A Short Note on Meshfree PDE-constrained Optimization using Moving Least Squares

This short note discusses meshfree approach to PDE-constrained optimization. For meshfree discretization of partial differential equations, “Moving Least Squares (MLS)” was adopted, and for searching the best solution which minimize the evaluation function, “Constrained Least Squares (CLS)” was used. The combination of MLS and CLS was applied to tidal motion and found to work well for solving PDEs-described system. On the other hand, when this method is applied to a kind of data assimilation problem where the observation data have some random error, it was found that the raw PED model is too sensitive to the data error and leads to an over-fitted result which is full of short wavelength-noise. To overcome this difficulty, a Laplacian filter was introduced and found that it is effective to supress the undesirable growth of short wave component, and gives reasonable fitting result.