Fast Analysis and Optimization of Sparsely Distributed Partial Modification Problems

This article addresses the efficient numerical analysis of sparsely distributed small modifications in a large structure, that is, the sparsely distributed partial modification problem (SDPMP). A hierarchical matrix (H-matrix)-based fast direct solver is adopted to solve the SDPMP in a very short time for each new set of modifications. The main idea of this method is to reuse the parts of the compressed inverse matrix operator that do not change and only recompute the modified parts. In contrast with already existing algorithms designed to tackle a single localized modification, this new approach is still very fast for sparsely distributed small modifications, so it becomes a very attractive option to solve optimization problems efficiently, due to the very cheap analysis of the modified structure at each iteration of the optimization procedure. The efficiency of the solution of SDPMP is demonstrated for several cases involving passive and active metasurfaces and a reconfigurable antenna. As a practical example, a programmable metasurface reflector is optimized with particle swarm optimization to obtain a prescribed reflection pattern.