Nov
13
awarded  Yearling
Nov
13
awarded  Yearling
Oct
10
awarded  Notable Question
Aug
5
awarded  Notable Question
May
8
awarded  Caucus
2018
Dec
1
awarded  Popular Question
Jun
4
awarded  Popular Question
May
20
awarded  Notable Question
Jan
21
awarded  Notable Question
2017
Dec
11
awarded  Popular Question
May
22
awarded  Notable Question
May
15
comment Is it necessary to invert precondition matrix for iterative solver?
@alephzero yes indeed it is super slow and expensive, that's why it didn't make sense to me, but now I understand what people mean when they write inverse in an equation thanks to the answers below.
May
14
accepted Is it necessary to invert precondition matrix for iterative solver?
May
13
comment Is it necessary to invert precondition matrix for iterative solver?
Now, when using an iterative solver to solve for $x$ in the equation $Ax = b$ we can use a preconditioner such that $M^{-1}Ax = M^{-1}b$. In the slides I saw it is possible to use the $LU$ factorization as preconditioner, i.e. $M = LU$, so the iterative solver will solve the following $(LU)^{-1}Ax = (LU)^{-1}b$ but in the commets above someone said we do not actually compute the inverse, therefore, how to use it in an iterative solver?
May
13
comment Is it necessary to invert precondition matrix for iterative solver?
Uhm ... I think this is not what I'm missing. So, suppose we have the following equation to solve $Ax = b$ where we are solving for $x$. I know it is possible to find an $LU$ factorization for $A$, i.e. $A = LU$ so that we can transform the original problem to $LUx = b$. That I understand and know how to solve. Is this what you tried to describe?
May
12
comment Is it necessary to invert precondition matrix for iterative solver?
@GoHokies ok I see, I've thought about it the last hour and still don't see the complete picture of the preconditioner. I've read the wikipedia article trying to put all the pieces together for the CG with preconditioner but I don't know how to solve for $z = M^{-1}r$ Could you maybe give an example and put it as an answer to this question?
May
12
comment Is it necessary to invert precondition matrix for iterative solver?
@GoHokies I just see $L^{-1}$ and $U^{-1}$ I can't see how using the non-inverted versions come into the solver :/
May
12
comment Is it necessary to invert precondition matrix for iterative solver?
@BillGreene I think that is what messes with my head, so if we do not compute the inverse how do we actually apply it when solving the system? Does it depend on the iterative solver? Any resource you can point me to where I it explains how the LU (without invert) is used to solve the problem?
May
12
asked Is it necessary to invert precondition matrix for iterative solver?
2016
Dec
22
awarded  Popular Question