Development of efficient data assimilation methods for solute transport problems

Abstract

The solution of marine contaminant transport problems is a significant research topic in civil engineering. Typically, the problem is represented as a partial differential equation described by the non-stationary, advection-diffusion operator. The underlying equation is approximated in space and time, and the state of the approximate numerical model is obtained as a solution of the corresponding linear algebraic system. However, for several reasons, numerical models do not necessarily replicate the process investigated exactly. In fact, their application and deployment generate modelling errors and discrepancies, a well-known and challenging problem to solve, which cannot be ignored in practice. At the same time, the rapid development of measuring devices allows easier collection of data of the physical process. Typically, observations are prone to be contaminated by errors too, those generated by noise or other physical reasons. Moreover, observations are usually quite sparse in time and space. The combination and compromise between a numerical model and observations are integrated using data assimilation techniques. The development of efficient methods of data assimilation techniques in terms of estimation quality and computation speed is the main concern of this research. The traditional algorithms of data assimilation such as minimax or Kalman filters are often used to quantify uncertainties represented by the model and observation errors. They construct an analysis state and propagate in time, taking into account model dynamics and observed information. The numerical algorithms of these filters are computationally expensive as they require multiplication and inversion of matrices of the size equal to the number of degrees of freedom of the system. Moreover, traditional filters are not scalable with respect to the number of discretisation nodes. In this research, a combination of traditional filters with domain decomposition techniques are investigated to assess reduction of computational costs. The application of decomposition to the assimilation problem facilitates the reformulation of the global problem as a set of local subproblems coupled by continuity or transmission conditions. To solve the decomposed assimilation problem, two new approaches are considered. The first one discretises the transmission conditions directly and yields a system of differential-algebraic equations. The latter is solved by using a modified version of the minimax filter. The second approach imposes transmission conditions into the variational formulation of the local subproblems. A set of local differential problems is solved by the iterative method of Schwarz. This approach is further extended to Kalman and ensemble filters using their equivalence with the minimax filter. The efficiency of the proposed methods is examined using numerical experiments with different configurations including simulations with constant velocity field, periodic velocity field and velocity field generated by TELEMAC 2D for a tidal basin. The quality of estimates of the localised filters is assessed against both traditional filters and true solutions. The computational efficiency of the localised filters is evaluated, compared to the existing methods and discussed. Finally, scalability properties of the proposed algorithms are presented.