Hyperellipsoid Density Sampling: Exploitative Sequences to Accelerate High-Dimensional Numerical Optimization

Julian G. Soltes

REVIEW 3 major objections 5 minor 1 cited by

Hyperellipsoid Density Sampling: Exploitative Sequences to Accelerate High-Dimensional Numerical Optimization

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Hyperellipsoid Density Sampling (HDS) replaces uniform quasi-Monte Carlo sequences with a non-uniform, cluster-driven sequence that improves high-dimensional optimization results, reporting an overall 15% gain in final solution quality for

desk verdict New heuristic, unsupported 'exploitative' claim, and internally inconsistent numbers — a legitimate empirical question that needs major cleanup before the mechanism can be believed. read the letter →

arxiv 2511.07836 v5 pith:QZNG76IL submitted 2025-11-11 math.NA cs.LGcs.NAcs.NE

Julian G. Soltes This is my paper

classification math.NAcs.LGcs.NAcs.NE MSC 65C0565K10

keywords hyperellipsoiddensitysamplingquasi-MonteCarlodifferentialevolutionglobaloptimizationCEC2017benchmarkclusteringprincipalcomponentanalysisnon-uniform

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

Hyperellipsoid Density Sampling (HDS) replaces the uniform Sobol sequence used to initialize differential evolution with a non-uniform, data-driven sequence. The paper claims that on the 29 CEC2017 benchmark functions, across 10 to 100 dimensions, this change improves final solution fitness by an average of 15% (geometric mean improvement factor 1.15x) while increasing total run time by only about 5%. The method builds hyperellipsoids from a clustered Sobol sequence, using KMeans, hierarchical clustering, and PCA to locate and shape denser sampling regions. A sympathetic reader would care because it offers a practical, drop-in alternative to QMC initialization that exploits structure in the search space without needing prior knowledge about the optimum, and the optional Gaussian weights let users inject known regions of interest.

What carries the argument

The central object is the hyperellipsoid sequence: K hyperellipsoids fitted to clusters of a Sobol sequence define a non-uniform density. The ellipsoid count comes from a dendrogram cut of hierarchical clustering; centers come from KMeans; axes come from PCA variances; directions are uniform on the unit sphere via Marsaglia's polar method; and radial distances follow q^{1/D} from a 1D Sobol sequence for uniform hypervolume coverage. Void filling with a KNN BallTree adds explorative samples outside the ellipsoids.

What would settle it

Compare HDS to a control that uses the same ellipsoidal sampling procedure but with cluster assignments randomly shuffled (or ellipsoid centers drawn uniformly) without any reference to the Sobol sequence's clustering; if the control matches HDS's 1.15x improvement over plain Sobol, the claimed link between cluster density and optima location is not supported. Alternatively, compute the correlation between HDS's local sample density and the objective function's landscape on the CEC2017 functions; near-zero correlation would undermine the 'promising regions' interpretation.

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Extended reading notes

Core claim

HDS generates samples by clustering an initial Sobol point set, using Agglomerative Hierarchical Clustering to pick the number of ellipsoids, KMeans to assign cluster centers and sample counts, and PCA to set each ellipsoid's orientation and axis lengths. Samples are drawn uniformly inside unit hyperspheres, scaled radially with a one-dimensional Sobol sequence and a chi-squared critical value, then transformed into the ellipsoids. The resulting sequence concentrates density near the cluster centroids, and when used as the initial population for differential evolution on CEC2017 benchmarks, it yields statistically significant improvements in final solution error (p<0.05) across most dimensio

Load-bearing premise

The paper assumes that the cluster structure of a uniform Sobol sequence contains information about where the objective's optima are likely to be, even though the sequence itself has no knowledge of the objective; the improved results are attributed to these statistically promising regions rather than to the optimizer's reaction to any clustered non-uniform population.

Editorial extensions

If this is right

If HDS's improvement is real, any population-based optimizer that starts from a QMC sample can likely adopt HDS initialization and gain similar quality improvements at modest cost.
The optional Gaussian weighting extends the same mechanism to targeted sampling, potentially useful in Bayesian optimization, experimental design, and active learning, not just global optimization.
The reported low Centered-L2 discrepancy means HDS sequences have high local uniformity; this suggests the method may also serve as a quasi-random sampler in settings where local coverage matters more than global discrepancy.
The method's improvement is largest in 10D (37%) and smallest in 30D (3% in one table) — the dimension-dependence hints that the advantage may be tied to how DE's search behavior interacts with non-uniform initial density.

Reading between the lines

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

Since the initial Sobol sequence is uniform, the cluster size asymmetries that drive HDS's density bias are finite-sample fluctuations. The paper does not test whether these fluctuations correlate with actual objective landscape; if they don't, the improvement may come from DE's exploitation of clustered starting points rather than from locating genuinely promising regions.
A direct test would compare HDS against a version where ellipsoid locations are randomly permuted while preserving the same density shape; if random placement yields similar gains, the 'data-driven' component is not doing the work.
The Gaussian-weight mode, untested in the paper, could be evaluated on problems with known optima to see whether injecting prior information yields the expected density shift and further improvements.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version: the HDS pipeline — clustering a Sobol sequence with MiniBatchKMeans, picking the cluster count via AHC, fitting PCA ellipsoids, and sampling with Marsaglia polar coordinates — is a new combination, and the benchmark is substantial (11,400 DE runs). But the paper's central claim, that HDS "exploits statistically promising regions," is not supported by the experiments as run. All trials used the optional Gaussian weights off, so the clustering sees only a uniform Sobol sequence. Cluster sizes of a uniform point set are near-equal up to finite-sample noise, and PCA axes are approximately isotropic. There is no input from the objective function anywhere in the construction. So the allocation of samples to different ellipsoids is essentially driven by random fluctuations in one Sobol draw, not by any learned density. The reported 1.15x gain may be real, but it remains unexplained and might come from boundary handling, void filling, or the specific interaction of a fixed Sobol draw with the CEC2017 shift vectors.

The paper does some things well. The run-time overhead is modest (about 5% at N=1000), the sample-size and dimension scaling are reported, and the author explicitly notes in Future Works that parameters are empirical and trials were done without weights. That candor helps, but it undercuts the "exploitative" framing.

The soft spots are real and not minor. The reported numbers are internally inconsistent: the first abstract says gains from 37% (10D) to 11% (100D); the body abstract says 3% (30D) to 37% (10D); the conclusion says the low end is 3% in 50D. Table 1 shows 1.11 at 50D and 1.11 at 100D for N=1000, matching none of those. The N=64, D=100 row reports p=0.16 while the 95% CI (1.04,1.13) excludes 1 — a statistical report that is just wrong. Section 3.3 says L2-Star discrepancies are "lower" for HDS, while Table 5 shows they are orders of magnitude higher. The tuning constants C_D, K_init, N_init, alpha, delta are all chosen empirically, with no sensitivity analysis and no evidence they weren't tuned on the same benchmark. No working code or data is linked; the github URL is unverified.

Who is this for? A developer looking for a cheap initialization heuristic for DE might find it worth testing, but only as an empirical twist, not a principled method. A sampling theorist will find the mechanism unsupported and the statistics sloppy.

My recommendation: it deserves a serious referee, because the empirical question — can a non-uniform QMC-like initialization help DE on CEC2017 — is legitimate and the results are not obviously false, just misinterpreted and misreported. But I'd send it back for major revision: corrected numbers, a clear explanation of what HDS actually does without weights (ideally compared to randomly placed ellipsoids), and either code/data or a sensitivity analysis.

Referee Report

3 major / 5 minor

Summary. The paper proposes Hyperellipsoid Density Sampling (HDS), a non-uniform QMC-style initialization scheme for high-dimensional optimization. HDS generates an initial Sobol sequence, clusters it via MiniBatchKMeans, determines the number of ellipsoids via hierarchical clustering, fits PCA-based ellipsoids, and samples from these ellipsoids after radial scaling and boundary rejection. The authors evaluate HDS as an initial population for differential evolution on the CEC2017 benchmark suite in 10, 30, 50, and 100 dimensions, reporting that HDS improves final solution quality by a geometric mean factor of 1.15× compared to plain Sobol at N=1000, with statistically significant gains in many configurations. The paper also reports run-time ratios and discrepancy metrics.

Significance. If the reported improvements are genuine and mechanism is as claimed, HDS would be a useful, low-cost drop-in replacement for uniform QMC initialization in evolutionary optimization, with broad applicability to any optimizer that consumes an initial sample sequence. The manuscript's strengths include a concrete algorithm, an open-source implementation, and a benchmark comparison with statistical reporting. However, the central scientific claim—that the method 'exploits statistically promising regions'—depends on an information pathway that is absent from the reported experiments: with no Gaussian weights or function evaluations entering the construction, the clustering of a uniform Sobol point set carries no objective-specific signal. The empirical claims are also weakened by internal inconsistencies in the reported gain figures and a statistically contradictory p-value/confidence-interval pair. These issues are load-bearing because they affect what the paper can claim about HDS's mechanism and its measured advantage.

major comments (3)

[§2.2.2–2.2.8, Future Work] The mechanism by which HDS identifies 'promising regions' is not established. All reported DE trials omit Gaussian weights, so the only input to MiniBatchKMeans is a uniform Sobol sequence in the unit hypercube. For a uniform point set, cluster sizes n_k are approximately N_init/K up to O(sqrt(N_init)) fluctuations, and PCA axes of locally uniform points are approximately isotropic. Thus the allocation N_k = round(N·n_k / sum n_i) and the subsequent ellipsoid geometry are driven by finite-sample noise of one Sobol draw, not by any property of the objective. Since no function evaluations enter the pipeline, there is no pathway from the objective landscape to sample density. The reported 1.15× gain may plausibly arise from an uncontrolled side effect (e.g., void filling or accidental alignment with a specific CEC2017 shifting). To substantiate the 'exploitative' claim, the authors should e
[Abstract, body abstract, §5 Conclusion, Table 1] The reported performance gains are internally inconsistent. The front-page abstract states gains 'ranging from 37% in 10D to 11% in 100D'; the body abstract states 'from 3% in 30D to 37% in 10D'; the Conclusion states 'from a high of 37% in 10D to a low of 3% in 50D'. Table 1 gives ratios 1.37 (10D), 1.03 (30D), 1.11 (50D), and 1.11 (100D) for N=1000. The Conclusion's '3% in 50D' directly contradicts Table 1's 11% for 50D, and the two abstracts disagree about which dimension has 3%. Since the quantitative performance claim is the paper's headline result, this inconsistency must be corrected and the summary statistics unified.
[Table 1, §3.1] The N=64, D=100 row reports a 95% CI of (1.04, 1.13) for the improvement factor, which excludes 1, yet also reports p=1.6e-1. Under standard definitions, a 95% CI excluding 1 is incompatible with a non-significant p-value at the 0.05 level unless an unusual paired/bootstrap convention is used; no such convention is stated. This row is also cited as evidence of 'struggling to show statistical significance in 10D and 100D', while the preceding text claims HDS 'consistently outperformed Sobol in all tested dimensions.' The inconsistency affects the paper's statistical credibility and must be resolved, e.g., by reporting the test used, the CI construction, and a corrected table.

minor comments (5)

[§3.3, Table 5] The sentence 'the L2-Star discrepancies for HDS are lower than those of Sobol' is the reverse of what Table 5 shows: HDS L2-Star values (e.g., 48×10^-17 for N=10) are far larger than Sobol's (2.3×10^-23). The text should read 'higher', consistent with the expected uniformity trade-off.
[§3.1, Table 1] The text says 'HDS consistently outperformed the Sobol QMC in all of the tested dimensions', but Table 1's N=64, D=10 row has ratio 0.97 and p=0.15, indicating a 3% degradation. Please qualify this statement.
[References] The CEC2017 benchmark functions are cited only through a general survey [5]. Please cite the original CEC2017 technical report or benchmark paper so readers can identify the exact function definitions.
[§2.2.10] The formula C_D = 0.55 - 0.01·ln(D) is described as 'empirically tuned'. Since this is a free parameter, please report the tuning procedure and data used, or at least state that it was selected on a separate validation set.
[Table 4] The text states 'the HDS trials were 4.90% slower overall for the 1000 sample size trials', but the overall average run time ratio is reported as 0.951, implying HDS is 4.9% faster, not slower. Please reconcile this wording.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is an empirical benchmark study, and no prediction reduces by construction to its fitted inputs or to a self-citation chain.

full rationale

The paper does not present a derivation in which an output quantity equals its input by construction. Its central evidence is an empirical comparison of HDS versus Sobol on CEC2017 with DE. The HDS construction (Sections 2.2.2–2.2.14) takes an initial Sobol sequence and transforms it using KMeans, hierarchical clustering, PCA, rescaling, and void filling; no step fits a parameter to the objective-function values used in the evaluation. The paper explicitly notes in Future Works that "Many components of the HDS workflow are derived empirically" and calls for "rigorously defining the parameters," which is an admitted heuristic limitation rather than a circular step. The concern that a uniform Sobol sequence, clustered without objective information, may not actually mark "statistically promising regions" is a substantive scientific objection, but it is not circularity: the paper does not define promising regions in terms of the optimization outcome it later claims to predict. All cited prior results are external QMC and DE references; there is no load-bearing self-citation chain or imported uniqueness theorem. The discrepancy section contains a contradictory statement about L2-Star values versus Table 5, but that is an internal consistency/correctness issue, not circularity. Accordingly, no circular step can be exhibited under the hard-evidence rule, and the appropriate score is 0.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The method is a heuristic assembly with several hand-set constants (C_D, K_init, N_init, alpha, delta) and an unsupported assumption that clustering a uniform sequence reveals promising regions. There are no invented physical entities, but the method's central mechanism is an ad hoc postulate rather than a derived property.

free parameters (5)

C_D (global scaling constant) = 0.55 - 0.01 ln(D)
Described as 'empirically tuned, dimension-variant constant' in Section 2.2.10; controls radial extent of all ellipsoids and is not derived from theory.
K_init = 100 (default), scaled down for D near 1000
Number of initial KMeans centroids; chosen as 'stable value' balancing spatial info and cost; no independent justification.
N_init = 2^ceil(log2(200 D)), capped at 2^15
Initial QMC sample count; heuristic scaling with dimension; no theoretical basis.
alpha (chi-square confidence) = 0.9999
Confidence level for radial scaling, chosen by hand; affects ellipsoid sizes and rejection rate.
AHC cut-off distance delta = chosen per run by 'largest dissimilarity'
Determines number of ellipsoids K; selection rule is qualitative and may vary stochastically.

assumptions (3)

domain assumption CEC2017 benchmark functions are a valid proxy for high-dimensional optimization problems
The paper's only evidence is performance on these benchmarks; no real-world problems are tested.
standard math PCA eigenvalues represent meaningful spread of each cluster
Uses variance along principal components to set ellipsoid axis lengths; mathematically valid, but the clusters themselves have no semantic meaning.
ad hoc to paper A uniform Sobol sequence's local density after clustering indicates promising regions
This is the core exploitative claim; without Gaussian weights, cluster density of a uniform sequence should be statistically flat, making this an unsupported heuristic.

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Cite this review

Pith. "Pith review of Hyperellipsoid Density Sampling: Exploitative Sequences to Accelerate High-Dimensional Numerical Optimization." pith.science (2026). https://pith.science/paper/QZNG76IL

@misc{pith2026251107836,
  author       = {Pith},
  title        = {Pith review of: Hyperellipsoid Density Sampling: Exploitative Sequences to Accelerate High-Dimensional Numerical Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QZNG76IL}},
  note         = {Machine review of arXiv:2511.07836}
}

read the original abstract

The curse of dimensionality remains a persistent challenge in modern optimization problems. Expanding the search space into higher dimensions exponentiates the difficulty of finding optimal solutions, rendering traditional algorithms inefficient. An efficient sampling strategy is presented to accelerate high-dimensional optimization as an alternative to uniform quasi-Monte Carlo (QMC) methods. This method, referred to as Hyperellipsoid Density Sampling (HDS), generates sequences by defining multiple hyperellipsoids throughout the search space. HDS utilizes three types of unsupervised learning algorithms to bypass high-dimensional geometric calculations, producing a non-uniform sample sequence that exploits statistically promising regions of the parameter space. The ability to influence its distribution towards regions of interest makes HDS versatile for applications beyond global optimization, where models benefit from samples focused in specific regions. HDS was evaluated against Sobol, a highly uniform QMC sampling method, using differential evolution (DE) on the challenging set of 29 CEC2017 benchmark test functions. The results show statistically significant improvements in final solution geometric mean error (p<0.05), with average performance gains ranging from 37% in 10D to 11% in 100D. This paper demonstrates the efficacy of HDS as a robust alternative to uniform QMC sampling in high-dimensional optimization.

Figures

Figures reproduced from arXiv: 2511.07836 by the authors.

**Figure 2.** 2D HDS sequence with weights applied towards [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Dendrogram used to identify number of ellipsoids [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: HDS samples in 2D for number of ellipsoids [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]

**Figure 6.** Figure 6: HDS vs Sobol geometric mean final solution im [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]

**Figure 5.** Figure 5: Sample distributions for HDS sequences in 2D (left) [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 8.** Figure 8: Ratio of Sobol and HDS DE full optimization run [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]

**Figure 9.** Figure 9: L2-Star (left) and Centered L2 (right) Discrepancies [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Quasi-Monte Carlo Initialization for Meta-Reinforcement Learning
cs.LG 2026-07 conditional novelty 5.0 of 10

Quasi-Monte Carlo weight priors give a small convergence boost over orthogonal initialization on similar continuous-control tasks, but hurt on dissimilar tasks.

Reference graph

Works this paper leans on

8 extracted references · 2 canonical work pages · cited by 1 Pith paper

[1]

Ivanovska

Emil Atanassov and S. Ivanovska. 2022. On the Use of Sobol’ Sequence for High Dimensional Simulation. InLecture Notes in Computer Science. 646–652. doi:10. 1007/978-3-031-08760-8_53

2022
[2]

Jian Chen, Hui Jiang, and Nicholas Kirk. 2025. High-Dimensional Quasi-Monte Carlo via combinatorial discrepancy.arXiv.org(2025). arXiv:2508.18426 https: //arxiv.org/abs/2508.18426 Preprint submitted August 25, 2025

arXiv 2025
[3]

Thies Hinz, Nestor Navarro-Guerrero, Sebastian Magg, and Stefan Wermter. 2018. Speeding up the hyperparameter optimization of deep convolutional neural net- works.International Journal of Computational Intelligence and Applications17, 02 (2018), 1850008. doi:10.1142/s1469026818500086

work page doi:10.1142/s1469026818500086 2018
[4]

Stephen Joe and F. Y. Kuo. 2008. Constructing Sobol Sequences with Better Two-Dimensional Projections.SIAM Journal on Scientific Computing30, 5 (2008), 2635–2654. doi:10.1137/070709359

doi:10.1137/070709359 2008
[5]

Naser, Mohammed K

Mohammad Z. Naser, Mohammed K. Al-Bashiti, Abdul T. G. Tapeh, Ammar Naser, V. Kodur, Rami Hawileeh, Jaafar Abdalla, Nasim Khodadadi, Amir H. Gandomi, and Alireza Dehghan Eslamlou. 2025. A review of benchmark and test functions for global optimization algorithms and metaheuristics.Wiley Interdisciplinary Reviews Computational Statistics17, 2 (2025). doi:10...

doi:10.1002/wics.70028 2025
[6]

Dong Peng, Zhiwei Gui, and Haijun Wu. 2023. Interpreting the Curse of Di- mensionality from Distance Concentration and Manifold Effect.arXiv.org(2023). arXiv:2401.00422 https://arxiv.org/abs/2401.00422 Preprint submitted December 31, 2023

arXiv 2023
[7]

Zoe Stokes, Arghya Mandal, and W. K. Wong. 2020. Using Differential Evolution to design optimal experiments.Chemometrics and Intelligent Laboratory Systems 199 (2020), 103955. doi:10.1016/j.chemolab.2020.103955

arXiv 2020
[8]

Xiaoqun Wang and Ian H. Sloan. 2007. Low discrepancy sequences in high dimensions: How well are their projections distributed?J. Comput. Appl. Math. 213, 2 (2007), 366–386. doi:10.1016/j.cam.2007.01.005 7

work page doi:10.1016/j.cam.2007.01.005 2007

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Ivanovska

Emil Atanassov and S. Ivanovska. 2022. On the Use of Sobol’ Sequence for High Dimensional Simulation. InLecture Notes in Computer Science. 646–652. doi:10. 1007/978-3-031-08760-8_53

2022

[2] [2]

Jian Chen, Hui Jiang, and Nicholas Kirk. 2025. High-Dimensional Quasi-Monte Carlo via combinatorial discrepancy.arXiv.org(2025). arXiv:2508.18426 https: //arxiv.org/abs/2508.18426 Preprint submitted August 25, 2025

arXiv 2025

[3] [3]

Thies Hinz, Nestor Navarro-Guerrero, Sebastian Magg, and Stefan Wermter. 2018. Speeding up the hyperparameter optimization of deep convolutional neural net- works.International Journal of Computational Intelligence and Applications17, 02 (2018), 1850008. doi:10.1142/s1469026818500086

work page doi:10.1142/s1469026818500086 2018

[4] [4]

Stephen Joe and F. Y. Kuo. 2008. Constructing Sobol Sequences with Better Two-Dimensional Projections.SIAM Journal on Scientific Computing30, 5 (2008), 2635–2654. doi:10.1137/070709359

doi:10.1137/070709359 2008

[5] [5]

Naser, Mohammed K

Mohammad Z. Naser, Mohammed K. Al-Bashiti, Abdul T. G. Tapeh, Ammar Naser, V. Kodur, Rami Hawileeh, Jaafar Abdalla, Nasim Khodadadi, Amir H. Gandomi, and Alireza Dehghan Eslamlou. 2025. A review of benchmark and test functions for global optimization algorithms and metaheuristics.Wiley Interdisciplinary Reviews Computational Statistics17, 2 (2025). doi:10...

doi:10.1002/wics.70028 2025

[6] [6]

Dong Peng, Zhiwei Gui, and Haijun Wu. 2023. Interpreting the Curse of Di- mensionality from Distance Concentration and Manifold Effect.arXiv.org(2023). arXiv:2401.00422 https://arxiv.org/abs/2401.00422 Preprint submitted December 31, 2023

arXiv 2023

[7] [7]

Zoe Stokes, Arghya Mandal, and W. K. Wong. 2020. Using Differential Evolution to design optimal experiments.Chemometrics and Intelligent Laboratory Systems 199 (2020), 103955. doi:10.1016/j.chemolab.2020.103955

arXiv 2020

[8] [8]

Xiaoqun Wang and Ian H. Sloan. 2007. Low discrepancy sequences in high dimensions: How well are their projections distributed?J. Comput. Appl. Math. 213, 2 (2007), 366–386. doi:10.1016/j.cam.2007.01.005 7

work page doi:10.1016/j.cam.2007.01.005 2007