|Abstract as per original application
In shape optimization problems, the aim is to find a class of shapes that optimizes some prescribed goals under a partial differential equation (PDE) constraint. One example is drag minimization, seeking the shape of an impermeable object exhibiting the least drag while flowing through a fluid. The classical formulation for such problems involves representing the unknown shape as a mathematical surface, and seeks to derive optimality conditions (equations/inequalities involving nonlinear PDEs). The optimal shapes can then be realized numerically by solving the optimality conditions. However, shape optimization problems are often severely ill-posed (as the existence of optimal shapes is not guaranteed in general) and can be computationally intensive (due to the requirement of a new approximation of the computational domain at every numerical iteration).
We propose to use a phase field approach to overcome the above problems, and for drag minimization, it has several important mathematical and numerical advantages over the classical approach; namely the phase field method can be justified (in the sense that existence of optimal shapes can be proved rigorously) and in addition both the PDE constraint and the optimality conditions are solved on the same fixed domain throughout the optimization procedure. Furthermore, by sending a small parameter to zero, the optimality conditions from the classical formulation are recovered. Hence, one may view the phase field approach as a consistent approximation for shape optimization problems.
The goal of this proposal is to develop phase field approximations to study nonlinear inverse problems that can be casted as shape optimization problems. In light of the mathematical advantages outlined above, we believe that the proposed phase field approach may offer novel and alternative resolution strategies for such problems. Firstly, we consider an abstract shape optimization problem with the aim of providing analytical results for the PDE constraint and establish the existence of optimal shapes, the derivation of optimality conditions, as well as the connection with the classical formulation. Then, we apply this theory to two important examples of inverse problems: inverse acoustic scattering and electrical impedance tomography. Numerical simulations will be carried out to compare the proposed phase field approach with current state-of-the-art methods, and successful investigations will serve to enhance the applicability of the phase field approach.
形狀優化問題的目標是找到一類在偏微分方程（PDE）約束下優化某些規定目標的形狀。一個例子是尋求在流體內表現出最小阻力的物體的形狀。針對這些問題的經典方法涉及將未知的形狀表示為數學表面，並尋求導出最優性條件（涉及非線性PDE的方程/不等式）。然後，通過求解最優性條件，可以在數值上實現最佳形狀。然而，形狀優化問題往往是嚴重不適（因為通常不能保證最優形狀的存在）並且可以是計算密集型的（由於在每次數值迭代時需要新的計算域）。我們建議使用相場模式方法來克服上述問題，並且對於拖曳最小化的形狀優化問題，它比傳統方法具有幾個重要的數學和數值優勢: 相場模式方法可以有進一步分析（可以嚴格證明最佳形狀的存在）並且另外通過優化過程在相同的計算域上解決PDE約束和最優性條件。此外，通過發送一個小參數為零，就獲得了經典方法的最優性條件。因此，可以將相場模式方法視為形狀優化問題的一致近似。本計畫提議的目標是開發相場模式近似以研究可以作為形狀優化問題的非線性反問題。鑑於上述的數學優勢，我們認為所提出的相場模式方法可以為這些問題提供新穎的解決方案策略。首先，我們考慮一個抽象的形狀優化問題，目的是為PDE約束提供分析結果，建立最優形狀的存在，最優性條件的推導，以及與經典方法的聯繫。然後，我們將這個理論應用於兩個重要的反問題：逆聲散射和電阻抗斷層成像。 將進行數值模擬以將所提出的相場模式方法與當前最先進的方法進行比較，並且成功的研究將有助於增強相場模式方法的適用性。