Data assimilation in price formation

We consider the problem of estimating the density of buyers and vendors in a nonlinear parabolic price formation model using measurements of the price and the transaction rate. Our approach is based on a work by Puel et al., see \cite{Puel2002}, and results in a optimal control problem. We analyse this problems and provide stability estimates for the controls as well as the unknown density in the presence of measurement errors. Our analytic findings are supported with numerical experiments

Variational data assimilation for two interface problems

“Variational data assimilation (VDA) is a process that uses optimization techniques to determine an initial condition of a dynamical system such that its evolution best fits the observed data. In this dissertation, we develop and analyze the variational data assimilation method with finite element discretization for two interface problems, including the Parabolic Interface equation and the Stokes-Darcy equation with the Beavers-Joseph interface condition. By using Tikhonov regularization and formulating the VDA into an optimization problem, we establish the existence, uniqueness and stability of the optimal solution for each concerned case. Based on weak formulations of the Parabolic Interface equation and Stokes-Darcy equation, the dual method and Lagrange multiplier rule are utilized to derive the first order optimality system (OptS) for both the continuous and discrete VDA problems, where the discrete data assimilations are built on certain finite element discretization in space and the backward Euler scheme in time. By introducing auxiliary equations, rescaling the optimality system, and employing other subtle analysis skills, we present the finite element convergence estimation for each case with special attention paid to recovering the properties missed in between the continuous and discrete OptS. Moreover, to efficiently solve the OptS, we present two classical gradient methods, the steepest descent method and the conjugate gradient method, to reduce the computational cost for well-stabilized and ill-stabilized VDA problems, respectively. Furthermore, we propose the time parallel algorithm and proper orthogonal decomposition method to further optimize the computing efficiency. Finally, numerical results are provided to validate the proposed methods”--Abstract, page iii