Pith. sign in

REVIEW 4 major objections 6 minor 38 references

On the Incorporation of Box-Constraints for Ensemble Kalman Inversion

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A projected ensemble Kalman inversion with variance inflation converges in mean square to the smoothed constrained optimum.

desk verdict The algorithmic idea is plausible and the numerics are informative, but the main convergence theorem has a load-bearing algebraic error and the advertised analysis is not established. read the letter →

arxiv 1908.00696 v2 pith:WDG7ISHE submitted 2019-08-02 math.NA cs.NAmath.OC

Neil K. Chada , Claudia Schillings , Simon Weissmann This is my paper · ORCID
classification math.NAcs.NAmath.OC MSC 37C1049M1565M3265N20
keywords box-constrainedoptimizationensembleKalmaninversionBayesianinverseproblemsconvergenceanalysisvarianceinflationprojectedgradientflowbarriermethods
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

Ensemble Kalman inversion (EKI) is a cheap, derivative-free workhorse for inverse problems, but its standard form ignores constraints such as physical bounds on parameters. The paper proposes a box-constrained variant: project the ensemble onto the box, smooth the projection by log-barrier terms, and inflate the empirical covariance by $\varepsilon I$. Its main analytical claim is that, for linear forward operators with $A^\top\Gamma^{-1}A$ positive definite, this smoothed projected EKI converges in mean square to the KKT point of the smoothed constrained least-squares problem; simple projection without inflation is shown not to have this descent property. This matters because it gives a complete convergence guarantee for a constrained EKI, with rates for ensemble collapse and residual decay, and numerical experiments on PDE inverse problems support the theory.

What carries the argument

The load-bearing object is the smoothed projected preconditioned gradient flow (3.13): $du^{(j)}_t/dt = -\iota D(u_t)\nabla\Phi(u^{(j)}_t) + \sum_i h_i(u)^{-1}\nabla h_i(u)$, with $D(u)=C(u)+\varepsilon I$. The empirical covariance $C(u)$ is the usual ensemble covariance; the sum over $h_i$ is the log-barrier term keeping particles inside the box; $\iota$ is the barrier weight. The crucial identity is that $\varepsilon I$ makes $D(u)$ diagonal on the active-set index set $I_+(u)$, so the projected step becomes a genuine descent step. The proof runs a Lyapunov argument with $V(u)=\frac1J\sum_j \frac12|u^{(j)}-u^*_\iota|^2$ and shows $\frac{d}{dt}V(u_t)<0$, yielding the mean-square convergence.

What would settle it

Compute the Lyapunov derivative along the smoothed flow (3.13) for the two-dimensional quadratic of Example 2.8, with $J=2$, one particle on the boundary $x_2=0$, small $\varepsilon>0$, and the first term in the proof of Theorem 3.4 evaluated using $C(u)$ with ensemble-mean deviations rather than deviations from $u^*_\iota$. If that first term is positive at any finite time, the claimed strict monotonicity of $V$ is contradicted; the same check can be run on the 15-observation linear elliptic example, where $A^\top\Gamma^{-1}A$ is semidefinite and the theorem's assumptions are not met.

Watch

Extended reading notes

Core claim

At the paper's center is the observation that the EKI update is a preconditioned gradient flow—preconditioned by the empirical covariance $C(u)$—and that projecting this flow onto a box does not automatically preserve descent. To fix this, the paper transforms the preconditioner so that it is diagonal on the active set of constraints; the same effect is achieved by additive variance inflation, replacing $C(u)$ by $C(u)+\varepsilon I$. The main theorem (Theorem 3.4) states that for the smoothed flow (3.13)—where log-barrier terms replace the hard projection—the ensemble-mean square distance to the KKT point $u^*_\iota$ of the smoothed barrier problem tends to zero as $t\to\infty$, under $A^\top\Gamma^{-1}A>0$, $\varepsilon>0$, and feasible initial data. The analysis also proves monotone decrease of the ensemble spread and, with time-decaying $\varepsilon(t)=1/(t^\alpha+R)$, rates $O(t^{-(1-\alpha)})$ for both spread and residual.

Load-bearing premise

The proof's load-bearing premise is that the ensemble's sample covariance can be expanded around the target optimum even though, by definition, it is expanded around the ensemble mean; these two centers match only in the limit, and the argument also assumes the matrix $A^\top\Gamma^{-1}A$ is positive definite, which the paper notes typically fails when there are far more unknowns than observations.

Editorial extensions

If this is right

  • For linear forward problems with $A^\top\Gamma^{-1}A$ positive definite, the smoothed projected EKI with variance inflation converges in mean square to the KKT point $u^*_\iota$ of the smoothed barrier problem (Theorem 3.4).
  • The ensemble spread is non-increasing in time; with decaying variance inflation $\varepsilon(t)=1/(t^\alpha+R)$ it decays as $O(t^{-(1-\alpha)})$, and the mean-square residual to $u^*_\iota$ decays at the same rate when $\alpha>1/2$ (Propositions 3.3, 3.6, Corollary 3.7).
  • Simple projection of the EKI update to the box does not guarantee descent; the variance-inflated version fixes this by making the preconditioner diagonal on the active set (Example 2.8, Remark 3.2).
  • The same transformed formulation applies to the ensemble square-root filter variant (ESRF), with the Lyapunov analysis adapted to include the ensemble mean (Remark 3.8).
  • For nonlinear forward problems, the paper proposes a Jacobian-based variance inflation approximation and demonstrates the improvement numerically, though the convergence theorems are linear-only (Section 4.2).

Reading between the lines

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

  • The paper's proof only covers $A^\top\Gamma^{-1}A$ positive definite, while the authors note this is atypical when there are far more unknowns than observations; a likely extension is a convergence analysis in observation space under semidefinite $A$, which the numerics suggest but the theorems do not establish.
  • The variance-inflation mechanism effectively adds $\varepsilon I$ to the empirical covariance, which parallels Levenberg–Marquardt-style regularization; this suggests the inflation schedule $\varepsilon(t)$ could be tuned adaptively, trading convergence rate against feasibility.
  • Because the smoothed KKT point $u^*_\iota$ approximates the true constrained minimizer only to $O(m/\iota)$, the paper's guarantee is for an approximate problem; a joint limit $\iota\to\infty$ with $t\to\infty$ would be needed for a theorem about the original box-constrained problem.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 manuscript proposes a box-constrained variant of ensemble Kalman inversion (EKI) by combining projection onto a box with variance inflation, motivated by the continuous-time preconditioned gradient-flow interpretation. It derives a continuous-time limit, replaces the discontinuous projected flow by a log-barrier smoothed flow, and claims a complete convergence analysis for linear forward problems: Proposition 3.3 bounds ensemble collapse, Theorem 3.4 proves convergence to the KKT point of the smoothed problem under A^T Γ^-1 A positive definite, and Corollaries 3.5 and 3.7 give residual convergence and rates. Numerical experiments on linear and nonlinear elliptic PDE inverse problems compare the projected EKI with and without variance inflation.

Significance. Conditional on correctness, the paper would be a useful contribution: it would provide the first convergence guarantee for a box-constrained EKI method, and the numerical comparison between naive projection and variance-inflated projection illustrates a genuine algorithmic insight. The authors appropriately connect the method to Bertsekas' projected-gradient framework and provide numerical evidence on PDE-based inverse problems. However, the central convergence theorem contains a load-bearing algebraic error in its Lyapunov proof, so the main theoretical claim is not established in the present form.

major comments (4)
  1. [§3.3.3, Theorem 3.4 proof] The first displayed identity in the proof of Theorem 3.4 is false. The preconditioner C(u) is defined in Section 2.1 with deviations u^(k)-ū from the ensemble mean, but the proof replaces this center by the target point u*_ι in both factors. With the correct center, the term is 1/J Σ_j,k ⟨u^(j)-u*_ι, u^(k)-ū⟩⟨u^(k)-ū, A^T Γ^-1 A (u^(j)-u*_ι)⟩, not the double sum with u^(k)-u*_ι in both slots. Since the centers differ, the quadratic form is not sign-controlled and can be positive; consequently the claimed inequality dV/dt < 0 does not follow.
  2. [§3.3.3, Theorem 3.4 proof] The proof also drops the affine part of the gradient. Let M = A^T Γ^-1 A and let u*_ls be the unconstrained least-squares minimizer. Because u*_ι is the KKT point of the smoothed barrier problem, ∇Φ(u*_ι) = M(u*_ι - u*_ls) is generally nonzero when constraints are active. Therefore ∇Φ(u^(j)) = M(u^(j) - u*_ι) + M(u*_ι - u*_ls), and the second term is silently discarded in the ε-term of the Lyapunov derivative. The contribution ε⟨u^(j)-u*_ι, M(u*_ι - u*_ls)⟩ is not sign-controlled, so the ε-term is not nonnegative as claimed.
  3. [§3.3.3, Proposition 3.3 and Proposition 3.6] The same type of algebraic issue affects the ensemble-collapse results. In Proposition 3.3 the first inner product is claimed to be 'straightforwardly' nonpositive, but C(u)M is not symmetric and the expression 1/J Σ_j,k ⟨e_j,e_k⟩⟨e_k,M e_j⟩ with e_j = u^(j)-ū is not nonnegative for general positive definite M. Proposition 3.6 and Corollary 3.7 reuse this Lyapunov argument, so the claimed rates O(t^{-(1-α)}) are not supported by the proof as written.
  4. [Assumptions versus numerical claims] All convergence theorems require A^T Γ^-1 A to be positive definite, a condition the authors themselves note in Remark 2.2 is typically false when n ≫ K. The numerical section includes low-dimensional observations for which A^T Γ^-1 A is only positive semidefinite, so the presented theorems do not cover those experiments; the phrase 'complete convergence analysis' therefore overstates the scope of the results.
minor comments (6)
  1. [§2.1, tensor product definition] The definition z1⊗z2(q) := ⟨z2,q⟩_{H2}·z2 appears to have a typo; the final factor should be z1, not z2.
  2. [§3.1, equation (3.2)] The Neumann expansion in equation (3.2) has all plus signs; the correct expansion of (h^{-1}Γ + C)^{-1} alternates in sign, starting with hΓ^{-1} - h^2 Γ^{-1} C Γ^{-1} + ... . The leading-order limit is unaffected, but the displayed series is incorrect.
  3. [§3.3.2, equation (3.12)] The last line of equation (3.12) repeats i = 1,...,m; it should read i = m+1,...,n.
  4. [§3.3.1, Remark 3.2] In the displayed computation of (C(u)+εI)_{ik}, the equality to 0 omits the εδ_{ik} contribution; only the off-diagonal part vanishes.
  5. [§3.3.3, Proposition 3.6 proof] The Grönwall-type estimate is misstated: from dV/dt ≤ -ε σ_min V one obtains V(t) ≤ V(0) exp(-σ_min ∫_0^t ε(s)ds), not the displayed integral inequality V(u0) ≥ ∫_0^t σ_min ε(s)ds V(ut).
  6. [General presentation] There are several typos, for example 'we we choose' in Section 4.1, and references [14] and [30] are listed as 'In preparation'; these should be updated or removed before any resubmission.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence analysis extends independent continuous-time EKI and projected-gradient results without re-using its own conclusions as inputs.

full rationale

The paper's central derivation is the smoothed, variance-inflated projected EKI flow (3.13) and the claimed mean-square convergence to the KKT point u*_ι of the barrier problem (2.13). The continuous-time limit of the unconstrained EKI is imported from Schillings & Stuart [32], a published, externally reviewed, parameter-free result, and the projected-gradient/barrier framework is taken from Bertsekas [6,7] and Boyd-Vandenberghe [11]; none of these are unverified self-citations that smuggle in the paper's conclusion. The variance-inflation preconditioner D(u)=C(u)+εI is proposed from data-assimilation practice and Remark 3.2 verifies its structural property; it is not fitted to the target theorem. No parameter is tuned to a subset of data and then renamed a prediction, and no quantity is defined in terms of the convergence claim. The numerical section benchmarks against truth and against KKT points computed with fmincon, i.e. external comparators. The proof of Theorem 3.4 does contain a genuine algebraic gap: the displayed equality centers the empirical covariance at u*_ι rather than at the ensemble mean and drops the affine term involving ∇Φ(u*_ι), so the negativity of dV/dt is not established. That is a correctness flaw in the derivation, not a circular dependency: the claim is not equivalent to an input by construction, and the self-citations present in the paper are contextual or independently supported. Therefore no circularity is found.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. Its theoretical claim rests on the strong observability assumption, the log-barrier smoothing of the projected flow, the choice of variance inflation parameters, and a Taylor approximation in the nonlinear experiments. The central proof error is not an axiom choice but an invalid algebraic step in the Lyapunov argument.

free parameters (4)
  • ε (variance inflation constant) = ε(t) = 1/(t^α + R), α = 0.75, R = 1 in numerics; ε > 0 in theory
    Introduced to restore descent in the projected preconditioned flow. Theorem 3.4 states any ε > 0 works, which is not established; the proof error concerns the covariance term, not the inflation itself.
  • α = 0.75
    Time-decay exponent for the inflation schedule in numerical experiments. Theory requires α ∈ (0,1), and Corollary 3.7 requires α ∈ (1/2,1).
  • R = 1
    Offset in the inflation schedule ε(t) = 1/(t^α + R), chosen by hand in the experiments.
  • ι (log-barrier parameter) = Not specified in numerics; taken large in theory
    Smoothing parameter for the barrier approximation. Theory proves convergence to u*_ι for fixed ι with accuracy O(1/ι), while the actual projected algorithm corresponds to the ι → ∞ limit, which is not directly analyzed.
assumptions (4)
  • domain assumption A^T Γ^-1 A is positive definite (strong observability).
    Required in Theorem 3.4 and Proposition 3.6. The paper notes in Remark 2.2 that this fails for typical underdetermined inverse problems with n ≫ K, so the central convergence result excludes the main application regime.
  • ad hoc to paper The discontinuous projected flow (3.4) can be replaced by the smoothed log-barrier flow (3.13), and convergence of the smoothed flow transfers to the algorithm.
    Section 3.1 and Remark 2.4. All theoretical results are for the smoothed system with fixed barrier parameter ι, not for the unsmoothed projected algorithm actually implemented in the numerics.
  • standard math The objective and the barrier function are strictly convex, yielding a unique global minimizer u*_ι of (2.13).
    Used in Theorem 2.5 and Theorem 3.4. This follows from A^T Γ^-1 A positive definite and convex log-barriers, but the positive definiteness is a genuine restriction.
  • domain assumption In the nonlinear setting, G(u) - G(ū) ≈ DG(ū)(u - ū).
    Introduced in Section 4.2.1 to define variance inflation for nonlinear forward models. No validity guarantee is given beyond small ensemble spread.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the Incorporation of Box-Constraints for Ensemble Kalman Inversion." pith.science (2026). https://pith.science/paper/WDG7ISHE

@misc{pith2026190800696,
  author       = {Pith},
  title        = {Pith review of: On the Incorporation of Box-Constraints for Ensemble Kalman Inversion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDG7ISHE}},
  note         = {Machine review of arXiv:1908.00696}
}
read the original abstract

The Bayesian approach to inverse problems is widely used in practice to infer unknown parameters from noisy observations. In this framework, the ensemble Kalman inversion has been successfully applied for the quantification of uncertainties in various areas of applications. In recent years, a complete analysis of the method has been developed for linear inverse problems adopting an optimization viewpoint. However, many applications require the incorporation of additional constraints on the parameters, e.g. arising due to physical constraints. We propose a new variant of the ensemble Kalman inversion to include box constraints on the unknown parameters motivated by the theory of projected preconditioned gradient flows. Based on the continuous time limit of the constrained ensemble Kalman inversion, we discuss a complete convergence analysis for linear forward problems. We adopt techniques from filtering which are crucial in order to improve the performance and establish a correct descent, such as variance inflation. These benefits are highlighted through a number of numerical examples on various inverse problems based on partial differential equations.

Figures

Figures reproduced from arXiv: 1908.00696 by the authors.

Figure 1
Figure 1. Varying contour lines of the function Φ(x) defined in Example 2.8, with both the preconditioned descent direction in the unconstrained case and the projected preconditioned descent direction. For α > 0, the next iteration is given by x k+1 = P(x k − αDk∇Φ(x k )) =  1 + α 0  , where P is the projection onto R × R≤0. Then, Φ(x k+1) = (1 + α) 2 + 1 + α > 2 = Φ(x k ), i.e. for all α > 0 the function value of the objec… view at source ↗
Figure 2
Figure 2. Transformed EnKF estimation in comparison to the EnKF estimation and the projected EnKF estimation. J = 5 particles have been simulated [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Ensemble spread in the transformed EnKF in compar￾ison to the EnKF and the projected EnKF. J = 5 particles have been simulated. Our first set of experiments for the linear PDE are shown in Figures 2 - 4, where we assume that we have full observations. The left hand side image of [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: KKT-Residuals and difference of the misfit functional and the global minimum in the transformed EnKF in comparison to the projected EnKF. J = 5 particles have been simulated. For the projected EnKF we notice a similar performance, however it takes into con￾sideration t…
Figure 5
Figure 5. Figure 5: Transformed EnKF estimation in comparison to the EnKF estimation and the projected EnKF estimation. J = 5 particles have been simulated. The second set of experiments in the linear setting are shown in Figures 5 - 6, where we assume that we have 15 low dimensional obse…
Figure 6
Figure 6. Figure 6: Difference of the misfit functional and the global min￾imum in the transformed EnKF in comparison to the EnKF and the projected EnKF. J = 5 particles have been simulated. κ ∈ L∞(D) to solve (4.3) −∇ · (κ∇p) = f, x ∈ D, (4.4) p = 0, x ∈ ∂D, where ∇· denotes the divergen…
Figure 7
Figure 7. Figure 7: Transformed EnKF parameter estimation in compari￾son to the EnKF estimation and the projected EnKF estimation. J = 5 particles have been simulated. For the non-linear experiments [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: Transformed EnKF observation estimation in compar￾ison to the EnKF estimation and the projected EnKF estimation. J = 5 particles have been simulated [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: Ensemble spread in the transformed EnKF in compar￾ison to the EnKF and the projected EnKF. J = 5 particles have been simulated [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Difference of the misfit functional and the global min￾imum in the transformed EnKF in comparison to the projected EnKF. J = 5 particles have been simulated. deriving a continuum limit and a gradient flow structure. The key insight from this work is that the projected…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    N. Amor, G. Rasool, and N. C. Bouaynaya , Constrained state estimation - a review , ArXiv preprint, arxiv:807.03463, (2018)

    work page 2018

  2. [2]

    J. L. Anderson , An adaptive covariance inflation error correction algorithm for ensemble filters, Tellus A, 59 (2007), pp. 210–224

    work page 2007

  3. [3]

    , Spatially and temporally varying adaptive covariance inflation for ensemble filters , Tellus A, 61 (2009), pp. 72–83

    work page 2009

  4. [4]

    Bergemann and S

    K. Bergemann and S. Reich , A localization technique for ensemble Kalman filters , Quar- terly Journal of the Royal Meteorological Society, 136 (2010), pp. 701–707

    work page 2010

  5. [5]

    1636–1643

    , A mollified ensemble Kalman filter , Quarterly Journal of the Royal Meteorological Society, 136 (2010), pp. 1636–1643

    work page 2010

  6. [6]

    Bertsekas, Projected newton methods for optimization problems with simple constraints , SIAM Journal on Control and Optimization, 20 (1982), pp

    D. Bertsekas, Projected newton methods for optimization problems with simple constraints , SIAM Journal on Control and Optimization, 20 (1982), pp. 221–246

    work page 1982

  7. [7]

    D. P. Bertsekas , Nonlinear programming, Athena Scientific, 2nd ed., Sept. 2008

    work page 2008

  8. [8]

    On the stability of matrix-valued Riccati diffusions

    A.N. Bishop and P. Del Moral, On the stability of matrix-valued Riccati diffusions, arXiv e-print, arXiv:1808.00235 (2018)

    work page Pith review arXiv 2018

Show all 38 references
  1. [9]

    Bl ¨omker, C

    D. Bl ¨omker, C. Schillings, and P. W acker , A strongly convergent numerical scheme from ensemble kalman inversion , SIAM Journal on Numerical Analysis, 56 (2018), pp. 2537 – 2562

    2018

  2. [10]

    Bl ¨omker, C

    D. Bl ¨omker, C. Schillings, P. W acker, and S. Weissmann , Well posedness and conver- gence analysis of the ensemble kalman inversion , Inverse Problems, 35 (2019), p. 085007

    2019

  3. [11]

    Boyd and L

    S. Boyd and L. V andenberghe , Convex Optimization , Cambridge University Press, New York, NY, USA, 2004

    2004

  4. [12]

    N. K. Chada, M. A. Iglesias, L. Roininen, and A. M. Stuart , Parameterizations for ensemble Kalman inversion , Inverse Problems, 34 (2018), p. 055009

    2018

  5. [13]

    X. Chen, J. D. Lee, X. T. Tong, and Y. Zhang , Statistical Inference for Model Parameters in Stochastic Gradient Descent , arXiv e-prints, (2016), p. arXiv:1610.08637

    2016 arXiv

  6. [14]

    E. E. S. D. Albers, P-A. Blacquart and A. M. Stuart , Ensemble Kalman methods with constraints, In preparation, (2019)

    2019

  7. [15]

    H. Engl, M. Hanke, and A. Neubauer , Regularization of Inverse Problems , Mathematics and Its Applications, Springer Netherlands, 2000

    2000

  8. [16]

    Evensen , The ensemble Kalman filter: theoretical formulation and practical implemen- tation, Ocean Dynamics, 53 (2003), pp

    G. Evensen , The ensemble Kalman filter: theoretical formulation and practical implemen- tation, Ocean Dynamics, 53 (2003), pp. 343–367. 22 N. K. CHADA, C. SCHILLINGS, AND S. WEISSMANN

    2003

  9. [17]

    Evensen , Data Assimilation: The Ensemble Kalman Filter , Earth and Environmental Science, Springer Berlin Heidelberg, 2009

    G. Evensen , Data Assimilation: The Ensemble Kalman Filter , Earth and Environmental Science, Springer Berlin Heidelberg, 2009

    2009

  10. [18]

    M. A. Iglesias , A regularizing iterative ensemble Kalman method for PDE-constrained in- verse problems, Inverse Problems, 32 (2016), p. 025002

    2016

  11. [19]

    M. A. Iglesias, K. Law, and A. M. Stuart , Ensemble Kalman methods for inverse prob- lems, Inverse Problems, 29 (2013), p. 045001

    2013

  12. [20]

    Kaipio and E

    J. Kaipio and E. Somersalo , Statistical and Computational Inverse Problems , Applied Mathematical Sciences, Springer New York, 2006

    2006

  13. [21]

    Kandepu, L

    R. Kandepu, L. Imsland, and B. A. Foss , Constrained state estimation using the unscented Kalman filter, 16th Mediterranean Conference on Control and Automation, (2008)

    2008

  14. [22]

    Kelly, K

    D. Kelly, K. Law, and A. M. Stuart , Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time , Nonlinearity, 27 (2014), p. 2579

    2014

  15. [23]

    Kelly, A

    D. Kelly, A. J. Majda, and X. T. Tong , Concrete ensemble Kalman filters with rigorous catastrophic filter divergence, Proceedings of the National Academy of Sciences, (2015)

    2015

  16. [24]

    Kwiatkowski and J

    E. Kwiatkowski and J. Mandel , Convergence of the square root ensemble Kalman filter in the large ensemble limit , SIAM/ASA Journal on Uncertainty Quantification, 3 (2015), pp. 1–17

    2015

  17. [25]

    K. Law, A. M. Stuart, and K. Zygalakis , Data Assimilation: A Mathematical Introduc- tion, Texts in Applied Mathematics, Springer International Publishing, 2016

    2016

  18. [26]

    Le Gland, V

    F. Le Gland, V. Monbet, and V.-D. Tran , Large sample asymptotics for the ensemble Kalman filter, Research Report RR-7014, INRIA, 2009

    2009

  19. [27]

    Li and A

    G. Li and A. C. Reynolds , Iterative ensemble Kalman filters for data assimilation , Society of Petroleum Engineers, (2009)

    2009

  20. [28]

    D. M. Livings, S. L. Dance, and N. K. Nichols , Unbiased ensemble square root filters , Physica D: Nonlinear Phenomena, 237 (2008), pp. 1021 – 1028

    2008

  21. [29]

    B. K. P. Hungerlander and F. Rendl , Regularization of inverse problems via box con- strained minimization, ArXiv preprint arXiv:1807.11316, (2018)

    2018 arXiv

  22. [30]

    Reich , Data assimilation, Acta Numerica, in preparation, (2019)

    S. Reich , Data assimilation, Acta Numerica, in preparation, (2019)

    2019

  23. [31]

    Reich and C

    S. Reich and C. J. Cotter , Probabilistic Forecasting and Bayesian Data Assimilation, 2014

    2014

  24. [32]

    Schillings and A

    C. Schillings and A. M. Stuart , Analysis of the ensemble Kalman filter for inverse prob- lems, SIAM Journal on Numerical Analysis, 55 (2017), pp. 1264–1290

    2017

  25. [33]

    Schmidt, D

    M. Schmidt, D. Kim, and S. Sra , Projected newton-type methods in machine learning , (2013)

    2013

  26. [34]

    Shikhman and O

    V. Shikhman and O. Stein , Constrained optimization: Projected gradient flows , Journal of Optimization Theory and Applications, 140 (2009), pp. 117–130

    2009

  27. [35]

    Simon , Kalman filtering with state constraints: a survey of linear and nonlinear algo- rithms, 4 (2009), pp

    D. Simon , Kalman filtering with state constraints: a survey of linear and nonlinear algo- rithms, 4 (2009), pp. 1303–1318

    2009

  28. [36]

    A. M. Stuart, Inverse problems: A bayesian perspective, Acta Numerica, 19 (2010), pp. 451– 559

    2010

  29. [37]

    M. K. Tippett, J. L. Anderson, C. H. Bishop, T. M. Hamill, and J. S. Whitaker , Ensemble square root filters , Monthly Weather Review, 131 (2003), pp. 1485–1490

    2003

  30. [38]

    W ang, Y

    D. W ang, Y. Chen, and X. Cai , State and parameter estimation of hydrologic models using the constrained ensemble Kalman filter , Monthly Resources Research, 45 (2009). Department of Statistics and Applied Probability, National University of Singa- pore, 119077, Singapore E-ma...

    2009

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.