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

Abstract

The dense matrix resulting from an integral equation (IE) based solution of Maxwell's equations can be compactly represented by an ${\cal H}^2$-matrix. Given a general dense ${\cal H}^2$-matrix, prevailing fast direct solutions involve approximations whose accuracy can only be indirectly controlled. In this work, we propose new accuracy-controlled direct solution algorithms, including both factorization and inversion, for solving general ${\cal H}^2$-matrices, which does not exist prior to this work. Different from existing direct solutions, where the cluster bases are kept unchanged in the solution procedure thus lacking explicit accuracy control, the proposed new algorithms update the cluster bases and their rank level by level based on prescribed accuracy, without increasing computational complexity. Zeros are also introduced level by level such that the size of the matrix blocks computed at each tree level is the rank at that level, and hence being small. The proposed new direct solution has been applied to solve electrically large volume IEs whose rank linearly grows with electric size. A complexity of $O(NlogN)$ in factorization and inversion time, and a complexity of $O(N)$ in storage and solution time are both theoretically proven and numerically demonstrated. For constant-rank cases, the proposed direct solution has a strict $O(N)$ complexity in both time and memory. Rapid direct solutions of millions of unknowns can be obtained on a single CPU core with directly controlled accuracy.