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An inverse problem formulation of the immersed boundary method
We formulate the immersed-boundary method (IBM) as an inverse problem. A
control variable is introduced on the boundary of a larger domain that
encompasses the target domain. The optimal control is the one that minimizes
the mismatch between the state and the desired boundary value along the
immersed target-domain boundary. We begin by investigating a na\"ive problem
formulation that we show is ill-posed: in the case of the Laplace equation, we
prove that the solution is unique but it fails to depend continuously on the
data; for the linear advection equation, even solution uniqueness fails to
hold. These issues are addressed by two complimentary strategies. The first
strategy is to ensure that the enclosing domain tends to the true domain as the
mesh is refined. The second strategy is to include a specialized parameter-free
regularization that is based on penalizing the difference between the control
and the state on the boundary. The proposed inverse IBM is applied to the
diffusion, advection, and advection-diffusion equations using a high-order
discontinuous Galerkin discretization. The numerical experiments demonstrate
that the regularized scheme achieves optimal rates of convergence and that the
reduced Hessian of the optimization problem has a bounded condition number as
the mesh is refined