Multigrid for Atmospheric Data Assimilation: Analysis

Abstract

Multigrid methods are known to provide the most efficient solvers to many well-posed boundary-value PDE problems. In the case of ill-determined problems they can supply several additional advantages. Unlike evolution problems with well-posed initial conditions which can be solved by direct marching in time, when only scattered data are known, each datum affects both earlier and later solution values, so simple marching cannot be used, and fast solvers would again require multigrid methods. Multigrid solver can provide natural regularization to the ill-posed problem, since the main ill-posedness is the long term and long range influence of fine-scale oscillations, while the multiscale large-scale interactions are mediated by coarse grids that omit those oscillations. As a model problem we treat a hyperbolic PDE: the wave equation with only approximately known coefficients. The results of a detailed Fourier analysis, comparing full-flow control with initialvalue control are presented.