Hi all,
Commercial solvers like COMSOL and Fluent uses Algebraic multigrid methods to solve the PDEs. AMG method uses V, F, W cycles.
I was just going through a link,but i could not understand the method at all.
Can anyone please tell me a easy to read materials for beginners or explain it ?
Thanks
Vivek
Vivek - AMG uses relaxation methods to reduce 'high-frequency error,' which is generally error associated with large eigenvalues of the original matrix. Relaxation methods are extremely slow to capture 'low-frequency error,' or modes associated with very small eigenvalues of the original matrix. This is where multigrid comes in - a 'coarse-grid-correction' is used to remove low-frequency error.
This can be conceptualized in a geometric setting - low frequency modes are, e.g., the single sin hump (smallest eigenfunction of Laplacian). Relaxation is very slow to remove error like this. But, one also does not need a super fine grid to capture a single sin hump - this should only require a few discrete points to capture. Thus on an appropriate coarse grid, this error can be removed. On the original 'fine' grid with a small step size, high frequency error, e.g. sin(1000x), will be accounted for by relaxation. Note, you could not discretize sin(1000x) with only a few grid points.
Now algebraic multigrid doesn't have an actual grid, per-se. Not in a geometric setting. In algebraic multigrid, progressively smaller matrices (as opposed to actual geometric grids) are used to account for algebraically smooth modes of the operator A, which is another word for modes with small eigenvalues.
You can look at the book Multigrid Tutorial by McCormick et al. published by SIAM.
Have a look at youtube. There is a set of three very good videos that explain the multigrid method. Here is the link to the first one: https://www.youtube.com/watch?v=lQgX7QV37qo
First of all, sorry to say that normally there is NOT a simple or easy material to read... But I have a comprehensive material to give you.. Good Luck
The basic idea of this method is explained in my book " Classical and modern engineering methods in fluid flow and heat transfer" pp.262-263. There is also the reference of original article. I hope it will help. Good luck.
Dear Dr. Abram Dorfman, Can you please upload that original article ??
Dear Muhammad, The article is: 6 Acharya, S., Baliga, B. R., Karki, K., Murthy, J. Y., Prakash, C. and Vanka, S. P., 2007, Pressure-based finite-volume methods in computational fluid dynamic, ASME J. Heat Transfer 129: 407-424.
Hi all,
I was just going through a paper 'An Introduction to Algebraic
Multigrid' by R. D. Falgout. So many doubts are there.
In normal algebraic linear system solver, one solves a system of equations through Gauss-Seidel method or realxation methods or direct solvers.
How does AMG method differs from the above common way of solving? How does AMG reduces the computation time to O(N), with N unknowns?
What i have understood that AMG finds the smooth errors. Smooth errors are obtained by doing Gauss-Seidel iteration. In the region where geometric smoothness is there, AMG allows the coarse grid and fine else where.
Are these grids same as the grids obtained by meshing the geometry? or, these are imaginary grids based on the smoothness errors, the information is transferred to these coarse grids from the real mesh grids, equations are solved in these coarse grids and then information is transferred back to the real mesh elements?
Please read this and respond.
Thanks,
Vivek
Vivek - AMG uses relaxation methods to reduce 'high-frequency error,' which is generally error associated with large eigenvalues of the original matrix. Relaxation methods are extremely slow to capture 'low-frequency error,' or modes associated with very small eigenvalues of the original matrix. This is where multigrid comes in - a 'coarse-grid-correction' is used to remove low-frequency error.
This can be conceptualized in a geometric setting - low frequency modes are, e.g., the single sin hump (smallest eigenfunction of Laplacian). Relaxation is very slow to remove error like this. But, one also does not need a super fine grid to capture a single sin hump - this should only require a few discrete points to capture. Thus on an appropriate coarse grid, this error can be removed. On the original 'fine' grid with a small step size, high frequency error, e.g. sin(1000x), will be accounted for by relaxation. Note, you could not discretize sin(1000x) with only a few grid points.
Now algebraic multigrid doesn't have an actual grid, per-se. Not in a geometric setting. In algebraic multigrid, progressively smaller matrices (as opposed to actual geometric grids) are used to account for algebraically smooth modes of the operator A, which is another word for modes with small eigenvalues.
Vivek,
the Multigrid Tutorial (mentioned earlier) is available on Researchgate.
Another good, but very extensive book about the topic is "Multigrid" by Trottenberg, et. al.
https://www.amazon.com/Multigrid-Ulrich-Trottenberg/dp/012701070X/ref=sr_1_1?ie=UTF8&qid=1481201033&sr=8-1&keywords=multigrid+trottenberg
Article A Multigrid Tutorial, 2nd edition (with corrections)
Try mgnet.org.
There are a lot of materials to start understanding the heuristics of the method.
Best regards.
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