A fast method for variable-order space-fractional diffusion equations

Hong Wang; Jinhong Jia; Xiangcheng Zheng

arxiv: 1907.02697 · v2 · pith:BUEL2MK7new · submitted 2019-07-05 · 🧮 math.NA · cs.NA

A fast method for variable-order space-fractional diffusion equations

Jinhong Jia , Xiangcheng Zheng , Hong Wang This is my paper

Pith reviewed 2026-05-25 02:18 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords variable-order fractional diffusionspace-fractional equationsToeplitz matrix approximationfast collocation methoddivided-and-conquerasymptotic consistencynumerical PDEindirect collocation

0 comments

The pith

A sum of Toeplitz matrices scaled by diagonals approximates the stiffness matrix of variable-order space-fractional diffusion equations with asymptotic consistency.

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

The paper develops a fast divided-and-conquer indirect collocation method for the homogeneous Dirichlet problem of variable-order space-fractional diffusion equations. The space-dependent order destroys the Toeplitz structure that usually permits fast methods, so the authors replace the stiffness matrix with an approximation formed as a sum of Toeplitz matrices each multiplied by a diagonal matrix. They prove the approximation is asymptotically consistent with the true matrix. The resulting scheme uses O(kN log² N) memory and O(k N log³ N) operations. A reader would care because standard dense-matrix methods become prohibitive for large grids while this approach keeps the underlying collocation accurate.

Core claim

The central claim is that the coefficient matrix from the collocation discretization can be replaced by a sum of Toeplitz matrices multiplied by diagonal matrices without losing asymptotic consistency with the original variable-order operator, thereby recovering fast matrix-vector multiplication via the FFT while solving the homogeneous Dirichlet problem at the reduced memory and time costs stated above.

What carries the argument

The Toeplitz-diagonal sum approximation to the stiffness matrix, which restores fast FFT-based arithmetic while accounting for the spatially varying fractional order.

If this is right

Matrix-vector products can be performed in O(N log N) time per summand using the FFT, yielding the overall O(k N log³ N) complexity after the divided-and-conquer assembly.
The method inherits the convergence rate of the underlying indirect collocation scheme because the approximation error vanishes asymptotically.
Storage is reduced from O(N²) to O(kN log² N) because only the generating vectors of the Toeplitz blocks and the diagonal scalings need to be kept.
The same decomposition applies uniformly across all time steps when the spatial operator is time-independent.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same matrix-structure recovery may apply to other variable-coefficient fractional operators that destroy Toeplitz symmetry, such as those with spatially varying diffusivity.
Combining the approximation with existing fast preconditioners for constant-order fractional problems could further reduce iteration counts inside the solver.
The O(k N log³ N) scaling suggests the method remains practical when the variable order itself is approximated by a moderate number k of smooth basis functions.

Load-bearing premise

The space-dependent variable order permits an accurate and asymptotically consistent decomposition of the stiffness matrix into a sum of Toeplitz matrices scaled by diagonal matrices without introducing errors that invalidate the collocation solution.

What would settle it

Direct computation of the relative Frobenius-norm difference between the true stiffness matrix and its Toeplitz-diagonal approximation on successively refined grids; the difference must tend to zero if the consistency claim holds.

read the original abstract

We develop a fast divided-and-conquer indirect collocation method for the homogeneous Dirichlet boundary value problem of variable-order space-fractional diffusion equations. Due to the impact of the space-dependent variable order, the resulting stiffness matrix of the numerical approximation does not have a Toeplitz-like structure. In this paper we derive a fast approximation of the coefficient matrix by the means of a sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires $O(kN\log^2 N)$ memory and $O(k N\log^3 N)$ computational complexity with $N$ and $k$ being the numbers of unknowns and the approximants, respectively. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's new contribution is a sum-of-Toeplitz-times-diagonal approximation to the stiffness matrix for variable-order fractional diffusion that restores fast matrix-vector products, but the boundary consistency claim under homogeneous Dirichlet conditions needs explicit verification.

read the letter

The main thing to know is that this work takes the standard fast Toeplitz techniques for constant-order space-fractional diffusion and extends them to the variable-order case by writing the coefficient matrix as a sum of Toeplitz matrices each multiplied by a diagonal. That decomposition is the actual new step, and the paper claims it is asymptotically consistent while delivering the stated memory and flop counts with k terms. The method is built around indirect collocation on the homogeneous Dirichlet problem, and they include numerical tests to show the expected speed-up. The approach sits squarely on top of earlier constant-order results, which is the right way to proceed rather than starting from scratch. The numerics appear to confirm practical efficiency for the examples they ran. The soft spot is exactly the one flagged in the stress-test note. Because the order varies in space, the underlying operator loses translation invariance, so any decomposition introduces a perturbation. It is not automatic that this perturbation remains small enough in the discrete norms once the boundary rows are zeroed to enforce the Dirichlet conditions. The abstract asserts asymptotic consistency, but without seeing a uniform error bound that covers the boundary points, it is unclear whether the collocation solution retains its expected convergence rate. If the full derivation only controls the interior and treats the boundaries separately, that would be a real gap. This paper is for people who already work on fast solvers for fractional PDEs and want to handle variable orders without losing the Toeplitz structure entirely. A reader focused on matrix approximations or anomalous diffusion numerics would find the specific decomposition useful to examine. It deserves a serious referee because the idea is a direct, concrete extension with measurable complexity gains, even if the boundary analysis requires extra scrutiny. I would send it to review and ask the referees to check whether the consistency error is controlled uniformly up to the boundaries.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a fast divided-and-conquer indirect collocation method for the homogeneous Dirichlet problem of variable-order space-fractional diffusion equations. Because the space-dependent order destroys the Toeplitz structure of the stiffness matrix, the authors derive an approximation of this matrix as a sum of Toeplitz matrices each multiplied by a diagonal matrix. They claim this approximation is asymptotically consistent with the original operator, yielding O(k N log² N) memory and O(k N log³ N) computational complexity (N unknowns, k approximants), and support the claims with numerical experiments.

Significance. If the asymptotic consistency of the Toeplitz-diagonal decomposition holds with a boundary-uniform error bound, the method would supply a practical fast solver for variable-order fractional diffusion problems whose dense matrices otherwise preclude large-scale computation. The work correctly identifies the loss of translation invariance and supplies explicit complexity bounds together with numerical verification of efficiency.

major comments (2)

[consistency analysis section] The central claim of asymptotic consistency (abstract and the consistency analysis section) does not supply an explicit bound on the perturbation induced by the Toeplitz-diagonal decomposition that remains uniform up to the boundary points once the homogeneous Dirichlet conditions are enforced by zeroing the first and last rows/columns. Without such a bound, it is unclear whether the collocation solution retains its expected convergence rate in the discrete H¹ or L² norm.
[section deriving the fast matrix-vector product] The error analysis for the divided-and-conquer indirect collocation scheme (the section deriving the fast matrix-vector product) treats the approximation error as asymptotically negligible, yet provides no quantitative estimate of how the variable-order function affects the consistency error near x=0 and x=1. This estimate is load-bearing for the claimed O(k N log³ N) complexity to hold without additional terms that could dominate for non-constant order.

minor comments (2)

[introduction] Notation for the variable order function α(x) and the number of approximants k should be introduced once in the introduction and used consistently thereafter.
[numerical experiments section] The numerical experiments section would benefit from a table reporting observed convergence rates versus the theoretical rates predicted by the consistency analysis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on the consistency and error analysis. We address the two major comments below.

read point-by-point responses

Referee: [consistency analysis section] The central claim of asymptotic consistency (abstract and the consistency analysis section) does not supply an explicit bound on the perturbation induced by the Toeplitz-diagonal decomposition that remains uniform up to the boundary points once the homogeneous Dirichlet conditions are enforced by zeroing the first and last rows/columns. Without such a bound, it is unclear whether the collocation solution retains its expected convergence rate in the discrete H¹ or L² norm.

Authors: We agree that an explicit boundary-uniform perturbation bound after row/column zeroing would make the argument clearer. In the revised manuscript we will add a lemma in the consistency analysis section that derives such a bound, showing that the Toeplitz-diagonal approximation error remains controlled uniformly up to the boundary points and does not degrade the expected convergence rates. revision: yes
Referee: [section deriving the fast matrix-vector product] The error analysis for the divided-and-conquer indirect collocation scheme (the section deriving the fast matrix-vector product) treats the approximation error as asymptotically negligible, yet provides no quantitative estimate of how the variable-order function affects the consistency error near x=0 and x=1. This estimate is load-bearing for the claimed O(k N log³ N) complexity to hold without additional terms that could dominate for non-constant order.

Authors: We will strengthen the error analysis in that section by adding a quantitative estimate that bounds the contribution of the variable-order function to the local consistency error near the endpoints. The new estimate will confirm that this term remains of lower order and does not alter the overall O(k N log³ N) complexity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is constructive and self-contained

full rationale

The paper derives an approximation of the stiffness matrix as a sum of Toeplitz matrices scaled by diagonals for the variable-order operator, then proves asymptotic consistency with the original collocation problem. This is a standard constructive approximation followed by error analysis, not a reduction to fitted parameters, self-definitions, or self-citation chains. No load-bearing steps reduce by construction to the inputs; the consistency claim is independent of the final complexity bounds. The approach relies on external matrix techniques without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review limits identification of specific parameters or entities; the method appears to rest on standard numerical analysis assumptions for collocation and fractional operators rather than new postulates.

pith-pipeline@v0.9.0 · 5663 in / 1056 out tokens · 21635 ms · 2026-05-25T02:18:37.449082+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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A, Meerschaert, M

Schumer, R., Benson, D. A, Meerschaert, M. M., Wheatcraf t, S. W.: Eulerian deriva- tion of the fractional advection-dispersion equation. J. C ontaminant Hydrology.48, 69– 88(2001)

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Sun, H., Chang, A., Zhang, Y., Chen, W.: A review on variab le-order fractional dif- ferential equations: mathematical foundations, physical models, numerical methods and applications. Fract. Calc. Appl. Anal. 22, 27–59 (2019)

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Physica A: Stat

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W ang, H., Du, N.: A superfast-preconditioned iterative method for steady-state space- fractional diﬀusion equations. J. Comput. Phys. 240, 49–57(2013)

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W ang, H., W ang, K., Sircar, T.: A direct O(N log2 N ) ﬁnite diﬀerence method for frac- tional diﬀusion equations. J. Comput. Phys. 229, 8095-8104(2010)

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SIAM Sci

Zeng, F., Zhang, Z., Karniadakis, G.: A generalized spec tral collocation method with tunable accuracy for variable-order fractional diﬀerenti al equations. SIAM Sci. Comp. 37, A2710–A2732(2015)

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Zhao, Z, Jin, X., Lin, M.: Preconditioned iterative meth ods for space-time fractional advection-diﬀusion equations. J. Comput. Phys. 319, 266279(2016)

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submitted

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SIAM Numer

Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical metho ds for the variable-order frac- tional advection-diﬀusion equation with a nonlinear sourc e term. SIAM Numer. Anal. 47, 1760–1781(2009)

work page 2009

[1] [1]

Bai, Z., Lu, K., Pan, J.: Diagonal and Toeplitz splitting i teration methods for diagonal- plus-Toeplitz linear systems from spatial fractional diﬀu sion equations. Numer. Lin. Al- gebra Appl. 24, e2093(2017)

work page 2017

[2] [2]

Bear, J.: Some experiments on dispersion. J. Geophys. Res . 66, 2455-2467(1961)

work page 1961

[3] [3]

Elsevier, Ne w York(1972)

Bear, J.:Dynamics of Fluids in Porous Media. Elsevier, Ne w York(1972)

work page 1972

[4] [4]

M., Wheatcraft, S

Benson, D., Schumer, R., Meerschaert, M. M., Wheatcraft, S. W.: Fractional dispersion, L´ evy motions, and the MADE tracer tests. Transport in Porou s Media. 42, 211-240(2001)

work page 2001

[5] [5]

Bertaccini, D., Durastante, F.: Block structured precon ditioners in tensor form for the all-at-once solution of a ﬁnite volume fractional diﬀusion equation. Appl. Math. Lett. 95, 92-97(2019)

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Chen, S., Liu, F., Burrage, K.: Numerical simulation of a n ew two-dimensional variable- order fractional percolation equation in non-homogeneous porous media. Comput. Math. Appl.68, 2133–2141(2014)

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A., Lynch, V

Del-Castillo-Negrete, D., Carreras, B. A., Lynch, V. E.: Fractional diﬀusion in plasma turbulence. Phys. Plasmas. 11, 3854(2004)

work page 2004

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Boletn de la Sociedad Matemtica Mexicana

Del-Castillo-Negrete, D.: Front propagation in reactio n-diﬀusion systems with anomalous diﬀusion. Boletn de la Sociedad Matemtica Mexicana. 20, 87-105(2014)

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Deng, W.: Finite element method for the space and time frac tional Fokker-Planck equa- tion. SIAM J. Numer. Anal.. 47, 204–226(2008)

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Princeton University Press, Princeton, NJ, 2002

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[11] [11]

J., Heuer, N., Roop, J

Ervin, V. J., Heuer, N., Roop, J. P.: Regularity of the sol ution to 1-D fractional order diﬀusion equations. Math. Comput. 87, 2273-2294(2018)

work page 2018

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J., Roop, J

Ervin, V. J., Roop, J. P.: Variational formulation for th e stationary fractional advection dispersion equation. Numer. Meth. PDEs. 22, 558-576(2005)

work page 2005

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K., W ang, H.: A divided-and-conquer fast ﬁn ite diﬀerence method for space-time fractional partial diﬀerential equation

Fu, H., Ng, M. K., W ang, H.: A divided-and-conquer fast ﬁn ite diﬀerence method for space-time fractional partial diﬀerential equation. C omput. Math. Appl.. 73(6),1233- 1242(2017)

work page 2017

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Jin, X., Lin, F., Zhao, Z.: Preconditioned iterative met hods for two-dimensional space- fractional diﬀusion equations. Commun. Comput. Phys. 18, 469488(2015)

work page 2015

[15] [15]

K., Sun, H.: A fast direct method for block tr iangular Toeplitz-like with tri-diagonal block systems from time-fractional partial d iﬀerential equations

Ke, R., Ng, M. K., Sun, H.: A fast direct method for block tr iangular Toeplitz-like with tri-diagonal block systems from time-fractional partial d iﬀerential equations. J. Comput. Phys.303(C), 203-211(2015)

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Kilbas, A., Srivastava, H., Trujillo, J.: Theory and app lications of fractional diﬀerential equations. 204. Elsevier B.V., 2006

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Q.: Numerical approximation of nonlinear fractional diﬀer- ential equations with subdiﬀusion and superdiﬀusion

Li, C., Zhao, Z., Chen, Y. Q.: Numerical approximation of nonlinear fractional diﬀer- ential equations with subdiﬀusion and superdiﬀusion. Comp ut. Math. Appl. 62, 855–875 (2011)

work page 2011

[18] [18]

Li, Y., Chen, H., W ang, H.: A mixed-type Galerkin variati onal formulation and fast algorithms for variable-coeﬃcient fractional diﬀusion eq uations. Math. Methods Appl. Sci. DOI: 10.1002/mma.4367(2017)

work page doi:10.1002/mma.4367(2017 2017

[19] [19]

Lin, F., Yang, S., Jin, X.: Preconditioned iterative met hods for fractional diﬀusion equation. J. Comput. Phys. 256 , 109117(2014)

work page 2014

[20] [20]

K., Sun, H.: Eﬃcient preconditioner of one -sided space fractional diﬀu- sion equation[J]

Lin, X., Ng, M. K., Sun, H.: Eﬃcient preconditioner of one -sided space fractional diﬀu- sion equation[J]. BIT Numer. Math.(2018)

work page 2018

[21] [21]

K., Sun, H.: A Splitting Preconditioner fo r Toeplitz-Like Linear Systems Arising from Fractional Diﬀusion Equations

Lin, X., Ng, M. K., Sun, H.: A Splitting Preconditioner fo r Toeplitz-Like Linear Systems Arising from Fractional Diﬀusion Equations. SIAMX, 38, 1580–1614(2017)

work page 2017

[22] [22]

Liu, F., Anh, V., Turner, I.: Numerical solution of the sp ace fractional Fokker-Planck equation. J. Comput. Appl. Math. 166, 209–219 (2004)

work page 2004

[23] [23]

M, Sikorskii, A.: Stochastic Models for Fractional Calculus

Meerschaert, M. M, Sikorskii, A.: Stochastic Models for Fractional Calculus. De Gruyter Studies in Mathematics, 2011

work page 2011

[24] [24]

Metzler, R., Klafter, J.: The restaurant at the end of the random walk: recent develop- ments in the description of anomalous transport by fraction al dynamics. J. Phys. A Math. Gen.,37, R161–R208(2004)

work page 2004

[25] [25]

K., W ang, H.: Fast preconditioned iterati ve methods for ﬁnite volume discretization of steady-state space-fractional diﬀusio n equations

Pan, J., Ng, M. K., W ang, H.: Fast preconditioned iterati ve methods for ﬁnite volume discretization of steady-state space-fractional diﬀusio n equations. Numer. Algorithms, 74, 153–173(2017) 18 Jia, Zheng and W ang

work page 2017

[26] [26]

Podlubny, I.: Fractional Diﬀerential Equations.Acade mic Press, New York, 1999

work page 1999

[27] [27]

P.: Computational aspects of FEM approximation of fractional advection dis- persion equations on bounded domains in R2

Roop, J. P.: Computational aspects of FEM approximation of fractional advection dis- persion equations on bounded domains in R2. J. Comput. Appl. Math. 193, 243–268(2006)

work page 2006

[28] [28]

A, Meerschaert, M

Schumer, R., Benson, D. A, Meerschaert, M. M., Wheatcraf t, S. W.: Eulerian deriva- tion of the fractional advection-dispersion equation. J. C ontaminant Hydrology.48, 69– 88(2001)

work page 2001

[29] [29]

Sun, H., Chang, A., Zhang, Y., Chen, W.: A review on variab le-order fractional dif- ferential equations: mathematical foundations, physical models, numerical methods and applications. Fract. Calc. Appl. Anal. 22, 27–59 (2019)

work page 2019

[30] [30]

Physica A: Stat

Sun, H., Chen, W., Chen, Y.: Variable-order fractional d iﬀerential operators in anoma- lous diﬀusion modeling. Physica A: Stat. Mech. Appl. 388 , 4586–4592(2009)

work page 2009

[31] [31]

W ang, H., Du, N.: A superfast-preconditioned iterative method for steady-state space- fractional diﬀusion equations. J. Comput. Phys. 240, 49–57(2013)

work page 2013

[32] [32]

W ang, H., W ang, K., Sircar, T.: A direct O(N log2 N ) ﬁnite diﬀerence method for frac- tional diﬀusion equations. J. Comput. Phys. 229, 8095-8104(2010)

work page 2010

[33] [33]

SIAM Sci

Zeng, F., Zhang, Z., Karniadakis, G.: A generalized spec tral collocation method with tunable accuracy for variable-order fractional diﬀerenti al equations. SIAM Sci. Comp. 37, A2710–A2732(2015)

work page 2015

[34] [34]

Zhao, Z, Jin, X., Lin, M.: Preconditioned iterative meth ods for space-time fractional advection-diﬀusion equations. J. Comput. Phys. 319, 266279(2016)

work page 2016

[35] [35]

submitted

Zheng, X., W ang, H.: An optimal-order numerical approxi mation to variable-order space-fractional diﬀusion equations on uniform or graded m eshes. submitted

work page

[36] [36]

SIAM Numer

Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical metho ds for the variable-order frac- tional advection-diﬀusion equation with a nonlinear sourc e term. SIAM Numer. Anal. 47, 1760–1781(2009)

work page 2009