A Dual-Weighted Approach to Order Reduction in 4DVAR Data Assimilation

Abstract

Strategies to achieve order reduction in four-dimensional variational data assimilation (4DVAR) search for an optimal low-rank state subspace for the analysis update. A common feature of the reduction methods proposed in atmospheric and oceanographic studies is that the identification of the basis functions relies on the model dynamics only, without properly accounting for the specific details of the data assimilation system (DAS). In this study a general framework of the proper orthogonal decomposition (POD) method is considered and a cost-effective approach is proposed to incorporate DAS information into the orderreduction procedure. The sensitivities of the cost functional in 4DVAR data assimilation with respect to the time-varying model state are obtained from a backward integration of the adjoint model. This information is further used to define appropriate weights and to implement a dual-weighted proper orthogonal decomposition (DWPOD) method for order reduction. The use of a weighted ensemble data mean and weighted snapshots using the adjoint DAS is a novel element in reduced-order 4DVAR data assimilation. Numerical results are presented with a global shallow-water model based on the Lin-Rood flux-form semi-Lagrangian scheme. A simplified 4DVAR DAS is considered in the twin-experiment framework with initial conditions specified from the 40-yr ECMWF Re-Analysis (ERA-40) datasets. A comparative analysis with the standard POD method shows that the reduced DWPOD basis may provide an increased efficiency in representing an a priori specified forecast aspect and as a tool to perform reduced-order optimal control. This approach represents a first step toward the development of an order-reduction methodology that combines in an optimal fashion the model dynamics and the characteristics of the 4DVAR DAS.

A dual‐weighted trust‐region adaptive POD 4D‐VAR applied to a finite‐element shallow‐water equations model

“…However, increasing the dimension of the POD reduced order space from 45 to 75 can increase the computational cost of POD reduced order 4-D Var. This agrees with results obtained in [35] that for practical applications, the dual-weighted procedure may be of particular benefit for use only with small dimensional bases in the context of adaptive order reduction as the minimization approaches the optimal solution. For other beneficial effects of POD 4-D Var related to its use in the framework of second-order adjoint of a global shallow water equation models, see Daescu and Navon [34].…”

“…By implementing a dual-weighted POD (DWPOD) method [10,35] Assume that the cost functional J (y(t)) is defined explicitly in terms of each state y(t) at time step t. For any fixed time step <t, the model can be written as, ∀ <t, y(t) = M →t (y( )) = M ,t (y( )) such that implicitly, the cost functional J can be viewed as a function of the previous state y( ) to first-order approximation. The impact of small errors/perturbations y i in the state error at a snapshot time t i t on J may be estimated using the tangent linear model M(t i , t) and its adjoint model M T (t, t i ), where the brackets stand for the l 2 product.…”

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

“…In a series of papers (see e.g. [31,36,37]) Navon et al proposed a dual-weighted POD method, where the weights assigned to each snapshot were derived from an adjoint related to the optimality system of a variational data assimilation problem in meteorology. It is also known that for compressible flows the choice of inner product and weighting of the different flow variables (velocity, pressure, speed of sound) in the snapshot matrix can have a large effect on the stability and accuracy of the ROM [14,35].…”

A non-intrusive reduced-order model for compressible fluid and fractured solid coupling and its application to blasting

“…POD is the method most widely used to form reduced order models and it aims to represent a large system, with only a relatively small number of basis functions, and is optimal in the sense that they minimize the L 2 error to the training set. POD has been used successfully in various fields such as air pollution [1], ocean modelling [2], fluid mechanics [3,4,5], aerospace design [6], neutron photon transport [7], porous media [8], shape optimization [9] and shock problems [10]. Reduced order models (ROMs) can be derived by a combination of POD and Galerkin projection methods.…”