EFFICIENT NUMERICAL METHOD FOR SHAPE OPTIMIZATION PROBLEM CONSTRAINED BY STOCHASTIC ELLIPTIC INTERFACE EQUATION

This paper provides a computational framework for stochastic shape optimization problems. The primary control objective involves minimizing the expected value of a tracking cost function while adhering to constraints posed by a stochastic elliptic interface equation. This study incorporates stochastic shape variation and shape derivatives, enabling the establishment of a diminishing sequence of admissible interfaces. The finite element method is employed for discretizing both state and adjoint systems, yielding mesh displacement directions. Notably, the mesh manipulation technique utilized within the optimization process is subject to regularization through cubic spline interpolation employing a centripetal scheme. To ensure the fidelity of the interface curve and prevent undesirable twisting, the convex hull technique is incorporated, thus upholding mesh displacement consistency. To mitigate the computational demands associated with uncertainty quantification, the sparse grid collocation method is enlisted. This technique facilitates the matching of probability distributions, particularly in scenarios involving relatively high- dimensional problems. Moreover, for extensive-scale sampling optimization, a stochastic sampling-based descent method is integrated. The effectiveness and efficiency of the proposed algorithms are underscored through a series of numerical experiments, which serve to validate their performance.