A Metric Tensor Approach to Data Assimilation on Adaptive Moving Meshes

Abstract

Data assimilation (DA) combines noisy, partial data with imperfect physical models in order to make better predictions about the past, current, or future state of some dynamical system. Often the partial differential equations (PDEs) used in these physical models require fine spatial resolution in some areas, and less resolution in others. As such, the physical models benefit from the use of adaptive moving mesh methods, where the location of the spatial grid points can change in time. The combination of data assimilation procedures with adaptive moving mesh techniques is nontrivial when an ensemble-based DA procedure is employed. Ensemble DA procedures use an ensemble of solutions to the physical model, and the ensemble mean and covariance must be calculated each time an observation is incorporated into the model. Therefore, special care must be taken in the case where each ensemble member is allowed to evolve on its own independent mesh. This thesis combines ensemble data assimilation techniques with adaptive moving mesh methods through a novel adaptive common mesh. Each ensemble member evolves independently on its own adaptive mesh through the use of a monitor function, or metric tensor. The information from these ensemble metric tensors is used to define an adaptive common mesh at each observation time-step through the metric tensor intersection. Unlike previous works, this method easily generalizes to higher spatial dimensions. One benefit to this approach is that the common mesh can also be concentrated near observation locations by modifying the monitor function that controls the movement of the adaptive common mesh. This greatly reduces the need to interpolate observation values, which can cause significant errors in the DA predictions. This gives the user a flexible framework where the meshing scheme can either be chosen so that the adaptive mesh sufficiently supports all ensemble members, is concentrated near the observations, or some combination thereof, depending on what is best for their problem. In addition to the framework for the adaptive common mesh, we also provide a new localization scheme based off the metric tensor intersection. This domain-based localization scheme, through the use of a tuning parameter, gives a wider radius of influence to observations that occur in smooth parts of the numerical solution and a smaller radius of influence to the observations that occur near sharp interfaces or gradients. Through various test problems in one and two spatial dimensions, we show that this localization scheme is competitive with two of the standard domain-based localization schemes widely used. Though the use of an independent adaptive mesh for each ensemble member can be advantageous, it also can be computationally expensive. Therefore, we also develop a method to have all ensemble members reside on the same mesh, eliminating much of the interpolation. We use a sample of the ensemble members to “look ahead” at each observation step and determine what the next common mesh would be if each of the sample representatives evolved on its own independent mesh. If that new mesh differs significantly from the current mesh, we then use the new mesh for all of the ensemble members. If, on the other hand, it is similar (within some tolerance) to the current mesh, we continue to update via DA and integrate in time on the current mesh. The efficacy of these methods is demonstrated in both one and two spatial dimensions with the inviscid Burgers equation, in which a steep shock forms and moves across the spatial domain. This is a common test problem for DA with adaptive meshes, because it requires a finer resolution near the shock, and can use a coarser mesh in areas away from the shock. Finally, we consider a more systematic approach to determining the common mesh. A variational approach can be used to optimize the RMSE, the norm of the innovation, or some other measure of DA success, subject to a linear combination of constraints, such as the equidistribution of arclength or the proximity of common mesh points to the observation locations.