LEMDA: Multiscale Lagrangian-Eulerian DA

• LEMDA is a framework that recovers turbulent flow fields by combining Lagrangian particle trajectories with continuum Eulerian grid statistics.
• It employs closed-form analytic updates, akin to Kalman–Bucy equations, to efficiently handle high-dimensional and nonlinear observational challenges.
• The multiscale approach enables recovery of coarse large-scale structures and fine-scale features, with applications in sea-ice dynamics and turbulent flow systems.

Lagrangian–Eulerian Multiscale Data Assimilation (LEMDA) is a framework for recovering turbulent flow fields from Lagrangian tracer data by combining a Lagrangian data-assimilation component that uses individual particle trajectories with an Eulerian data-assimilation component that uses continuum, grid-based statistics of particles obtained from a Boltzmann kinetic description (Deng et al., 2024). It was developed to address strong nonlinearity in the observational process, high dimensionality, and settings in which trajectories are influenced by external forces or collisions. Within the framework, the posterior evolution can be written using closed analytic formulae, Eulerian DA recovers large-scale structures, and Lagrangian DA resolves small-scale features in each grid cell via parallel computing (Deng et al., 2024).

1. Foundational definition and conceptual scope

LEMDA is organized around a dual representation of the same physical system. The Lagrangian side uses particle trajectories as observations. The Eulerian side replaces direct use of all trajectories by continuum quantities such as particle density, momentum, and related grid-based statistics. The 2024 formulation explicitly starts from exploiting the Boltzmann kinetic description of the particle dynamics to derive a set of continuum equations, which characterize the statistical quantities of particle motions at fixed grids and serve as Eulerian observations (Deng et al., 2024).

The original framework is motivated by three difficulties in standard Lagrangian DA: the strong nonlinearity of the observational process, the high dimensionality of the underlying state, and the presence of external forces or collisions, such as those arising in sea-ice floe applications (Deng et al., 2024). In this setting, “multiscale” refers both to spectral scale separation and to a division of labor between coarse Eulerian recovery of large-scale flow and local Lagrangian recovery of fine-scale features. A later physical-domain formulation describes the original LEMDA vision as combining classical Lagrangian DA, an Eulerian continuum description derived from a Boltzmann or kinetic description, Eulerian DA for large-scale flow, and local Lagrangian DA within each grid cell to recover fine-scale features (Yun et al., 18 Sep 2025).

The same general objective appears in related multiscale DA work that does not use the name LEMDA explicitly. In particular, the effective-average-action formulation of multiscale variational DA introduces coarse-grained sensitivity, or effective gradient, for slow variables in the presence of fast fluctuations. That work is explicitly described as highly relevant to LEMDA’s core goal of obtaining scale-consistent, physically meaningful gradients and updates when a model has both slow and fast degrees of freedom, or Lagrangian-style advected quantities and Eulerian fields (Sugiura, 2015).

2. Core mathematical formulation

The Lagrangian particle model in the baseline framework is written on a doubly periodic square domain. For tracer $l$, with position $\mathbf{x}_l$ and velocity $\mathbf{v}_l$, the dynamics are

$$\frac{d\mathbf{x}_l}{dt} = \mathbf{v}_l + \sigma_x \dot{\mathbf{W}}_l, \qquad \frac{d\mathbf{v}_l}{dt} = \beta(\mathbf{u}(\mathbf{x}_l,t)-\mathbf{v}_l) + \mathbf{f}_l,$$

where $\mathbf{u}(\mathbf{x},t)$ is the underlying flow field, $\beta$ is a drag coefficient, $\sigma_x \dot{\mathbf{W}}_l$ is Gaussian noise, and $\mathbf{f}_l$ denotes external forcing such as collision effects (Deng et al., 2024).

The Eulerian flow is represented spectrally: $$\mathbf{u}(\mathbf{x},t) = \sum_{\mathbf{k}\in\mathcal{K},\ \alpha\in\mathcal{A}} \hat{u}_{\mathbf{k},\alpha}(t)\, e^{i\mathbf{k}\cdot \mathbf{x}}\, \mathbf{r}_{\mathbf{k},\alpha},$$ with Fourier coefficients $\hat{u}_{\mathbf{k},\alpha}$ evolving according to complex Ornstein–Uhlenbeck dynamics,

$\mathbf{x}_l$0

The coefficients $\mathbf{x}_l$1, $\mathbf{x}_l$2, $\mathbf{x}_l$3, and $\mathbf{x}_l$4 are calibrated from empirical mean, variance, decorrelation time, and phase decorrelation statistics of the modes (Deng et al., 2024).

The Eulerianization step introduces a Boltzmann-type kinetic density $\mathbf{x}_l$5 in phase space. The Boltzmann equation is

$\mathbf{x}_l$6

where $\mathbf{x}_l$7 is a collision or interaction operator satisfying mass conservation. From $\mathbf{x}_l$8, one defines number density $\mathbf{x}_l$9, mean velocity $\mathbf{v}_l$0, and momentum density $\mathbf{v}_l$1. Integrating the kinetic equation yields the continuity equation

$\mathbf{v}_l$2

and the momentum equation

$\mathbf{v}_l$3

These continuum relations supply Eulerian observations on fixed grids (Deng et al., 2024).

Both the Lagrangian and Eulerian components are then embedded in a conditional-Gaussian system of the form

$\mathbf{v}_l$4

$\mathbf{v}_l$5

where $\mathbf{v}_l$6 denotes observed variables and $\mathbf{v}_l$7 denotes unobserved variables to be inferred. In the Lagrangian component, $\mathbf{v}_l$8 is the tracer-position process. In the Eulerian component, $\mathbf{v}_l$9 is the vector of continuum observables, such as momentum components on the grid (Deng et al., 2024).

3. Analytic posterior evolution and reduced-order inference

A defining property of LEMDA is that the coupled model is nonlinear in the joint variables but linear in the unobserved state conditioned on the observed path. As a result, the conditional distribution

$$\frac{d\mathbf{x}_l}{dt} = \mathbf{v}_l + \sigma_x \dot{\mathbf{W}}_l, \qquad \frac{d\mathbf{v}_l}{dt} = \beta(\mathbf{u}(\mathbf{x}_l,t)-\mathbf{v}_l) + \mathbf{f}_l,$$0

is Gaussian, and its mean $$\frac{d\mathbf{x}_l}{dt} = \mathbf{v}_l + \sigma_x \dot{\mathbf{W}}_l, \qquad \frac{d\mathbf{v}_l}{dt} = \beta(\mathbf{u}(\mathbf{x}_l,t)-\mathbf{v}_l) + \mathbf{f}_l,$$1 and covariance $$\frac{d\mathbf{x}_l}{dt} = \mathbf{v}_l + \sigma_x \dot{\mathbf{W}}_l, \qquad \frac{d\mathbf{v}_l}{dt} = \beta(\mathbf{u}(\mathbf{x}_l,t)-\mathbf{v}_l) + \mathbf{f}_l,$$2 satisfy closed analytic equations of Kalman–Bucy type: $$\frac{d\mathbf{x}_l}{dt} = \mathbf{v}_l + \sigma_x \dot{\mathbf{W}}_l, \qquad \frac{d\mathbf{v}_l}{dt} = \beta(\mathbf{u}(\mathbf{x}_l,t)-\mathbf{v}_l) + \mathbf{f}_l,$$3

$$\frac{d\mathbf{x}_l}{dt} = \mathbf{v}_l + \sigma_x \dot{\mathbf{W}}_l, \qquad \frac{d\mathbf{v}_l}{dt} = \beta(\mathbf{u}(\mathbf{x}_l,t)-\mathbf{v}_l) + \mathbf{f}_l,$$4

This gives an exact and efficient way of carrying out DA and avoids using ensemble approximations and the associated tunings (Deng et al., 2024).

The same conditional-Gaussian line of development appears in analytically tractable continuous-time Lagrangian DA with linear stochastic forecast models. There, the posterior distribution is available in closed form, which enables approximate filters, uncertainty quantification, and randomized selection of a small number of tracers at each time step to reduce computational cost while retaining DA accuracy (Chen et al., 2022). Within LEMDA, this lineage supports the view that analytic posterior updates are not merely a formal property but a practical mechanism for scaling Lagrangian inference.

The 2024 LEMDA paper also derives an effective reduced-order Lagrangian DA scheme. The starting point is the block decomposition of the posterior covariance into tracer-velocity, cross, and flow-mode blocks. Because the forecast model for spectral modes is independent across $$\frac{d\mathbf{x}_l}{dt} = \mathbf{v}_l + \sigma_x \dot{\mathbf{W}}_l, \qquad \frac{d\mathbf{v}_l}{dt} = \beta(\mathbf{u}(\mathbf{x}_l,t)-\mathbf{v}_l) + \mathbf{f}_l,$$5, the flow-mode covariance block is approximated as nearly diagonal under mean-field assumptions. The diagonal entry $$\frac{d\mathbf{x}_l}{dt} = \mathbf{v}_l + \sigma_x \dot{\mathbf{W}}_l, \qquad \frac{d\mathbf{v}_l}{dt} = \beta(\mathbf{u}(\mathbf{x}_l,t)-\mathbf{v}_l) + \mathbf{f}_l,$$6 of the flow covariance satisfies

$$\frac{d\mathbf{x}_l}{dt} = \mathbf{v}_l + \sigma_x \dot{\mathbf{W}}_l, \qquad \frac{d\mathbf{v}_l}{dt} = \beta(\mathbf{u}(\mathbf{x}_l,t)-\mathbf{v}_l) + \mathbf{f}_l,$$7

with quasi-equilibrium solution

$$\frac{d\mathbf{x}_l}{dt} = \mathbf{v}_l + \sigma_x \dot{\mathbf{W}}_l, \qquad \frac{d\mathbf{v}_l}{dt} = \beta(\mathbf{u}(\mathbf{x}_l,t)-\mathbf{v}_l) + \mathbf{f}_l,$$8

This approximation precomputes the flow-mode covariance and significantly reduces the computational burden of the Lagrangian component while preserving nearly identical skill for moderate to large tracer counts (Deng et al., 2024).

4. Multiscale coupling between Lagrangian and Eulerian representations

The multiscale organization of LEMDA is explicit. In the spectral formulation, the flow is decomposed into coarse and fine components,

$$\frac{d\mathbf{x}_l}{dt} = \mathbf{v}_l + \sigma_x \dot{\mathbf{W}}_l, \qquad \frac{d\mathbf{v}_l}{dt} = \beta(\mathbf{u}(\mathbf{x}_l,t)-\mathbf{v}_l) + \mathbf{f}_l,$$9

where $\mathbf{u}(\mathbf{x},t)$0 contains low-wavenumber modes and $\mathbf{u}(\mathbf{x},t)$1 contains high-wavenumber modes. Eulerian DA is applied first on a coarse grid to recover the large-scale modes. Then, within each grid cell, a local Lagrangian DA problem is solved for the fine-scale modes, taking the large-scale component as known forcing. The small-scale reconstructions are independent across cells and therefore parallelizable (Deng et al., 2024).

A closely related physical-domain reformulation replaces Fourier space with the physical domain. Its stated motivation is the ability to deal in non-periodic system and a more intuitive representation of localised phenomena or time-dependent problems. In that formulation, the Eulerian backbone is a two-layer quasi-geostrophic model on a regular grid, the data-assimilation backbone is a Conditional Gaussian Nonlinear System, and “multiscale” is interpreted primarily as vertical layering and horizontal scale separation between grid-scale flow and particle-scale tracers (Yun et al., 18 Sep 2025). The particle observation operator is written as trajectory advection by geostrophic velocity,

$\mathbf{u}(\mathbf{x},t)$2

which makes the Lagrangian–Eulerian coupling explicit in physical space (Yun et al., 18 Sep 2025).

The effective-average-action formulation offers a different multiscale interpretation of the same problem. There, the effective average action $\mathbf{u}(\mathbf{x},t)$3 is a coarse-grained cost functional and its gradient

$\mathbf{u}(\mathbf{x},t)$4

is the coarse-grained sensitivity. In the slow–fast decomposition, the sensitivity with respect to slow variables is computed as an average over fast fluctuations. This is described as conceptually similar to multiscale DA frameworks, including LEMDA, in which fast processes are marginalized to obtain consistent slow-variable updates (Sugiura, 2015). A plausible implication is that LEMDA can be viewed not only as a data-structuring strategy but also as a scale-consistent inference principle.

5. Computational variants and implementation pathways

Several subsequent formulations instantiate LEMDA-like ideas with different numerical architectures. One direction uses ensemble methods for particle-based simulations. In that setting, two methodologies are introduced: an intermediary Eulerian transformation that combines a projection with a remeshing process, and a purely Lagrangian scheme for situations where remeshing is not appropriate. These are evaluated in one-dimensional advection–diffusion and in a two-dimensional incompressible Vortex-In-Cell setting, and they are presented as feasible ways to bring DA into particle-based discretizations that do not naturally share a fixed Eulerian state space (Duvillard et al., 2024).

A second direction is domain decomposition for drifting sea-ice floe dynamics. There, the Eulerian domain is partitioned into subdomains, an ensemble transform Kalman filter is imposed in each subdomain to recover local fine-scale ocean features, and a Gaussian-weighted blending step reconstructs a globally consistent flow field across subdomain boundaries. The reported outcome is consistently better skill scores characterised by normalised root mean square error and pattern correlation coefficients than a global and expensive DA baseline (Li et al., 20 Feb 2026). This architecture is directly compatible with LEMDA’s local-Lagrangian/global-Eulerian multiscale logic.

A third direction is data-driven latent conditional-Gaussian reduction. The Lagrangian conditional Gaussian Koopman network embeds Eulerian flow dynamics into a low-dimensional latent space governed by a nonlinear stochastic system with conditional Gaussian structure, enabling analytic posterior updates without ensemble forecasting. The framework adds tracer homogenization to enforce permutation equivariance, Fourier positional encoding to reconstruct local flow features at moving tracer locations, and an SVD-inspired low-rank parameterization of the latent transition operator (Wang et al., 14 Mar 2026). In LEMDA terms, this supplies a reduced-order Eulerian core and a learned Lagrangian–Eulerian observation map.

A fourth direction focuses on joint observability of flow fields and particle properties from Lagrangian particle tracking. The neural data-assimilation framework couples an Eulerian flow representation with Lagrangian particle models and establishes empirical existence proofs of joint observability in a turbulent boundary layer, homogeneous isotropic turbulence with inertial particles, and a compressible shock-dominated flow. It reports simultaneous recovery of carrier fields and particle properties under the governing equations of disperse multiphase flow (Zhou et al., 1 Oct 2025). This suggests an extension of LEMDA from state recovery alone to joint state–parameter inference.

6. Applications, empirical behavior, and limitations

The baseline LEMDA paper reports skilful numerical results for the Lagrangian component, the Eulerian component, and the combined multiscale configuration. It states that the Lagrangian DA has advantages when a moderate number of particles is used, while the Eulerian DA can effectively save computational costs when the number of particle observations becomes large. It also states that the Eulerian DA is valuable when particles collide, such as using sea-ice floe trajectories as observations, because averaging in each grid cell smooths out collision-induced noise in the observation process (Deng et al., 2024).

The physical-domain extension studies Arctic sea-ice floe trajectories and recovery of ocean eddies with a two-layer quasi-geostrophic model and CGNS. Performance is evaluated using normalised root mean square error and pattern correlation. In the reported grid-search experiments, RMSE decreases with increasing $\mathbf{u}(\mathbf{x},t)$5, roughly converging to $\mathbf{u}(\mathbf{x},t)$6, nearly independent of grid size, while pattern correlation increases with both $\mathbf{u}(\mathbf{x},t)$7 and spatial resolution and reaches up to $\mathbf{u}(\mathbf{x},t)$8 for $\mathbf{u}(\mathbf{x},t)$9. The computational cost scales approximately as $\beta$0, and the heaviest case takes about 30 minutes on a modest laptop (Yun et al., 18 Sep 2025).

The same literature is explicit about limitations. In the baseline framework, standard Lagrangian DA remains sensitive to strong observational nonlinearity, high dimensionality, and imperfect modeling of external forces and collisions (Deng et al., 2024). In the physical-domain formulation, particle trajectories are traced using assimilated Eulerian fields, but their trajectories are not yet inverted to update $\beta$1 and $\beta$2, so the full LEMDA coupling is still under development; that work also notes computational cost, complexity in deriving CGNS matrices, and proposes deploying neural networks to accelerate recovery of local particle information for the fine scale (Yun et al., 18 Sep 2025). In neural joint-observability studies, sparse seeding, high noise, and large Stokes number can render the inverse problem ill-conditioned, while some particle properties, such as density in the compressible case, are only weakly observable (Zhou et al., 1 Oct 2025).

Taken together, these formulations define LEMDA less as a single algorithm than as a technical program: derive or learn an Eulerian representation from Lagrangian observations, preserve multiscale structure across representations, and exploit conditional-Gaussian, reduced-order, localized, or domain-decomposed updates to make the resulting inference problem tractable.