Perhaps the most widely used technique for obtaining accurate numerical solutions of multiscale, multiphysics problems is operator decomposition. The general approach is to decompose a given model into components involving simpler physics over a relatively limited range of scales, and then to seek the solution of the entire system by using numerical solutions of the individual components. This approach has many appealing aspects, and in particular, provides a natural way to tackle problems encompassing multiple time and length scales. However, multiscale operator decomposition affects both accuracy and stability in both obvious and subtle ways that are difficult to quantify accurately.
In this talk, I will describe a powerful approach for quantifying sensitivity, error, and uncertainty based on duality and adjoint operators. This approach provides a way to quantify the effects of stability, which is the key to accurate error estimation. I will explain my view of stability, describe the connection to duality and adjoints, and apply these ideas to a variety of multiscale, multiphysics problems.