Controlled Accuracy Direct H2-Matrix Arithmetic for Fast and Large-Scale Solutions of Maxwell’s Equations

An H2-matrix is a structured matrix but general enough for representing both PDE and IE operators in electromagnetic analysis. An example is the matrix resulting from a fast multiple method (FMM) based representation of integral equation operators. Prevailing arithmetic of H2-matrix in direct solutions involves theoretical approximations since formatted multiplications and additions are performed. In this short course, we will present novel error-controlled fast direct solution and matrix-matrix product algorithms for general H2-matrices, where the underlying multiplications and additions are carried out as they are with theoretically controlled accuracy. Applications to the analysis of large-scale integrated circuits and system problems have demonstrated a clear O(N) complexity in direct solutions with controlled accuracy.