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On the recursive structure of multigrid cycles

Or Avnat, Irad Yavneh

TL;DR

A simple model is presented for predicting the approximate run-time of the $\kappa$-cycle, which is useful in pre-selecting an appropriate cycle counter for a given problem on a given platform.

Abstract

A new fixed (non-adaptive) recursive scheme for multigrid algorithms is introduced. Governed by a positive parameter $κ$ called the cycle counter, this scheme generates a family of multigrid cycles dubbed $κ$-cycles. The well-known $V$-cycle, $F$-cycle, and $W$-cycle are shown to be particular members of this rich $κ$-cycle family, which satisfies the property that the total number of recursive calls in a single cycle is a polynomial of degree $κ$ in the number of levels of the cycle. This broadening of the scope of fixed multigrid cycles is shown to be potentially significant for the solution of some large problems on platforms, such as graphics processing units, where the overhead induced by numerous sequential calls to the coarser levels may be relatively significant. In cases of problems for which the convergence of standard $V$-cycles or $F$-cycles (corresponding to $κ=1$ and $κ=2$, respectively) is particularly slow, and yet the cost of $W$-cycles is very high due to the large number of coarse-level calls (which is exponential in the number of levels), intermediate values of $κ$ may prove to yield significantly faster run-times. This is demonstrated in examples where $κ$-cycles are used for the solution of rotated anisotropic diffusion problems, both as a stand-alone solver and as a preconditioner. Moreover, a simple model is presented for predicting the approximate run-time of the $κ$-cycle, which is useful in pre-selecting an appropriate cycle counter for a given problem on a given platform. Implementing the $κ$-cycle requires making just a small change in the classical multigrid cycle.

On the recursive structure of multigrid cycles

TL;DR

A simple model is presented for predicting the approximate run-time of the -cycle, which is useful in pre-selecting an appropriate cycle counter for a given problem on a given platform.

Abstract

A new fixed (non-adaptive) recursive scheme for multigrid algorithms is introduced. Governed by a positive parameter called the cycle counter, this scheme generates a family of multigrid cycles dubbed -cycles. The well-known -cycle, -cycle, and -cycle are shown to be particular members of this rich -cycle family, which satisfies the property that the total number of recursive calls in a single cycle is a polynomial of degree in the number of levels of the cycle. This broadening of the scope of fixed multigrid cycles is shown to be potentially significant for the solution of some large problems on platforms, such as graphics processing units, where the overhead induced by numerous sequential calls to the coarser levels may be relatively significant. In cases of problems for which the convergence of standard -cycles or -cycles (corresponding to and , respectively) is particularly slow, and yet the cost of -cycles is very high due to the large number of coarse-level calls (which is exponential in the number of levels), intermediate values of may prove to yield significantly faster run-times. This is demonstrated in examples where -cycles are used for the solution of rotated anisotropic diffusion problems, both as a stand-alone solver and as a preconditioner. Moreover, a simple model is presented for predicting the approximate run-time of the -cycle, which is useful in pre-selecting an appropriate cycle counter for a given problem on a given platform. Implementing the -cycle requires making just a small change in the classical multigrid cycle.

Paper Structure

This paper contains 14 sections, 9 theorems, 29 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction
  2. The k-cycle
  3. k-cycle definition
  4. Theoretical properties
  5. Number of operations in a geometric multigrid k-cycle
  6. Predicting k-cycle run-time on GPU processors
  7. The model
  8. Estimating the model parameters
  9. Values for NgpuCalls and Nops in our implementation
  10. Calculating the turning point
  11. Relative prediction errors
  12. Numerical results
  13. Conclusions and further research
  14. Appendix

Key Result

Proposition 1

\newlabelprop:VFW0 $$

Figures (3)

  • Figure 1: The $\kappa$-cycle is illustrated with 5 levels and $\kappa=1,2,3,5$. Circles denote relaxations, corresponding to lines 2 and 8 of Algorithm \ref{['alg:kappa_cycle']}, while red X's denote recursive calls.
  • Figure 1: Measured and predicted run-times per cycle for $\kappa=1,2,3,4,\infty$. The vertical lines in panel (a) show the turning points, where red and green curves cross in panels (b)-(f).
  • Figure 1: Solve run-times

Theorems & Definitions (15)

  • Proposition 1
  • Proof 1
  • Proposition 2
  • Proof 2
  • Corollary 3
  • Proposition 4
  • Proof 3
  • Corollary 5
  • Proposition 6
  • Proof 4: Sketch of Proof
  • ...and 5 more