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arxiv: 2604.12688 · v1 · submitted 2026-04-14 · 🧮 math.NA · cs.NA

Statistical finite elements for sequential data synthesis in solid dynamics

Igor Kavrakov , Yaswanth Sai Jetti , Ahmet Oguzhan Yuksel , Fehmi Cirak This is my paper

Pith reviewed 2026-05-10 14:49 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords statistical finite elementsBayesian filteringelastodynamicsdata assimilationperturbation approximationNewmark schememodel misspecificationstochastic PDEs
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The pith

A Bayesian filtering extension to statistical finite elements assimilates observational data sequentially into elastodynamic models by advancing the discretized state with a stochastic Newmark scheme.

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

The paper extends the statistical finite element framework to handle time-dependent solid mechanics problems where measurements arrive over time. It formulates the problem as an incremental Bayesian update: at each step a prior distribution over the displacement and velocity fields is generated by solving a probabilistic forward problem that includes random fields for forcing, material parameters, and model error. This prior is non-Gaussian because of the stochastic PDEs, so a perturbation expansion is used to produce an approximate Gaussian that still yields closed-form expressions for the posterior mean and covariance after conditioning on the new observations. Hyperparameters controlling the strength of model discrepancy are chosen by maximizing the marginal likelihood of the data seen so far. A reader would care because the resulting procedure supplies both a running estimate of the structural state and a quantitative measure of remaining uncertainty without having to restart the entire simulation when fresh measurements appear.

Core claim

Observational data are assimilated while the state of the spatially discretised finite element problem is advanced in time using the stochastic variant of the explicit Newmark scheme. The prior probability density of the state is obtained by solving an incremental probabilistic forward problem in which spatio-temporal Gaussian random fields for forcing, model misspecification and material parameters are specified via their stochastic PDE formulation. The resulting non-Gaussian prior is approximated by a perturbation approach, producing a Gaussian posterior with closed-form mean and covariance at every time step. The hyperparameters of the misspecification random field are calibrated by maxim

What carries the argument

The perturbation approximation that converts the non-Gaussian prior density arising from the stochastic PDEs into a Gaussian form, enabling closed-form Bayesian updates of the finite-element state within the statistical finite element framework.

If this is right

  • At each time step the updated state estimate is Gaussian with an explicit mean vector and covariance matrix that incorporate both model uncertainty and fresh observations.
  • Hyperparameters of the model-misspecification random field are determined automatically by maximising the marginal likelihood of the data observed up to the current time.
  • The same sequential procedure applies without modification to one- and two-dimensional elastodynamic problems supplied with synthetic sensor data.
  • Uncertainties from random forcing, parameter variability and structural model error propagate forward in time and are reduced at each measurement instant.

Where Pith is reading between the lines

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

  • Because the posterior remains Gaussian after each update, the method could be embedded inside existing explicit time integrators with only modest extra linear-algebra cost per step.
  • The same perturbation device might be reused for other time-dependent problems whose stochastic PDEs produce mildly non-Gaussian priors, such as diffusion or wave propagation outside solid mechanics.
  • In engineering practice the calibrated misspecification field supplies a running diagnostic of where the finite-element model systematically deviates from reality.
  • The closed-form posterior covariance could serve as the basis for adaptive mesh refinement or sensor placement decisions that minimise future uncertainty.

Load-bearing premise

The perturbation expansion remains accurate enough to represent the non-Gaussian prior induced by the stochastic PDEs for forcing, material parameters and model discrepancy.

What would settle it

A low-dimensional elastodynamic test problem in which the exact posterior can be computed by direct sampling or quadrature; the perturbation-based mean and covariance would then be compared directly to the sampled statistics for increasing levels of stochastic forcing or material variability.

Figures

Figures reproduced from arXiv: 2604.12688 by Ahmet Oguzhan Yuksel, Fehmi Cirak, Igor Kavrakov, Yaswanth Sai Jetti.

Figure 1
Figure 1. Figure 1: Accuracy of the perturbation-based stochastic model for a single-degree-of-freedom (SDOF) oscillator compared with Monte Carlo (MC) [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Convergence plots for the relative perturbation approximate standard deviation [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Graphical model of statFEM for solid dynamics with augmented-state filtering. The large circles represent random variables that are [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Normalized marginal negative log likelihood [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Schematic of a bar with axial harmonic loading. The red dots indicate the locations of sensor measurements. [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Forward predictive displacement at the tip of the bar for two correlation lengths of the material field. [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Marginal likelihood as a function of the misspecification standard deviation [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Posterior predictive displacement and updated Young’s modulus field obtained from the augmented state and parameter estimation [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Posterior predictive displacement and updated Young’s modulus field obtained from the augmented state and parameter estimation [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Posterior predictive displacement without updating the Young’s modulus field. [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Schematic of the anti-plane shear problem: (a) geometry and boundary conditions; the red dots indicate the locations of sensor mea [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Forward predictive displacement response at probe locations [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Marginal likelihood as a function of the misspecification standard deviation [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Posterior displacement response at probe locations [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Posterior inference of the shear modulus field in the anti-plane problem at the final time. [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Posterior displacement response at probe locations [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
read the original abstract

We present an approach for synthesising observational data with elastodynamic finite element models by extending the statistical finite element method (statFEM) framework. The proposed formulation adopts a Bayesian filtering approach to account for uncertainties in the data, the finite element model, and the discrepancies between the model and the physical system. Observational data are assimilated while the state of the spatially discretised finite element problem is advanced in time using the stochastic variant of the explicit Newmark scheme. The prior probability density of the state is obtained by solving an incremental probabilistic forward problem, and the corresponding posterior density is obtained by conditioning on the data available at each time step. In the probabilistic forward problem, spatio-temporal Gaussian random fields representing the forcing, model misspecification, and material parameters are specified via their stochastic PDE formulation. The resulting non-Gaussian prior probability density is approximated using a perturbation approach, yielding a Gaussian posterior with closed-form mean and covariance. The hyperparameters of the random field representing model misspecification are calibrated by maximising the marginal likelihood of the data. The proposed approach is illustrated on one- and two-dimensional elastodynamic examples with synthetic data.

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

2 major / 2 minor

Summary. The paper extends the statistical finite element method (statFEM) to sequential data assimilation in elastodynamics. It advances the spatially discretized state using a stochastic explicit Newmark scheme while assimilating observations via Bayesian filtering. Spatio-temporal Gaussian random fields for forcing, model misspecification, and material parameters are defined through stochastic PDEs; the resulting non-Gaussian prior is approximated by perturbation to obtain a closed-form Gaussian posterior at each step. Hyperparameters of the misspecification field are calibrated by maximizing the marginal likelihood, and the method is demonstrated on 1D and 2D synthetic elastodynamic examples.

Significance. If the perturbation approximation remains accurate under time stepping, the approach would offer an efficient, closed-form route to uncertainty-aware data synthesis in finite-element solid dynamics, extending statFEM to sequential settings with explicit treatment of model discrepancy and parameter uncertainty. The combination of stochastic PDE priors, Newmark propagation, and marginal-likelihood calibration is technically novel and could support engineering applications where full Monte-Carlo propagation is prohibitive.

major comments (2)
  1. [§3 (probabilistic forward problem) and §5 (numerical examples)] The central claim that the perturbation expansion produces a sufficiently accurate Gaussian prior (and thus reliable closed-form posterior) at each time step is load-bearing, yet the manuscript provides no quantitative validation of the approximation error. In particular, material parameters enter the discrete stiffness and mass matrices nonlinearly, so even a first-order perturbation truncates higher-order moments that the explicit Newmark integrator can amplify over many steps. No a-posteriori error indicator, convergence study with respect to perturbation order, or comparison against Monte-Carlo propagation of the full non-Gaussian process appears in the numerical examples.
  2. [§5] The synthetic-data illustrations in §5 report only qualitative agreement between assimilated and reference solutions. Quantitative error norms (e.g., L2 or energy-norm discrepancies versus the true synthetic trajectory), calibration diagnostics for the marginal-likelihood hyperparameters, and baseline comparisons (deterministic Newmark, ensemble Kalman filter, or full Monte-Carlo statFEM) are absent, leaving the practical accuracy of the Gaussian posterior unquantified.
minor comments (2)
  1. [§3] Notation for the stochastic Newmark update and the perturbation expansion should be collected in a single table or appendix for readability.
  2. [§3.2] The manuscript should state explicitly whether the perturbation is first-order or higher and whether the resulting covariance is exact or approximate under the linearised map.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments that highlight important aspects of the perturbation approximation and the numerical validation. We respond to each major comment below and will revise the manuscript accordingly to strengthen the quantitative support for the claims.

read point-by-point responses

  1. Referee: [§3 (probabilistic forward problem) and §5 (numerical examples)] The central claim that the perturbation expansion produces a sufficiently accurate Gaussian prior (and thus reliable closed-form posterior) at each time step is load-bearing, yet the manuscript provides no quantitative validation of the approximation error. In particular, material parameters enter the discrete stiffness and mass matrices nonlinearly, so even a first-order perturbation truncates higher-order moments that the explicit Newmark integrator can amplify over many steps. No a-posteriori error indicator, convergence study with respect to perturbation order, or comparison against Monte-Carlo propagation of the full non-Gaussian process appears in the numerical examples.

    Authors: We agree that the accuracy of the first-order perturbation approximation for the non-Gaussian prior is a central assumption, particularly because material parameters enter the discrete operators nonlinearly and the explicit Newmark scheme can propagate higher-order effects over multiple steps. The manuscript justifies the approximation theoretically for small uncertainties and illustrates the overall procedure on synthetic data, but does not provide direct quantitative checks against the full non-Gaussian process. In the revised manuscript we will add, in §5, a comparison of the perturbation-derived mean and covariance against Monte-Carlo estimates obtained by sampling the stochastic forward problem for the 1D example; we will also include a simple a-posteriori indicator based on the size of the neglected second-order terms and examine its growth over time steps. revision: yes

  2. Referee: [§5] The synthetic-data illustrations in §5 report only qualitative agreement between assimilated and reference solutions. Quantitative error norms (e.g., L2 or energy-norm discrepancies versus the true synthetic trajectory), calibration diagnostics for the marginal-likelihood hyperparameters, and baseline comparisons (deterministic Newmark, ensemble Kalman filter, or full Monte-Carlo statFEM) are absent, leaving the practical accuracy of the Gaussian posterior unquantified.

    Authors: We concur that the current numerical examples emphasize visual agreement and would be strengthened by quantitative metrics. The manuscript demonstrates the assimilation procedure on 1D and 2D synthetic trajectories but reports only qualitative results. In the revision we will augment §5 with L2 and energy-norm error tables comparing the posterior mean and covariance to the known synthetic reference at selected time instants, together with the evolution of the marginal-likelihood-optimized hyperparameters. We will also add brief comparisons against a deterministic Newmark integrator and an ensemble Kalman filter on the same examples to quantify the benefit of the proposed closed-form Gaussian posterior. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained in standard Bayesian and FE techniques

full rationale

The paper advances the elastodynamic state via the stochastic explicit Newmark scheme, obtains the prior by solving an incremental probabilistic forward problem with Gaussian random fields for forcing/misspecification/parameters, approximates the resulting non-Gaussian density by perturbation to yield a closed-form Gaussian posterior, and calibrates misspecification hyperparameters by maximising the marginal likelihood. None of these steps reduces by construction to a fitted quantity renamed as a prediction, nor relies on a self-citation chain whose cited result is itself unverified; the perturbation and filtering steps are independent approximations grounded in external Bayesian and numerical methods rather than tautological redefinitions of the inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on modeling uncertainties as Gaussian random fields via stochastic PDEs and on the validity of the perturbation approximation to obtain closed-form Gaussian posteriors; these are standard domain assumptions rather than new invented entities.

free parameters (1)
  • hyperparameters of the model-misspecification random field
    Calibrated by maximising the marginal likelihood of the observational data at each time step.
axioms (2)
  • domain assumption Spatio-temporal Gaussian random fields for forcing, model misspecification, and material parameters can be specified via their stochastic PDE formulation.
    Used to define the prior probability density in the incremental probabilistic forward problem.
  • ad hoc to paper The perturbation approach yields a sufficiently accurate Gaussian approximation to the non-Gaussian prior density.
    Invoked to obtain closed-form expressions for the posterior mean and covariance.

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Works this paper leans on

66 extracted references · 66 canonical work pages

  1. [1]

    Willcox, B

    K. Willcox, B. Segundo, The role of computational science in digital twins, Nature Computational Science 4 (2024) 147–149

    work page 2024

  2. [2]

    D. Wagg, K. Worden, R. Barthorpe, P. Gardner, Digital twins: state-of-the-art and future directions for modeling and simulation in engineering dynamics applications, ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering 6 (2020) 030901

    work page 2020

  3. [3]

    L. Sun, Z. Shang, Y . Xia, S. Bhowmick, S. Nagarajaiah, Review of bridge structural health monitoring aided by big data and artificial intelligence: from condition assessment to damage detection, Journal of Structural Engineering 146 (2020) 04020073:1–04020073:22

    work page 2020

  4. [4]

    Girolami, E

    M. Girolami, E. Febrianto, G. Yin, F. Cirak, The statistical finite element method (statFEM) for coherent synthesis of observation data and model predictions, Computer Methods in Applied Mechanics and Engineering 375 (2021) 113533:1–113533:32

    work page 2021

  5. [5]

    Duffin, E

    C. Duffin, E. Cripps, T. Stemler, M. Girolami, Statistical finite elements for misspecified models, Proceedings of the National Academy of Sciences 118 (2) (2021) e2015006118

    work page 2021

  6. [6]

    Febrianto, L

    E. Febrianto, L. Butler, M. Girolami, F. Cirak, Digital twinning of self-sensing structures using the statistical finite element method, Data- Centric Engineering 3 (2022) e31

    work page 2022

  7. [7]

    ¨O. D. Akyildiz, C. Duffin, S. Sabanis, M. Girolami, Statistical finite elements via Langevin dynamics, SIAM/ASA Journal on Uncertainty Quantification 10 (2022) 1560–1585

    work page 2022

  8. [8]

    K.-J. Koh, F. Cirak, Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysis, Computer Methods in Applied Mechanics and Engineering 417 (2023) 116358

    work page 2023

  9. [9]

    Glyn-Davies, C

    A. Glyn-Davies, C. Duffin, I. Kazlauskaite, M. Girolami,¨O. D. Akyildiz, Statistical finite elements via interacting particle Langevin dynamics, SIAM/ASA Journal on Uncertainty Quantification 13 (2025) 1200–1227

    work page 2025

  10. [10]

    Narouie, H

    V . Narouie, H. Wessels, F. Cirak, U. R ¨omer, Mechanical state estimation with a polynomial-chaos-based statistical finite element method, Computer Methods in Applied Mechanics and Engineering 441 (2025) 117970

    work page 2025

  11. [11]

    Karvonen, F

    T. Karvonen, F. Cirak, M. Girolami, Error analysis for a statistical finite element method, Journal of Multivariate Analysis 210 (2025) 105468

    work page 2025

  12. [12]

    V . K. Narouie, J.-H. Urrea-Quintero, F. Cirak, H. Wessels, Unsupervised constitutive model discovery from sparse and noisy data, Computer Methods in Applied Mechanics and Engineering 452 (2026) 118722

    work page 2026

  13. [13]

    Kaipio, E

    J. Kaipio, E. Somersalo, Statistical and computational inverse problems, Springer, 2006

    work page 2006

  14. [14]

    C. M. Bishop, Pattern Recognition and Machine Learning, Springer, 2006

    work page 2006

  15. [15]

    E. Bach, R. Baptista, D. Sanz-Alonso, A. Stuart, Machine learning for inverse problems and data assimilation, arXiv preprint arXiv:2410.10523

    work page arXiv

  16. [16]

    J. L. Beck, Bayesian system identification based on probability logic, Structural Control and Health Monitoring 17 (2010) 825–847

    work page 2010

  17. [17]

    S ¨arkk¨a, L

    S. S ¨arkk¨a, L. Svensson, Bayesian filtering and smoothing, vol. 17, Cambridge University Press, 2023

    work page 2023

  18. [18]

    R. E. Kalman, A new approach to linear filtering and prediction problems, Transactions of the ASME: Journal of Basic Engineering 82 (1960) 35–45

    work page 1960

  19. [19]

    A. H. Jazwinski, Stochastic processes and filtering theory, Courier Corporation, 2007

    work page 2007

  20. [20]

    E. A. Wan, R. Van Der Merwe, The unscented Kalman filter for nonlinear estimation, in: Proceedings of the IEEE 2000 adaptive systems for signal processing, communications, and control symposium (Cat. No. 00EX373), 153–158, 2000. 26

    work page 2000

  21. [21]

    G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, Journal of Geophysical Research: Oceans 99 (1994) 10143–10162

    work page 1994

  22. [22]

    Doucet, N

    A. Doucet, N. De Freitas, N. Gordon, An introduction to sequential Monte Carlo methods, in: Sequential Monte Carlo Methods in Practice, Springer New York, 3–14, 2001

    work page 2001

  23. [23]

    Corigliano, S

    A. Corigliano, S. Mariani, Parameter identification in explicit structural dynamics: performance of the extended Kalman filter, Computer methods in applied mechanics and engineering 193 (2004) 3807–3835

    work page 2004

  24. [24]

    Lourens, E

    E. Lourens, E. Reynders, G. De Roeck, G. Degrande, G. Lombaert, An augmented Kalman filter for force identification in structural dynamics, Mechanical systems and signal processing 27 (2012) 446–460

    work page 2012

  25. [25]

    S. E. Azam, E. Chatzi, C. Papadimitriou, A dual Kalman filter approach for state estimation via output-only acceleration measurements, Mechanical systems and signal processing 60 (2015) 866–886

    work page 2015

  26. [26]

    K. Maes, A. W. Smyth, G. De Roeck, G. Lombaert, Joint input-state estimation in structural dynamics, Mechanical Systems and Signal Processing 70 (2016) 445–466

    work page 2016

  27. [27]

    Ebrahimian, R

    H. Ebrahimian, R. Astroza, J. P. Conte, C. Papadimitriou, Bayesian optimal estimation for output-only nonlinear system and damage identi- fication of civil structures, Structural Control and Health Monitoring 25 (4) (2018) e2128

    work page 2018

  28. [28]

    Caglio, H

    L. Caglio, H. Stang, E. Katsanos, Bayesian model updating and nonlinear response estimation of structures subjected to unknown loads via FE-aided Kalman Filtering, Mechanical Systems and Signal Processing 241 (2025) 113505

    work page 2025

  29. [29]

    S ¨arkk¨a, A

    S. S ¨arkk¨a, A. Solin, Applied stochastic differential equations, vol. 10, Cambridge University Press, 2019

    work page 2019

  30. [30]

    Burrage, I

    K. Burrage, I. Lenane, G. Lythe, Numerical methods for second-order stochastic differential equations, SIAM Journal on Scientific Computing 29 (1) (2007) 245–264

    work page 2007

  31. [31]

    Mannella, Quasisymplectic integrators for stochastic differential equations, Physical Review E 69 (4)

    R. Mannella, Quasisymplectic integrators for stochastic differential equations, Physical Review E 69 (4)

    work page

  32. [32]

    M. C. Kennedy, A. O’Hagan, Bayesian calibration of computer models, Journal of the Royal Statistical Society: Series B (Statistical Method- ology) 63 (2001) 425–464

    work page 2001

  33. [33]

    Jiang, Z

    C. Jiang, Z. Hu, Y . Liu, Z. P. Mourelatos, D. Gorsich, P. Jayakumar, A sequential calibration and validation framework for model uncertainty quantification and reduction, Computer Methods in Applied Mechanics and Engineering 368 (2020) 113172

    work page 2020

  34. [34]

    N ´ovoa, A

    A. N ´ovoa, A. Racca, L. Magri, Inferring unknown unknowns: Regularized bias-aware ensemble Kalman filter, Computer Methods in Applied Mechanics and Engineering 418 (2024) 116502

    work page 2024

  35. [35]

    Lindgren, H

    F. Lindgren, H. Rue, J. Lindstr ¨om, An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach, Journal of the Royal Statistical Society Series B: Statistical Methodology 73 (4) (2011) 423–498

    work page 2011

  36. [36]

    Zhang, J

    H. Zhang, J. Guilleminot, L. J. Gomez, Stochastic modeling of geometrical uncertainties on complex domains, with application to additive manufacturing and brain interface geometries, Computer Methods in Applied Mechanics and Engineering 385 (2021) 114014:1–114014:22

    work page 2021

  37. [37]

    Lindgren, D

    F. Lindgren, D. Bolin, H. Rue, The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running, Spatial Statistics (2022) 100599:1–100599:28

    work page 2022

  38. [38]

    R. G. Ghanem, P. D. Spanos, Stochastic finite elements: a spectral approach, Courier Corporation, 2003

    work page 2003

  39. [39]

    Xiu, Numerical methods for stochastic computations: a spectral method approach, Princeton University Press, 2010

    D. Xiu, Numerical methods for stochastic computations: a spectral method approach, Princeton University Press, 2010

    work page 2010

  40. [40]

    Sudret, A

    B. Sudret, A. Der Kiureghian, Stochastic finite element methods and reliability: a state-of-the-art report, Tech. Rep. UCB/SEMM-2000/08, Department of Civil & Environmental Engineering, University of California, Berkeley, 2000

    work page 2000

  41. [41]

    H. G. Matthies, A. Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, Computer Methods in Applied Mechanics and Engineering 194 (2005) 1295–1331

    work page 2005

  42. [42]

    W. K. Liu, T. Belytschko, A. Mani, Random field finite elements, International Journal for Numerical methods in Engineering 23 (10) (1986) 1831–1845

    work page 1986

  43. [43]

    S ¨arkk¨a, A

    S. S ¨arkk¨a, A. Solin, On continuous-discrete cubature Kalman filtering, IFAC Proceedings V olumes 45 (16) (2012) 1221–1226

    work page 2012

  44. [44]

    D. J. MacKay, Comparison of approximate methods for handling hyperparameters, Neural computation 11 (5) (1999) 1035–1068

    work page 1999

  45. [45]

    Sudret, Polynomial chaos expansions and stochastic finite element methods, Taylor & Francis, 2015

    B. Sudret, Polynomial chaos expansions and stochastic finite element methods, Taylor & Francis, 2015

    work page 2015

  46. [46]

    Arnst, R

    M. Arnst, R. Ghanem, C. Soize, Identification of Bayesian posteriors for coefficients of chaos expansions, Journal of Computational Physics 229 (9) (2010) 3134–3154

    work page 2010

  47. [47]

    C. V . Mai, M. D. Spiridonakos, E. N. Chatzi, B. Sudret, Surrogate modelling for stochastic dynamical systems by combining NARX models and polynomial chaos expansions, 2016

    work page 2016

  48. [48]

    Lacour, G

    M. Lacour, G. Bal, N. Abrahamson, Dynamic stochastic finite element method using time-dependent generalized polynomial chaos, Interna- tional Journal for Numerical and Analytical Methods in Geomechanics 45 (3) (2021) 293–306, ISSN 1096-9853

    work page 2021

  49. [49]

    Bernard, G

    P. Bernard, G. Fleury, Stochastic Newmark scheme, Probabilistic Engineering Mechanics 17 (1) (2002) 45–61

    work page 2002

  50. [50]

    D. Roy, M. K. Dash, Explorations of a family of stochastic Newmark methods in engineering dynamics, Computer Methods in Applied Mechanics and Engineering 194 (45) (2005) 4758–4796

    work page 2005

  51. [51]

    Sarkka, A

    S. Sarkka, A. Solin, J. Hartikainen, Spatiotemporal learning via infinite-dimensional Bayesian filtering and smoothing: a look at Gaussian process regression through Kalman filtering, IEEE Signal Processing Magazine 30 (4) (2013) 51–61

    work page 2013

  52. [52]

    Weiland, M

    T. Weiland, M. Pf ¨ortner, P. Hennig, Flexible and efficient probabilistic PDE solvers through Gaussian Markov random fields, 2025

    work page 2025

  53. [53]

    Tondo, I

    G. Tondo, I. Kavrakov, G. Morgenthal, Efficient dynamic modal load reconstruction using physics-informed Gaussian processes based on frequency-sparse Fourier basis functions, Mechanical Systems and Signal Processing 225 (2025) 112295

    work page 2025

  54. [54]

    S. Chu, J. Guilleminot, C. Kelly, B. Abar, K. Gall, Stochastic modeling and identification of material parameters on structures produced by additive manufacturing, Computer Methods in Applied Mechanics and Engineering 387 (2021) 114166

    work page 2021

  55. [55]

    Ben-Yelun, A

    I. Ben-Yelun, A. O. Yuksel, F. Cirak, Robust topology optimisation of lattice structures with spatially correlated uncertainties, Structural and Multidisciplinary Optimization 67 (2) (2024) 16

    work page 2024

  56. [56]

    Torquato, Random heterogeneous materials: microstructure and macroscopic properties, vol

    S. Torquato, Random heterogeneous materials: microstructure and macroscopic properties, vol. 16, Springer, 2002

    work page 2002

  57. [57]

    Savvas, G

    D. Savvas, G. Stefanou, M. Papadrakakis, Determination of RVE size for random composites with local volume fraction variation, Computer Methods in Applied Mechanics and Engineering 305 (2016) 340–358

    work page 2016

  58. [58]

    Y . S. Jetti, M. Ostoja-Starzewski, Correlation structures of statistically isotropic stiffness and compliance TRFs through upscaling, Computer 27 Methods in Applied Mechanics and Engineering 432 (2024) 117356

    work page 2024

  59. [59]

    Joshi, P

    A. Joshi, P. Thakolkaran, Y . Zheng, M. Escande, M. Flaschel, L. De Lorenzis, S. Kumar, Bayesian-EUCLID: Discovering hyperelastic material laws with uncertainties, Computer Methods in Applied Mechanics and Engineering 398 (2022) 115225

    work page 2022

  60. [60]

    Sarkka, A

    S. Sarkka, A. Nummenmaa, Recursive noise adaptive Kalman filtering by variational Bayesian approximations, IEEE Transactions on Auto- matic Control 54 (3) (2009) 596–600

    work page 2009

  61. [61]

    Course, P

    K. Course, P. B. Nair, State estimation of a physical system with unknown governing equations., Nature 622 (2023) pages261–267

    work page 2023

  62. [62]

    M. D. Hoffman, D. D. Blei, C. Wang, J. Paisley, Stochastic variational inference, Journal of Machine Learning Research 14 (2013) 1303–1347

    work page 2013

  63. [63]

    K. B. Petersen, M. S. Pedersen, The matrix cookbook, Technical University of Denmark 7 (15) (2008) 510

    work page 2008

  64. [64]

    Grønbech-Jensen, O

    N. Grønbech-Jensen, O. Farago, A simple and effective Verlet-type algorithm for simulating Langevin dynamics, Molecular Physics 111 (8) (2013) 983–991

    work page 2013

  65. [65]

    Melchionna, Design of quasisymplectic propagators for Langevin dynamics, The Journal of Chemical Physics 127 (4) (2007) 044108

    S. Melchionna, Design of quasisymplectic propagators for Langevin dynamics, The Journal of Chemical Physics 127 (4) (2007) 044108

    work page 2007

  66. [66]

    Grønbech-Jensen, Complete set of stochastic Verlet-type thermostats for correct Langevin simulations, Molecular Physics 118 (8) (2020) e1662506, ISSN 0026-8976

    N. Grønbech-Jensen, Complete set of stochastic Verlet-type thermostats for correct Langevin simulations, Molecular Physics 118 (8) (2020) e1662506, ISSN 0026-8976. 28

    work page 2020