ekalmX x Ensemble Kalman Processes

Subject: Iterative ensemble methods for derivative-free parameter estimation and calibration of expensive black-box simulators.

Date: 2026-04-03

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1 Scope

Ensemble Kalman Processes (EKP) are iterative ensemble methods that solve inverse problems without requiring gradients of the forward model. They are the tool of choice when the simulator \(G\) is expensive, non-differentiable, or only available as a black box (climate models, PDE solvers, lab equipment).

All methods in this catalog implement AbstractProcess (see api/components.md). The caller provides forward evaluations \(\{G(\theta^{(j)})\}\) at each iteration; the process provides the ensemble update rule.

In scope: EKI, ETKI, EKS/ALDI, UKI, GNKI, SparseEKI, TEKI Out of scope: Sequential data assimilation filters (AbstractSequentialFilter), smoothers, neural-operator surrogates (those are upstream of \(G\)).

Key insight: Every method in this catalog shares the same outer loop -- only the update rule and state representation differ. The AbstractProcess protocol (init / update) captures this cleanly: the caller owns the forward model calls, the process owns the ensemble algebra.

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2 Common Mathematical Framework

The EKP framework (Iglesias, Law & Stuart 2013) treats parameter estimation as an iterative filtering problem in artificial time.

Forward model. \(G: \mathbb{R}^p \to \mathbb{R}^d\) maps parameters \(\theta\) to observables.

Observations. \(y = G(\theta^\dagger) + \eta\), where \(\eta \sim \mathcal{N}(0, \Gamma)\) is observation noise.

Ensemble. \(\{\theta_n^{(j)}\}_{j=1}^J\), with forward evaluations \(\{G(\theta_n^{(j)})\}\).

Empirical statistics.

\[\bar\theta_n = \frac{1}{J}\sum_j \theta_n^{(j)}, \qquad \bar G_n = \frac{1}{J}\sum_j G(\theta_n^{(j)})\]

\[C_n^{\theta G} = \frac{1}{J-1}\sum_j (\theta_n^{(j)} - \bar\theta_n)(G(\theta_n^{(j)}) - \bar G_n)^\top \quad \text{(cross-covariance)}\]

\[C_n^{GG} = \frac{1}{J-1}\sum_j (G(\theta_n^{(j)}) - \bar G_n)(G(\theta_n^{(j)}) - \bar G_n)^\top \quad \text{(obs-space covariance)}\]

\[C_n^{\theta\theta} = \frac{1}{J-1}\sum_j (\theta_n^{(j)} - \bar\theta_n)(\theta_n^{(j)} - \bar\theta_n)^\top \quad \text{(parameter covariance)}\]

Generic update.

\[\theta_{n+1}^{(j)} = \theta_n^{(j)} + \Delta t_n \cdot C_n^{\theta G}\bigl(C_n^{GG} + \Delta t_n^{-1}\,\Gamma\bigr)^{-1}\bigl(y - G(\theta_n^{(j)})\bigr)\]

The step size \(\Delta t_n\) plays the role of a learning rate. When \(\Delta t_n \to 0\) the update vanishes; when \(\Delta t_n \to \infty\) the noise covariance \(\Gamma\) is ignored and the update becomes a full Kalman step.

Connection to gradient flow. In the continuous-time limit (\(\Delta t \to 0\)), EKI implements preconditioned gradient descent on the data-misfit functional \(\frac{1}{2}\|y - G(\theta)\|_{\Gamma^{-1}}^2\), where the preconditioner is the ensemble-estimated covariance \(C^{\theta\theta}\). The ensemble provides a derivative-free approximation to the Jacobian \(\nabla_\theta G\) via the cross-covariance \(C^{\theta G}\).

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3 Gap Catalog

Gap 1: EKI -- Ensemble Kalman Inversion

Ref: Iglesias, Law & Stuart (2013)

Purpose: Standard inversion (optimization). Produces a point estimate of \(\theta\) by iterating the ensemble update until convergence.

Update:

\[\theta_{n+1}^{(j)} = \theta_n^{(j)} + \Delta t_n \cdot C_n^{\theta G}\bigl(C_n^{GG} + \Delta t_n^{-1}\,\Gamma\bigr)^{-1}\bigl(y - G(\theta_n^{(j)})\bigr)\]

Behavior: The ensemble collapses to a point estimate as \(n \to \infty\). The ensemble mean \(\bar\theta_n\) converges to a regularized least-squares solution. Ensemble spread vanishes -- no posterior uncertainty.

Complexity: \(O(J^2 d + J p)\) per step (\(J\) = ensemble size, \(d\) = obs dimension, \(p\) = parameter dimension).

class EKI(AbstractProcess):
    scheduler: AbstractScheduler
    prior_mean: Optional[Float[Array, "N_p"]] = None
    prior_cov: Optional[AbstractLinearOperator] = None

When to use: Default choice for calibration when only a point estimate is needed. Works well even when \(J \ll p\) (underdetermined regime).

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Gap 2: ETKI -- Ensemble Transform Kalman Inversion

Purpose: Output-scalable variant of EKI. Performs the update via a transform matrix in ensemble space rather than observation space.

Math: Instead of forming the \(d \times d\) matrix \((C^{GG} + \Delta t^{-1}\Gamma)\), ETKI works in the \(J \times J\) ensemble subspace:

\[T = \bigl(I_J + \Delta t \cdot Y^\top \Gamma^{-1} Y\bigr)^{-1}\]

where \(Y \in \mathbb{R}^{d \times J}\) is the matrix of centered forward evaluations. The ensemble is updated via \(\Theta_{n+1} = \Theta_n \cdot W\) for an appropriate weight matrix \(W\) derived from \(T\).

Complexity: \(O(J^2 d)\) per step instead of \(O(d^3)\). The key advantage is avoiding the \(d \times d\) solve when \(d \gg J\).

class ETKI(AbstractProcess):
    scheduler: AbstractScheduler

When to use: High-dimensional observation spaces (\(d \gg J\)), e.g. satellite retrievals, image-based observations, full-field comparisons.

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Gap 3: EKS / ALDI -- Ensemble Kalman Sampler

Ref: Garbuno-Inigo, Hoffmann, Li & Stuart (2020)

Purpose: Posterior sampling. Unlike EKI, the ensemble does NOT collapse -- it produces approximate posterior samples at stationarity.

Update (ALDI form):

\[d\theta^{(j)} = C_n^{\theta G}\,\Gamma^{-1}(y - G(\theta^{(j)}))\,dt - \frac{d+1}{J}(\theta^{(j)} - \bar\theta)\,dt + \sqrt{2\,C_n^{\theta\theta}}\,dW^{(j)}\]

The first term drives particles toward the data. The second term is a finite-sample drift correction. The third term is Langevin noise scaled by the ensemble covariance, ensuring ergodicity -- the ensemble explores the posterior rather than collapsing.

Behavior: Ensemble spread is maintained. At stationarity, particles are approximate samples from the posterior \(p(\theta | y)\). Requires careful step-size control (use EKSStableScheduler).

class EKS_Process(AbstractProcess):
    scheduler: AbstractScheduler  # use EKSStableScheduler

When to use: When posterior uncertainty quantification is needed, not just a point estimate. Climate sensitivity studies, model structural uncertainty, Bayesian model evidence.

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Gap 4: UKI -- Unscented Kalman Inversion

Ref: Huang, Schneider & Stuart (2022)

Purpose: Parametric inversion with uncertainty. Maintains an explicit mean \(\mu_n\) and covariance \(\Sigma_n\) (not a random ensemble). Uses \(2p+1\) deterministic sigma points for quadrature.

Update:

\[\text{Sigma points:} \quad \hat\theta^0 = \hat\mu, \quad \hat\theta^{\pm j} = \hat\mu \pm c_j\,[\sqrt{\hat\Sigma}]_j, \quad j = 1,\dots,p\]

\[C^{\theta y} = \sum_j W_j\,(\hat\theta^{(j)} - \hat\mu)(\hat y^{(j)} - \hat y^0)^\top\]

\[C^{yy} = \sum_j W_j\,(\hat y^{(j)} - \hat y^0)(\hat y^{(j)} - \hat y^0)^\top + \Sigma_\nu\]

\[\mu_{n+1} = \hat\mu + C^{\theta y}(C^{yy})^{-1}(y - \hat y^0)\]

\[\Sigma_{n+1} = \hat\Sigma - C^{\theta y}(C^{yy})^{-1}(C^{\theta y})^\top\]

Behavior: Deterministic -- no sampling noise. Does not suffer from ensemble collapse. The covariance \(\Sigma_n\) provides calibrated uncertainty estimates (converges to the posterior covariance in the linear-Gaussian case).

Complexity: \(O(p^2 d)\) per step. Requires \(2p+1\) forward evaluations per iteration (fixed, not tunable).

class UKI(AbstractProcess):
    scheduler: AbstractScheduler
    alpha: float = 1.0   # sigma-point spread
    beta: float = 2.0    # higher-order moment weighting
    kappa: float = 0.0   # secondary scaling

    def init(self, mean, covariance, obs, noise_cov) -> UKIState: ...
    def update(self, state: UKIState, forward_evals) -> UKIState: ...

When to use: Moderate parameter dimension (\(p \lesssim 100\)) where calibrated uncertainty is important. Better UQ than EKI, deterministic (reproducible), no ensemble-size tuning.

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Gap 5: GNKI -- Gauss-Newton Kalman Inversion

Purpose: Faster convergence via explicit Jacobian estimation from ensemble perturbations.

Math: The ensemble-estimated Jacobian is:

\[\tilde J_n \approx C_n^{\theta G}(C_n^{\theta\theta})^{-1}\]

This is used in a Gauss-Newton update:

\[\delta\theta = \bigl(\tilde J_n^\top\,\Gamma^{-1}\,\tilde J_n + \Sigma_{\text{prior}}^{-1}\bigr)^{-1}\tilde J_n^\top\,\Gamma^{-1}(y - G(\theta))\]

The update includes both a data-fit term and a prior-pull term:

\[\theta_{n+1}^{(j)} = \theta_n^{(j)} + \alpha\bigl\{K_n(y - G(\theta_n^{(j)})) + (I - K_n\tilde J_n)(m_0 - \theta_n^{(j)})\bigr\}\]

Behavior: Faster convergence than EKI for well-conditioned problems. In the linear-Gaussian case, GNKI recovers the exact posterior mean and covariance. Requires \(J > p\) for \(C^{\theta\theta}\) to be invertible.

class GNKI(AbstractProcess):
    scheduler: AbstractScheduler

When to use: Well-conditioned inverse problems where fast convergence matters. Requires \(J > p\) (cannot operate in the underdetermined regime unlike EKI).

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Gap 6: SparseEKI -- Sparse Ensemble Kalman Inversion

Ref: Schneider, Stuart & Wu

Purpose: Variable selection and sparse parameter recovery. Adds an \(L_1\) sparsity penalty to the EKI update.

Math: Modified update with proximal operator for \(L_1\) regularization:

\[\theta_{n+1}^{(j)} = \text{prox}_{\lambda\|\cdot\|_1}\bigl(\theta_n^{(j)} + \Delta t \cdot K_n(y - G(\theta_n^{(j)}))\bigr)\]

where \(\text{prox}_{\lambda\|\cdot\|_1}(z)_i = \text{sign}(z_i)\max(|z_i| - \lambda, 0)\) is the soft-thresholding operator.

Behavior: Drives inactive parameters to exactly zero. Useful when the true parameter vector is sparse or when performing feature/variable selection. The proximal step is applied element-wise after each EKI update.

class SparseInversion(AbstractProcess):
    scheduler: AbstractScheduler
    penalty: str = "l1"       # "l0" | "l1"
    penalty_weight: float = 0.1

When to use: High-dimensional parameter spaces where most parameters are expected to be zero or near-zero. Source localization, sensor placement, sparse physics discovery.

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Gap 7: TEKI -- Tikhonov-Regularized Ensemble Kalman Inversion

Ref: Chada, Chen & Stuart

Purpose: Prevents ensemble collapse toward degenerate solutions by adding prior regularization directly to the EKI update.

Math: The forward map, data, and noise covariance are augmented:

\[\tilde G(\theta) = \begin{bmatrix} G(\theta) \\ \theta \end{bmatrix}, \qquad \tilde y = \begin{bmatrix} y \\ m_0 \end{bmatrix}, \qquad \tilde\Gamma = \begin{bmatrix} \Gamma & 0 \\ 0 & C_0 \end{bmatrix}\]

The standard EKI update is then applied in the augmented space. The identity block in \(C^{\theta\tilde G}\) pulls particles back toward the prior mean \(m_0\), preventing the ensemble from drifting arbitrarily far from the prior.

Behavior: Converges to the MAP estimate (not just MLE). More stable than vanilla EKI when the problem is ill-posed. The regularization strength is controlled by the prior covariance \(C_0\).

class TEKI(AbstractProcess):
    scheduler: AbstractScheduler
    prior_mean: Float[Array, "N_p"]
    prior_cov: AbstractLinearOperator

When to use: Ill-posed inverse problems where vanilla EKI diverges or produces physically implausible solutions. When the MAP estimate (incorporating prior information) is preferred over the MLE.

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4 Shared Infrastructure

All EKP methods share common infrastructure -- they differ only in the update rule and state representation.

Component	Source	Notes
Ensemble statistics	ekalmX._src.ensemble	\(\bar\theta\), \(C^{\theta G}\), \(C^{GG}\), \(C^{\theta\theta}\)
FixedScheduler	ekalmX.schedulers	Constant \(\Delta t\)
DataMisfitController	ekalmX.schedulers	Adaptive \(\Delta t\); terminates when algo_time reaches 1.0
EKSStableScheduler	ekalmX.schedulers	Stability-aware for EKS/ALDI Langevin dynamics
gaussx operators	gaussx	Structured covariance: DiagonalLinearOperator, LowRankUpdate, Kronecker
optax integration	optax	Learning rate schedules, gradient clipping for hybrid methods
ProcessState	ekalmX.types	Particles, forward evals, obs, noise_cov, step, algo_time
UKIState	ekalmX.types	Mean, covariance (parametric state for UKI)

Scheduler selection guidance:

• FixedScheduler(dt=1.0) -- simple, requires manual tuning.
• DataMisfitController(target_misfit=1.0) -- adaptive, standard choice for EKI/ETKI/GNKI. Adjusts \(\Delta t\) so the data misfit decreases at a controlled rate.
• EKSStableScheduler(max_dt=1.0) -- required for EKS/ALDI to prevent divergence in the Langevin dynamics.

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5 Comparison Table

Method	Convergence	Output	Obs scaling	Ensemble size	UQ	Best for
EKI	Collapse \(\to\) point	MLE / regularized LS	\(O(d^3)\) or \(O(J^2 d)\)	\(J \geq p+1\) (flexible)	None (spread \(\to 0\))	Default calibration
ETKI	Collapse \(\to\) point	MLE	\(O(J^2 d)\)	\(J \geq p+1\)	None	High-dim observations
EKS/ALDI	Ergodic (no collapse)	Posterior samples	\(O(J^2 d)\)	\(J \geq p+1\) (larger = better)	Full posterior	Uncertainty quantification
UKI	Parametric convergence	MAP + covariance	\(O(p^2 d)\)	\(2p+1\) (fixed)	Calibrated \(\Sigma\)	Moderate \(p\), needs UQ
GNKI	Fast (Gauss-Newton)	MAP + spread	\(O(p^3)\)	\(J > p\) (required)	Via ensemble spread	Well-conditioned, fast convergence
SparseEKI	Collapse + sparsity	Sparse point estimate	\(O(J^2 d)\)	\(J \geq p+1\)	None	Variable selection
TEKI	Collapse \(\to\) MAP	MAP estimate	\(O((d+p)^3)\)	\(J \geq p+1\)	None	Ill-posed problems

Decision tree:

1. Need posterior samples? \(\to\) EKS/ALDI
2. Need calibrated covariance? \(\to\) UKI (if \(p \lesssim 100\))
3. Sparse parameters? \(\to\) SparseEKI
4. Ill-posed / need prior regularization? \(\to\) TEKI
5. High-dim observations (\(d \gg J\))? \(\to\) ETKI
6. Well-conditioned, want fast convergence? \(\to\) GNKI (if \(J > p\))
7. Otherwise \(\to\) EKI (simple, robust default)

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6 References

1. Iglesias, M. A., Law, K. J. H., & Stuart, A. M. (2013). Ensemble Kalman methods for inverse problems. Inverse Problems, 29(4), 045001.
2. Huang, D. Z., Schneider, T., & Stuart, A. M. (2022). Iterated Kalman methodology for inverse problems. Journal of Computational Physics, 463, 111262.
3. Garbuno-Inigo, A., Hoffmann, F., Li, W., & Stuart, A. M. (2020). Interacting Langevin diffusions: gradient structure and ensemble Kalman sampler. SIAM Journal on Applied Dynamical Systems, 19(1), 412-441.
4. Chada, N. K., Chen, Y., & Stuart, A. M. (2019). Tikhonov regularization within ensemble Kalman inversion. SIAM Journal on Numerical Analysis.
5. Kovachki, N. B., & Stuart, A. M. (2019). Ensemble Kalman inversion: a derivative-free technique for machine learning tasks. Inverse Problems, 35(9), 095005.
6. Schneider, T., Stuart, A. M., & Wu, J.-L. (2022). Ensemble Kalman inversion for sparse learning of dynamical systems from time-averaged data. Journal of Computational Physics.
7. Calvello, E., Reich, S., & Stuart, A. M. (2022). Ensemble Kalman methods: a mean field perspective. arXiv:2209.11371.