A NUMERICAL GRADIENT DESCENT METHOD FORTHE GENERALIZED CHEEGER PROBLEM ALONGCUBIC SPLINE SHAPE EVOLUTION PATHS

Abstract

This paper introduces a new evolutionary gradient descent
optimization method based on cubic splines for solving the generalized Cheeger problem arising from non-uniform mass and shear strength distributions, relevant to geophysical events such as avalanches and landslides. The goal is to minimize a functional involving surface and boundary integrals to determine the safety factor and collapse domain. The evolving shape is represented using cubic spline interpolation, restricting the search to smooth geometries with minimal bending energy. The gradient of the
objective functional is computed using finite differences, and the shape is iteratively updated while preserving topology. Grid refinement confirms the method’s consistency, and the monotonic decay of the gradient norm verifies numerical stability and convergence. Two numerical experiments with different mass and shear strength distributions demonstrate the robustness, efficiency, and physical relevance of the proposed framework, offering an accurate and flexible approach for geometric optimization under complex material distributions.