Accuracy directly controlled fast direct solutions of general H2-matrices and its application to electrically large electromagnetic analysis

The dense matrix resulting from an integral equation (IE) based solution of Maxwell's equations can be compactly represented by an H²-matrix. Given a general H²-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 H²-matrices. Different from existing direct solutions, where the cluster bases are kept unchanged in the solution procedure thus lacking an 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 small, being the rank at that level. The proposed new direct solution has been applied to solve electrically large volume IEs. 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. Rapid direct solutions of millions of unknowns can be obtained on a single CPU core with directly controlled accuracy.