Robust Adaptive Meshing, Mesh Density Functions, and Nonlocal Observations for Ensemble Based Data Assimilation

TL;DR

This work addresses the challenge of performing ensemble-based data assimilation on adaptive spatial meshes without incurring excessive interpolation costs. It introduces robust look-ahead meshes formed via metric tensor intersections, fixing a common nonuniform mesh across all ensemble members during a forecast interval and incorporating near-future observations, aided by quasi-Lagrangian moving mesh DG and embedded-time stepping. The authors develop and deploy nonlocal observation metrics, Hessian-based mesh densities, and goal-oriented tensor formulations to steer refinement toward observation supports and DA performance, while providing a detailed computational framework (MMDG, SSP RK, and interpolation). Numerical experiments on 2D Burgers’, 2D Shallow Water, and coupled KS systems demonstrate stable, accurate DA with reduced inter-mesh interpolation and improved vectorization, illustrating the practical impact for high-dimensional, multi-physics DA problems. The work lays out a modular path for extending adaptive meshing in DA to more complex, multi-component systems and nonlocal observational operators.

Abstract

Adaptive spatial meshing has proven invaluable for the accurate, efficient computation of solutions of time dependent partial differential equations. In a DA context the use of adaptive spatial meshes addresses several factors that place increased demands on meshing; these include the location and relative importance of observations and the use of ensemble solutions. To increase the efficiency of adaptive meshes for data assimilation, robust look ahead meshes are developed that fix the same adaptive mesh for all ensemble members for the entire time interval of the forecasts and that incorporates the observations at the next analysis time. This allows for increased vectorization of the ensemble forecasts while minimizing interpolation of solutions between different meshes. The techniques to determine these robust meshes are based upon combining metric tensors or mesh density functions to define nonuniform meshes. We illustrate the robust ensemble look ahead meshes using traveling wave solutions of a bistable reaction-diffusion equation. Observation operators based on convolution type integrals and their associated metric tensors are derived. These further the goals of making efficient use of adaptive meshes in ensemble based DA techniques, developing and employing robust meshes that are effective for a range of similar behaviors in both the ensembles and the observations, and the integration with advanced numerical PDE techniques (a quasi-Lagrangian moving mesh DG technique employing embedded pairs for time stepping). Numerical experiments with different observation scenarios are presented for a 2D inviscid Burgers' equation, a multi-component system, a 2D Shallow Water model, and for a coupled system of two 1D Kuramoto-Sivashinsky equations.

Figures (15)

• Figure 1: Combining meshes to form a robust, commmon, look ahead mesh.
• Figure 2: Timesteps of the small ensemble employed when forming the LAH mesh and timesteps of the full ensemble when using the LAH as a fixed mesh. $\mathbf{P}_0$ is the covariance matrix of the initial perturbations of the ensemble members.
• Figure 3: Snapshots of moving observations for observation scenarios I and II. Green denotes current observation locations and red denotes the locations at the previous observation time.
• Figure 4: 2D Burgers: $N_u=1861$, Remesh after Analysis $\Delta t = 1$
• Figure 5: 2D Burgers: $N_u=1861$, Remesh after Analysis $\Delta t = 1$

Theorems & Definitions (1)

• Remark 1