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arxiv: 2606.11554 · v1 · pith:4JFX2SA2new · submitted 2026-06-10 · 🧮 math.AP · cs.NA· math.NA

Recovering the initial condition and physical coefficients in a nonlinear PDE model of cell invasion

Beiji Chen , Kui Ren This is my paper

Pith reviewed 2026-06-27 09:27 UTC · model grok-4.3

classification 🧮 math.AP cs.NAmath.NA
keywords inverse problemCarleman estimatesreaction-diffusion equationcell invasionstability estimatenonlinear PDEuniquenessnumerical reconstruction
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The pith

Carleman estimates establish global uniqueness with Lipschitz stability for reaction coefficients and logarithmic stability for the initial condition in a nonlinear cell invasion PDE model.

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

The paper shows that two spatially varying reaction coefficients and the unknown initial condition can be simultaneously recovered from observations in a nonlinear density-dependent reaction-diffusion model of cell invasion. Carleman estimates deliver a global uniqueness result together with a Lipschitz-type stability estimate for the coefficients and only logarithmic stability for the initial condition. A two-stage numerical algorithm is introduced that employs a time-shift strategy to decouple the coefficient reconstruction from the initial-condition reconstruction. Numerical experiments illustrate that the method achieves feasible, accurate, and robust recovery. These results address the inverse problem of identifying parameters that control proliferation and saturation in tumor-growth models.

Core claim

Using Carleman estimates, we establish a global uniqueness result together with a Lipschitz-type stability estimate for the reaction coefficients and a weaker, logarithmic stability estimate for the initial condition.

What carries the argument

Carleman estimates that produce the weighted energy inequalities required to obtain uniqueness and the stated stability rates for the inverse problem.

If this is right

  • The reaction coefficients admit Lipschitz-stable recovery from the data.
  • The initial condition admits only logarithmic-stable recovery.
  • The time-shift strategy in the two-stage algorithm separates coefficient reconstruction from initial-condition reconstruction.
  • Numerical tests confirm feasibility, accuracy, and robustness of the inversion procedure.

Where Pith is reading between the lines

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

  • Logarithmic stability for the initial condition indicates that early-time measurements carry disproportionate weight in practical recovery.
  • The same Carleman technique might be tested on other nonlinear reaction-diffusion inverse problems arising in biological modeling.
  • Robustness observed in the numerical experiments suggests the algorithm could tolerate moderate measurement noise without additional regularization.
  • The decoupling achieved by the time-shift may extend to higher-dimensional or multi-species versions of the model.

Load-bearing premise

The observation data must satisfy the conditions that allow the Carleman estimates to produce the weighted energy inequalities needed for global uniqueness and the reported stability rates.

What would settle it

Finding two different pairs of reaction coefficients and initial conditions that generate identical observation data would disprove the uniqueness claim.

Figures

Figures reproduced from arXiv: 2606.11554 by Beiji Chen, Kui Ren.

Figure 1
Figure 1. Figure 1: Reconstruction results for the coefficients. Top row: [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Reconstruction results for the coefficients. Top row: [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reconstruction failure for the coefficients. From left to right: ground truth of [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Reconstruction of the initial condition given by a superposition of four Gaussian [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Reconstruction of the initial condition consisting of a circle and a square. From [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Reconstruction of the initial condition of a localized wave packet. From left [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
read the original abstract

This paper investigates an inverse problem for the simultaneous reconstruction of two spatially varying reaction coefficients, the local proliferation rate and the competition (saturation) coefficient, together with the unknown initial condition, in a nonlinear, density-dependent reaction-diffusion model motivated by cell invasion and tumor growth dynamics. Using Carleman estimates, we establish a global uniqueness result together with a Lipschitz-type stability estimate for the reaction coefficients and a weaker, logarithmic stability estimate for the initial condition. For the numerical reconstructions, we develop a two-stage algorithm employing a time-shift strategy to decouple the coefficient and the initial condition. Numerical experiments are presented to illustrate the feasibility, accuracy, and robustness of the proposed inversion 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.

Referee Report

1 major / 1 minor

Summary. The manuscript addresses an inverse problem for simultaneous recovery of two spatially varying reaction coefficients (proliferation and competition) and the initial condition in a nonlinear density-dependent reaction-diffusion PDE for cell invasion. It claims global uniqueness together with Lipschitz stability for the coefficients and logarithmic stability for the initial condition via Carleman estimates, and proposes a two-stage numerical algorithm with a time-shift strategy to decouple the unknowns, illustrated by numerical experiments.

Significance. If the global Carleman estimates hold as stated, the work would supply rigorous uniqueness and stability results for a practically relevant nonlinear inverse problem in mathematical biology, together with a concrete reconstruction procedure. The explicit separation of coefficient and initial-data recovery via time-shift is a constructive feature.

major comments (1)
  1. [Carleman estimate and stability proof (section containing the difference equation and weighted energy identity)] The central stability claim rests on absorbing the nonlinear remainder (u1 + u2)(u1 - u2) times the coefficient difference into the Carleman-weighted energy identity. No explicit a priori L^∞ bound on the solutions, independent of the unknown coefficients, is recorded; without it the absorption constant depends on solution size and the claimed global Lipschitz stability reduces to a local result around a reference solution.
minor comments (1)
  1. [Abstract] The abstract does not specify the type or spatial-temporal domain of the observation data used to obtain the Carleman estimates; this information is required to confirm that the weighted estimates close globally.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting this technical point in the stability analysis. We address the comment below.

read point-by-point responses

  1. Referee: [Carleman estimate and stability proof (section containing the difference equation and weighted energy identity)] The central stability claim rests on absorbing the nonlinear remainder (u1 + u2)(u1 - u2) times the coefficient difference into the Carleman-weighted energy identity. No explicit a priori L^∞ bound on the solutions, independent of the unknown coefficients, is recorded; without it the absorption constant depends on solution size and the claimed global Lipschitz stability reduces to a local result around a reference solution.

    Authors: We agree that the absorption of the nonlinear remainder requires an a priori L^∞ bound on the solutions that is independent of the unknown coefficients a and b. The original manuscript does not record such an explicit bound. In the revised version we will add a preliminary lemma establishing that, for admissible coefficients belonging to a fixed bounded subset of L^∞ (with a ≥ a₀ > 0 and b ≥ b₀ > 0) and initial data 0 ≤ u₀ ≤ 1, the solutions satisfy a uniform L^∞ bound independent of the particular choice of a and b inside that class. This bound is obtained via the maximum principle and will be inserted before the Carleman estimate. The main stability theorem will be restated to make the dependence on the admissible class explicit, thereby preserving the global character of the result within the natural bounded-coefficient class used for inverse problems of this type. revision: yes

Circularity Check

0 steps flagged

No circularity: uniqueness and stability derived from standard Carleman estimates applied to the given nonlinear PDE.

full rationale

The paper's central claims rest on applying Carleman estimates to obtain global uniqueness and stability for the reaction coefficients and initial condition in the nonlinear reaction-diffusion model. The abstract and description indicate use of established Carleman machinery without any reduction of the result to fitted parameters, self-definitions, or load-bearing self-citations that loop back to the target statements. No quoted equations or steps in the provided material exhibit a prediction or uniqueness result that is equivalent to its inputs by construction. The derivation is self-contained against external mathematical benchmarks (Carlemann theory), qualifying for the default non-circularity outcome.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the validity of the nonlinear PDE as a model for cell invasion and on the technical applicability of Carleman estimates to the inverse problem; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The nonlinear density-dependent reaction-diffusion PDE accurately captures the cell invasion dynamics under consideration.
    The entire inverse problem is posed for this specific model class.
  • domain assumption Carleman estimates hold for the linearized or appropriately weighted versions of the nonlinear system under the regularity assumed for the coefficients.
    This is the key technical tool invoked for uniqueness and stability.

pith-pipeline@v0.9.1-grok · 5640 in / 1315 out tokens · 26482 ms · 2026-06-27T09:27:41.537528+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

300 extracted references · 38 canonical work pages · 5 internal anchors

  1. [1]

    Identification of Nonconcave Aggregate Production Functions in Spatial

    Pan, Yinxi and Wang, Qi and Zhang, Lu , journal=. Identification of Nonconcave Aggregate Production Functions in Spatial

  2. [2]

    Tinsley and Babu

    Oden, J. Tinsley and Babu. Predictive Computational Science: Computer Predictions in the Presence of Uncertainty , booktitle =. 2017 , pages =

    2017

  3. [3]

    and Quaranta, Vito and Evans, Katherine J

    Yankeelov, Thomas E. and Quaranta, Vito and Evans, Katherine J. and Rericha, Erin C. , title =. Cancer Research , volume =

  4. [4]

    The case of recurrent epidemics caused by respiratory syncytial virus , author=

    Parameter estimation of some epidemic models. The case of recurrent epidemics caused by respiratory syncytial virus , author=. Bulletin of Mathematical Biology , volume=

  5. [5]

    Inverse Problems and Imaging , volume=

    Spatiotemporal monitoring of epidemics via solution of a coefficient inverse problem , author=. Inverse Problems and Imaging , volume=. 2025 , publisher=

    2025

  6. [6]

    Inverse problem for coefficient identification in

    Marinov, Tchavdar T and Marinova, Rossitza S and Omojola, Joe and Jackson, Michael , journal=. Inverse problem for coefficient identification in. 2014 , publisher=

    2014

  7. [7]

    Mathematical modeling of the

    Kabanikhin, Sergey Igorevich and Krivorotko, Olga Igorevna , journal=. Mathematical modeling of the. 2020 , publisher=

    2020

  8. [8]

    2022 , month =

    Yi-Hsuan Lin and Hongyu Liu and Xu Liu and Shen Zhang , title =. 2022 , month =. doi:10.1088/1361-6420/ac91ee , url =

    work page doi:10.1088/1361-6420/ac91ee 2022

  9. [9]

    2013 , publisher=

    Nonlinear Functional Analysis , author=. 2013 , publisher=

    2013

  10. [10]

    Payne, L. E. , title =

  11. [11]

    1995 , publisher=

    Linear and quasilinear parabolic problems , author=. 1995 , publisher=

    1995

  12. [12]

    Transportation research part C: emerging technologies , volume=

    Fourier neural operator for learning solutions to macroscopic traffic flow models: Application to the forward and inverse problems , author=. Transportation research part C: emerging technologies , volume=

  13. [13]

    Swanson and Carly Bridge and J.D

    Kristin R. Swanson and Carly Bridge and J.D. Murray and Ellsworth C. Alvord , keywords =. Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion , journal =. 2003 , issn =

    2003

  14. [14]

    Murray, J. D. , title =

  15. [15]

    2022 , author =

    Inverse problems for the fractional Laplace equation with lower order nonlinear perturbations , journal =. 2022 , author =

    2022

  16. [16]

    Nonlinear Analysis: Theory, Methods and Applications , volume =

    Uniqueness and stability in determining the heat radiative coefficient, the initial temperature and a boundary coefficient in a parabolic equation , author =. Nonlinear Analysis: Theory, Methods and Applications , volume =

  17. [17]

    2012 , author =

    Data regularization for a backward time-fractional diffusion problem , journal =. 2012 , author =

    2012

  18. [18]

    arXiv preprint arXiv:2509.07458 , year=

    Unveiling Biological Models Through Turing Patterns , author=. arXiv preprint arXiv:2509.07458 , year=

    work page arXiv

  19. [19]

    arXiv preprint arXiv:2512.22946 , year=

    Determining habitat anomalies in cross-diffusion predator-prey chemotaxis models , author=. arXiv preprint arXiv:2512.22946 , year=

    work page arXiv

  20. [20]

    Journal of Computational Physics , volume=

    Numerical solution of an inverse boundary value problem for the heat equation with unknown inclusions , author=. Journal of Computational Physics , volume=

  21. [21]

    Applied Mathematical Letters , volume=

    Jin Cheng and Jijun Liu , title=. Applied Mathematical Letters , volume=

  22. [22]

    Inverse Problems and Imaging , volume=

    Simultaneously identifying piecewise smooth conductivity and initial value for a heat conduction equation , author=. Inverse Problems and Imaging , volume=

  23. [23]

    Inverse Problems and Imaging , volume =

    Kaltenbacher, Barbara and Rundell, William , title =. Inverse Problems and Imaging , volume =. 2020 , doi =

    2020

  24. [24]

    Journal of Mathematical Analysis and Applications , volume =

    Kaltenbacher, Barbara and Rundell, William , title =. Journal of Mathematical Analysis and Applications , volume =. 2021 , doi =

    2021

  25. [25]

    Annali di Matematica Pura ed Applicata , volume =

    Del Santo, Daniele and Prizzi, Martino , title =. Annali di Matematica Pura ed Applicata , volume =. 2025 , doi =

    2025

  26. [26]

    Journal of Mathematical Analysis and Applications , volume =

    Wei, Ya-Fang and Zheng, Guo-Jie and Han, Zhong-Jie and Zhao, Zhi-Xue , title =. Journal of Mathematical Analysis and Applications , volume =. 2026 , doi =

    2026

  27. [27]

    Applied Mathematics and Computation , volume =

    Conte, Martina and Surulescu, Christina , title =. Applied Mathematics and Computation , volume =. 2021 , doi =

    2021

  28. [28]

    and Kuang, Yang and Kashkynbayev, Ardak , title =

    Tursynkozha, Aisha and Harris, Duane C. and Kuang, Yang and Kashkynbayev, Ardak , title =. Mathematical Biosciences , volume =. 2025 , doi =

    2025

  29. [29]

    and Maini, Philip K

    Colson, Chlo\'e and S\'anchez-Gardu\ no, Faustino and Byrne, Helen M. and Maini, Philip K. and Lorenzi, Tommaso , title =. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume =. 2021 , doi =

    2021

  30. [30]

    IEEE Transactions on Medical Imaging , volume =

    Scheufele, Klaudius and Subramanian, Shashank and Biros, George , title =. IEEE Transactions on Medical Imaging , volume =. 2021 , doi =

    2021

  31. [31]

    2023 , note =

    Liang, Baoshan and Lozenski, Luke and Villa, Umberto and Faghihi, Danial , title =. 2023 , note =

    2023

  32. [32]

    Research in the Mathematical Sciences , volume =

    Kow, Pu-Zhao and Wang, Jenn-Nan , title =. Research in the Mathematical Sciences , volume =. 2023 , doi =

    2023

  33. [33]

    2022 , note =

    Lin, Ching-Lung and Lin, Yi-Hsuan and Uhlmann, Gunther , title =. 2022 , note =

    2022

  34. [34]

    Inverse Problems , volume =

    Egger, Herbert and Pietschmann, Jan-Frederik and Schlottbom, Matthias , title =. Inverse Problems , volume =. 2017 , doi =

    2017

  35. [35]

    2020 , note =

    Kian, Yavar and Uhlmann, Gunther , title =. 2020 , note =

    2020

  36. [36]

    2024 , note =

    Choy, Jason and Kian, Yavar , title =. 2024 , note =

    2024

  37. [37]

    Inverse Problems , volume =

    Ait Ben Hassi, El Mustapha and Chorfi, Salah-Eddine and Maniar, Lahcen , title =. Inverse Problems , volume =. 2022 , doi =

    2022

  38. [38]

    Discrete and Continuous Dynamical Systems - Series S , volume =

    Martinez, Patrick and Vancostenoble, Judith , title =. Discrete and Continuous Dynamical Systems - Series S , volume =. 2021 , doi =

    2021

  39. [39]

    2024 , note =

    Schmitz, Lina Sophie and Walker, Christoph , title =. 2024 , note =

    2024

  40. [40]

    , title =

    Klibanov, Michael V. , title =. 2024 , note =

    2024

  41. [41]

    2024 , note =

    Karakazian, Hagop and Sayah, Toni and Triki, Faouzi , title =. 2024 , note =

    2024

  42. [42]

    Mathematics , volume =

    Lukyanenko, Dmitry and Yeleskina, Tatyana and Prigorniy, Igor and Isaev, Temur and Borzunov, Andrey and Shishlenin, Maxim , title =. Mathematics , volume =. 2021 , number =

    2021

  43. [43]

    Journal of Computational and Applied Mathematics , volume =

    The. Journal of Computational and Applied Mathematics , volume =. 2024 , issn =. doi:https://doi.org/10.1016/j.cam.2024.115827 , url =

    work page doi:10.1016/j.cam.2024.115827 2024

  44. [44]

    2015 , issn =

    Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology , journal =. 2015 , issn =

    2015

  45. [45]

    2006 , volume =

    Cristofol, Michel and Gaitan, Patricia and Ramoul, Hichem , title =. 2006 , volume =

    2006

  46. [46]

    doi: 10.1017/S0962492919000059

    Arridge, Simon and Maass, Peter and Öktem, Ozan and Schönlieb, Carola-Bibiane , year=. Solving inverse problems using data-driven models , volume=. doi:10.1017/S0962492919000059 , journal=

    work page doi:10.1017/s0962492919000059

  47. [47]

    Inverse Problems , volume=

    The quadratic Wasserstein metric for inverse data matching , author=. Inverse Problems , volume=. 2020 , publisher=

    2020

  48. [48]

    Chewi, Sinho , title =

  49. [49]

    Handbook of Uncertainty Quantification , editor =

    Marzouk, Youssef and Moselhy, Tarek and Parno, Matthew and Spantini, Alessio , title =. Handbook of Uncertainty Quantification , editor =

  50. [50]

    and Wibisono, Andre , title =

    Vempala, Santosh S. and Wibisono, Andre , title =. Advances in Neural Information Processing Systems (NeurIPS) , year =

  51. [51]

    Bou-Rabee, Nawaf and Eberle, Andreas and Zimmer, Raphael , title =. Ann. Appl. Probab. , volume =

  52. [52]

    Eberle, Andreas and Guillin, Arnaud and Zimmer, Raphael , title =. Ann. Probab. , volume =

  53. [53]

    High-dimensional

    Monmarch. High-dimensional. Electron. J. Stat. , volume =

  54. [54]

    , title =

    Ma, Yi-An and Chen, Yuansi and Jin, Chi and Flammarion, Nicolas and Jordan, Michael I. , title =. Bernoulli , volume =

  55. [55]

    Electron

    Bou-Rabee, Nawaf and Schuh, Katharina , title =. Electron. J. Probab. , volume =

  56. [56]

    Leimkuhler, Benedict and Matthews, Charles and Stoltz, Gabriel , title =. IMA J. Numer. Anal. , volume =

  57. [57]

    and Papaspiliopoulos, Omiros , title =

    Titsias, Michalis K. and Papaspiliopoulos, Omiros , title =. J. R. Stat. Soc. Ser. B Stat. Methodol. , volume =

  58. [58]

    Adaptive. Ann. Appl. Probab. , volume =

  59. [59]

    arXiv preprint arXiv:2202.13230 , year =

    Riou-Durand, Lionel and Vogrinc, Jure , title =. arXiv preprint arXiv:2202.13230 , year =

    work page arXiv

  60. [60]

    Fang, Yi and Sanz-Serna, J. M. and Skeel, Robert D. , title =. J. Chem. Phys. , volume =

  61. [61]

    Model reduction algorithms for optimal control and importance sampling of diffusions , journal =

    Hartmann, Carsten and Sch. Model reduction algorithms for optimal control and importance sampling of diffusions , journal =

  62. [62]

    Variational characterization of free energy: theory and algorithms , journal =

    Hartmann, Carsten and Richter, Lorenz and Sch. Variational characterization of free energy: theory and algorithms , journal =

  63. [63]

    Solving high-dimensional

    N. Solving high-dimensional. Partial Differ. Equ. Appl. , volume =

  64. [64]

    and Hartmann, Carsten , title =

    Zhang, Wei and Sahai, Tuhin and Marzouk, Youssef M. and Hartmann, Carsten , title =. SIAM J. Sci. Comput. , volume =

  65. [65]

    and Parrinello, Michele , title =

    Branduardi, Davide and Gervasio, Francesco L. and Parrinello, Michele , title =. J. Chem. Phys. , volume =

  66. [66]

    WIREs Comput

    Barducci, Alessandro and Bonomi, Massimiliano and Parrinello, Michele , title =. WIREs Comput. Mol. Sci. , volume =

  67. [67]

    Advances in Neural Information Processing Systems (NeurIPS) , year =

    Ge, Rong and Lee, Holden and Risteski, Andrej , title =. Advances in Neural Information Processing Systems (NeurIPS) , year =

  68. [68]

    and Deem, Michael W

    Earl, David J. and Deem, Michael W. , title =. Phys. Chem. Chem. Phys. , volume =

  69. [69]

    Non-reversible parallel tempering: a scalable highly parallel

    Syed, Saifuddin and Bouchard-C. Non-reversible parallel tempering: a scalable highly parallel. J. R. Stat. Soc. Ser. B Stat. Methodol. , volume =

  70. [70]

    Parallel tempering with a variational reference , booktitle =

    Surjanovic, Nikola and Syed, Saifuddin and Bouchard-C. Parallel tempering with a variational reference , booktitle =

  71. [71]

    Accelerating Langevin Sampling with Birth-death

    Lu, Yulong and Lu, Jianfeng and Nolen, James , title =. arXiv preprint arXiv:1905.09863 , year =

    work page internal anchor Pith review Pith/arXiv arXiv 1905

  72. [72]

    Birth--death dynamics for sampling: global convergence, approximations and their asymptotics , journal =

    Lu, Yulong and Slep. Birth--death dynamics for sampling: global convergence, approximations and their asymptotics , journal =

  73. [73]

    Advances in Neural Information Processing Systems (NeurIPS) , year =

    Liu, Qiang and Wang, Dilin , title =. Advances in Neural Information Processing Systems (NeurIPS) , year =

  74. [74]

    , title =

    Bernton, Espen and Heng, Jeremy and Doucet, Arnaud and Jacob, Pierre E. , title =. arXiv preprint arXiv:1912.13170 , year =

    work page arXiv 1912

  75. [75]

    SIAM/ASA J

    Reich, Sebastian and Weissmann, Simon , title =. SIAM/ASA J. Uncertain. Quantif. , volume =

  76. [76]

    Bierkens, Joris and Fearnhead, Paul and Roberts, Gareth , title =. Ann. Statist. , volume =

  77. [77]

    The bouncy particle sampler: A non-reversible rejection-free

    Bouchard-C. The bouncy particle sampler: A non-reversible rejection-free. J. Amer. Statist. Assoc. , volume =

  78. [78]

    Piecewise-Deterministic Markov Chain Monte Carlo

    Vanetti, Paul and Bouchard-C. Piecewise deterministic. arXiv preprint arXiv:1707.05296 , year =

    work page internal anchor Pith review Pith/arXiv arXiv

  79. [79]

    , title =

    Fearnhead, Paul and Bierkens, Joris and Pollock, Murray and Roberts, Gareth O. , title =. Statist. Sci. , volume =

  80. [80]

    Andrieu, Christophe and Livingstone, Samuel , title =. Ann. Statist. , volume =

Showing first 80 references.