Interpolated Discrepancy Data Assimilation for PDEs with Sparse Observations

TL;DR

This work tackles data assimilation for dissipative PDEs with sparse observations by introducing Interpolated Discrepancy Data Assimilation (IDDA), which embeds the interpolated discrepancy both as a forcing term and inside the nonlinear operator. Theoretical results establish exponential convergence under explicit bounds linking observation spacing $h$, nudging strength $\lambda$, and diffusion $\mu$, with a decay rate $\gamma = \lambda\alpha - \frac{L^2 C^2 h^2}{2\mu}$. Numerical experiments on viscous Burgers', KPP–Burgers', Kuramoto–Sivashinsky, and 2D Navier–Stokes demonstrate faster, more stable convergence than interpolated AOT under sparse data and show compatibility with explicit time stepping. The findings offer a practical, robust upgrade for operational systems constrained by sparse sensor coverage, with clear guidelines on parameter windows and interpolation choices.

Abstract

Sparse sensor networks in weather and ocean modeling observe only a small fraction of the system state, which destabilizes standard nudging-based data assimilation. We introduce Interpolated Discrepancy Data Assimilation (IDDA), which modifies how discrepancies enter the governing equations. Rather than adding observations as a forcing term alone, IDDA also adjusts the nonlinear operator using interpolated observational information. This structural change suppresses error amplification when nonlinear effects dominate. We prove exponential convergence under explicit conditions linking error decay to observation spacing, nudging strength, and diffusion coefficient. The key requirement establishes bounds on nudging strength relative to observation spacing and diffusion, giving practitioners a clear operating window. When observations resolve the relevant scales, error decays at a user-specified rate. Critically, the error bound scales with the square of observation spacing rather than through hard-to-estimate nonlinear growth rates. We validate IDDA on Burgers flow, Kuramoto-Sivashinsky dynamics, and two-dimensional Navier-Stokes turbulence. Across these tests, IDDA reaches target accuracy faster than standard interpolated nudging, remains stable in chaotic regimes, avoids non-monotone transients, and requires minimal parameter tuning. Because IDDA uses standard explicit time integration, it fits readily into existing simulation pipelines without specialized solvers. These properties make IDDA a practical upgrade for operational systems constrained by sparse sensor coverage.

Figures (5)

• Figure 1: Comparison for viscous Burgers' equation.
• Figure 2: Comparison for the KPP--Burgers equation.
• Figure 3: Comparison for KS equation.
• Figure 4: Performance comparison of IDDA and AOT for the two-dimensional Navier--Stokes equations with $N_s = 400$ observations, $\lambda = 2$, and artificial diffusion $\eta = h$. Panel (a) shows vorticity field reconstructions demonstrating IDDA’s superior alignment with the reference solution. Panel (b) quantifies this improvement through spatial error distributions and convergence rates.
• Figure 5: Convergence behavior for the two-dimensional Navier--Stokes equations with varying observation density, interpolation radius, and artificial diffusion. The left panel shows convergence rates versus the number of observation points $N_s$ (ranging from 100 to 1000) for both AOT and IDDA with $\lambda = 2$, $\rho = 5h$, and $\eta = h$. IDDA achieves rates close to the theoretical target when $N_s \ge 400$. The middle panel shows the dependence of convergence rate on the RBF support radius $r=\rho h$ (ranging from $h$ to $10h$); IDDA consistently reaches the target rate for $\rho \ge 2$. The right panel examines the effect of the artificial diffusion coefficient $\eta = kh$ (ranging from $0$ to $2h$) on convergence, where AOT is augmented with matching artificial diffusion for a fair comparison. Across all tested parameters, IDDA maintains faster and stable convergence than AOT.

Theorems & Definitions (10)

• Lemma 2.5: Interpolation Inner Product Bound
• Proof 1
• Remark 2.6: Operator Classification
• Theorem 2.7: IDDA Exponential Convergence
• Proof 2: Proof of Theorem \ref{['thm:IDDA_convergence']}
• Remark 2.8: Artificial Dissipation
• Remark 2.9: Time-Dependent Nudging Parameter
• Remark 2.10: Time-Discrete Observations
• Theorem A.1: AOT Exponential Convergence
• Proof 3: Proof of Theorem \ref{['thm:AOT_convergence']}