Using Jacobi iterations and blocking for solving sparse triangular systems in incomplete factorization preconditioning

Question

1. What are the contributions mentioned in the paper "Using jacobi iterations and blocking for solving sparse triangular systems in incomplete factorization preconditioning" ?

2. How many iterations is required for the Jacobi sweep?

3. Why does RCM ordering preserve existing supervariable structure?

4. How many iterations does a Jacobi sweep take?

Accepted Answer

The authors also show that by using block Jacobi iterations, they can extend the range of problems for which such an approach can be effective.

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As the number of Jacobi sweeps increases, the number of iterations approaches 1338, which is the number required if exact triangular solves are used, corresponding to the300 conventional approach.

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RCM ordering tends to preserve any existing supervariable structure because variables belonging to a super-150 variable will tend to stay numbered together.

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A single scalar Jacobi sweep on an IC(0) factor takes 6.5 milliseconds for this problem, which is more than 33× faster than the cuSPARSE triangular solve, taking 216.7 milliseconds.

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In their application in which the authors want to solve the system Ax = b, R is an incomplete factor of A and, in particular, in the symmetric case, R = L or LT , where L is an incomplete Cholesky factor of A.

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Although increasing the priority of an item in a heap can be done efficiently, in time proportional to log2 of the number of elements in the250 heap, not all priority queues implement this operation.

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For some “easy” problems, the convergence rate may be degraded by as much as 60 to65 100% but this can be compensated for by the additional parallelism in the solves.

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|S l j | ≤ m thenNli = merge adjacency lists Nli and Nl j Nl j = ∅ if (k, v) ∈ Nli is the result of combining two or more adjacencies thenAdd (li, k) to heap H with priority v end if lk = i for all k ∈ S l j S li = merge(S li , S l j )

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The improved speed for each triangular solve is because, for some problems, particularly with high levels of fill in the triangular factors, level scheduling is unable to reveal sufficient parallelism to fully exploit the GPU hardware.

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This research was supported by the Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Numbers DE-SC-0016564 and DE-SC-0016513.

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One major alternative is to compute and use sparse ap-45 proximate inverses of the incomplete factors, so that preconditioning reduces to SpMV operations [8, 9].

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In the symmetric case, the amount of parallelism in level-scheduled sparse triangular solves can be increased by using multicolor reordering of the rows and columns of60 the original matrix and computing the incomplete Cholesky factors of this reordered matrix [14].

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based on the encouraging results in [7] for using Jacobi itera-55 tions to solve triangular systems, Huckle and coauthors [13] have suggested the use of stationary iterations for solving sparse triangular systems based on sparse approximate inverses.

Accepted Answer

160 When identifying supervariables, techniques based on hashing are available to rapidly determine if the structure of two columns differ [24, 25] (further discussion of efficient algorithms for identifying supervariables is given in [26]).

Using Jacobi iterations and blocking for solving sparse triangular systems in incomplete factorization preconditioning

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Most frequently asked questions

1. What are the contributions mentioned in the paper "Using jacobi iterations and blocking for solving sparse triangular systems in incomplete factorization preconditioning" ?

2. How many iterations is required for the Jacobi sweep?

3. Why does RCM ordering preserve existing supervariable structure?

4. How many iterations does a Jacobi sweep take?

5. What is the Jacobi iteration for solving the system Ax = b?

6. What is the common way to increase the priority of a heap?

7. What is the way to solve symmetric problems?

8. What is the result of combining two or more adjacencies?

9. Why is the speed of the triangular solve improved?

10. What was the funding for this research?

11. What is the main alternative to sparse triangular solves?

12. What is the way to solve sparse triangular problems?

13. What is the way to solve sparse triangular systems?

14. What is the way to identify supervariables?

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Citations

Linear solvers for power grid optimization problems: a review of GPU-accelerated linear solvers.

Variable-size batched Gauss–Jordan elimination for block-Jacobi preconditioning on graphics processors

Efficient Block Algorithms for Parallel Sparse Triangular Solve

Nonlinear Equation Solving: A Faster Alternative to Feedforward Computation

MPI+OpenMP реализация метода сопряженных градиентов с факторизованными неявными предобусловливателями

References

A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs

The university of Florida sparse matrix collection

Iterative Solution Methods

Reducing the bandwidth of sparse symmetric matrices

A flexible inner-outer preconditioned GMRES algorithm

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