2025/04/09 by Gang Bao, Bao, Gang, Jun Lai +3
Decision Sciences · Engineering · Mathematics · #30B20 #35J05 #45A05 #49J50 #78M50 #FOS: Mathematics #Microwave Imaging and Scattering Analysis #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.2504.06621
openalex publication_date 2025/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Shape derivative is an important analytical tool for studying scattering problems involving perturbations in scatterers. Many applications, including inverse scattering, optimal design, and uncertainty quantification, are based on shape derivatives. However, computing high order shape derivatives is challenging due to the complexity of shape calculus. This work introduces a comprehensive method for computing shape Taylor expansions in two dimensions using recurrence formulas. The approach is developed under sound-soft, sound-hard, impedance, and transmission boundary conditions. Additionally, we apply the shape Taylor expansion to uncertainty quantification in wave scattering, enabling high order moment estimation for the scattered field under random boundary perturbations. Numerical examples are provided to illustrate the effectiveness of the shape Taylor expansion in achieving high order approximations.