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Fast automatically differentiable matrix functions and applications in molecular simulations

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Differentiating a conformally mapped contour-integral quadrature yields accurate gradients and Jacobians of matrix functions such as the logarithm and $p$-th roots, at costs that scale with matrix structure, and makes free-energy barrier…

desk verdict A practical AD-through-contour-integral package with plausible complexity gains and a working silicon application, but derivative accuracy is tested only against finite differences of the same quadrature, leaving the m/M dependence and convergence of d(f_N)/du untested. read the letter →

arxiv 2412.12598 v3 pith:YXBXAFO7 submitted 2024-12-17 physics.comp-ph

Tina Torabi , Timon S Gutleb , Christoph Ortner This is my paper · ORCID
classification physics.comp-ph MSC 65F60
keywords matrixfunctionsautomaticdifferentiationcontourintegrationconformalmapslogarithmvibrationalentropyfreeenergysurfacesilicondefects
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 aims to show that derivatives of analytic matrix functions $f(X(u))$—including the matrix logarithm, $p$-th roots, and trace-logarithms—can be computed accurately and cheaply by differentiating a conformally mapped contour-integral quadrature instead of using finite differences or eigendecompositions. The motivating case is the vibrational entropy $S(u)=-\frac{1}{2}\operatorname{Trace}\log^+(F H(u) F)$ of a crystal, whose gradient is required to search for free-energy barriers in defect migration. By applying forward or reverse automatic differentiation to the quadrature, the paper obtains gradients and Jacobians whose accuracy is controlled by the number of quadrature points $\ell$ and whose cost scales with matrix structure, for example $O(\ell b p n^2)$ in reverse mode for banded matrices with bandwidth $b$ and $p$ outputs. If the method holds, finite-temperature free-energy-surface calculations become practical at system sizes where direct differentiation of the matrix function is prohibitive, and the paper demonstrates the point on vacancy and interstitial migration in silicon.

What carries the argument

The load-bearing object is the Cauchy integral representation $f(X)=\frac{1}{2\pi i}\oint_C f(z)(zI-X)^{-1}\,dz$ together with the conformal map that sends the doubly connected slit domain to an annulus through logarithmic, Jacobi-elliptic, and Möbius transformations. After the change of variable $w=\sqrt{z}$ for functions with a branch cut on $(-\infty,0]$, the trapezoidal rule yields the quadratures (9) and (11), whose convergence is bounded by $O(e^{-\pi^2 N/\log(M/m+3)})$ where $m$ and $M$ are the spectral endpoints of $X$. The paper differentiates these quadrature sums through automatic differentiation, with Lemma 2.1 providing the resolvent-integral form of the parameter derivative as justification; the computational mechanism is simply that the conformal map, the resolvent solves, and the quadrature weights are all composed from differentiable elementary operations, so the same code path that evaluates $f_N(X(u))$ also evaluates its gradient or Jacobian in forward or reverse mode.

What would settle it

Take a small symmetric matrix $X(u)$ with a known eigendecomposition and spectral endpoints that move strongly with $u$, and compare $\partial f_N(X(u))/\partial u$ from forward-mode differentiation of Eq. (11) with the exact derivative obtained from the closed-form derivative of $X^{1/p}$ or $\log X$; if the difference grows with $dM/du$ in a way not bounded by the function error $\|f(X)-f_N(X)\|$, the central derivative-accuracy claim would be refuted.

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Extended reading notes

Core claim

The paper claims that differentiating the quadrature approximation $f_N(X(u))$ reproduces the derivative of $f(X(u))$ to the accuracy needed in applications, and supports that claim with comparisons against second-order centered finite differences. The theoretical starting point is the Cauchy integral formula and its parameter derivative $\frac{\partial f(X)}{\partial u_k} = \frac{1}{2\pi i}\oint_C f(z)(zI-X)^{-1}\frac{\partial X}{\partial u_k}(zI-X)^{-1}\,dz$; in practice the paper differentiates the explicit trapezoidal conformal-map quadrature (Eqs. (9) and (11)) through automatic differentiation, so forward mode propagates directional derivatives and reverse mode propagates adjoints through the same resolvent solves. This avoids forming the full $n\times n\times m$ Jacobian tensor $\partial X/\partial u$ when only a scalar output is needed. In the silicon application the method produces a 0K vacancy migration energy of $0.515$ eV, matching the literature value of $0.52$ eV, and temperature-dependent free-energy barriers that decrease as entropic contributions are included.

Load-bearing premise

The load-bearing premise is that differentiating the finite $N$-point quadrature sum gives the true derivative of the matrix function, including when the spectral endpoints $m$ and $M$ that set the quadrature nodes themselves depend on $u$; the paper relies on this without proving a separate convergence bound for the derivative.

Editorial extensions

If this is right

  • For a scalar output and a banded matrix, reverse mode costs $O(\ell b n^2)$, so gradient-based searches on a free-energy surface can scale to much larger supercells than the 64-atom cells shown.
  • The convergence estimate $O(e^{-\pi^2 N/\log(M/m+3)})$ means the number of quadrature points needed for a target accuracy grows only logarithmically with the condition number $M/m$, so ill-conditioned spectra remain tractable.
  • Because the 0K vacancy migration energy comes out at $0.515$ eV against a literature value of $0.52$ eV, and the barriers shift with temperature, free-energy-surface nudged-elastic-band paths can be computed with the same workflow as ordinary potential-energy paths.
  • For full Jacobians of $f$, reverse mode costs $O(\ell n^4)$ in the banded one-dimensional case, so the paper's tables imply that forward mode remains preferable for many-output problems and full Jacobians are only competitive when the number of outputs $p$ is small.

Reading between the lines

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

  • A natural extension would be to prove that $\partial f_N/\partial u$ inherits the quadrature's exponential convergence; the paper's numerical validation compares against finite differences, and a closed-form derivative bound would make the accuracy guarantee independent of finite-difference step-size tuning.
  • Because the quadrature nodes depend on the spectral endpoints $m$ and $M$, and those endpoints are only estimated in practice, an untested source of error is whether $dm/du$ and $dM/du$ contribute to the gradient; differentiating through the eigenvalue estimates and comparing with treating them as constants would settle the size of that term.
  • The same differentiate-the-quadrature strategy should extend to second derivatives, giving Hessians of the vibrational entropy and hence variational transition-state-theory prefactors without finite differences, even though the paper only demonstrates first derivatives.
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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. The paper proposes a method for computing Jacobians and gradients of analytic matrix functions f(X(u)) by applying automatic differentiation through the conformally mapped contour-integral quadrature of Hale, Higham, and Trefethen. The authors present an exact contour-integral formula for derivatives (Lemma 2.1), derive complexity estimates for dense, banded, and 2D/3D sparse matrices, and implement forward and reverse mode AD in Julia. They validate the approach on toy problems and use it to compute free-energy barriers for vacancy and interstitial migration in silicon, where the quantity of interest is the gradient of Trace log^+(F H(u) F). The central claim is that differentiating the truncated quadrature yields accurate derivatives at a cost that exploits sparsity, e.g., O(ℓ b p n^2) in reverse mode for banded systems.

Significance. If fully established, the method is practically useful: it provides a way to differentiate matrix functions such as the logarithm and p-th roots without dense eigendecompositions, with complexity that exploits structural sparsity. The derivative formula is exact and standard, the complexity tables are plausible, no fitted parameters enter the migration-barrier calculations, and the companion code is provided. However, the numerical evidence currently validates only the derivative of the truncated quadrature, not the convergence of that derivative to the true derivative, and the treatment of spectral bounds under differentiation is unspecified. These gaps are load-bearing for the paper's central accuracy claim.

major comments (3)
  1. [§2.3, Figs. 4b/6b] Figures 4b and 6b compare the AD gradient of the truncated quadrature f_N(X(u)) with centered finite differences of the same f_N. This only demonstrates that differentiation through the quadrature is consistent with the quadrature itself; it does not test whether d f_N/du approximates d f/du. Please add either a theorem bounding ||d/du f(X(u)) - d/du f_N(X(u))|| in terms of N and the spectral condition (for example, by applying the Hale–Higham–Trefethen convergence analysis to the derivative integrand (zI-X)^(-1) X_u (zI-X)^(-1), whose singularities are also confined to the spectrum of X), or a numerical test against an independent high-accuracy reference such as an eigendecomposition on small matrices. This is load-bearing because the silicon application in Section 5 needs the gradient of Trace log^+(F H(u) F), not merely the gradient of its quadrature approximation.
  2. [§2.2–2.3, Eqs. (9),(11)] The quadrature nodes z(t_j) and weights are defined through the conformal map, which depends on the spectral bounds m and M. The manuscript nowhere states whether m and M are treated as constants when differentiating f_N(X(u)). If they are constant, the derivative omits dm/du and dM/du terms, and the resulting gradient is valid only if the same fixed contour encloses σ(X(u)) for every u in the parameter set considered; no such spectral-inclusion condition is given. If instead m and M are recomputed for each u, then the AD trace must differentiate through the eigenvalue-extremum estimation (or its proxy), which is neither described nor analyzed. Please specify the convention used in the code and experiments, and add the required condition or analysis.
  3. [§5.3.2, Eq. (29)] The VTST rate expression contains an inconsistent sign. The text states E_ξξ(ξ_saddle) < 0 and concludes that HTST overestimates the rate, which requires the variational correction to reduce k relative to k_HTST; however, the printed equation shows exp(β T^2 S_ξ^2/(2E_ξξ)) (equivalently exp(β T^2 T_e)) with the opposite sign, and the following display then writes exp(-β T^2/(2 T_e)). Please correct the signs and define T_e unambiguously, since the direction of the entropy correction is one of the reported application results.
minor comments (5)
  1. [Eq. (6)] The derivative du/dt is stated as sn(t); the derivative of sn(t) is cn(t)dn(t). The subsequent formulas use cn(t)dn(t), so this is a typo but should be corrected.
  2. [Appendix 8.2] The sentence 'centered finite difference or automatic differentiation [58]' cites [58], which is the ComplexElliptic.jl package; a differentiation reference such as [51] or [36] is intended.
  3. [Eqs. (16) and (54)] The toy potential in Eq. (16) contains the term (1/2)|u_i-u_j|^3 while the appendix version in Eq. (54) writes δ|u_i-u_j|^3; the coefficient and the parameter δ should be harmonized.
  4. [§2.1] The sentence 'Suppose that f is analytic on a closed set D, where D is an open set containing the spectrum' is self-contradictory; rephrase as analytic on an open set containing a closed neighborhood of the spectrum.
  5. [Eq. (29)] The notation T_e is introduced immediately after its use in the exponential; define it before the formula and ensure it is nonnegative as an 'effective temperature'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the contour-integral derivatives follow from the defining formula and the application benchmarks are external.

full rationale

The paper's central derivative construction is not circular: Eq. (9) and Eq. (11) are quadrature approximations of the Cauchy integral (1), and Lemma 2.1 differentiates that integral with respect to the parameter-dependent matrix X(u). The AD gradients are therefore derivatives of the approximant f_N(X(u)), not fitted quantities. No parameter is calibrated to a target output: the quadrature count ℓ is a convergence knob, and the reported migration barriers at 0 K are compared to the literature value 0.52 eV from Spiewak and Kurzydlowski, not to any fitted value. The entropy model S(u) = -1/2 Trace log+(F H(u) F) is attributed to both external reviews (Fultz; Lapointe et al.) and the authors' prior work, so no load-bearing claim rests on a self-citation alone. The main caveat—that m and M enter the conformal-map nodes and the paper does not analyze dm/du, dM/du, nor prove convergence of d(f_N)/du to df/du—is an accuracy/robustness gap and is flagged in the paper's own assumption that m and M 'would typically only be estimated in practice.' It is not an equivalence between an input and an output, so it does not constitute circularity under the stated criteria. The finite-difference checks in Sections 4.2, 4.3, and 5.2 validate the AD of the quadrature against finite differences of the same quadrature; that is a self-consistency check, not a fitted-input prediction, and the external literature comparison in Section 5.3 provides independent anchoring.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

No fitted parameters enter the matrix-function identity; the only tunable is quadrature count. The method relies on standard analytic functional calculus, the Hale-Higham-Trefethen conformal map, and the harmonic vibrational entropy model from prior literature. The spectral bounds m and M are assumed known and are not reported for the silicon application.

free parameters (3)
  • quadrature point count = 10, 15, 20, 25, 35 across experiments
    Hand-chosen convergence parameter controlling the number of complex linear solves; the central accuracy and cost claims depend on it, but it is not fitted to a target result.
  • finite difference step size h = swept over a range near 1e-5 to 1e-1
    Used only for validation against AD gradients; chosen by hand and not part of the proposed method.
  • spectral bounds m and M = not reported for silicon runs
    The conformal map and quadrature nodes depend on these bounds; in practice they are estimated, and the paper does not report values or check that estimates are conservative for the defect configurations.
assumptions (6)
  • standard math Cauchy integral representation of analytic matrix functions
    Invoked in Eq. (1), Section 2.1, as the starting point for evaluation and differentiation.
  • domain assumption Conformal map to a doubly connected domain and exponential convergence of the trapezoidal rule
    Taken from Hale et al. (2008), used in Eqs. (5)-(11) without independent verification for the silicon spectra.
  • domain assumption Spectrum of X lies in a positive interval [m, M] with no large gaps
    Required for the conformal map construction in Section 2.2; assumed known and not verified in the application.
  • domain assumption Vibrational entropy S(u) = -1/2 Trace log^+(F H(u) F) is a valid free-energy model
    Taken from vibrational thermodynamics literature (refs. [62, 8]); the F preconditioner is assumed to create the spectral gap for log^+.
  • domain assumption Stillinger-Weber potential and NEB on the free energy surface give correct minimum free-energy paths
    Used throughout Section 5; the potential is chosen for convenience and the convergence of NEB on the FES is not quantified.
  • domain assumption Variational transition state theory expansion (Eqs. 22-28) is valid
    Borrowed from VTST literature; the paper uses it to estimate migration rates without deriving it.

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Pith. "Pith review of Fast automatically differentiable matrix functions and applications in molecular simulations." pith.science (2026). https://pith.science/paper/YXBXAFO7

@misc{pith2026241212598,
  author       = {Pith},
  title        = {Pith review of: Fast automatically differentiable matrix functions and applications in molecular simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YXBXAFO7}},
  note         = {Machine review of arXiv:2412.12598}
}
abstract

We describe efficient differentiation methods for computing Jacobians and gradients of a large class of matrix functions including the matrix logarithm $\log(A)$ and $p$-th roots $A^{\frac{1}{p}}$. We exploit contour integrals and conformal maps as described by (Hale et al., SIAM J. Numer. Anal. 2008) for evaluation and differentiation and analyze the computational complexity as well as numerical accuracy compared to high accuracy finite difference methods. As a demonstrator application we compute properties of structural defects in silicon crystals at positive temperatures, requiring efficient and accurate gradients of matrix trace-logarithms.

Figures

Figures reproduced from arXiv: 2412.12598 by the authors.

Figure 1
Figure 1. Figure (a) depicts the spectrum of X along with the appropriate contour C. Figure (b) demonstrates mapping the entire doubly connected domain of analyticity Ξ, onto an annulus region X through a conformal map. In this figure, the interval of singularities corresponds to the outer boundary of the annulus, while the interval including the spectrum aligns with the dashed inner boundary circle. We then apply the trapezo… view at source ↗
Figure 2
Figure 2. Steps of the conformal map shown in detail. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Computational graphs for the forward and reverse modes of differentiation for the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: b shows the error between the second-order centered finite difference method and reverse AD using the contour integral and conformal map approach, plotted against step size for various quadrature points ℓ = 10, 15, 20, 25. The errors decrease as ℓ increases, with highe…
Figure 5
Figure 5. Figure 5: Figure (a) shows obtained evaluation time for computing the Jacobian of matrix X with respect to m-vector u using a contour integral quadrature with ℓ = 25 quadrature points. Figure (b) shows convergence of a second order centered finite difference computation of the J…
Figure 6
Figure 6. Figure 6: Figure (a) shows obtained evaluation time for computing the gradient of vibrational entropy in (18) with respect to a displacement vector u of length m. Both methods used contour integral quadrature with ℓ = 20 quadrature points. Figure (b) shows convergence of a secon…
Figure 7
Figure 7. Figure 7: (a) Illustration of vacancy migration in silicon. The orange atom in (1) migrates to occupy the adjacent vacancy site, resulting in configuration (2). (b) Free-energy profiles for vacancy migration in silicon at various temperatures, illustrating the influence of tempe…
Figure 8
Figure 8. Figure 8: (a) Illustration of interstitial migration in silicon, with the red atom indicating the interstitial defect migrating from position 1 to 2. (b) Evolution of the free-energy profiles for the migration of an interstitial defect in Silicon. We generated a bulk silicon sys…
Figure 9
Figure 9. Figure 9: Estimated migration rates obtained from the variational free energy barriers for vacancy [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: Jacobi Elliptic Functions sn(u, k), cn(u, k), dn(u, k) for m = 0.7. 8.2 Strategies for Utilizing Matrix Sparsity Leveraging potential sparsity of the matrix X can significantly enhance the efficiency of computing its Jacobian. Consider the toy model presented in Secti…
Figure 11
Figure 11. Figure 11: The process of compressing the Jacobian matrix of the Hessian of toy model 16 with [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]

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Works this paper leans on

70 extracted references · 48 canonical work pages

  1. [1]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun, eds.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications Inc., 1965, p. 1046

    work page 1965

  2. [2]

    Anomalous entropy-driven kinetics of dislocation nucleation

    S. Bagchi and D. Perez. Anomalous entropy-driven kinetics of dislocation nucleation. 2024. URL: https://arxiv.org/ abs/2402.00810

    work page Pith review arXiv 2024

  3. [3]

    Variational transition state theory: theoretical framework and recent developments

    J. L. Bao and D. G. Truhlar. “Variational transition state theory: theoretical framework and recent developments”. In: Chem. Soc. Rev. 46 (24 2017), pp. 7548–7596. DOI: 10.1039/C7CS00602K

    doi:10.1039/c7cs00602k 2017

  4. [4]

    Batatia et al

    I. Batatia et al. Equivariant Matrix Function Neural Networks. 2024. URL: https://arxiv.org/abs/2310.10434

    arXiv 2024

  5. [5]

    Automatic differentiation in machine learning: a survey

    A. G. Baydin et al. “Automatic differentiation in machine learning: a survey”. In:J. Mach. Learn. Res. 18.1 (Jan. 2017), pp. 5595–5637. ISSN : 1532-4435

    work page 2017

  6. [6]

    Matrix functions in network analysis

    M. Benzi and P. Boito. “Matrix functions in network analysis”. In:GAMM-Mitt. 43.3 (2020), e202000012. DOI: 10.1002/ gamm.202000012

    work page 2020

  7. [7]

    Forward- or reverse-mode automatic differentiation: What’s the difference?

    B. van den Berg et al. “Forward- or reverse-mode automatic differentiation: What’s the difference?” In: Science of Computer Programming 231 (2024), p. 103010. ISSN : 0167-6423. DOI: https : / / doi . org / 10 . 1016 / j.scico.2023.103010

    arXiv 2024

  8. [8]

    Thermodynamic Limit of the Transition Rate of a Crystalline Defect

    J. Braun, M. H. Duong, and C. Ortner. “Thermodynamic Limit of the Transition Rate of a Crystalline Defect”. In: Arch. Ration. Mech. Anal. 238.3 (2020), pp. 1413–1474

    work page 2020

Show all 70 references
  1. [9]

    Reactivity of Single-Atom Alloy Nanoparticles: Modeling the Dehydrogenation of Propane

    R. J. Bunting et al. “Reactivity of Single-Atom Alloy Nanoparticles: Modeling the Dehydrogenation of Propane”. In: J. Am. Chem. Soc. 145.27 (2023), pp. 14894–14902. DOI: 10.1021/jacs.3c04030

    2023 doi

  2. [10]

    P. F. Byrd and M. D. Friedman. Handbook of Elliptic Integrals for Engineers and Scientists. 2nd ed. Grundlehren der mathematischen Wissenschaften. Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg, 1971, pp. XVI, 360. ISBN : 978-3-642-65140-3. DOI: 10.1007/978-3-642-65138-0

    1971 doi

  3. [11]

    Caillard and J.-L

    D. Caillard and J.-L. Martin. Thermally activated mechanisms in crystal plasticity. Pergamon materials series 8. Amsterdam ; Boston, Mass: Pergamon, 2003. ISBN : 9780080427034

    2003

  4. [12]

    Constrained reaction coordinate dynamics for the simulation of rare events

    E. Carter et al. “Constrained reaction coordinate dynamics for the simulation of rare events”. In: Chem. Phys. Lett. 156.5 (1989), pp. 472–477. ISSN : 0009-2614. DOI: 10.1016/S0009-2614(89)87314-2

    1989 doi

  5. [13]

    QM/MM Methods for Crystalline Defects. Part 1: Locality of the Tight Binding Model

    H. Chen and C. Ortner. “QM/MM Methods for Crystalline Defects. Part 1: Locality of the Tight Binding Model”. In: Multiscale Model. Sim. 14.1 (2016), pp. 232–264. DOI: 10.1137/15M1022628

    2016 doi

  6. [14]

    Locating landmarks on high-dimensional free energy surfaces

    M. Chen, T. -Q. Yu, and M. E. Tuckerman. “Locating landmarks on high-dimensional free energy surfaces”. In: Proceedings of the National Academy of Sciences 112.11 (2015), pp. 3235–3240. DOI: 10.1073/pnas.1418241112

    2015 doi

  7. [15]

    Activation energies for vacancy migration, clustering and annealing in silicon

    P. G. Coleman. “Activation energies for vacancy migration, clustering and annealing in silicon”. In: J. Phys. Conf. Ser. 265.1 (Jan. 2011), p. 012001. DOI: 10.1088/1742-6596/265/1/012001

    2011 doi

  8. [16]

    A survey of direct methods for sparse linear systems

    T. A. Davis, S. Rajamanickam, and W. M. Sid-Lakhdar. “A survey of direct methods for sparse linear systems”. In: Acta Numerica 25 (2016), pp. 383–566. DOI: 10.1017/S0962492916000076

    2016 doi

  9. [17]

    J. E. Dennis and R. B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Society for Industrial and Applied Mathematics, 1996. DOI: 10.1137/1.9781611971200

    1996 doi

  10. [18]

    Free energy calculations: an efficient adaptive biasing potential method

    B. M. Dickson et al. “Free energy calculations: an efficient adaptive biasing potential method”. en. In:J. Phys. Chem. B 114.17 (May 2010), pp. 5823–5830

    2010

  11. [19]

    https : / / dlmf

    NIST Digital Library of Mathematical Functions. https : / / dlmf . nist . gov/, Release 1.2.1 of 2024-06-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V . Saunders, H. S. Cohl, and M. A. McClain, eds

    2024

  12. [20]

    I. S. Duff, A. M. Erisman, and J. K. Reid. Direct Methods for Sparse Matrices. Oxford University Press, Jan. 2017. ISBN : 9780198508380. DOI: 10.1093/acprof:oso/9780198508380.001.0001

    2017

  13. [21]

    Stability analysis of linear time-invariant dynamic systems using the matrix sign function and the Adomian decomposition method

    H. Fatoorehchi and S. Djilali. “Stability analysis of linear time-invariant dynamic systems using the matrix sign function and the Adomian decomposition method”. In: Int. J. Dyn. Control 11 (July 2022). DOI: 10.1007/s40435-022-00989-3

    2022 doi

  14. [22]

    Vibrational thermodynamics of materials

    B. Fultz. “Vibrational thermodynamics of materials”. In: Prog. Mater. Sci. 55.4 (2010), pp. 247–352. DOI: 10.1016/j. pmatsci.2009.05.002

    2010 doi

  15. [23]

    What Color Is Your Jacobian? Graph Coloring for Computing Derivatives

    A. Gebremedhin, F. Manne, and A. Pothen. “What Color Is Your Jacobian? Graph Coloring for Computing Derivatives”. In: SIAM Rev. 47 (2005), pp. 629–705. DOI: 10.1137/S0036144504444711

    2005 doi

  16. [24]

    Breakdown of the Arrhenius Law in Describing Vacancy Formation Energies: The Importance of Local Anharmonicity Revealed by Ab initio Thermodynamics

    A. Glensk et al. “Breakdown of the Arrhenius Law in Describing Vacancy Formation Energies: The Importance of Local Anharmonicity Revealed by Ab initio Thermodynamics”. In: Phys. Rev. X 4 (1 Feb. 2014), p. 011018. DOI: 10.1103/ PhysRevX.4.011018

    2014

  17. [25]

    G. H. Golub and C. F. Van Loan. Matrix computations. eng. Baltimore, 2013 - 2013

    2013

  18. [26]

    Ab initio up to the melting point: Anharmonicity and vacancies in aluminum

    B. Grabowski et al. “Ab initio up to the melting point: Anharmonicity and vacancies in aluminum”. In: Phys. Rev. B 79 (13 Apr. 2009), p. 134106. DOI: 10.1103/PhysRevB.79.134106

    2009 doi

  19. [27]

    On the numerical stability of algorithmic differentiation

    A. Griewank, K. Kulshreshtha, and A. Walther. “On the numerical stability of algorithmic differentiation”. In:Computing 94 (2012), pp. 125–149

    2012

  20. [28]

    Griewank and A

    A. Griewank and A. Walther. Evaluating Derivatives. Second. Society for Industrial and Applied Mathematics, 2008. DOI: 10.1137/1.9780898717761. 29

    2008 doi

  21. [29]

    On Automatic Differentiation

    A. Griewank. “On Automatic Differentiation”. In: Mathematical Programming. Ed. by M. Iri and K. Tanabe. Dordrecht: Kluwer Academic Publishers, 1989, pp. 83–108. ISBN : 079230490X

    1989

  22. [30]

    Dual-number transformation and its applications to robotics

    Y .-L. Gu and J. Y . S. Luh. “Dual-number transformation and its applications to robotics”. In: IEEE Journal on Robotics and Automation 3 (1987), pp. 615–623

    1987

  23. [31]

    Computing Aα, log(A), and Related Matrix Functions by Contour Integrals

    N. Hale, N. J. Higham, and L. N. Trefethen. “Computing Aα, log(A), and Related Matrix Functions by Contour Integrals”. In: SIAM J. Numer. Anal. 46.5 (2008), pp. 2505–2523. DOI: 10.1137/070700607

    2008 doi

  24. [32]

    Theory of the backpropagation neural network

    R. Hecht-Nielsen. “Theory of the backpropagation neural network”. In: International 1989 Joint Conference on Neural Networks (1989), 593–605 vol.1

    1989

  25. [33]

    Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points

    G. Henkelman and H. Jónsson. “Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points”. In: J. Chem. Phys. 113.22 (Dec. 2000), pp. 9978–9985. ISSN : 0021-9606. DOI: 10.1063/1. 1323224

    doi

  26. [34]

    N. J. Higham. Functions of Matrices. Society for Industrial and Applied Mathematics, 2008. DOI: 10 . 1137 / 1 . 9780898717778

    2008

  27. [35]

    A Hitchhiker’s Guide to Automatic Differentiation

    P. Hoffmann. “A Hitchhiker’s Guide to Automatic Differentiation”. In: Numerical Algorithms 72 (July 2016). DOI: 10. 1007/s11075-015-0067-6

    2016

  28. [36]

    Don’t Unroll Adjoint: Differentiating SSA-Form Programs

    M. Innes. “Don’t Unroll Adjoint: Differentiating SSA-Form Programs”. In: CoRR abs/1810.07951 (2018). URL: http: //arxiv.org/abs/1810.07951

    2018 arXiv

  29. [37]

    PSelInv—A Distributed Memory Parallel Algorithm for Selected Inversion: The Symmetric Case

    M. Jacquelin, L. Lin, and C. Yang. “PSelInv—A Distributed Memory Parallel Algorithm for Selected Inversion: The Symmetric Case”. In: ACM Trans. Math. Softw. 43.3 (Dec. 2016). ISSN : 0098-3500. DOI: 10.1145/2786977

    2016 doi

  30. [38]

    M. Kubale. Graph Colorings. Contemporary Mathematics; v. 352. American Mathematical Society, 2004. ISBN : 9780821834589

    2004

  31. [39]

    Escaping free-energy minima

    A. Laio and M. Parrinello. “Escaping free-energy minima”. en. In: Proc. Natl. Acad. Sci. U. S. A. 99.20 (Oct. 2002), pp. 12562–12566

    2002

  32. [40]

    Machine learning surrogate models for strain-dependent vibrational properties and migration rates of point defects

    C. Lapointe et al. “Machine learning surrogate models for strain-dependent vibrational properties and migration rates of point defects”. In: Phys. Rev. Mater. 6 (11 Nov. 2022), p. 113803.DOI: 10.1103/PhysRevMaterials.6.113803

    2022 doi

  33. [41]

    Lewars.Computational Chemistry: Introduction to the Theory and Applications of Molecular and Quantum Mechanics

    E. Lewars.Computational Chemistry: Introduction to the Theory and Applications of Molecular and Quantum Mechanics. Springer, Jan. 2011, pp. 1–664. ISBN : 978-90-481-3860-9. DOI: 10.1007/978-90-481-3862-3

    2011 doi

  34. [42]

    Atomistic analysis of the vacancy diffusion mechanism

    S. List and H. Ryssel. “Atomistic analysis of the vacancy diffusion mechanism”. In: 1996 International Conference on Simulation of Semiconductor Processes and Devices. SISPAD ’96 (IEEE Cat. No.96TH8095). 1996, pp. 27–28. DOI: 10.1109/SISPAD.1996.865257

    1996

  35. [43]

    A. M. Mathai. A handbook of generalized special functions for statistical and physical sciences. eng. Oxford, 1993

    1993

  36. [44]

    EmpiricalPotentials.jl

    C. Ortner et al. “EmpiricalPotentials.jl”. URL: https://github.com/JuliaMolSim/EmpiricalPotentials.jl

  37. [45]

    SaddleSearch.jl

    C. Ortner et al. “SaddleSearch.jl”. URL: https://github.com/cortner/SaddleSearch.jl

  38. [46]

    Entropic Stabilization of Nanoscale V oids in Materials under Tension

    D. Perez et al. “Entropic Stabilization of Nanoscale V oids in Materials under Tension”. In:Phys. Rev. Lett. 110 (20 May 2013), p. 206001. DOI: 10.1103/PhysRevLett.110.206001

    2013 doi

  39. [47]

    Automatic Differentiation, C++ Templates, and Photogrammetry

    D. Piponi. “Automatic Differentiation, C++ Templates, and Photogrammetry”. In: Journal of Graphics Tools 9 (Jan. 2004). DOI: 10.1080/10867651.2004.10504901

    2004

  40. [48]

    Diffusion on Demand to Control Precipitation Aging: Application to Al-Mg-Si Alloys

    S. Pogatscher et al. “Diffusion on Demand to Control Precipitation Aging: Application to Al-Mg-Si Alloys”. In: Phys. Rev. Lett. 112 (22 June 2014), p. 225701. DOI: 10.1103/PhysRevLett.112.225701

    2014 doi

  41. [49]

    Correlation between self-diffusion in Si and the migration mechanisms of vacancies and self-interstitials: An atomistic study

    M. Posselt, F. Gao, and H. Bracht. “Correlation between self-diffusion in Si and the migration mechanisms of vacancies and self-interstitials: An atomistic study”. In: Phys. Rev. B 78 (3 July 2008), p. 035208. DOI: 10.1103/PhysRevB.78.035208

    2008 doi

  42. [50]

    A Discrete Dislocation Model of Creep in Single Crystals

    M. Rajaguru and S. M. Keralavarma. “A Discrete Dislocation Model of Creep in Single Crystals”. In: TMS 2016 145th Annual Meeting & Exhibition. Cham: Springer International Publishing, 2016, pp. 351–358. ISBN : 978- 3-319-48254-5

    2016

  43. [51]

    Forward-Mode Automatic Differentiation in Julia

    J. Revels, M. Lubin, and T. Papamarkou. “Forward-Mode Automatic Differentiation in Julia”. In:arXiv:1607.07892 [cs.MS] (2016). URL: https://arxiv.org/abs/1607.07892

    2016 arXiv

  44. [52]

    Entropic effect on the rate of dislocation nucleation

    S. Ryu, K. W. Kang, and W. Cai. “Entropic effect on the rate of dislocation nucleation”. In: Proceedings of the National Academy of Sciences 108 (2010), pp. 5174–5178

    2010

  45. [53]

    Formation and migration energies of the vacancy in Si calculated using the HSE06 range-separated hybrid functional

    P. ´Spiewak and K. J. Kurzydłowski. “Formation and migration energies of the vacancy in Si calculated using the HSE06 range-separated hybrid functional”. In: Phys. Rev. B 88 (19 Nov. 2013), p. 195204. DOI: 10.1103/PhysRevB.88.195204

    2013 doi

  46. [55]

    Mapping saddles and minima on free energy surfaces using multiple climbing strings

    G. Shrivastav, E. Vanden-Eijnden, and C. F. Abrams. “Mapping saddles and minima on free energy surfaces using multiple climbing strings”. In: J. Chem. Phys. 151.12 (Sept. 2019), p. 124112. ISSN : 0021-9606. DOI: 10.1063/1.5120372

    2019 doi

  47. [56]

    Unsupervised Calculation of Free Energy Barriers in Large Crystalline Systems

    T. D. Swinburne and M.-C. Marinica. “Unsupervised Calculation of Free Energy Barriers in Large Crystalline Systems”. In: Phys. Rev. Lett. 120 (13 Mar. 2018), p. 135503. DOI: 10.1103/PhysRevLett.120.135503

    2018 doi

  48. [57]

    cppduals: a nestable vectorized templated dual number library for C++11

    M. Tesch. “cppduals: a nestable vectorized templated dual number library for C++11”. In:Journal of Open Source Software 4.43 (2019), p. 1487. DOI: 10.21105/joss.01487

    2019 doi

  49. [58]

    T. Torabi. tinatorabi/ComplexElliptic.jl: ComplexElliptic (v1.0). 2024. DOI: 10.5281/zenodo.12622677. 30

    2024 doi

  50. [59]

    Animations of migration paths for defects in silicon crystals at 100K

    T. Torabi, T. S. Gutleb, and C. Ortner. “Animations of migration paths for defects in silicon crystals at 100K”. In:FigShare (Sept. 2024). DOI: https://doi.org/10.6084/m9.figshare.27003448.v3

    2024 doi

  51. [60]

    Vibrational entropy: spatial decomposition and surrogate models

    T. Torabi. “Vibrational entropy: spatial decomposition and surrogate models”. MA thesis. University of British Columbia,

  52. [61]

    EntropyGrad.jl

    T. Torabi and T. S. Gutleb. “EntropyGrad.jl”. URL: https://doi.org/10.5281/zenodo.14510426

    doi

  53. [62]

    Surrogate models for vibrational entropy based on a spatial decomposition

    T. Torabi, Y . Wang, and C. Ortner. “Surrogate models for vibrational entropy based on a spatial decomposition”. In: arXiv preprint arXiv:2402.12744 (2024)

    2024 arXiv

  54. [63]

    Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling

    G. Torrie and J. Valleau. “Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling”. In: J. Comput. Phys. 23.2 (1977), pp. 187–199. ISSN : 0021-9991. DOI: 10.1016/0021-9991(77)90121-8

    1977 doi

  55. [64]

    Theory of Grain Boundary Migration Rates

    D. Turnbull. “Theory of Grain Boundary Migration Rates”. In: JOM 3.8 (Aug. 1951), pp. 661–665. ISSN : 1543-1851. DOI: 10.1007/BF03397362

    1951 doi

  56. [65]

    Direct Transformation of Vacancy V oids to Stacking Fault Tetrahedra

    B. P. Uberuaga et al. “Direct Transformation of Vacancy V oids to Stacking Fault Tetrahedra”. In:Phys. Rev. Lett. 99 (13 Sept. 2007), p. 135501. DOI: 10.1103/PhysRevLett.99.135501

    2007 doi

  57. [66]

    Origins and implications of temperature-dependent activation energy barriers for dislocation nucleation in face-centered cubic metals

    D. Warner and W. Curtin. “Origins and implications of temperature-dependent activation energy barriers for dislocation nucleation in face-centered cubic metals”. In: Acta Materialia 57.14 (2009), pp. 4267–4277. ISSN : 1359-6454. DOI: https://doi.org/10.1016/j.actamat.2009.05.024

    2009 doi

  58. [67]

    A Simple Automatic Derivative Evaluation Program

    R. Wengert. “A Simple Automatic Derivative Evaluation Program”. In:Communications of the ACM 7.8 (1964), pp. 463– 464

    1964

  59. [68]

    F. White. JuliaDiff/ChainRules.jl: v1.61.0. 2024. DOI: 10.5281/zenodo.10604851

    2024 doi

  60. [69]

    E. T. Whittaker and G. N. Watson.A Course of Modern Analysis. 4th ed. Cambridge Mathematical Library. Cambridge University Press, 1996

    1996

  61. [70]

    Generalized stacking fault energies for embedded atom FCC metals

    J. A. Zimmerman, H. Gao, and F. F. Abraham. “Generalized stacking fault energies for embedded atom FCC metals”. In: Model. Simul. Mater. Sc. 8.2 (Mar. 2000), p. 103. DOI: 10.1088/0965-0393/8/2/302. 31

    doi

  62. [2024]

    DOI: http://dx.doi.org/10.14288/1.0442021

    doi

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.