סמינר מקוון עם ארקדי שרשבסקי
M.Sc. student under the supervision of Prof. Amir Boag and Dr. Yaniv Brick
Recent years have seen an increasing interest in the development of fast direct integral equation solvers. These do not rely on the convergence of iterative procedures for obtaining the solution. Instead, they compute a compressed factorized form of the impedance matrix resulting from the discretization of an underlying integral equation. The compressed form can then be applied to multiple right-hand sides, at a relatively low additional cost. The most common class of direct integral equation solvers exploits the rank-deficiency of off-diagonal blocks of the impedance matrix, in order to express them in a compressed manner. However, such rank deficiency is inherent to problems of small size compared to the wavelength as well as to problems of reduced dimensionality, e.g., elongated and quasi-planar problems.
The presented work proposes a class of Generalized Source Integral Equation (GSIE) formulations, which aim to extend the range of problems exhibiting inherent rank-deficiency. The new formulation effectively reduces the problem’s dimensionality and, thus, allows for efficient low-rank matrix compression. When the formulation is used with a hierarchical matrix compression and factorization algorithm, a fast direct solver is obtained. The computational bottlenecks introduced by the proposed generalized formulation are reduced by using non-uniform sampling-based techniques. These techniques are described in detail for one choice of generalized sources. The formulation’s properties and limitations are studied and its use for the development of a fast direct solver is showcased.