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An Empirical Study of Conjugate Gradient Preconditioners for Solving Symmetric Positive Definite Systems of Linear Equations

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This benchmark of 108 preconditioner configurations on 79 symmetric positive definite matrices finds that incomplete Cholesky with AMD ordering yields the most consistent work reductions, with algebraic multigrid reliable but rarely…

desk verdict A careful, honestly reported benchmark with a genuinely new AMD-over-RCM result, but the abstract's blanket claim about SPD systems outruns the filtered test set. read the letter →

arxiv 2505.20696 v1 pith:2XN2KFBT submitted 2025-05-27 math.NA cs.NA

Marc A. Tunnell , David F. Gleich This is my paper · ORCID
classification math.NAcs.NA MSC 65F0865F1065F50
keywords preconditionedconjugategradientincompleteCholeskyalgebraicmultigridperformanceprofilessymmetricpositivedefinitesystemsmatrixorderingapproximateminimumdegreeblack-boxpreconditioners
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks which black-box preconditioners actually speed up preconditioned conjugate gradient on real symmetric positive definite (SPD) systems, and it answers with a wide benchmark: 79 matrices, 10 preconditioner classes, and 108 configurations measured by floating-point work to reach a relative residual of $10^{-10}$. The central finding is that incomplete symmetric factorizations, especially thresholded incomplete Cholesky (IC) with approximate minimum degree (AMD) ordering, give the most consistent large speedups, while algebraic multigrid is reliable but rarely the fastest. The paper also shows that preconditioner construction cost is a deciding factor: once build effort is counted, the advantage of IC over both plain CG and direct solvers narrows, though tuned IC still wins on most matrices. The practical upshot is that IC should be the default first try for SPD systems where plain CG converges.

What carries the argument

The load-bearing measurement is the performance profile: for each preconditioner configuration, the fraction of test matrices where the total work, $\text{work} = (5n + \text{nonzeros} + \text{preconditioner})\cdot\text{iterations} + \text{preconditioner}$, is reduced by at least a factor $\tau$ relative to the diagonally scaled control, summarized by the area under the curve (AUC) in log2 space. Within that metric, the decisive object is the simplicial incomplete Cholesky factorization, which approximates $A \approx LL^T$ by keeping only entries above a drop tolerance; its best configuration combines threshold $10^{-8}$ with approximate minimum degree reordering. This object carries the argument because it alone combines large AUC with speedups of 2x or more on most of the test set, and because its generation-failure rate motivates the paper's call for fallback mechanisms.

What would settle it

Take the SPD matrices of dimension at least 10,000 from the same public sparse-matrix collection the paper used, keep only the ones that fail the $10n$ unpreconditioned-CG filter, and run the same 108 configurations with the same flop-based work metric. If IC/MIC's generation-success rate drops below the roughly 80% seen here, or if smoothed aggregation AMG achieves a higher AUC on that cohort, the paper's broad preference for incomplete Cholesky would not carry over to hard SPD systems.

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Extended reading notes

Core claim

The paper's claim is that, over the class of SPD systems it can study, incomplete Cholesky is the most effective preconditioner class even when no per-matrix tuning is allowed. Under a single-best configuration, IC with drop tolerance $10^{-8}$ and AMD ordering achieves an AUC of 0.64, cuts total work by at least a factor of 2 on 78% of the 79 matrices, and reaches at least 8x on 62%; modified IC is close behind with AUC 0.61. Smoothed aggregation AMG, the best multigrid option, has AUC 0.36 and reaches 2x on only 38% of matrices, though it generates successfully on essentially all problems while IC and MIC fail to factor about 16-19% of the time. Classical SGS and SSOR preconditioners do not pay off, and unpreconditioned CG already beats an estimated direct solve by 2x on roughly 47% of matrices. The authors further claim that including construction cost shrinks but does not eliminate IC's edge, and that AMD ordering outperforms both RCM and natural orderings for incomplete factorizations, contradicting the usual preference for RCM.

Load-bearing premise

The whole comparison is run on a filtered set of SPD matrices where plain conjugate gradient converges within $10n$ iterations, so the conclusions about 'SPD systems' depend on that slice being representative; if the harder excluded matrices were added, incomplete factorizations would likely fail more often and the ranking could shift.

Editorial extensions

If this is right

  • A user facing an SPD system where plain CG converges should try thresholded incomplete Cholesky with AMD ordering before anything else: the single-best IC configuration, drop tolerance $10^{-8}$ with AMD, gives at least a 2x work reduction on 78% of the benchmark matrices.
  • Symmetric Gauss-Seidel and SSOR are a poor investment as black-box preconditioners; even their tuned variants produce large speedups on under 10% of problems.
  • Algebraic multigrid is the most dependable fallback because it generates on 100% of test matrices and reaches parity with the baseline on 89-96%, even though its top speedups are smaller than IC's.
  • Including preconditioner construction cost changes the practical message: with a single fixed configuration, only about 28% of problems retain a 4x total-work improvement, so per-matrix tuning or a build-cost-aware choice matters.
  • Even the unpreconditioned baseline is competitive with direct solves on a large share of the test set, so direct solvers should not be assumed superior without measurement.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the $10n$ convergence filter likely biases the test set toward matrices where IC works, so the paper's headline claim should be read as 'IC is best among SPD systems that are already tractable for CG,' not as a universal SPD statement.
  • Editorial inference: the work metric counts floating-point operations but not memory traffic or parallelism; a wall-clock study on multicore machines could narrow or reverse IC's margin over direct solvers, since sparse factorization is more parallelizable than the serial PCG recurrence.
  • Editorial inference: the AMD-over-RCM result suggests a testable extension: compare IC with nested-dissection orderings at larger scales, where the authors' own data shows IC generation failing on 12 of the 16 largest matrices.
  • Editorial inference: the success-parity numbers imply a natural hybrid strategy\u2014start with IC/AMD, fall back to AMG when generation or convergence fails\u2014that would likely combine IC's speed with AMG's near-universal reliability on this test set.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper presents an empirical benchmark of preconditioners for solving large sparse SPD systems with PCG. Using 79 SuiteSparse matrices of dimension at least 10,000, the authors evaluate 10 preconditioner classes in 108 configurations, reporting float-arithmetic work to reduce the relative residual to 1e-10 relative to a Jacobi-scaled unpreconditioned baseline and to an estimated sparse direct solve. The main reported findings are that incomplete Cholesky and modified incomplete Cholesky give the largest work reductions when they can be generated; AMG methods are reliable and occasionally strong; classical SGS/SSOR and SSPAI give little benefit; and including factorization construction costs substantially narrows the advantage over direct methods. The paper also studies the effect of AMD/RCM/natural orderings and of fill-in on incomplete factorizations, and it contains extensive per-matrix convergence plots in an appendix.

Significance. If the headline claims are taken as claims about the studied cohort, the paper is a useful and unusually transparent contribution: it covers a broad set of preconditioner families and configurations, documents the exact work metric, distinguishes single-best from per-matrix tuned choices, separates solve work from construction work, and provides reproducible code and data. The finding that AMD ordering consistently outperforms RCM for incomplete factorizations, contrary to some classical guidance, is interesting and worth reporting. However, the abstract and key findings generalize to 'symmetric positive definite systems' as a whole, while the experimental design explicitly filters to matrices where unpreconditioned CG already converges in at most 10n iterations and where the preconditioners can be generated. The in-sample selection of the 'single-best' configuration also makes the headline numbers optimistic. These issues are fixable with a revision that narrows the claims and adds robustness analysis.

major comments (3)
  1. [Abstract and §3] The abstract's headline claim, 'symmetric positive definite systems are mostly likely to benefit from incomplete symmetric factorizations,' is not supported for SPD systems in general. Section 5.1 restricts the test set to matrices for which unpreconditioned CG converges within 10n iterations, and footnote 1 says the remaining SPD matrices are still being studied. Section 3 then explicitly warns that concluding IC/MIC are the preferred preconditioners 'is premature' because they fail to generate on roughly 15–20% of problems and on 12 of the 16 largest matrices; Table 7 shows the best IC configuration has an 84% success rate. The evidence supports a claim about the filtered cohort, not about all SPD systems. The abstract and Section 3 should be reworded to state this conditionality, and the paper should at least discuss how the ranking might change if the excluded hard systems were included.
  2. [§2.2, §A.1, Table 7] The 'single-best configuration' is selected and evaluated on the same 79 matrices. Section 2.2 says the single-best configuration is chosen by computing the AUC over all matrices, and Section A.1 defines that AUC over the test set; the same data are then used to report the configuration's parity and 2x/4x/8x rates in Table 7 and Figure 4a. This is in-sample model selection and makes the headline numbers optimistic as estimates of how a fixed configuration would perform on new SPD systems. The authors should use a split or cross-validation for configuration selection, or otherwise quantify the selection bias, before presenting the single-best configuration as a practical recommendation.
  3. [Table 7 and §2.2] The central ranking of preconditioners rests on the AUC and on the geometric mean, where Table 7 defines failure to generate or converge as 4x control work. No sensitivity analysis is reported for this ad-hoc penalty, and the AUC can be dominated by a few very large speedups, so it is unclear whether the ordering 'IC best, AMG second' is robust. The paper should report additional distributional summaries (e.g., median work reduction, quantiles, and success rates alone) and show how the ranking changes if the 4x penalty is varied or if failed generations are treated as missing rather than as 4x control work.
minor comments (5)
  1. [Throughout] There are several typos and small wording errors: 'preconditoners' and 'mostly likely' in the abstract, 'availalbe' in Section 3.4, 'matric' in Section 5.3, 'T refethen' in Appendix B headings, and 'MC' instead of 'MIC' in Section 7.1.
  2. [§2.2] The sentence 'The best configurations for each method are summarized in Table 2' should refer to Table 7, since Table 2 lists preconditioner classes rather than configurations.
  3. [Table 7 caption] The caption defines 'Succ.' as the fraction of problems in which the preconditioner was successfully generated, but the geometric-mean definition also treats convergence failure as 4x control work; the caption should clarify whether 'Succ.' includes convergence failures after successful generation or only generation failures.
  4. [§2.3 and §2.6] The paper explicitly limits generation-cost comparisons to methods for which the cost can be estimated, and it correctly notes that AMG construction costs are not included. It would help to add a sentence in Section 3 making clear that the 'including generation cost' conclusions do not apply to AMG or to SSPAI.
  5. [§7.4] The table footnote for the AMG smoothed-aggregation entry states that one iteration of each cycle was best on 2.5% of the test set, but the wording is slightly confusing about whether the percentage refers to being tied for best; consider rewording for clarity.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the benchmark is an empirical measurement with disclosed in-sample selection and one minor non-load-bearing self-citation.

full rationale

This paper is an empirical benchmark, not a derivation chain, so most circularity patterns do not apply. The central claim that IC/MIC preconditioners give the largest work reduction is a measured ranking over a disclosed test set of 79 SPD matrices, computed from actual PCG convergence runs; it is not defined in terms of a fitted constant, nor does any equation make the conclusion true by construction. The single-best and tuned-best configurations are selected from the same test data (Section 2.2, Figures 4a/4b), but the paper explicitly labels them as in-sample selections rather than predictions, and it reports both to show sensitivity. The test-set filter requiring unpreconditioned CG convergence within 10n iterations (Section 5.1) and the footnote that the remaining SPD matrices are still being studied are scope limitations that affect generalizability, but they do not make the reported ranking circular: the inclusion criterion is about the unpreconditioned baseline, not about the success or ranking of the preconditioners being compared. The only self-citation is Gleich's GenericArpack.jl (reference in Section 4.2), used to compute the operator 2-norm for normwise backward error checking; that is a tool for convergence measurement, not a load-bearing premise for the IC-versus-AMG conclusion. No step reduces a prediction to its input by construction, no fitted parameter is renamed as a prediction, and no uniqueness theorem or ansatz is imported from the authors' prior work to force the choice of preconditioner. Accordingly, the appropriate finding is no significant circularity, with the score reflecting only the minor, non-load-bearing self-citation.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The benchmark introduces no fitted scientific constants or invented entities. The main methodological debt is the in-sample selection of the best configuration per preconditioner, the filtered test set, and several domain assumptions about the validity of flop-based and fill-based cost models. These are disclosed in Section 3.2 and Section A.2.

free parameters (5)
  • IC drop tolerance (best) = 1e-8 (with AMD ordering)
    Selected by maximizing AUC over the 79 test matrices; headline IC metrics are for this in-sample best, so the reported performance is optimistic.
  • AMG Smoothed Aggregation cycle (best) = V-cycle, 1 iteration
    Selected in-sample by AUC; represents the tuned best for the SA method.
  • SSOR relaxation (best) = omega = 1.2, 1 iteration
    Selected in-sample by AUC; classical methods show low sensitivity, but the best config is still data-dependent.
  • SSPAI fill multiplier (best) = 2x average column count
    Selected in-sample from the grid {0.5, 1, 2, 3}x.
  • Failure penalty in geometric mean = 4x control
    Hand-chosen constant defining the work assigned to failure scenarios in the Geo metric of Table 7; affects aggregate comparisons.
assumptions (6)
  • domain assumption The 79 selected SuiteSparse matrices are representative of real-world SPD systems despite the filter that unpreconditioned CG converges within 10n iterations.
    The filter in Section 5.1 excludes the hardest systems, and the paper's general conclusions assume this subset is representative.
  • domain assumption Floating-point operation count is an adequate proxy for computational cost.
    The work formula in Section 2 intentionally ignores memory, parallelism, and hardware; the paper acknowledges this is a proxy.
  • domain assumption The fill-based generation cost estimate approximates actual preconditioner construction work.
    Section A.2 uses the sum of squares of factor column counts as a lower bound; the paper notes it omits ordering time and dropped entries.
  • domain assumption The seeded right-hand side generation produces convergence behavior representative of generic RHS vectors.
    Section 5.4 and Table 4 rely on a single seeded RHS; Table 4 shows up to 17x iteration-count variation for some matrices.
  • ad hoc to paper Failure to generate or converge is assigned 4x control work in the geometric mean metric.
    Table 7 defines this constant; it is a modeling choice made by the authors.
  • standard math Standard PCG, SPD matrix theory, and Dolan-More performance profile methodology are accepted background.
    Used throughout Sections 4 and 2.2.

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Cite this review

Pith. "Pith review of An Empirical Study of Conjugate Gradient Preconditioners for Solving Symmetric Positive Definite Systems of Linear Equations." pith.science (2026). https://pith.science/paper/2XN2KFBT

@misc{pith2026250520696,
  author       = {Pith},
  title        = {Pith review of: An Empirical Study of Conjugate Gradient Preconditioners for Solving Symmetric Positive Definite Systems of Linear Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2XN2KFBT}},
  note         = {Machine review of arXiv:2505.20696}
}
read the original abstract

Despite hundreds of papers on preconditioned linear systems of equations, there remains a significant lack of comprehensive performance benchmarks comparing various preconditioners for solving symmetric positive definite (SPD) systems. In this paper, we present a comparative study of 79 matrices using a broad range of preconditioners. Specifically, we evaluate 10 widely used preconditoners across 108 configurations to assess their relative performance against using no preconditioner. Our focus is on preconditioners that are commonly used in practice, are available in major software packages, and can be utilized as black-box tools without requiring significant \textit{a priori} knowledge. In addition, we compare these against a selection of classical methods. We primarily compare them without regards to effort needed to compute the preconditioner. Our results show that symmetric positive definite systems are mostly likely to benefit from incomplete symmetric factorizations, such as incomplete Cholesky (IC). Multigrid methods occasionally do exceptionally well. Simple classical techniques, symmetric Gauss Seidel and symmetric SOR, are not productive. We find that including preconditioner construction costs significantly diminishes the advantages of iterative methods compared to direct solvers; although, tuned IC methods often still outperform direct methods. Additionally, ordering strategies such as approximate minimum degree significantly enhance IC effectiveness. We plan to expand the benchmark with larger matrices, additional solvers, and detailed metrics to provide actionable information on SPD preconditioning.

Figures

Figures reproduced from arXiv: 2505.20696 by the authors.

Figure 1
Figure 1. Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Performance profiles of the PCG method with various preconditioners applied to [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (93 more)
Figure 4
Figure 4. Figure 4: The single-best configuration selects the best configuration over all matrices [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Including generation work. Performance profiles of the PCG method with various preconditioners applied to all matrices in the test set where work includes an approximation to the amount of work required to generate the preconditioner. Work Reduction Factor τ (Relative …
Figure 6
Figure 6. Figure 6: Including generation work. The single-best configuration selects the best configu￾ration over all matrices whereas the tuned-best configuration selects the best run for each matrix within a given preconditioner group. Both include an approximation the the amount of wor…
Figure 7
Figure 7. Figure 7: Focusing on ordering. Performance profiles of CG with various preconditioners applied to all matrices in the test set, grouped by ordering. We expect no impact to Neumann, SSPAI, AMG RS, AMG SA, and Laplacians. This largely follows although we see some modest effects f…
Figure 8
Figure 8. Figure 8 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Comparison to direct solve. The Single-best configuration selects the best configu￾ration over all matrices whereas the tuned-best configuration selects the best run for each matrix within a given preconditioner group. Plotted relative to solving directly. (log-linear …
Figure 10
Figure 10. Figure 10: Comparison to direct solve including generation cost. Performance profiles of the PCG method with various preconditioners applied to all matrices in the test set, grouped by ordering. This is relative to the work required to solve the system with a direct method and i…
Figure 11
Figure 11. Figure 11: Comparison to direct solve including generation cost. The Single-best configuration selects the best configuration over all matrices whereas the tuned-best configuration selects the best run for each matrix within a given preconditioner group. Plotted relative to solv…
Figure 12
Figure 12. Figure 12: The single-best configuration selects the best configuration over matrices of [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]
Figure 13
Figure 13. Figure 13: Including generation work. The single-best configuration selects the best config￾uration over matrices of dimension 250k or higher, whereas the tuned-best configuration selects the best run for each matrix within a given preconditioner group. Plotted relative to the u…
Figure 14
Figure 14. Figure 14: The Single-best configuration selects the best configuration over matrices that [PITH_FULL_IMAGE:figures/full_fig_p030_14.png]
Figure 15
Figure 15. Figure 15: Factorization Fill-in versus Work Reduction Factor relative to the work reduction [PITH_FULL_IMAGE:figures/full_fig_p031_15.png]
Figure 16
Figure 16. Figure 16: Factorization Fill-in versus Work Reduction Factor relative to the work reduction [PITH_FULL_IMAGE:figures/full_fig_p031_16.png]
Figure 17
Figure 17. Figure 17: Kernel density estimate plot of the Spearman correlation coefficients of Factor [PITH_FULL_IMAGE:figures/full_fig_p032_17.png]
Figure 18
Figure 18. Figure 18: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p042_18.png]
Figure 19
Figure 19. Figure 19: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p043_19.png]
Figure 20
Figure 20. Figure 20: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p044_20.png]
Figure 21
Figure 21. Figure 21: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p045_21.png]
Figure 22
Figure 22. Figure 22: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p046_22.png]
Figure 23
Figure 23. Figure 23: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p047_23.png]
Figure 24
Figure 24. Figure 24: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p048_24.png]
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Figure 25. Figure 25: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p049_25.png]
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Figure 26. Figure 26: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p050_26.png]
Figure 27
Figure 27. Figure 27: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p051_27.png]
Figure 28
Figure 28. Figure 28: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p052_28.png]
Figure 29
Figure 29. Figure 29: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p053_29.png]
Figure 30
Figure 30. Figure 30: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p054_30.png]
Figure 31
Figure 31. Figure 31: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p055_31.png]
Figure 32
Figure 32. Figure 32: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p056_32.png]
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Figure 33. Figure 33: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p057_33.png]
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Figure 34. Figure 34: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p058_34.png]
Figure 35
Figure 35. Figure 35: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p059_35.png]
Figure 36
Figure 36. Figure 36: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p060_36.png]
Figure 37
Figure 37. Figure 37: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p061_37.png]
Figure 38
Figure 38. Figure 38: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p062_38.png]
Figure 39
Figure 39. Figure 39: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p063_39.png]
Figure 40
Figure 40. Figure 40: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p064_40.png]
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Figure 41. Figure 41: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p065_41.png]
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Figure 42. Figure 42: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p066_42.png]
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Figure 43. Figure 43: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p067_43.png]
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Figure 44. Figure 44: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p068_44.png]
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Figure 45. Figure 45: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p069_45.png]
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Figure 46. Figure 46: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p070_46.png]
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Figure 47. Figure 47: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p071_47.png]
Figure 48
Figure 48. Figure 48: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p072_48.png]
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Figure 49. Figure 49: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p073_49.png]
Figure 50
Figure 50. Figure 50: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p074_50.png]
Figure 51
Figure 51. Figure 51: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p075_51.png]
Figure 52
Figure 52. Figure 52: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p076_52.png]
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Figure 53. Figure 53: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p077_53.png]
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Figure 54. Figure 54: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p078_54.png]
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Figure 55. Figure 55: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p079_55.png]
Figure 56
Figure 56. Figure 56: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p080_56.png]
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Figure 57. Figure 57: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p081_57.png]
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Figure 58. Figure 58: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p082_58.png]
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Figure 59. Figure 59: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p083_59.png]
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Figure 60. Figure 60: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p084_60.png]
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Figure 61. Figure 61: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p085_61.png]
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Figure 62. Figure 62: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p086_62.png]
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Figure 63. Figure 63: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p087_63.png]
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Figure 64. Figure 64: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p088_64.png]
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Figure 65. Figure 65: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p089_65.png]
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Figure 66. Figure 66: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p090_66.png]
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Figure 67. Figure 67: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p091_67.png]
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Figure 68. Figure 68: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p092_68.png]
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Figure 69. Figure 69: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p093_69.png]
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Figure 70. Figure 70: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p094_70.png]
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Figure 71. Figure 71: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p095_71.png]
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Figure 72. Figure 72: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p096_72.png]
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Figure 73. Figure 73: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p097_73.png]
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Figure 74. Figure 74: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p098_74.png]
Figure 75
Figure 75. Figure 75: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p099_75.png]
Figure 76
Figure 76. Figure 76: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p100_76.png]
Figure 77
Figure 77. Figure 77: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p101_77.png]
Figure 78
Figure 78. Figure 78: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p102_78.png]
Figure 79
Figure 79. Figure 79: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p103_79.png]
Figure 80
Figure 80. Figure 80: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p104_80.png]
Figure 81
Figure 81. Figure 81: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p105_81.png]
Figure 82
Figure 82. Figure 82: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p106_82.png]
Figure 83
Figure 83. Figure 83: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p107_83.png]
Figure 84
Figure 84. Figure 84: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p108_84.png]
Figure 85
Figure 85. Figure 85: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p109_85.png]
Figure 86
Figure 86. Figure 86: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p110_86.png]
Figure 87
Figure 87. Figure 87: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p111_87.png]
Figure 88
Figure 88. Figure 88: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p112_88.png]
Figure 89
Figure 89. Figure 89: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p113_89.png]
Figure 90
Figure 90. Figure 90: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p114_90.png]
Figure 91
Figure 91. Figure 91: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p115_91.png]
Figure 92
Figure 92. Figure 92: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p116_92.png]
Figure 93
Figure 93. Figure 93: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p117_93.png]
Figure 94
Figure 94. Figure 94: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p118_94.png]
Figure 95
Figure 95. Figure 95: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p119_95.png]
Figure 96
Figure 96. Figure 96: Convergence of the PCG method with various preconditioners applied to the [PITH_FULL_IMAGE:figures/full_fig_p120_96.png]

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