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Predictive Digital Twins with Quantified Uncertainty for Patient-Specific Decision Making in Oncology

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper develops an end-to-end Bayesian data-to-decisions pipeline that calibrates a 3D reaction-diffusion model of glioma growth to longitudinal MRI, producing probabilistic forecasts of tumor progression with quantified uncertainty…

desk verdict A solid Bayesian calibration pipeline for full 3D brain tumors with careful synthetic verification, but the clinical validation is partly circular through the shared ADC-to-cellularity mapping and should be reframed. read the letter →

arxiv 2505.08927 v1 pith:I7EKZB26 submitted 2025-05-13 cs.CE physics.comp-phphysics.med-ph

Graham Pash , Umberto Villa , David A. Hormuth II , Thomas E. Yankeelov , Karen Willcox This is my paper · ORCID
classification cs.CEphysics.comp-phphysics.med-ph MSC 65N2162F1592C5035Q92
keywords digitaltwinsuncertaintyquantificationBayesianinverseproblemsreaction-diffusiontumorgrowthgliomamagneticresonanceimagingLaplaceapproximationoptimalexperimentaldesign
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

Patient-specific digital twins for oncology require turning sparse, noisy MRI into trustworthy predictions of tumor growth with quantified uncertainty. This paper develops an end-to-end Bayesian pipeline that calibrates a three-dimensional reaction-diffusion model of high-grade glioma to longitudinal imaging data, using a scalable low-rank Laplace approximation to keep the high-dimensional posterior tractable. The authors verify the pipeline on a virtual patient with synthetic data, show that more frequent imaging improves predictive skill with diminishing returns, and validate on a cohort of glioma patients by withholding the final scan as a prediction target. Calibrated models reduce predictive variance and generally improve spatial and cellularity agreement relative to the prior, while exposing specific model-inadequacy issues in the non-enhancing tumor region. If the approach holds, routine MRI could support risk-informed decisions about when to image and how to tailor therapy for an individual patient.

What carries the argument

The central machinery is the coupling of three components: the ADC-based cellularity estimate $d(\bar{x},t)$ that turns MRI voxels into observations of tumor volume fraction; the reaction-diffusion forward model $\partial_t u - \nabla\cdot(D\nabla u) - \kappa u(1-u) = f_{\text{rt}}+f_{\text{ct}}$ with treatment terms; and the scalable Bayesian inversion that approximates the posterior over the spatial fields $m_D=\log D$ and $m_\kappa=\log\kappa$ as a Laplace approximation with a low-rank covariance update. The low-rank update is the load-bearing computational device: because the data-misfit Hessian has rapidly decaying eigenvalues, only the leading eigenpairs need to be computed, so the posterior can be sampled in a dimension-independent way and uncertainty can be pushed through the forward model to forecast tumor volume, cellularity, Dice, and concordance correlation.

What would settle it

Run the pipeline on a synthetic patient whose true cellularity field is known, then replace the true observations with ADC-mapped estimates carrying a known systematic error in the non-enhancing region; if posterior predictive intervals fail to cover the true tumor volume or the Dice gain over the prior shrinks, the ADC-as-truth assumption is falsified. A complementary check is to compare ADC-derived cellularity in non-enhancing regions against coregistered biopsy or histology.

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Extended reading notes

Core claim

The paper's central claim is that an end-to-end Bayesian data-to-decisions pipeline—from MRI acquisition to patient-specific brain geometry, to calibrated spatial parameter fields, to probabilistic forecasts of clinically relevant quantities—can be made computationally tractable at the scale of the human brain. The posterior distribution over the high-dimensional log-diffusivity and log-proliferation fields is approximated by a Laplace approximation centered at the MAP point, with covariance built as a low-rank correction to a Gaussian random field prior. On the virtual patient the pipeline reconstructs known spatial heterogeneity in the diffusion field; on the clinical cohort the posterior predictive distributions for Dice similarity and total tumor cellularity are substantially tighter than prior-based distributions and generally closer to the withheld observations. The paper asserts that this is the first demonstration of such an end-to-end Bayesian pipeline on clinical brain-tumor data with patient-specific anatomy, and it treats model inadequacy—particularly in the non-enhancing tumor region and in the fixed chemoradiation response model—as an explicit finding rather than a hidden failure.

Load-bearing premise

The load-bearing premise is that the ADC-derived cellularity estimate equals the tumor volume fraction the model evolves; if that equality is biased, especially in the non-enhancing tumor region, every calibration and validation result inherits the bias.

Editorial extensions

If this is right

  • Calibrated models yield posterior predictive distributions for tumor volume, cellularity, and spatial overlap that are significantly narrower and more accurate than prior-based forecasts, giving decision makers an explicit uncertainty budget.
  • Information gain from imaging grows with frequency but with diminishing returns, so the question of when to image a patient becomes a well-posed optimal experimental design problem.
  • The MAP and posterior computation is feasible in clinically relevant timeframes for realistic brain geometries, with median MAP times around 14 hours in this study.
  • The framework is forward-model agnostic, so richer mechanistic models—mass effect, multispecies, vascular, or metabolic—can be substituted into the same calibration and forecasting shell.

Reading between the lines

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

  • If the ADC-to-cellularity mapping is biased in the non-enhancing tumor region, both the calibration target and the validation truth share that bias, so the reported Dice and cellularity metrics cannot detect it; an independent histologic or alternative-modality ground truth would be needed.
  • The Laplace approximation is exact only for linear inverse problems, so in strongly nonlinear regimes the reported credible intervals likely understate total uncertainty; a synthetic study with large treatment effects and known truth could quantify the shortfall.
  • The optimal experimental design framing could be extended from imaging frequency to modality selection, trading the cost of each MRI sequence against its expected information gain.
  • If the same pipeline transfers to breast or prostate cancer, as the paper suggests, routine quantitative MRI could provide probabilistic forecasts for those settings with minimal methodological change.
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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

4 major / 6 minor

Summary. The paper develops an end-to-end Bayesian data-to-decisions pipeline for patient-specific oncology digital twins. The methodology combines a reaction-diffusion model of high-grade glioma growth with a scalable Bayesian inverse problem: spatially varying log-diffusivity and log-proliferation fields are inferred from longitudinal MRI-derived cellularity maps, using a Laplace approximation to the posterior with a low-rank correction to the prior covariance. The pipeline is verified on a virtual patient with synthetic observations (avoiding an inverse crime by solving on a coarser mesh), and the value of imaging frequency is assessed through predictive distributions of tumor volume and concordance correlation coefficient. The clinical demonstration calibrates the model on a subset of the IvyGAP cohort, withholds the last image for validation, and reports prior and posterior predictive distributions of Dice similarity coefficient and total tumor cellularity. The paper concludes that this is the first end-to-end Bayesian pipeline demonstrated on clinical data with complex brain anatomy.

Significance. If the results hold, the paper would make a useful contribution: it combines several existing components (PDE-constrained Bayesian inversion, Laplace approximation, low-rank covariance updates, parallel finite-element implementation) into a single framework that can be applied to patient-specific 3D brain geometries. The synthetic verification is careful: the inverse crime is avoided, noise and imaging frequency are controlled, and the reconstructed parameters and QoIs recover the truth. The public release of the code is a concrete strength. However, the clinical validation is the load-bearing part of the claimed contribution, and, as detailed in the major comments, the evidence currently supports a more modest statement: the pipeline demonstrates self-consistency of the ADC-to-cellularity processing chain and shows some improvement over the prior, but it does not yet establish that forecasts match true tumor burden, and for several patients the posterior is no better than the prior. The significance of the paper depends on the resolution of these validation concerns.

major comments (4)
  1. [Section 2.2, Eq. (6); Section 4.1; Section 5.2; Section 6] The ADC-to-cellularity mapping in Eq. (6) is the single most load-bearing assumption of the clinical validation, and the current study cannot detect a bias in this mapping. The same d(x,t) obtained from Eq. (6) is used as the calibration data, as the initial condition u0, and as the 'true' tumor u† against which Dice and total-cellularity metrics are computed in Section 5.2. The synthetic verification in Section 4.1 does not exercise this mapping: observations are generated by directly interpolating the model state u and adding 2% noise, with no simulation of ADC. Section 5.3 explicitly concedes that the non-enhancing tumor region is 'notoriously difficult to accurately estimate.' If Eq. (6) is biased there, the MAP calibration fits that bias and the validation compares the model to a reference generated by the same biased formula. Consequently, the 'robust agreement' claimed in Section 6 is currently evidence of self-consistency of the processing chain, not evidence of predictive accuracy against true tumor burden. A concrete remedy would be to validate the cellularity estimates against an independent reference (e.g., histology), or at minimum to perform a sensitivity analysis in which the ADC formula's parameters (ADCw, ADCmin, the treatment of edema) are perturbed and the resulting predictions are shown to be stable.
  2. [Appendix C, Tables C.4-C.7; Section 5.2] The claim that the posterior predictive distributions 'typically exhibit better spatial agreement' is contradicted by several entries in the summary tables. In the last-to-final Dice comparison (Table C.4), the posterior mean is worse than the prior mean for W03 (0.462 vs 0.566), W16 (0.870 vs 0.879), W36 (0.654 vs 0.671), and only marginally better for W43 (0.597 vs 0.594). In total-cellularity relative error (Table C.6), the posterior is worse than the prior for W16 (0.683 vs 0.046), W29 (1.971 vs 1.640), and W43 (2.766 vs 1.935). The text in Section 5.2 and the abstract's emphasis on 'robust agreement' overstate the cohort-level evidence. The authors should report per-patient results with error bars, state how many patients improve under each metric, and either provide a statistical test across the cohort or soften the conclusion accordingly.
  3. [Section 5.1, Table 3; Section 3.3] The prior hyperparameters in Table 3 are estimated from the same IvyGAP cohort that is subsequently used for the validation study: Section 5.1 states that 'an initial calibration of the cohort is performed' to determine the prior mean and variance. This introduces a form of data leakage that can inflate the apparent performance of the posterior relative to a fully out-of-sample prior. The comparison between prior and posterior predictive distributions is still meaningful if the prior is viewed as an empirical Bayes prior, but the paper should state this explicitly and discuss the effect on the claimed validation. If the prior means and variances are intended to represent literature-based knowledge, they should be fixed before any cohort data are used.
  4. [Section 3.4, Eqs. (15) and (21)] The Laplace approximation with a low-rank update (r=50 eigenpairs) is used to characterize the posterior and to generate all predictive intervals, but its accuracy is not assessed for the clinical problem. The virtual study evaluates the MAP point and QoI pushforwards, but it does not compare the Laplace approximation against a high-fidelity posterior sampler (e.g., MCMC) or against the full-rank Hessian. For a nonlinear reaction-diffusion problem with spatially varying coefficients, the Gaussian approximation may understate uncertainty or miss multimodality. Since the paper's contribution includes 'quantified uncertainty,' the authors should either provide such a comparison on at least one patient or explicitly qualify the intervals as approximate and justify the low-rank truncation level.
minor comments (6)
  1. [Section 2.1] In Section 4.1, the diffusion coefficient is reported as '0.03 mm 3/day' and '0.3 mm 3/day'; the correct physical units for a diffusion coefficient in 3D are mm^2/day, not mm^3/day.
  2. [Section 2.2] There is a typo: 'cebrospinal fluid' should be 'cerebrospinal fluid' in the description of prior work.
  3. [Section 4.4] There is a typo: 'Futhermore' should be 'Furthermore'.
  4. [Section 4.1] The sentence 'Since the UPENN-GBM dataset lacks ADC estimates, we follow [94] and take the tumor volume fraction is taken to be 0.8 and 0.16' is grammatically broken and also unclear about whether 0.8 and 0.16 are volume fractions or cellularity values; please rephrase.
  5. [Section 5.2] Figures 10(b) and 11(b) are described in the text as showing 'total tumor cellularity,' but the vertical axis labels are not visible in the captions or the figures; please ensure the ordinate is labeled on every panel.
  6. [Section 6] The claim that this is 'the first development and demonstration of such an end-to-end Bayesian pipeline on clinical data' should be softened or supported with a more explicit comparison to prior work, in particular the Bayesian personalization studies cited as [25]-[29], some of which include clinical patient data. If the novelty is the specific combination of high-dimensional spatial parameterization and human-brain geometry, that narrower claim should be stated.

Circularity Check

2 steps flagged · score 4.0 of 10

Held-out final images give the main prediction independent grounding, but the validation truth is produced by the same ADC-to-cellularity estimator used for calibration (Eq. 6), and cohort-fit priors are not documented as excluding the targets; the clinical 'validation' is partly self-consistency.

  1. self definitional [Section 2.2 (Eq. 6), Section 3.5, Section 5.2]
    "This approach generates observations of the tumor cellularity, d(¯x,t), every time the patient is imaged. … in this case the sets are the indicator functions for the predicted measurable tumor ˆuand the true, observed tumor u†."

    The calibration data are the ADC-derived cellularity estimates d from Eq. 6. In the clinical validation (Section 5.2), Dice and total-cellularity metrics are computed against the 'true, observed tumor u†', which is the same d at the withheld final time. Thus the same estimator that produces the calibration targets also produces the validation reference. Any bias in Eq. 6—especially in the non-enhancing region, which Section 5.3 calls 'notoriously difficult to accurately estimate'—enters both the fitted model and the validation truth, so the reported 'robust agreement' is a self-consistency check of the ADC-to-cellularity pipeline, not independent evidence that the model predicts true tumor burden. The synthetic study (Section 4.1) bypasses Eq.

  2. fitted input called prediction [Section 5.1, Table 3]
    "To determine the prior mean and variance for each of the parameters, an initial calibration of the cohort is performed where the parameters are modeled as scalar quantities. … For model validation, the last image in the dataset is withheld from the calibration and is set aside to serve as a prediction target."

    The posterior (Eq. 17) and hence the posterior predictive distributions used for validation (Figures 10 and 11) depend on the prior mean and covariance (m_pr, C_pr). These hyperparameters are estimated by a cohort-level calibration on the same patients being validated. The text does not state that the withheld final images were excluded from that cohort-level fit; if they were not, the 'prediction' of each final image is informed by that image through the prior, making the validation partly in-sample. This is data leakage rather than an algebraic identity, but it reduces the out-of-sample force of the clinical validation as described.

full rationale

The mathematical derivation is not circular: the MAP point and low-rank Laplace posterior (Eqs. 16–21) form a standard Bayesian inverse problem, the synthetic verification uses a known ground truth and a coarser mesh to avoid an inverse crime, and the clinical prediction uses a genuinely withheld final image. No uniqueness theorem or ansatz is smuggled in via self-citation; the cited prior works (e.g., [20, 29]) supply parameters and noise assumptions, not the core claim. The two concerns above are real but partial. First, calibrating to d and then validating against the same d-derivative 'truth' u† means the ADC-to-cellularity mapping is never independently tested; Section 5.3's own admission about the non-enhancing region makes this a substantive limitation. Second, the empirical-Bayes prior is fit on the validation cohort, and the paper does not record exclusion of the held-out images from that fit. These issues weaken the out-of-sample interpretation of the clinical validation but do not make the central methodological derivation reduce to its inputs by construction. Score 4 reflects a partially circular validation, not a fully forced result.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The inferential machinery rests on a standard reaction-diffusion tumor model, fixed therapy-response submodels, Gaussian noise, Gaussian field priors, a Laplace approximation, and an ADC-derived cellularity mapping. The main hand-set numbers are the prior hyperparameters, some fitted to the same cohort, the noise level, the measurability threshold, and fixed treatment parameters. No new physical entities are introduced.

free parameters (5)
  • Clinical prior hyperparameters = means: m_kappa=-1.230, m_D=-1.167, m_D,wm=-0.991, m_D,gm=-1.467; variances 0.040/0.115; rho 180/360 mm
    Set by scalar calibration on the IvyGAP cohort and used as Gaussian random field priors in Section 5.1 (Table 3); they directly shape posterior inference, so the clinical validation is partly informed by the same cohort.
  • Virtual study prior hyperparameters = m_D mean -1.30, var 0.05; m_kappa mean -1.00, var 0.02; rho 180 mm
    Chosen in Table 2 for the synthetic case; they affect the MAP point and posterior and are not derived from first principles.
  • Tumor measurability threshold = u_bar = 0.1
    Threshold defining measurable tumor for volume, cellularity, Dice, and CCC quantities of interest (Section 3.5); predictions and validation scores depend on it.
  • Fixed therapy model parameters = alpha_rt=0.025 Gy^-1; alpha/beta=10 Gy; alpha_ct=0.9 (virtual), 0.82 (clinical); beta_ct=1.8 h
    Taken from the literature or prior calibration (refs 20, 42, 96) and not calibrated per patient; the paper itself notes in Section 5.3 that the fixed treatment model produced unrealistic remission for patient W43.
  • Likelihood noise variance = sigma_noise^2 = 3.9e-3 (clinical); 2% Gaussian (virtual)
    The likelihood in Eq. (11) is weighted by this variance; the clinical value is imported from prior work (ref 29) rather than estimated from the cohort.
assumptions (7)
  • domain assumption The reaction-diffusion equation (1) with logistic growth and Neumann boundary conditions adequately represents spatiotemporal glioma progression.
    The entire calibration and prediction machinery operates on this PDE (Section 2.1). Its two spatially varying parameters are the objects of inference; if the model form is wrong, the inferred parameters and forecasts inherit the error.
  • domain assumption Radiotherapy acts instantaneously with a linear-quadratic surviving fraction, and chemotherapy follows a decaying-exponential model, Eqs (2)-(4).
    Treatment is part of the forward model for both synthetic and clinical studies; the paper acknowledges in Section 5.3 that this fixed model can be too strong.
  • domain assumption Observation noise is additive, zero-mean, spatially uniform Gaussian with known variance, Eq. (8) and Section 3.2.
    Defines the likelihood and hence the posterior; the clinical noise level is imported from prior work rather than estimated from the imaging pipeline.
  • domain assumption The parameters log(D) and log(kappa) are independent Gaussian random fields with Matern covariance defined by Eq. (13).
    The prior is essential for well-posedness of the Bayesian inverse problem and strongly shapes the posterior outside data-rich regions.
  • ad hoc to paper The Laplace approximation is an adequate surrogate for the posterior, Eq. (15).
    The approximation is exact only for linear inverse problems; for this nonlinear model its accuracy is not checked against MCMC or a coverage diagnostic before being used for the reported uncertainty intervals.
  • domain assumption The ADC-to-cellularity relation (Eq. 6) maps imaging data to the model state u and to validation truth.
    If this mapping is biased, calibration and validation are biased together; the paper itself discusses the difficulty in the non-enhancing tumor region in Section 5.3.
  • ad hoc to paper The low-rank approximation with r=50 eigenpairs captures the informative directions of the posterior covariance, Eq. (21).
    The truncation and oversampling factor of 10 are not justified by a convergence study in the paper.

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Pith. "Pith review of Predictive Digital Twins with Quantified Uncertainty for Patient-Specific Decision Making in Oncology." pith.science (2026). https://pith.science/paper/I7EKZB26

@misc{pith2026250508927,
  author       = {Pith},
  title        = {Pith review of: Predictive Digital Twins with Quantified Uncertainty for Patient-Specific Decision Making in Oncology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I7EKZB26}},
  note         = {Machine review of arXiv:2505.08927}
}
read the original abstract

Quantifying the uncertainty in predictive models is critical for establishing trust and enabling risk-informed decision making for personalized medicine. In contrast to one-size-fits-all approaches that seek to mitigate risk at the population level, digital twins enable personalized modeling thereby potentially improving individual patient outcomes. Realizing digital twins in biomedicine requires scalable and efficient methods to integrate patient data with mechanistic models of disease progression. This study develops an end-to-end data-to-decisions methodology that combines longitudinal non-invasive imaging data with mechanistic models to estimate and predict spatiotemporal tumor progression accounting for patient-specific anatomy. Through the solution of a statistical inverse problem, imaging data inform the spatially varying parameters of a reaction-diffusion model of tumor progression. An efficient parallel implementation of the forward model coupled with a scalable approximation of the Bayesian posterior distribution enables rigorous, but tractable, quantification of uncertainty due to the sparse, noisy measurements. The methodology is verified on a virtual patient with synthetic data to control for model inadequacy, noise level, and the frequency of data collection. The application to decision-making is illustrated by evaluating the importance of imaging frequency and formulating an optimal experimental design question. The clinical relevance is demonstrated through a model validation study on a cohort of patients with publicly available longitudinal imaging data.

Figures

Figures reproduced from arXiv: 2505.08927 by the authors.

Figure 1
Figure 1. Illustration of our digital twin workflow for a cancer patient. Observational data are integrated with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The computational pipeline: anatomic segmentation and mesh generation, cellularity estimation, and [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Snapshots of synthetic tumor progression for UPENN-GBM subject 101. Note the heterogeneous initial [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (12 more)
Figure 5
Figure 5. Figure 5: As expected, the magnitude of eigenvalues is larger when the patient is observed more [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 4
Figure 4. Figure 4: First column: axial slices showing white and gray matter segmentation with tumor state at final observation [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Spectral decay of the prior-preconditioned Hessian for various imaging frequencies. The larger eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Prior-preconditioned eigenvectors of the data-misfit Hessian, [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: Predictive distributions of (a) relative error in total tumor volume and (b) concordance correlation coefficient [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: Strong scaling of the forward solve obtaining a one-month prediction in (a) 42.2 seconds on one node for a [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: Imaging and treatment timelines for the IvyGAP cohort used for the model validation study. There is [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Prior and posterior predictive distributions of (a) Dice similarity coefficient and (b) total tumor cellularity [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: Posterior predictive distributions of (a) Dice similarity coefficient and (b) total tumor cellularity for the [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: Observed disease progression of IvyGAP patient W43 throughout the course of treatment. The top row [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: Predicted tumor progression from the first visit to the last visit for patient W43 using the MAP point. [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: Predicted final tumor state for IvyGAP patient W43 using the MAP point and prior mean both from [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]

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    Solve the forward model equation foru, given (m D,mκ)

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    Solve the adjoint model equation forp, given (m D,mκ) andu

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    Appendix A.4

    Evaluate the Fr´ echet derivativesδmDLg andδ mκLg, given (mD,mκ),u, andp. Appendix A.4. Hessian action We now form the Lagrangian for the Hessian,L H (where the superscriptHrefers to the role in deriving the Hessian action). We retain the notation that ˜·denotes a trial functi...

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    Solve the forward model foru, given (m D,mκ)

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    Solve the adjoint equation forp, given (m D,mκ) andu

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    Solve the incremental forward equation foreu, given (m D,mκ), (emD,emκ), andu

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    Solve the incremental adjoint equation forep, given (m D,mκ), (emD,emκ),u,p, andeu

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    36 Appendix B

    Evaluate the Hessian actions, given (m D,mκ), (emD,emκ),u,p,eu, andep. 36 Appendix B. Additional scaling studies Since the computational domain developed in Sec. 2.2 is assumed to be fixed at simulation time, we are primarily concerned with the strong scalability of the forwar...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.