pith. sign in

arxiv: 2604.20058 · v1 · submitted 2026-04-21 · 🧮 math.AP

A critical note on back-and-forth Data Assimilation Nudging Algorithm

Aseel Farhat , Edriss S. Titi , Collin Victor This is my paper

Pith reviewed 2026-05-10 01:20 UTC · model grok-4.3

classification 🧮 math.AP
keywords data assimilationback-and-forth nudgingdissipative systemssparse observationsinitial condition recoveryLorenz modelBurgers equationNavier-Stokes equations
0
0 comments X

The pith

Back-and-forth nudging cannot reliably recover initial conditions from sparse observations in dissipative systems.

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

The paper demonstrates that the Back-and-Forth Nudging algorithm, when combined with continuous data assimilation nudging, fails to uniquely recover initial conditions for several dissipative dynamical systems. Infinitely many distinct solutions can generate identical spatially sparse observational data, rendering the observations insufficient to identify the correct initial state. This non-uniqueness is constructed explicitly for the Lorenz 1963 model and one-dimensional equations including the heat equation, viscous linear transport, and both viscous and inviscid Burgers equations. Numerical simulations confirm the analytical findings. A Voigt-regularized variant of the algorithm is proposed to stabilize backward iterations, yet it still cannot recover unobserved fine spatial scales.

Core claim

For several dissipative systems, infinitely many distinct solutions share identical spatially sparse observational data. The BFN algorithm, which relies on nudging based on these observations, therefore cannot recover the true initial condition or differentiate between these solutions. This is demonstrated analytically for the Lorenz 1963 model, the heat equation, viscous linear transport, and viscous and inviscid Burgers equations, with supporting numerical simulations. A Voigt-regularized BFN is introduced to address ill-posedness in backward iterations for the 2D Navier-Stokes equations and viscous KdV equation, yet it remains limited in reconstructing the full state from sparse data.

What carries the argument

The explicit construction of infinitely many distinct solutions that produce identical spatially sparse observational data, which prevents any observation-dependent nudging scheme from enforcing uniqueness of the initial condition.

Load-bearing premise

Spatially sparse observational data is sufficient to uniquely determine the solution trajectory among different possible initial conditions in the dissipative systems considered.

What would settle it

Numerical simulation of the Lorenz model or Burgers equation showing whether two different initial conditions that are constructed to match at the sparse observation locations remain indistinguishable in their observed values while diverging in unobserved regions.

Figures

Figures reproduced from arXiv: 2604.20058 by Aseel Farhat, Collin Victor, Edriss S. Titi.

Figure 1
Figure 1. Figure 1: BFN error behavior for the Lorenz 1963 system. step is sufficiently small; notably, a smaller time step required a larger nudging parameter, which may increase the overall error. We now consider the case where components of the full true solution u(t) of Equa￾tion (2.1) is observed on different sub-intervals of the time interval [0, T], an experimental setup similar to that in [40]. We include this case to… view at source ↗
Figure 2
Figure 2. Figure 2: BFN error for the Lorenz 1963 system. iterations or fails entirely for certain values of γobs, as shown in Figure 2b. Specifically, for γobs ∈ [0, 1 2 ], the BFN algorithm Equations (2.4) and (2.5)fails to recover the initial condition or provide significant improvement over the initial guess ˜v 0 0 . 3. 1D PDEs Paradigm In this section, we outline specific counterexamples that demonstrate the inefficacy o… view at source ↗
Figure 3
Figure 3. Figure 3: L 2 per error with zero observations. (a) Inviscid Burgers equation (b) Viscous Burgers equation [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: L 2 per error with zero observations. The simulation time (T = 0.0001) is sufficiently short to ensure that the dynamics of the Burgers equation Equation (3.22) do not excite wavenumbers beyond M = 16. Thus, the reference solution u(x, t) is fully observed at every time step. The results are presented in Figures 9 to 14. As expected, the BFN algorithm suc￾cessfully recovers the initial condition, consisten… view at source ↗
Figure 5
Figure 5. Figure 5: L 2 per energy spectrum of the recovered initial condition using zero observations. (a) Inviscid Burgers equation (b) Viscous Burgers equation [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: L 2 per energy spectrum of the recovered initial condition using zero observations. (a) Inviscid Linear Transport equation (b) Viscous Linear Transport equation [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Recovered initial condition with zero observations. (a) Inviscid Burgers equation (b) Viscous Burgers equation [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Recovered initial condition with zero observations. to stabilize and regularize the backward evolution. In this section, we investigate two regularization variants: the established Diffusive BFN (dBFN) [7] and the newly pro￾posed Voigt-regularized BFN (VBFN). We apply these methods to genuinely multiscale [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: L 2 per error using full observations. (a) Inviscid Burgers equation (b) Viscous Burgers equation [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: L 2 per error using full observations. (a) Inviscid Linear Transport equation (b) Viscous Linear Transport equation [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: L 2 per energy spectrum of the recovered initial condition with full observations. (a) Inviscid Burgers equation (b) Viscous Burgers equation [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: L 2 per energy spectrum of the recovered initial condition using full observations. dynamics—specifically the 2D Navier-Stokes equations (NSE) and 1D viscous Korteweg￾de Vries (KdV) equations—to determine the extent to which regularization can mitigate the ill-posedness inherent in the backward step [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Recovered initial condition using full observations. (a) Inviscid Burgers equation (b) Viscous Burgers equation [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Recovered initial condition using full observations. (a) Inviscid Linear Transport equation (b) Viscous Linear Transport equation [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: L 2 per error with low-mode observations (|k| ≤ M). (a) Inviscid Burgers equation (b) Viscous Burgers equation [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: L 2 per error with low-mode observations (|k| ≤ M). 4.1. dBFN and VBFN for the 2D NSE. We consider the 2D NSE given by Equa￾tion (1.4) on the periodic domain Ω = [−π, π] 2 , subject to periodic boundary conditions. We will consider the case where the prescribed body force, f, is presumed to be constant in time. Applying the BFN algorithm to the 2D NSE presents significant challenges due to the ill-posed n… view at source ↗
Figure 17
Figure 17. Figure 17: L 2 per energy spectrum of the recovered initial condition using low-mode observations. (a) Inviscid Burgers equation (b) Viscous Burgers equation [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: L 2 per energy spectrum of the recovered initial condition using low-mode observations. (a) Inviscid Linear Transport equation (b) Viscous Linear Transport equation [PITH_FULL_IMAGE:figures/full_fig_p023_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Recovered initial condition using low-mode observations. (a) Inviscid Burgers equation (b) Viscous Burgers equation [PITH_FULL_IMAGE:figures/full_fig_p023_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Recovered initial condition using low-mode observations. of time renders the viscous term anti-diffusive, causing errors in high-frequency modes to grow exponentially and often resulting in numerical instability. To address this, we compare three variations of the backward nudging step: the Standard BFN, the Diffusive BFN (dBFN), and our newly proposed Voigt-regularized BFN (VBFN). The VBFN is [PITH_FULL… view at source ↗
Figure 21
Figure 21. Figure 21: Energy spectrum of the initial data. The vertical red line is the 2/3’s dealiasing cutoff at 2 3 N 2 = 341.3 [PITH_FULL_IMAGE:figures/full_fig_p026_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: L 2 error evolution for 2D NSE. The model discrepancy is further highlighted in the limiting case where the entire solution is observed. As seen in [PITH_FULL_IMAGE:figures/full_fig_p026_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Enstrophy error evolution for 2D NSE. For observations [PITH_FULL_IMAGE:figures/full_fig_p027_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: L 2 and Enstrophy error evolution for 2D NSE. The entire solu￾tion was observed [PITH_FULL_IMAGE:figures/full_fig_p027_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: L 2 error evolution for 2D NSE [PITH_FULL_IMAGE:figures/full_fig_p027_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Enstrophy error evolution for 2D NSE [PITH_FULL_IMAGE:figures/full_fig_p027_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Damped/forced KdV: Error evolution for four backward reconstructions. error stems from the fact that the backward model is no longer physically consistent with the forward dynamics; the stabilization terms introduce a model discrepancy (bias) that nudging cannot fully rectify. Notably, the Voigt-regularized BFN (Figure 27d) achieves a similar stabilizing effect but produces a qualitatively different error… view at source ↗
Figure 28
Figure 28. Figure 28: Viscous/forced KdV: Error evolution for three backward recon￾structions. 5. Conclusion In this work, we critically examined the efficacy of the back-and-forth nudging (BFN) algorithm across a range of dynamical systems, including the Lorenz 1963 system and various 1D and 2D partial differential equations (PDEs). Our investigation focused on the algorithm’s ability to reconstruct initial states from sparse… view at source ↗
read the original abstract

This work investigates the effectiveness of the Back-and-Forth Nudging (BFN) data assimilation algorithm, specifically its performance when employing the Azouani-Olson-Titi (AOT) continuous data assimilation downscaling nudging algorithm, for recovering initial conditions of dissipative dynamical systems. Contrary to previous reports in the literature, we show that, for several systems of interest, one can construct initial conditions that BFN cannot reliably recover. Our key finding is the construction of infinitely many distinct solutions for certain dissipative systems that share identical spatially sparse observational data. Since these observations are indistinguishable, no data assimilation method relying only on them can differentiate between these solutions or recover the correct initial condition. We illustrate these pathological initial conditions for the Lorenz 1963 model and several 1D partial differential equations: the heat equation, viscous linear transport, and viscous and inviscid Burgers equations. Our analytical results are supported by numerical simulations. To address the numerical ill-posedness of backward-in-time iterations, an essential step of BFN for dissipative models, we introduce a regularized backward step, the Voigt-regularized BFN (VBFN). We investigate its performance for the 2D Navier-Stokes equations and the viscous 1D KdV equation, comparing it with standard BFN and Diffusive BFN (dBFN). While VBFN improves numerical stability and reduces model bias relative to dBFN, it still cannot reconstruct the unobserved fine spatial scales of the reference solution. This reinforces our main conclusion: even with regularization, BFN-type algorithms are limited in recovering the full state, and in particular the initial data, from sparse spatial observations.

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

3 major / 2 minor

Summary. The paper claims that Back-and-Forth Nudging (BFN) and its variants cannot reliably recover initial conditions for dissipative systems (Lorenz 1963, heat, viscous transport, viscous/inviscid Burgers) under spatially sparse continuous-in-time observations, because one can construct infinitely many distinct solutions sharing identical data; the nudging term vanishes on this family since it depends only on pointwise differences at observed locations. Analytical constructions are asserted for each system and supported by numerical simulations. To mitigate backward-step ill-posedness, the authors introduce Voigt-regularized BFN (VBFN) and compare it with standard BFN and diffusive BFN on 2D Navier-Stokes and 1D viscous KdV, concluding that regularization improves stability but does not recover unobserved fine scales.

Significance. If the non-uniqueness constructions are made fully explicit, the work identifies a concrete limitation of nudging-based DA: the observation operator has a non-trivial kernel for standard sparse point measurements, so any method using only those data cannot select a unique initial condition. The direct kernel argument (superposition of null-space components) is a strength, as is the introduction of VBFN for numerical stability. This could prompt re-examination of observation density requirements in DA literature and encourage hybrid schemes that add constraints beyond pointwise nudging.

major comments (3)
  1. [Analytical constructions (heat equation)] § on heat-equation construction: the assertion that higher eigenmodes (odd with respect to interior observation points) lie in the kernel and can be superposed while preserving observations is central, yet no explicit non-trivial example is supplied (e.g., a concrete eigenfunction, observation point x0, and verification that the solution satisfies the PDE and matches data for all t). Without this, the claim that the nudging term vanishes identically remains difficult to verify.
  2. [Numerical results] Numerical simulations section: the manuscript states that simulations support the analytical non-uniqueness, but provides neither the precise definition of the discrete observation operator, the time-stepping scheme, nor quantitative error metrics (L2 or pointwise differences between recovered and reference trajectories) in the figures or tables. This weakens the evidential link between analysis and numerics.
  3. [VBFN experiments] VBFN comparison for 2D Navier-Stokes: the conclusion that VBFN 'still cannot reconstruct the unobserved fine spatial scales' is load-bearing for the broader claim, yet no spectral decomposition, energy spectrum plot, or explicit comparison of unresolved modes versus the reference solution is given to quantify the residual error.
minor comments (2)
  1. [Introduction / Preliminaries] The observation operator is used throughout but never given a compact mathematical definition (e.g., as a sum of Dirac deltas or interpolation operator) in the preliminaries; this notation should be fixed early for clarity.
  2. [Introduction] Several references to prior BFN and AOT papers are cited, but the manuscript would benefit from a short table contrasting the present kernel argument with earlier uniqueness results under denser observations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight opportunities to strengthen the explicitness of our constructions and the quantitative support for our numerical claims. We address each major comment below and will incorporate the suggested clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Analytical constructions (heat equation)] § on heat-equation construction: the assertion that higher eigenmodes (odd with respect to interior observation points) lie in the kernel and can be superposed while preserving observations is central, yet no explicit non-trivial example is supplied (e.g., a concrete eigenfunction, observation point x0, and verification that the solution satisfies the PDE and matches data for all t). Without this, the claim that the nudging term vanishes identically remains difficult to verify.

    Authors: We agree that an explicit, verifiable example would improve clarity. In the revision we will insert a concrete construction for the heat equation on [0,1] with a single interior observation at x0=1/2. The eigenfunctions sin(2kπx) (k=1,2,…) vanish identically at x=1/2, satisfy the homogeneous heat equation, and therefore produce a zero contribution to the nudging term for any choice of coefficients. Superposition with a reference solution yields an infinite family of distinct solutions that share identical observations at x0 for all t while satisfying the PDE. We will explicitly verify the PDE satisfaction and the vanishing of the observation mismatch. revision: yes

  2. Referee: [Numerical results] Numerical simulations section: the manuscript states that simulations support the analytical non-uniqueness, but provides neither the precise definition of the discrete observation operator, the time-stepping scheme, nor quantitative error metrics (L2 or pointwise differences between recovered and reference trajectories) in the figures or tables. This weakens the evidential link between analysis and numerics.

    Authors: We accept that the numerical section requires additional implementation details to make the link with the analysis fully transparent. The revised manuscript will specify the discrete observation operator (direct sampling at the observation grid points), the time-stepping method (including order and stability constraints), and will augment the figures with quantitative metrics such as L2-norm differences and pointwise errors between the recovered and reference trajectories over the assimilation window. revision: yes

  3. Referee: [VBFN experiments] VBFN comparison for 2D Navier-Stokes: the conclusion that VBFN 'still cannot reconstruct the unobserved fine spatial scales' is load-bearing for the broader claim, yet no spectral decomposition, energy spectrum plot, or explicit comparison of unresolved modes versus the reference solution is given to quantify the residual error.

    Authors: We agree that a quantitative demonstration of the unresolved scales would strengthen the claim. In the revision we will add an energy-spectrum comparison (log-log plot of kinetic energy versus wavenumber) between the VBFN reconstruction and the reference solution, together with a decomposition of the residual energy into resolved and unresolved modes. This will explicitly document the persistent gap at high wavenumbers. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claim rests on explicit constructions of distinct solutions to the heat equation, transport equation, Burgers equations, and Lorenz system that agree on the values of a spatially sparse observation operator for all time. These constructions follow directly from the kernel of the observation map (e.g., higher eigenmodes vanishing at interior points) and the fact that the nudging term depends only on pointwise differences at observed locations; the argument invokes no fitted parameters, no self-referential definitions, and no load-bearing citations to prior work by the same authors. The introduction of VBFN regularization is presented as an auxiliary numerical fix whose limitations are then verified by the same kernel argument, without circular reduction. The derivation chain is therefore self-contained and externally falsifiable by direct substitution into the PDEs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard mathematical properties of the listed dissipative systems (existence of solutions, dissipativity) and the definition of sparse observations; no free parameters, new entities, or ad-hoc axioms are introduced beyond those implicit in the prior data-assimilation literature.

axioms (1)
  • domain assumption The systems under study are dissipative and admit solutions that remain indistinguishable under spatially sparse observations.
    This property is invoked to construct the pathological initial conditions for Lorenz, heat, transport, and Burgers equations.

pith-pipeline@v0.9.0 · 5611 in / 1328 out tokens · 81389 ms · 2026-05-10T01:20:44.191120+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    Samira Amraoui, Didier Auroux, Jacques Blum, and Emmanuel Cosme. Back-and-forth nudging for the quasi-geostrophic ocean dynamics with altimetry: Theoretical convergence study and numerical experiments with the future SWOT observations.Discrete & Continuous Dynamical Systems-Series S, 16(2), 2023

    work page 2023

  2. [2]

    PhD thesis, Nice, 2003

    Didier Auroux.Etude de diff´ erentes m´ ethodes d’assimilation de donn´ ees pour l’environnement. PhD thesis, Nice, 2003

    work page 2003

  3. [3]

    Didier Auroux. The back and forth nudging algorithm applied to a shallow water model, compar- ison and hybridization with the 4D-VAR.International Journal for Numerical Methods in Fluids, 61(8):911–929, 2009

    work page 2009

  4. [4]

    Didier Auroux, Patrick Bansart, and Jacques Blum. An evolution of the back and forth nudging for geophysical data assimilation: application to Burgers equation and comparisons.Inverse Problems in Science and Engineering, 21(3):399–419, 2013

    work page 2013

  5. [5]

    Back and forth nudging algorithm for data assimilation problems

    Didier Auroux and Jacques Blum. Back and forth nudging algorithm for data assimilation problems. Comptes Rendus. Math´ ematique, 340(12):873–878, 2005

    work page 2005

  6. [6]

    A nudging-based data assimilation method: the Back and Forth Nudging (BFN) algorithm.Nonlinear Processes in Geophysics, 15(2):305–319, 2008

    Didier Auroux and Jacques Blum. A nudging-based data assimilation method: the Back and Forth Nudging (BFN) algorithm.Nonlinear Processes in Geophysics, 15(2):305–319, 2008

    work page 2008

  7. [7]

    Data assimilation for geophysical fluids: the diffusive back and forth nudging

    Didier Auroux, Jacques Blum, and Giovanni Ruggiero. Data assimilation for geophysical fluids: the diffusive back and forth nudging. InMathematical Paradigms of Climate Science, pages 139–174. Springer, 2016

    work page 2016

  8. [8]

    Didier Auroux and Ma¨ elle Nodet. The back and forth nudging algorithm for data assimilation prob- lems: theoretical results on transport equations.ESAIM: Control, Optimisation and Calculus of Variations, 18(2):318–342, 2012

    work page 2012

  9. [9]

    Abderrahim Azouani, Eric Olson, and Edriss S. Titi. Continuous data assimilation using general interpolant observables.J. Nonlinear Sci., 24(2):277–304, 2014. 33

    work page 2014

  10. [10]

    Abderrahim Azouani and Edriss S. Titi. Feedback control of nonlinear dissipative systems by finite determining parameters—a reaction-diffusion paradigm.Evol. Equ. Control Theory, 3(4):579–594, 2014

    work page 2014

  11. [11]

    Brooke Benjamin, Jerry L

    T. Brooke Benjamin, Jerry L. Bona, and John J. Mahony. Model equations for long waves in non- linear dispersive systems.Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 272(1220):47–78, 1972

    work page 1972

  12. [12]

    Convergence of a mobile data assimila- tion scheme for the 2D Navier-Stokes equations.Discrete and Continuous Dynamical Systems, 43(11):4042–4068, 2023

    Animikh Biswas, Zachary Bradshaw, and Michael Jolly. Convergence of a mobile data assimila- tion scheme for the 2D Navier-Stokes equations.Discrete and Continuous Dynamical Systems, 43(11):4042–4068, 2023

    work page 2023

  13. [13]

    Data assimilation using noisy time-averaged measurements.Physica D: Nonlinear Phenomena, 376:49–59, 2018

    Jordan Blocher, Vincent R Martinez, and Eric Olson. Data assimilation using noisy time-averaged measurements.Physica D: Nonlinear Phenomena, 376:49–59, 2018

    work page 2018

  14. [14]

    Canuto, M

    C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang.Spectral Methods. Scientific Computation. Springer-Verlag, Berlin, 2006. Fundamentals in single domains

    work page 2006

  15. [15]

    Yanping Cao, Evelyn M Lunasin, and Edriss S. Titi. Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models.Commun. Math. Sci., 4(1):823–848, 2006

    work page 2006

  16. [16]

    Martinez, and Collin Victor

    Elizabeth Carlson, Aseel Farhat, Vincent R. Martinez, and Collin Victor. On the nudging approach to continuous data assimilation in the limit of infinite error feedback gain.SIAM Journal on Control and Optimization, 2026. (Accepted)

    work page 2026

  17. [17]

    Jones, and Edriss S

    Bernardo Cockburn, Don A. Jones, and Edriss S. Titi. Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems.Mathematics of Computation, 66(219):1073–1087, 1997

    work page 1997

  18. [18]

    Yi Juan Du and Ming-Cheng Shiue. Analysis and computation of continuous data assimilation algorithms for Lorenz 63 system based on nonlinear nudging techniques.Journal of Computational and Applied Mathematics, 386:113246, 2021

    work page 2021

  19. [19]

    Cambridge University Press, Cambridge, 2001

    Ciprian Foias, Oscar Manley, Ricardo Rosa, and Roger Temam.Navier-Stokes Equations and Turbu- lence, volume 83 ofEncyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2001

    work page 2001

  20. [20]

    Ciprian Foia¸ s and Giovanni Prodi. Sur le comportement global des solutions non-stationnaires des ´ equations de Navier-Stokes en dimension 2.Rendiconti del Seminario Matematico della Universit` a di Padova, 39:1–34, 1967

    work page 1967

  21. [21]

    Determination of the solutions of the Navier-Stokes equations by a set of nodal values.Mathematics of Computation, 43(167):117–133, 1984

    Ciprian Foia¸ s and Roger Temam. Determination of the solutions of the Navier-Stokes equations by a set of nodal values.Mathematics of Computation, 43(167):117–133, 1984

    work page 1984

  22. [22]

    Ciprian Foia¸ s and Edriss S. Titi. Determining nodes, finite difference schemes and inertial manifolds. Nonlinearity, 4(1):135–153, 1991

    work page 1991

  23. [23]

    Masakazu Gesho, Eric Olson, and Edriss S. Titi. A computational study of a data assimilation algorithm for the two-dimensional Navier–Stokes equations.Commun. Comput. Phys., 19(4):1094– 1110, 2016

    work page 2016

  24. [24]

    PhD thesis, Masters Thesis, University of Nevada, Department of Mathematics and Statistics, 2007

    Kevin Hayden.Synchronization in the Lorenz System. PhD thesis, Masters Thesis, University of Nevada, Department of Mathematics and Statistics, 2007

    work page 2007

  25. [25]

    Kevin Hayden, Eric Olson, and Edriss S. Titi. Discrete data assimilation in the Lorenz and 2D Navier–Stokes equations.Phys. D, 240(18):1416–1425, 2011

    work page 2011

  26. [26]

    The initialization of numerical models by a dynamic- initialization technique.Monthly Weather Review, 104(12):1551–1556, 1976

    James E Hoke and Richard A Anthes. The initialization of numerical models by a dynamic- initialization technique.Monthly Weather Review, 104(12):1551–1556, 1976

    work page 1976

  27. [27]

    Holst and Edriss S

    Michael J. Holst and Edriss S. Titi. Determining projections and functionals for weak solutions of the Navier-Stokes equations. InThe Curth Edwards Memorial Volume, volume 204 ofContemporary Mathematics, pages 125–138. American Mathematical Society, 1997

    work page 1997

  28. [28]

    Jolly, Tural Sadigov, and Edriss S

    Michael S. Jolly, Tural Sadigov, and Edriss S. Titi. Determining form and data assimilation algo- rithm for weakly damped and driven Korteweg–de Vries equation—Fourier modes case.Nonlinear Analysis: Real World Applications, 36:287–317, 2017

    work page 2017

  29. [29]

    Jones and Edriss S

    Don A. Jones and Edriss S. Titi. Determining finite volume elements for the 2D Navier-Stokes equations.Physica D: Nonlinear Phenomena, 60(1–4):165–174, 1992

    work page 1992

  30. [30]

    Jones and Edriss S

    Don A. Jones and Edriss S. Titi. Upper bounds on the number of determining modes, nodes and vol- ume elements for the Navier-Stokes equations.Indiana University Mathematics Journal, 42(3):875– 887, 1993

    work page 1993

  31. [31]

    Kalantarov and Edriss S

    Varga K. Kalantarov and Edriss S. Titi. Global attractors and determining modes for the 3D Navier- Stokes-Voigt equations.Chinese Annals of Mathematics, Series B, 30(6):697–714, 2009

    work page 2009

  32. [32]

    Trefethen

    Aly-Khan Kassam and Lloyd N. Trefethen. Fourth-order time-stepping for stiff PDEs.SIAM J. Sci. Comput., 26(4):1214–1233, 2005

    work page 2005

  33. [33]

    Analysis of the 3DVAR filter for the partially observed Lorenz ’63 model, 2014

    Kody Law, Abhishek Shukla, and Andrew Stuart. Analysis of the 3DVAR filter for the partially observed Lorenz ’63 model, 2014

    work page 2014

  34. [34]

    Boris Levant, F´ abio Ramos, and Edriss S. Titi. On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model.Communications in Mathematical Sciences, 8(1):277–293, 2010

    work page 2010

  35. [35]

    Deterministic nonperiodic flow.Journal of atmospheric sciences, 20(2):130–141, 1963

    Edward N Lorenz. Deterministic nonperiodic flow.Journal of atmospheric sciences, 20(2):130–141, 1963. 34

    work page 1963

  36. [36]

    Symmetry analysis and conservation laws of a general- ization of the Kelvin-Voigt viscoelasticity equation.Symmetry, 11(7):840, 2019

    Almudena P M´ arquez and Mar´ ıa S Bruz´ on. Symmetry analysis and conservation laws of a general- ization of the Kelvin-Voigt viscoelasticity equation.Symmetry, 11(7):840, 2019

    work page 2019

  37. [37]

    Eric Olson and Edriss S. Titi. Determining modes for continuous data assimilation in 2D turbulence. J. Statist. Phys., 113(5-6):799–840, 2003. Progress in statistical hydrodynamics (Santa Fe, NM, 2002)

    work page 2003

  38. [38]

    The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers.Zap

    Anatolii Petrovich Oskolkov. The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers.Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38:98–136, 1973

    work page 1973

  39. [39]

    Synchronization in chaotic systems.Physical review letters, 64(8):821, 1990

    Louis M Pecora and Thomas L Carroll. Synchronization in chaotic systems.Physical review letters, 64(8):821, 1990

    work page 1990

  40. [40]

    Yu Chen Peng, Liang Chun Wu, and Ming-Cheng Shiue. Finite time synchronization of the continu- ous/discrete data assimilation algorithms for Lorenz 63 system based on the back and forth nudging techniques.Results in Applied Mathematics, 20:100407, 2023

    work page 2023

  41. [41]

    Springer Science & Business Media, 2013

    Roger Peyret.Spectral Methods for Incompressible Viscous Flow, volume 148. Springer Science & Business Media, 2013

    work page 2013

  42. [42]

    Fabio Ramos and Edriss S. Titi. Invariant measures for the 3D Navier–Stokes-Voigt equations and their Navier–Stokes limit.Discrete Contin. Dyn. Syst., 28(1):375–403, 2010

    work page 2010

  43. [43]

    Springer, Heidelberg, 2011

    Jie Shen, Tao Tang, and Li-Lian Wang.Spectral Methods, volume 41 ofSpringer Series in Compu- tational Mathematics. Springer, Heidelberg, 2011. Algorithms, analysis and applications

    work page 2011

  44. [44]

    Springer-Verlag, New York, second edition, 1997

    Roger Temam.Infinite-Dimensional Dynamical Systems In Mechanics and Physics, volume 68 of Applied Mathematical Sciences. Springer-Verlag, New York, second edition, 1997

    work page 1997

  45. [45]

    Titi and Collin Victor

    Edriss S. Titi and Collin Victor. On the inadequacy of nudging data assimilation algorithms for non-dissipative systems.Discrete and Continuous Dynamical Systems - S, 2025

    work page 2025

  46. [46]

    Trefethen.Spectral Methods in MATLAB, volume 10 ofSoftware, Environments, and Tools

    Lloyd N. Trefethen.Spectral Methods in MATLAB, volume 10 ofSoftware, Environments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000

    work page 2000

  47. [47]

    The Lorenz attractor exists.Comptes Rendus de l’Acad´ emie des Sciences - Series I - Mathematics, 328(12):1197–1202, 1999

    Warwick Tucker. The Lorenz attractor exists.Comptes Rendus de l’Acad´ emie des Sciences - Series I - Mathematics, 328(12):1197–1202, 1999. (Aseel F arhat) Department of Mathematics, University of Virginia, Charlottesville, V A 22904, USA Email address:af7py@virginia.edu (Edriss S. Titi) Department of Applied Mathematics and Theoretical Physics, University...

    work page 1999