Pith. sign in

REVIEW 3 major objections 5 minor 56 references

ACTest: A testing toolkit for analytic continuation methods and codes

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read ACTest builds benchmark datasets with known exact spectra for analytic continuation methods.

desk verdict A genuinely useful benchmarking toolkit for analytic continuation with a solid core, but the noise model, pass criterion, and demo-style example need tightening before the benchmarks can be trusted for realistic QMC data. read the letter →

arxiv 2411.16412 v1 pith:PFZ4DVAO submitted 2024-11-25 physics.comp-ph

Li Huang This is my paper · ORCID
classification physics.comp-ph
keywords analyticcontinuationspectralfunctionGreen'smaximumentropymethodquantumMonteCarlobenchmarkdatasetJuliatoolkitACT100
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

ACTest sets out to solve a practical problem in quantum many-body physics: analytic continuation methods that turn imaginary-time or Matsubara-frequency Green's functions into real-frequency spectral functions are numerous, but there has been no standard, quantitative way to compare them. The toolkit generates many synthetic spectral functions A(ω) from random superpositions of parameterized peaks, computes the corresponding exact Green's functions through Laplace transformation, and can add artificial noise to mimic quantum Monte Carlo data. Because the exact spectrum is known for every generated case, any analytic continuation method can be scored by the error of its reconstruction. ACTest ships a built-in standard dataset, ACT100, with 100 reproducible test cases covering fermionic and bosonic systems, positive and non-positive definite spectra, and various peak types. The paper also demonstrates the workflow by benchmarking the maximum entropy method as implemented in the ACFlow toolkit.

What carries the argument

The load-bearing mechanism is the pair of exact spectrum and exact Green's function manufactured from parameterized peaks. Each spectral function is a sum A(ω)=Σ p_i(ω) of Gaussian, Lorentzian, narrow-Gaussian (δ-like), rectangular, or rise-and-decay peaks with random parameters; the Green's function follows exactly via the Laplace transform G(x)=∫dω K(x,ω)A(ω), with fermionic, bosonic, and symmetric-bosonic kernels on either the imaginary-time or Matsubara axis. The noise model G_noisy = G_exact[1+δN_C(0,1)] adds controlled complex Gaussian noise to mimic quantum Monte Carlo data, and the error metric Err = ∫|A_true − A_calc| / ∫|A_true| together with the pass-rate statistic f scores a method per test case. The included ACT100 dataset fixes 100 such pairs with predefined parameters so results are reproducible.

What would settle it

Compare method rankings obtained on ACTest-generated data with rankings obtained on a set of small exactly solvable systems, such as a few-site Hubbard model solved by exact diagonalization where the exact spectral function is known; if the relative errors of the methods differ substantially at matched noise levels, the synthetic noise model does not fully capture what real data demand.

Watch

Extended reading notes

Core claim

The central claim is that a standard, reproducible benchmark for analytic continuation is feasible by construction: instead of relying on unknown real spectra, one builds the spectrum first as a sum of few randomly parameterized peaks (Gaussian, Lorentzian, δ-like, rectangular, or rise-and-decay), then obtains the exact Green's function by evaluating the Laplace-transform kernel K(x, ω) for the chosen imaginary-time or Matsubara grid. The exact A(ω) serves as ground truth, so an analytic continuation code can be scored objectively against it. The paper argues this enables a fair comparison of methods such as maximum entropy, stochastic analytic continuation, Nevanlinna continuation, and others, and it provides the ACT100 dataset to do so out of the box, together with integration with the ACFlow toolkit. Benchmark results on ACT100 for the maximum entropy method illustrate the workflow.

Load-bearing premise

The benchmarks inherit the assumption that adding uncorrelated, complex-valued Gaussian noise to an exact Green's function reproduces the noise found in real quantum Monte Carlo data, which is often correlated and, for imaginary-time data, real-valued.

Editorial extensions

If this is right

  • Any analytic continuation method or code can be scored on the same 100 ACT100 cases, making comparisons between methods quantitative and reproducible across groups.
  • Users can generate arbitrarily large datasets with user-selected peak types, grids, noise levels, and spectral-sign constraints, which can directly serve as training and validation data for machine-learning analytic continuation methods.
  • The benchmark workflow exposes where a method struggles: varying lpeak, mesh types, and offdiag settings can reveal whether errors are driven by peak sharpness, non-linear grids, or sign changes in the spectrum.
  • The demonstration with maximum entropy on 100-case batches shows that, at noise 10^-6, diagonal fermionic and bosonic cases pass about 80% of the time while off-diagonal cases pass near 100%, despite off-diagonal errors being slightly larger, and the toolkit reports error, pass rate, and runtime for every test.

Reading between the lines

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

  • The synthetic noise in Eq. (28) is uncorrelated, complex, and multiplicative; if real quantum Monte Carlo noise is correlated or has different statistics, ranks obtained on ACTest cases may not fully transfer to real data, and injecting correlated or real-QMC noise would be a natural stress test.
  • The pass-rate statistic f only separates methods with Err above and below 1, so a stricter error threshold or a rank-based metric would better discriminate among high-accuracy methods on ACT100.
  • Because ACTest generates data from the same kernel and peak families that some solvers assume, methods that implicitly favor Gaussian-like features could look better on ACT100 than on out-of-distribution real spectra; cross-validation with a different spectral family would test for such bias.
Share X Bluesky LinkedIn Reddit HN

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. The manuscript introduces ACTest, a Julia toolkit for generating benchmark datasets for analytic continuation problems. Spectral functions are built as superpositions of randomly parameterized peaks (Gaussian, Lorentzian, delta-like, rectangular, and rise-and-decay), and exact Green's functions are computed on imaginary-time or Matsubara-frequency grids using fermionic, bosonic, or symmetric-bosonic kernels. Users can add noise, generate arbitrarily large datasets, or use the built-in 100-case ACT100 set. The toolkit is interfaced with ACFlow and includes scripts for running MaxEnt and other solvers, comparing reconstructed spectra to true spectra, and reporting error statistics. The paper's central claim is that ACTest provides a fair, reproducible, quantitative basis for comparing analytic continuation methods and codes.

Significance. If the toolkit is used as described, it addresses a genuine gap: analytic continuation methods are often tested on hand-picked examples without a shared standard. The kernel formulas in Eqs. (10)-(27) are standard and appear correctly stated, the ACT100 dataset is reproducible, and the ACFlow integration enables immediate use. The open-source availability, modular design, and built-in dataset are concrete strengths. However, the manuscript's own benchmark validation is limited by the noise model in Eq. (28), the very low noise level and small grid used in the Section 4.2 example, and the loose pass criterion in Eq. (30). The paper therefore establishes ACTest more convincingly as a generator of exact synthetic data than as a simulator of realistic QMC inputs.

major comments (3)
  1. [§2.6, Eq. (28)] The noise model G_noisy = G_exact[1 + δ N_C(0,1)] uses complex-valued Gaussian noise for all grid types. Imaginary-time Green's functions G(τ) for a Hermitian Hamiltonian are real; adding a complex random component is nonphysical and changes the analytic continuation problem in a way that is not representative of QMC data. Even for Matsubara data, the covariance of real and imaginary parts is not that of typical QMC estimators. In addition, the multiplicative form makes the noise singular wherever G_exact crosses zero, which can occur for off-diagonal correlators. Please replace or supplement Eq. (28) with grid-appropriate noise models (real noise for G(τ); realistic covariance for G(iωn)) and document the choice.
  2. [§4.2, Fig. 3] The only reported benchmark uses ngrid = 10 Matsubara points and noise = 1e-6. Ten points is far fewer than typical QMC imaginary-time or Matsubara grids, and 1e-6 is well below realistic noise floors, which are often in the 10^-4 to 10^-2 range. The statement that changing the computational configurations would lead to similar conclusions is not supported by any experiment in the paper. As a result, the reported pass rates and timings do not demonstrate how ACTest-based benchmarks reflect realistic data. Please add at least one benchmark at a realistic noise level and grid size, or explicitly restrict the claims to near-exact synthetic data.
  3. [§2.7, Eq. (30)] The pass criterion θ(1 - Err) classifies any reconstruction with normalized L1 error below 1 as a pass. Since Err is normalized by ||A_true||_1 via Eq. (29), a reconstruction with 99% error passes, and a trivial zero reconstruction sits exactly at Err = 1. This makes 'pass rate' a very loose, low-discrimination metric, and the reported ~80% pass rates are hard to interpret. The continuous error distributions shown in the left panel of Fig. 3 are more informative; please report those as the primary accuracy measure and define a stricter success threshold, or justify why Err < 1 is a meaningful passing criterion.
minor comments (5)
  1. [§2.7, Eq. (30)] The typeset equation appears to contain a duplicated bracket, so it is unclear whether the factor (1 - Err) is squared or whether the second factor is a typographical artifact.
  2. [§4.2] The sentence 'The spirit of the maximum entropy entropy' contains a duplicated word 'entropy'.
  3. [§4.2, [Solver] block] The comment lines contain the typo 'Typer' and several oddly spaced words; please proofread the input-file examples.
  4. [§3.5, offdiag parameter] The parameter name 'offdiag' is used to switch between positive-definite and non-positive-definite spectra, but non-positive-definite spectra are not necessarily off-diagonal Green's functions; a comment or a clearer name would avoid confusion.
  5. [Program Summary and Data Availability] The Program Summary still contains placeholders such as 'to be added by Technical Editor,' and the Data Availability statement says data will be provided on request even though the repository is public; please reconcile these statements.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: ACTest's benchmark pipeline compares solvers against independently generated ground-truth spectra.

full rationale

The claimed derivation chain is a standard forward/inverse benchmark: ACTest randomly parameterizes peak shapes to construct A(ω) (Sec. 2.4), computes G(τ) or G(iωn) through the Laplace kernels (Sec. 2.5, Eq. (9)), optionally adds noise (Sec. 2.6), and then compares a solver's output against the known Atrue using Eqs. (29)-(30). No quantity used as input is also the quantity claimed as output: the ground truth is generated by random parameter sampling, not by fitting and not by the solver being tested. The only close-to-self-referential element is the integration with the author's own ACFlow toolkit [44], but this is an open-source, code-reproduced software dependency used as an illustrative MaxEnt example, not a load-bearing theorem or fitted input. The noise model and the relatively loose pass threshold (Err<1) are modeling and quality choices, not circular steps; they affect realism and discrimination but do not make the benchmark equivalent to its input. Hence no circularity.

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

The central claim, that the toolkit generates valid test cases, rests on standard Laplace-transform kernels plus several modeling choices. No new physical entities are introduced. The only hand-chosen parameters are benchmark configuration values and the pass threshold; none are fitted to data.

free parameters (3)
  • Pass threshold in Eq. (30) = Err < 1
    A test counts as passed when the relative L1 error is below 1. This loose threshold drives the reported pass rates and is a hand-chosen benchmark parameter.
  • Noise level δ in the example = 1e-6
    The Section 4 benchmark uses δ = 1e-6, far below representative QMC noise. It is a user-controlled parameter, not fitted.
  • Number of Matsubara points in the example = 10
    The Section 4 example uses ngrid = 10, a small number that keeps the test computationally cheap but may not reflect realistic data sizes.
assumptions (3)
  • domain assumption Lehmann-representation Laplace kernels (Eqs. 10-27)
    The toolkit assumes the standard kernels relating spectral functions to imaginary-time and Matsubara Green's functions. This is standard physics, but it is the foundation of the data generation.
  • ad hoc to paper Spectral functions are superpositions of Gaussian, Lorentzian, δ-like, rectangular, and Rise-And-Decay peaks
    This is a modeling choice for generating test data. It is not derived from a physical principle, and the representativeness of this class of spectra is assumed.
  • ad hoc to paper Multiplicative complex Gaussian noise model (Eq. 28)
    The noise is assumed to be G_noisy = G_exact[1 + δ N(0,1)] with complex Gaussian noise. This form is chosen for convenience and is not shown to match QMC error statistics; it is also nonphysical for real-valued imaginary-time Green's functions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of ACTest: A testing toolkit for analytic continuation methods and codes." pith.science (2026). https://pith.science/paper/PFZ4DVAO

@misc{pith2026241116412,
  author       = {Pith},
  title        = {Pith review of: ACTest: A testing toolkit for analytic continuation methods and codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PFZ4DVAO}},
  note         = {Machine review of arXiv:2411.16412}
}
abstract

ACTest is an open-source toolkit developed in the Julia language. Its central goal is to automatically establish analytic continuation testing datasets, which include a large number of spectral functions and the corresponding Green's functions. These datasets can be used to benchmark various analytic continuation methods and codes. In ACTest, the spectral functions are constructed by a superposition of randomly generated Gaussian, Lorentzian, $\delta$-like, rectangular, and Rise-And-Decay peaks. The spectra can be positive definite or non-positive definite. The corresponding energy grids can be linear or non-linear. ACTest supports both fermionic and bosonic Green's functions on either imaginary time or Matsubara frequency axes. Artificial noise can be superimposed on the synthetic Green's functions to simulate realistic Green's functions obtained by quantum Monte Carlo calculations. ACTest includes a standard testing dataset, namely ACT100. This built-in dataset contains 100 testing cases that cover representative analytic continuation scenarios. Now ACTest is fully integrated with the ACFlow toolkit. It can directly invoke the analytic continuation methods as implemented in the ACFlow toolkit for calculations, analyze calculated results, and evaluate computational efficiency and accuracy. ACTest comprises many examples and detailed documentation. The purpose of this paper is to introduce the major features and usages of the ACTest toolkit. The benchmark results on the ACT100 dataset for the maximum entropy method, which is probably the most popular analytic continuation method, are also presented.

Figures

Figures reproduced from arXiv: 2411.16412 by the authors.

Figure 1
Figure 1. The four frequency meshes supported in ACTest: linear, tangent, Lorentzian, and half-Lorentzian meshes. The parameters for these [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Four representative spectra in the ACT100 dataset. The vertical dashed lines denote the Fermi level. The raw data and figures are [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Tests of the maximum entropy method as implemented in the ACFlow package [ [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Selected analytic continuation results for Matsubara Green’s functions (fermionic systems). (Left) Spectra for diagonal Green’s function. [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Selected analytic continuation results for Matsubara Green’s functions (bosonic systems). (Left) Spectra for diagonal Green’s function. [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

56 extracted references · 5 canonical work pages

  1. [1]

    J. W. Negele, H. Orland, Quantum Many-Particle Systems, Perseus Books, 1998. doi:10.1201/9780429497926

    doi:10.1201/9780429497926 1998

  2. [2]

    Coleman, Introduction to Many-Body Physics, Cambridge University Press, 2016

    P. Coleman, Introduction to Many-Body Physics, Cambridge University Press, 2016. doi:10.1017/CBO9781139020916

    doi:10.1017/cbo9781139020916 2016

  3. [3]

    Georges, G

    A. Georges, G. Kotliar, W. Krauth, M. J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys. 68 (1996) 13–125. doi:10.1103/RevModPhys.68.13. URL https://link.aps.org/doi/10.1103/RevModPhys.68.13

    doi:10.1103/revmodphys.68.13 1996

  4. [4]

    Asakawa, Y

    M. Asakawa, Y . Nakahara, T. Hatsuda, Maximum entropy analysis of the spectral functions in lattice QCD, Prog. Part. Nucl. Phys. 46 (2) (2001) 459–508. doi:https://doi.org/10.1016/S0146-6410(01)00150-8 . URL https://www.sciencedirect.com/science/article/pii/S0146641001001508

    doi:10.1016/s0146-6410(01)00150-8 2001

  5. [5]

    H. B. Sch ¨uttler, D. J. Scalapino, Monte Carlo studies of the dynamical response of quantum many-body systems, Phys. Rev. B 34 (1986) 4744–4756. doi:10.1103/PhysRevB.34.4744. URL https://link.aps.org/doi/10.1103/PhysRevB.34.4744

    doi:10.1103/physrevb.34.4744 1986

  6. [6]

    H. B. Sch ¨uttler, D. J. Scalapino, Monte Carlo Studies of the Dynamics of Quantum Many-Body Systems, Phys. Rev. Lett. 55 (1985) 1204–

    work page 1985

  7. [7]

    Osolin, R

    v. Osolin, R. ˇZitko, Pad ´e approximant approach for obtaining finite-temperature spectral functions of quantum impurity models using the numerical renormalization group technique, Phys. Rev. B 87 (2013) 245135. doi:10.1103/PhysRevB.87.245135. URL https://link.aps.org/doi/10.1103/PhysRevB.87.245135

    doi:10.1103/physrevb.87.245135 2013

  8. [8]

    Sch ¨ott, I

    J. Sch ¨ott, I. L. M. Locht, E. Lundin, O. Grån¨as, O. Eriksson, I. Di Marco, Analytic continuation by averaging Pad´e approximants, Phys. Rev. B 93 (2016) 075104. doi:10.1103/PhysRevB.93.075104. URL https://link.aps.org/doi/10.1103/PhysRevB.93.075104

    doi:10.1103/physrevb.93.075104 2016

Show all 56 references
  1. [9]

    H. J. Vidberg, J. W. Serene, Solving the Eliashberg equations by means of N-point Pad ´e approximants, J. Low Temp. Phys. 29 (3) (1977) 179–192. doi:10.1007/BF00655090. URL https://doi.org/10.1007/BF00655090

    1977 doi

  2. [10]

    Mark ´o, U

    G. Mark ´o, U. Reinosa, Z. Sz ´ep, Pad ´e approximants and analytic continuation of Euclidean Φ-derivable approximations, Phys. Rev. D 96 (2017) 036002. doi:10.1103/PhysRevD.96.036002. URL https://link.aps.org/doi/10.1103/PhysRevD.96.036002

    2017 doi

  3. [11]

    Jarrell, J

    M. Jarrell, J. Gubernatis, Bayesian inference and the analytic continuation of imaginary-time quantum Monte Carlo data, Phys. Rep. 269 (3) (1996) 133–195. doi:https://doi.org/10.1016/0370-1573(95)00074-7 . URL https://www.sciencedirect.com/science/article/pii/0370157395000747

    1996

  4. [12]

    R. N. Silver, D. S. Sivia, J. E. Gubernatis, Maximum-entropy method for analytic continuation of quantum Monte Carlo data, Phys. Rev. B 41 (1990) 2380–2389. doi:10.1103/PhysRevB.41.2380. URL https://link.aps.org/doi/10.1103/PhysRevB.41.2380 21

    1990 doi

  5. [13]

    J. E. Gubernatis, M. Jarrell, R. N. Silver, D. S. Sivia, Quantum Monte Carlo simulations and maximum entropy: Dynamics from imaginary- time data, Phys. Rev. B 44 (1991) 6011–6029. doi:10.1103/PhysRevB.44.6011. URL https://link.aps.org/doi/10.1103/PhysRevB.44.6011

    1991 doi

  6. [14]

    Gunnarsson, M

    O. Gunnarsson, M. W. Haverkort, G. Sangiovanni, Analytical continuation of imaginary axis data using maximum entropy, Phys. Rev. B 81 (2010) 155107. doi:10.1103/PhysRevB.81.155107. URL https://link.aps.org/doi/10.1103/PhysRevB.81.155107

    2010 doi

  7. [15]

    K. S. D. Beach, Identifying the maximum entropy method as a special limit of stochastic analytic continuation (2004). arXiv:0403055

    2004

  8. [16]

    A. W. Sandvik, Stochastic method for analytic continuation of quantum Monte Carlo data, Phys. Rev. B 57 (1998) 10287–10290. doi: 10.1103/PhysRevB.57.10287. URL https://link.aps.org/doi/10.1103/PhysRevB.57.10287

    1998 doi

  9. [17]

    A. W. Sandvik, Constrained sampling method for analytic continuation, Phys. Rev. E 94 (2016) 063308. doi:10.1103/PhysRevE.94. 063308. URL https://link.aps.org/doi/10.1103/PhysRevE.94.063308

    2016 doi

  10. [18]

    H. Shao, Y . Q. Qin, S. Capponi, S. Chesi, Z. Y . Meng, A. W. Sandvik, Nearly Deconfined Spinon Excitations in the Square-Lattice Spin-1/2 Heisenberg Antiferromagnet, Phys. Rev. X 7 (2017) 041072. doi:10.1103/PhysRevX.7.041072. URL https://link.aps.org/doi/10.1103/PhysRevX.7.041072

    2017 doi

  11. [19]

    H. Shao, A. W. Sandvik, Progress on stochastic analytic continuation of quantum Monte Carlo data, Phys. Rep. 1003 (2023) 1–88. doi: https://doi.org/10.1016/j.physrep.2022.11.002. URL https://www.sciencedirect.com/science/article/pii/S0370157322003921

    2023 doi

  12. [20]

    Ghanem, E

    K. Ghanem, E. Koch, Extending the average spectrum method: Grid point sampling and density averaging, Phys. Rev. B 102 (2020) 035114. doi:10.1103/PhysRevB.102.035114. URL https://link.aps.org/doi/10.1103/PhysRevB.102.035114

    2020 doi

  13. [21]

    Ghanem, E

    K. Ghanem, E. Koch, Average spectrum method for analytic continuation: E fficient blocked-mode sampling and dependence on the dis- cretization grid, Phys. Rev. B 101 (2020) 085111. doi:10.1103/PhysRevB.101.085111. URL https://link.aps.org/doi/10.1103/PhysRevB.101.085111

    2020 doi

  14. [22]

    O. F. Syljuåsen, Using the average spectrum method to extract dynamics from quantum Monte Carlo simulations, Phys. Rev. B 78 (2008) 174429. doi:10.1103/PhysRevB.78.174429. URL https://link.aps.org/doi/10.1103/PhysRevB.78.174429

    2008 doi

  15. [23]

    Fuchs, T

    S. Fuchs, T. Pruschke, M. Jarrell, Analytic continuation of quantum Monte Carlo data by stochastic analytical inference, Phys. Rev. E 81 (2010) 056701. doi:10.1103/PhysRevE.81.056701. URL https://link.aps.org/doi/10.1103/PhysRevE.81.056701

    2010 doi

  16. [24]

    Vafayi, O

    K. Vafayi, O. Gunnarsson, Analytical continuation of spectral data from imaginary time axis to real frequency axis using statistical sampling, Phys. Rev. B 76 (2007) 035115. doi:10.1103/PhysRevB.76.035115. URL https://link.aps.org/doi/10.1103/PhysRevB.76.035115

    2007 doi

  17. [25]

    A. S. Mishchenko, N. V . Prokof’ev, A. Sakamoto, B. V . Svistunov, Diagrammatic quantum Monte Carlo study of the Fr¨ohlich polaron, Phys. Rev. B 62 (2000) 6317–6336. doi:10.1103/PhysRevB.62.6317. URL https://link.aps.org/doi/10.1103/PhysRevB.62.6317

    2000 doi

  18. [26]

    Goulko, A

    O. Goulko, A. S. Mishchenko, L. Pollet, N. Prokof’ev, B. Svistunov, Numerical analytic continuation: Answers to well-posed questions, Phys. Rev. B 95 (2017) 014102. doi:10.1103/PhysRevB.95.014102. URL https://link.aps.org/doi/10.1103/PhysRevB.95.014102

    2017 doi

  19. [27]

    Huang, S

    L. Huang, S. Liang, Stochastic pole expansion method for analytic continuation of the Green’s function, Phys. Rev. B 108 (2023) 235143. doi:10.1103/PhysRevB.108.235143. URL https://link.aps.org/doi/10.1103/PhysRevB.108.235143 22

    2023 doi

  20. [28]

    Huang, S

    L. Huang, S. Liang, Reconstructing lattice QCD spectral functions with stochastic pole expansion and Nevanlinna analytic continuation, Phys. Rev. D 109 (2024) 054508. doi:10.1103/PhysRevD.109.054508. URL https://link.aps.org/doi/10.1103/PhysRevD.109.054508

    2024 doi

  21. [29]

    Fei, C.-N

    J. Fei, C.-N. Yeh, E. Gull, Nevanlinna Analytical Continuation, Phys. Rev. Lett. 126 (2021) 056402. doi:10.1103/PhysRevLett.126. 056402. URL https://link.aps.org/doi/10.1103/PhysRevLett.126.056402

    2021 doi

  22. [30]

    Fei, C.-N

    J. Fei, C.-N. Yeh, D. Zgid, E. Gull, Analytical continuation of matrix-valued functions: Carath ´eodory formalism, Phys. Rev. B 104 (2021) 165111. doi:10.1103/PhysRevB.104.165111. URL https://link.aps.org/doi/10.1103/PhysRevB.104.165111

    2021 doi

  23. [31]

    Otsuki, M

    J. Otsuki, M. Ohzeki, H. Shinaoka, K. Yoshimi, Sparse modeling approach to analytical continuation of imaginary-time quantum Monte Carlo data, Phys. Rev. E 95 (2017) 061302. doi:10.1103/PhysRevE.95.061302. URL https://link.aps.org/doi/10.1103/PhysRevE.95.061302

    2017 doi

  24. [32]

    Motoyama, K

    Y . Motoyama, K. Yoshimi, J. Otsuki, Robust analytic continuation combining the advantages of the sparse modeling approach and the Pad ´e approximation, Phys. Rev. B 105 (2022) 035139. doi:10.1103/PhysRevB.105.035139. URL https://link.aps.org/doi/10.1103/PhysRevB.105.035139

    2022 doi

  25. [33]

    Huang, E

    Z. Huang, E. Gull, L. Lin, Robust analytic continuation of Green’s functions via projection, pole estimation, and semidefinite relaxation, Phys. Rev. B 107 (2023) 075151. doi:10.1103/PhysRevB.107.075151. URL https://link.aps.org/doi/10.1103/PhysRevB.107.075151

    2023 doi

  26. [34]

    Ying, Pole Recovery From Noisy Data on Imaginary Axis, J

    L. Ying, Pole Recovery From Noisy Data on Imaginary Axis, J. Sci. Comput. 92 (3) (2022) 107. doi:10.1007/s10915-022-01963-z . URL https://doi.org/10.1007/s10915-022-01963-z

    2022 doi

  27. [35]

    Ying, Analytic continuation from limited noisy Matsubara data, J

    L. Ying, Analytic continuation from limited noisy Matsubara data, J. Comput. Phys. 469 (2022) 111549. doi:https://doi.org/10. 1016/j.jcp.2022.111549. URL https://www.sciencedirect.com/science/article/pii/S0021999122006118

    2022

  28. [36]

    Zhang, E

    L. Zhang, E. Gull, Minimal pole representation and controlled analytic continuation of Matsubara response functions, Phys. Rev. B 110 (2024) 035154. doi:10.1103/PhysRevB.110.035154. URL https://link.aps.org/doi/10.1103/PhysRevB.110.035154

    2024 doi

  29. [37]

    Fournier, L

    R. Fournier, L. Wang, O. V . Yazyev, Q. Wu, Artificial Neural Network Approach to the Analytic Continuation Problem, Phys. Rev. Lett. 124 (2020) 056401. doi:10.1103/PhysRevLett.124.056401. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.056401

    2020 doi

  30. [38]

    Yoon, J.-H

    H. Yoon, J.-H. Sim, M. J. Han, Analytic continuation via domain knowledge free machine learning, Phys. Rev. B 98 (2018) 245101. doi: 10.1103/PhysRevB.98.245101. URL https://link.aps.org/doi/10.1103/PhysRevB.98.245101

    2018 doi

  31. [39]

    Huang, Y .-f

    D. Huang, Y .-f. Yang, Learned optimizers for analytic continuation, Phys. Rev. B 105 (2022) 075112. doi:10.1103/PhysRevB.105. 075112. URL https://link.aps.org/doi/10.1103/PhysRevB.105.075112

    2022 doi

  32. [40]

    Zhang, M

    R. Zhang, M. E. Merkel, S. Beck, C. Ederer, Training biases in machine learning for the analytic continuation of quantum many-body Green’s functions, Phys. Rev. Res. 4 (2022) 043082. doi:10.1103/PhysRevResearch.4.043082. URL https://link.aps.org/doi/10.1103/PhysRevResearch.4.043082

    2022 doi

  33. [41]

    J. Yao, C. Wang, Z. Yao, H. Zhai, Noise enhanced neural networks for analytic continuation, Mach. Learn.: Sci. Technol. 3 (2) (2022) 025010. doi:10.1088/2632-2153/ac6f44. URL https://dx.doi.org/10.1088/2632-2153/ac6f44

    2022 doi

  34. [42]

    Arsenault, R

    L.-F. Arsenault, R. Neuberg, L. A. Hannah, A. J. Millis, Projected regression method for solving Fredholm integral equations arising in the analytic continuation problem of quantum physics, Inv. Prob. 33 (11) (2017) 115007. doi:10.1088/1361-6420/aa8d93. 23 URL https://dx.doi.o...

    2017 doi

  35. [43]

    N. S. Nichols, P. Sokol, A. Del Maestro, Parameter-free di fferential evolution algorithm for the analytic continuation of imaginary time correlation functions, Phys. Rev. E 106 (2022) 025312. doi:10.1103/PhysRevE.106.025312. URL https://link.aps.org/doi/10.1103/PhysRevE.106.025312

    2022 doi

  36. [44]

    Huang, ACFlow: An open source toolkit for analytic continuation of quantum Monte Carlo data, Comput

    L. Huang, ACFlow: An open source toolkit for analytic continuation of quantum Monte Carlo data, Comput. Phys. Commun. 292 (2023) 108863. doi:https://doi.org/10.1016/j.cpc.2023.108863. URL https://www.sciencedirect.com/science/article/pii/S0010465523002084

    2023

  37. [45]

    Bergeron, A.-M

    D. Bergeron, A.-M. S. Tremblay, Algorithms for optimized maximum entropy and diagnostic tools for analytic continuation, Phys. Rev. E 94 (2016) 023303. doi:10.1103/PhysRevE.94.023303. URL https://link.aps.org/doi/10.1103/PhysRevE.94.023303

    2016 doi

  38. [46]

    Kaufmann, K

    J. Kaufmann, K. Held, ana cont: Python package for analytic continuation, Comput. Phys. Commun. 282 (2023) 108519. doi:https: //doi.org/10.1016/j.cpc.2022.108519. URL https://www.sciencedirect.com/science/article/pii/S0010465522002387

    2023

  39. [47]

    Nogaki, J

    K. Nogaki, J. Fei, E. Gull, H. Shinaoka, Nevanlinna.jl: A Julia implementation of Nevanlinna analytic continuation, SciPost Phys. Codebases (2023) 19doi:10.21468/SciPostPhysCodeb.19. URL https://scipost.org/10.21468/SciPostPhysCodeb.19

    2023 doi

  40. [48]

    R. Levy, J. LeBlanc, E. Gull, Implementation of the maximum entropy method for analytic continuation, Comput. Phys. Commun. 215 (2017) 149–155. doi:https://doi.org/10.1016/j.cpc.2017.01.018. URL https://www.sciencedirect.com/science/article/pii/S0010465517300309

    2017 doi

  41. [49]

    Iskakov, A

    S. Iskakov, A. Hampel, N. Wentzell, E. Gull, TRIQS/Nevanlinna: Implementation of the Nevanlinna Analytic Continuation method for noise- free data, Comput. Phys. Commun. 304 (2024) 109299. doi:https://doi.org/10.1016/j.cpc.2024.109299. URL https://www.sciencedirect.com/science/...

    2024

  42. [50]

    Krivenko, M

    I. Krivenko, M. Harland, TRIQS /SOM: Implementation of the stochastic optimization method for analytic continuation, Comput. Phys. Commun. 239 (2019) 166–183. doi:10.1016/j.cpc.2019.01.021. URL https://www.sciencedirect.com/science/article/pii/S0010465519300402

    2019 doi

  43. [51]

    Krivenko, A

    I. Krivenko, A. S. Mishchenko, TRIQS /SOM 2.0: Implementation of the stochastic optimization with consistent constraints for analytic continuation, Comput. Phys. Commun. 280 (2022) 108491. doi:10.1016/j.cpc.2022.108491. URL https://www.sciencedirect.com/science/article/pii/S00...

    2022

  44. [52]

    F. F. Assaad, M. Bercx, F. Goth, A. G ¨otz, J. S. Hofmann, E. Huffman, Z. Liu, F. P. Toldin, J. S. E. Portela, J. Schwab, The ALF (Algorithms for Lattice Fermions) project release 2.0. Documentation for the auxiliary-field quantum Monte Carlo code, SciPost Phys. Codebases (202...

    2022 doi

  45. [53]

    Neuhaus, N

    J. Neuhaus, N. S. Nichols, D. Banerjee, B. Cohen-Stead, T. A. Maier, A. D. Maestro, S. Johnston, SmoQyDEAC.jl: A di fferential evo- lution package for the analytic continuation of imaginary time correlation functions, SciPost Phys. Codebases (2024) 39 doi:10.21468/ SciPostPhys...

    2024 doi

  46. [54]

    Nakatsukasa, L

    Y . Nakatsukasa, L. N. Trefethen, An Algorithm for Real and Complex Rational Minimax Approximation, SIAM J. Sci. Comput. 42 (5) (2020) A3157–A3179. doi:10.1137/19M1281897. URL https://doi.org/10.1137/19M1281897

    2020 doi

  47. [55]

    Nakatsukasa, O

    Y . Nakatsukasa, O. S `ete, L. N. Trefethen, The AAA Algorithm for Rational Approximation, SIAM J. Sci. Comput. 40 (3) (2018) A1494– A1522. doi:10.1137/16M1106122. URL https://doi.org/10.1137/16M1106122 24

    2018 doi

  48. [1207]

    URL https://link.aps.org/doi/10.1103/PhysRevLett.55.1204

    doi:10.1103/PhysRevLett.55.1204. URL https://link.aps.org/doi/10.1103/PhysRevLett.55.1204

    doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.