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Stabilization of BiCGSTAB by the generalized residual cutting method

Toshihiko Abe

Abstract

The residual cutting (RC) method has been proposed as an outer-inner loop iteration for efficiently solving large and sparse linear systems of equations arising in solving numerically problems of elliptic partial differential equations. Then based on RC the generalized residual cutting (GRC) method has been introduced, which can be applied to more general sparse linear systems problems. In this paper, we show that GRC can stabilize the BiCGSTAB, which is also an iterative algorithm for solving large, sparse, and nonsymmetric linear systems, and widely used in scientific computing and engineering simulations, due to its robustness. BiCGSTAB converges faster and more smoothly than the original BiCG method, by reducing irregular convergence behavior by stabilizing residuals. However, it sometimes fails to converge due to stagnation or breakdown. We attempt to emphasize its robustness by further stabililizing it by GRC, avoiding such failures.

Stabilization of BiCGSTAB by the generalized residual cutting method

Abstract

The residual cutting (RC) method has been proposed as an outer-inner loop iteration for efficiently solving large and sparse linear systems of equations arising in solving numerically problems of elliptic partial differential equations. Then based on RC the generalized residual cutting (GRC) method has been introduced, which can be applied to more general sparse linear systems problems. In this paper, we show that GRC can stabilize the BiCGSTAB, which is also an iterative algorithm for solving large, sparse, and nonsymmetric linear systems, and widely used in scientific computing and engineering simulations, due to its robustness. BiCGSTAB converges faster and more smoothly than the original BiCG method, by reducing irregular convergence behavior by stabilizing residuals. However, it sometimes fails to converge due to stagnation or breakdown. We attempt to emphasize its robustness by further stabililizing it by GRC, avoiding such failures.

Paper Structure

This paper contains 10 sections, 2 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Modified Gram-Schmidt based GRC
  3. BiCGSTAB
  4. GRC-BiCGSTAB
  5. Numerical experiments
  6. Experimental setup
  7. Experimental results
  8. Problem 1
  9. Other test problems
  10. Discussion and conclusion

Figures (3)

  • Figure 1: Residual norms of BiCGSTAB and GRC-BiCGSTAB. 'BiCGSTAB' (blue) shows the original BiCGSTAB. 'GRC-BiCGSTAB' (green) shows the residual norm stabilized by GRC. 'GRC-BiCGSTAB (inner loop)' (purple) shows the residual norm of BiCGSTAB of the inner loop in GRC, where each residual at the beginning of the inner loop is the one from the GRC outer loop.
  • Figure 2: Result for other test problems
  • Figure 3: Result for other test problems