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- Parallel Conjugate gradiant solver - 2 Updates
- LU factorization and numerical stability - 1 Update
| Ramine <ramine@1.1>: Oct 23 02:02PM -0700 Hello, I have read this thesis in Mathematis and statistics about conjugate gradients solver with incomplete Cholesky factorization: http://repositories.tdl.org/ttu-ir/bitstream/handle/2346/47490/KENNEDY-THESIS.pdf?sequence=2 You will notice that the preconditionner with the incomplete Cholesky factorization will give around 2x to 3x improvement on the number of iterations... But as you have noticed i have not used a preconditionner for my conjugate gradiants system solver cause it's already a parallel solver: https://sites.google.com/site/aminer68/parallel-implementation-of-conjugate-gradient-linear-system-solver But if you want me to implement a precondtionner that's easy: So to solve a linear system of linear equations: A*x = b [1] we have to first use incomplete Cholesky factorization on A this will give us A := R*Transpose(R) Note: Tanspose(R) means the matrix transpose of R and Inverse(M) means the inverse of the matrix [1] equal to inverse(M)*A = inverse(M)*b M is equal to R*Transpose(R) So after resolving, this will give: Inverse(Transpose(R))*A*Inverse(R)*R*x = Inverse(Transpose(R))*b So we have to resolve the following system of equations: A1*x1=b1 where A1=Inverse(Transpose(R))*A*Inverse(R) and x1=R*x and b = Inverse(Transpose(R))*b So we have to apply the conjugate gradients solver to the system: A1*x1=b1 And after that we have to resolve the system R*x=x1 to find the vector x So we have to parallelize the calculation of the determinant of a matrix to calculate the inverse of a matrix and we have to parallelize the calculation of the incomplete Cholesky factorization, and that's not so difficult for me. That's how the system will be preconditionned to accelerate the convergence and that's will give a 2x to 3x improvement on the number of iterations.. And as you have noticed also that i am using arrays to implement my parallel conjugate gradiant solver, but arrays do take too much memory even if the matrix is more sparse, but i have decided to implement it like that with arrays, so a matrix of 100000 by 100000 elements will take around 64 Gbytes of memory, so you have to have a computer with 32 Gbytes of memory or more to be able to solve large industrial problems with my parallel Parallel Conjugate gradient linear system solver. To use much less memory i have to use linklists , but linklists are not parallel freindly with my parallel algorithm, they will be too CPU expensive , hence too slow, so this is why i have decided to keep using arrays in my parallel conjugate gradiant algorithm Thank you, Amine Moulay Ramdane. |
| Ramine <ramine@1.1>: Oct 23 02:33PM -0700 Hello, To solve large insdustrial problems , you have to compile my Parallel conjugate gradient linear system solver library to a 64bit binary form and you have to have a 32Gbytes of memory or more in your computer as i have just explained to you You can download my Parallel Conjugate gradient system solver library from: https://sites.google.com/site/aminer68/parallel-implementation-of-conjugate-gradient-linear-system-solver Thank you, Amine Moulay Ramdane. |
| Ramine <ramine@1.1>: Oct 23 01:03PM -0700 Hello, As you have noticed i have implemented a multiple linear regression program here: https://sites.google.com/site/aminer68/multiple-linear-regression And since i am using an optimized LU factorization method that uses SSE2 instructions to compute the inverse matrix, this will give a more numerically stable method to compute the inverse. Cause LU factorization is a more numerically stable method, read here: http://www.metacademy.org/graphs/concepts/lu_factorization Amine Moulay Ramdane. |
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