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.

中文翻译：

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浸入边界法的逆问题公式

我们将浸入边界法 (IBM) 表述为一个逆问题。在包含目标域的更大域的边界上引入控制变量。最优控制是使状态与沿浸入目标域边界的所需边界值之间的失配最小化的控制。我们首先研究一个我们证明是不适定的朴素问题公式：在拉普拉斯方程的情况下，我们证明解是唯一的，但它不能连续依赖于数据；对于线性平流方程，甚至解决方案的唯一性也不能成立。这些问题可以通过两个互补的策略来解决。第一个策略是确保随着网格细化，封闭域趋向于真实域。第二种策略是包含一个专门的无参数正则化，它基于惩罚边界上的控制和状态之间的差异。使用高阶不连续 Galerkin 离散化将建议的逆 IBM 应用于扩散、对流和对流-扩散方程。数值实验表明，正则化方案实现了最佳收敛速度，并且优化问题的简化 Hessian 在网格细化时具有有界条件数。