Weak-norm posterior contraction rate of the 4DVAR method for linear severely ill-posed problems

Inspired by the artificial dynamic, we solve the linear severely ill-posed problems based on the 4DVAR method arising in data assimilation. To obtain the consistency of the posterior distribution in the weak norm by a fractional order of the forward operator, we are interested in asymptotic behavior of the weak norm squared posterior contraction (SPCL) function. By assuming exponentially decaying spectrum of the forward operator, the estimate of the SPCL function is established for a Sobolev-like source condition. Such a result is further extended to a general source condition. In both cases, we verify that the severe ill-posedness of the inverse problem can be reduced to moderate ill-posedness by the weak norm.