Erik S Van Vleck | Searcharxiv

arXiv2021

A Metric Tensor Approach to Data Assimilation with Adaptive Moving Meshes

Adaptive moving spatial meshes are useful for solving physical models given by time-dependent partial differentialequations. However, special consideration must be given when combining adaptive meshing procedures with ensemble-based data assimilation (DA) techniques. In particular, we focus on the case where each ensemble member evolvesindependently upon its own mesh and is interpolated to a common mesh for the DA update. This paper outlines aframework to develop time-dependent reference meshes using locations of observations and the metric tensors (MTs)or monitor functions that define the spatial meshes of the ensemble members. We develop a time-dependent spatiallocalization scheme based on the metric tensor (MT localization). We also explore how adaptive moving mesh tech-niques can control and inform the placement of mesh points to concentrate near the location of observations, reducingthe error of observation interpolation. This is especially beneficial when we have observations in locations that wouldotherwise have a sparse spatial discretization. We illustrate the utility of our results using discontinuous Galerkin(DG) approximations of 1D and 2D inviscid Burgers equations. The numerical results show that the MT localizationscheme compares favorably with standard Gaspari-Cohn localization techniques. In problems where the observationsare sparse, the choice of common mesh has a direct impact on DA performance. The numerical results also demonstratethe advantage of DG-based interpolation over linear interpolation for the 2D inviscid Burgers equation.

math.NA↗

arXiv2020

Particle filters for data assimilation based on reduced order data models

We introduce a framework for Data Assimilation (DA) in which the data is split into multiple sets corresponding to low-rank projections of the state space. Algorithms are developed that assimilate some or all of the projected data, including an algorithm compatible with any generic DA method. The major application explored here is PROJ-PF, a projected Particle Filter. The PROJ-PF implementation assimilates highly informative but low-dimensional observations. The implementation considered here is based upon using projections corresponding to Assimilation in the Unstable Subspace (AUS). In the context of particle filtering, the projected approach mitigates the collapse of particle ensembles in high dimensional DA problems while preserving as much relevant information as possible, as the unstable and neutral modes correspond to the most uncertain model predictions. In particular we formulate and numerically implement a projected Optimal Proposal Particle Filter (PROJ-OP-PF) and compare to the standard optimal proposal and to the Ensemble Transform Kalman Filter.

stat.CO↗