Accuracy Directly Controlled Fast Direct Solution of General ${\mathcal{ H}}^{2}$ -Matrices and Its Application to Solving Electrodynamic Volume Integral Equations

The dense matrix resulting from an integral equation (IE)-based solution of Maxwell’s equations can be compactly represented by an <inline-formula> <tex-math notation="LaTeX">${\mathcal{ H}}^{2}$ </tex-math></inline-formula>-matrix. Given a general dense <inline-formula> <tex-math notation="LaTeX">${\mathcal{ H}}^{2}$ </tex-math></inline-formula>-matrix, prevailing fast direct solutions involve approximations whose accuracy can only be indirectly controlled. In this paper, we propose new direct solution algorithms whose accuracy is directly controlled, including both factorization and inversion, for solving general <inline-formula> <tex-math notation="LaTeX">${\mathcal{ H}}^{2}$ </tex-math></inline-formula>-matrices. Different from the recursive inverse performed in existing <inline-formula> <tex-math notation="LaTeX">${\mathcal{ H}}^{2}$ </tex-math></inline-formula>-based direct solutions, this new direct solution is a one-way traversal of the cluster tree from the leaf level all the way up to the root level. The underlying multiplications and additions are carried out as they are without using formatted multiplications and additions whose accuracy cannot be directly controlled. The cluster bases and their rank of the original matrix are also updated level by level based on prescribed accuracy, without increasing computational complexity, to take into account the contributions of fill-ins generated during the direct solution procedure. For constant-rank <inline-formula> <tex-math notation="LaTeX">${\mathcal{ H}}^{2}$ </tex-math></inline-formula>-matrices, the proposed direct solution has a strict <inline-formula> <tex-math notation="LaTeX">$O(N)$ </tex-math></inline-formula> complexity in both time and memory. For rank that linearly grows with the electrical size, the complexity of the proposed direct solution is <inline-formula> <tex-math notation="LaTeX">$O(N\text {log}N)$ </tex-math></inline-formula> in factorization and inversion time, and <inline-formula> <tex-math notation="LaTeX">$O(N)$ </tex-math></inline-formula> in solution time and memory for solving volume IEs (VIEs). Rapid direct solutions of electrodynamic VIEs involving millions of unknowns have been obtained on a single CPU core with directly controlled accuracy. Comparisons with state-of-the-art <inline-formula> <tex-math notation="LaTeX">${\mathcal{ H}}^{2}$ </tex-math></inline-formula>-based direct VIE solvers have also demonstrated the advantages of the proposed direct solution in accuracy control, as well as achieving better accuracy with much less CPU time.