.spilu(A)，则ilu并不是任何矩阵，而是一个包含很多东西的对象.

For the iterative solvers in the scipy.sparse.linalg such as bicg, gmres, etc, there is an option to add the precondioner for the matrix A. However, the documentation is not very clear about what I should give as the preconditioner. If I use ilu = sp.sparse.linalg.spilu(A), ilu is not any matrices but an object that encompasses many things.

有人在此处询问了类似的问题，但我没有为我工作(Python 3.7，scipy版本1.1.0)

Someone asked about a similar question here for Python 2.7, but I doesn't work for me (Python 3.7, scipy version 1.1.0)

所以我的问题是如何为这些迭代算法合并不完整的LU预条件器?

So my question is how to incorporate incomplete LU preconditioner for those iterative algorithms?

推荐答案

作为前提条件， gmres 接受

• 稀疏矩阵
• 致密矩阵
• 线性算子

在您的情况下，预处理器来自因式分解，因此必须作为线性运算符传递.

In your case, the preconditioner comes from a factorization, thus it has to be passed as a linear operator.

因此，您可能需要根据通过spilu获得的ILU因式分解显式定义一个线性算子.大致情况:

Thus, you may want to explicitly define a linear operator from the ILU factorization you obtained via spilu. Something along the lines:

sA_iLU = sparse.linalg.spilu(sA)
M = sparse.linalg.LinearOperator((nrows,ncols), sA_iLU.solve)

在这里，sA是CSC格式的稀疏矩阵，并且M现在将是要提供给迭代求解器的前置条件线性运算符.

Here, sA is a sparse matrix in a CSC format, and M will now be the preconditioner linear operator that you will supply to the iterative solver.

基于您提到的问题的完整示例 :

A complete example based on the question you mentioned:

import numpy as np
from scipy import sparse
from scipy.sparse import linalg

A = np.array([[ 0.4445,  0.4444, -0.2222],
              [ 0.4444,  0.4445, -0.2222],
              [-0.2222, -0.2222,  0.1112]])
sA = sparse.csc_matrix(A)

b = np.array([[ 0.6667], 
              [ 0.6667], 
              [-0.3332]])

sA_iLU = sparse.linalg.spilu(sA)
M = sparse.linalg.LinearOperator((3,3), sA_iLU.solve)

x = sparse.linalg.gmres(A,b,M=M)

print(x)

注意:

• 我实际上是以一个密集矩阵为例，但是从您的情况下，从一个代表性的稀疏矩阵开始会更有意义.
• 线性运算符M的大小是硬编码的.
• 尚未对ILU进行任何配置，但使用默认设置.
• 这几乎是建议的

• I am actually using a dense matrix as an example, while it would make more sense to start from a representative sparse matrix in your case.
• size of the linear operator M is hardcoded.
• ILU has not been configured anyhow, but with the defaults.
• this is pretty much what has been suggested in the comments to that aforementioned answer, however, I had to do small tweaks to make it Python3-compatible.

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