An inverse problem formulation of the immersed boundary method

TL;DR

This paper reformulates the immersed boundary method as an inverse problem, introducing control variables and regularization strategies to address ill-posedness, and demonstrates optimal convergence in numerical experiments.

Contribution

It presents a novel inverse formulation of IBM with regularization techniques, improving stability and convergence for boundary value problems.

Findings

Regularized inverse IBM achieves optimal convergence rates.

The reduced Hessian maintains a bounded condition number with mesh refinement.

The method effectively handles diffusion, advection, and advection-diffusion equations.

Abstract

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.