Least square-control optimization formulation

by Yi Zhang

This is a general optimization application to the domain decomposition methods. Trying to build a constrained least squares problem and searching for the minimization of the square norm objective functional, subject to constraints. In DD application, the square norm is usually a meaure of difference between subdomain solutions, and the constraints require the local solution of local subdomain problems. The boundary data on the overlapping/interface part of the domain is unknown, and determined to minimize the square norm. So, basically the method consists of three factors: square norm measure the sub-solution distance, constraints ensure the local solution and the subdomain data as to-be-determined parameter in optimization problem. This type of methods was first applied to DD by Lions and Glowinsky, who came up with an overlapping formulation coupling the Laplace equation with Navier-Stokes equation in flow problem. For those heterogeneous-type problem, general well-poseness results are not known.