Application of the Orthogonal QD Algorithm with Shift to Singular Value Decomposition for Large Sparse Matrices

In semiconductor manufacturing process, lithography simulation modeling is known as an ill-posed problem. A normal solution of the problem is generally insignificant due to measurement constraints. In order to alleviate the difficulty, we introduced a regularization method using a preconditioning technique which consists of scaling and uniformization based on prior information. By regularizing the solution to prior knowledge, an accurate model can be achieved because the solution using truncated singular value decomposition from a few larger singular values becomes a reasonable solution based on the physically appropriate prior knowledge. The augmented implicitly restarted Lanczos bidiagonalization (AIRLB) algorithm is suitable for the purpose of the truncated singular value decomposition from a few larger singular values. Thus, the AIRLB algorithm is important for obtaining the solution in lithography simulation modeling. In this paper, we propose techniques for improving the AIRLB algorithm for the truncated singular value decomposition of large matrices. Specifically, we implement the improvement of the AIRLB algorithm by Ishida et al. Furthermore, instead of using the QR algorithm, we use the orthogonal-qd-with-shift algorithm for the singular value decomposition of the inner small matrix. Several numerical experiments demonstrate that, compared with AIRLB using the original QR algorithm, the proposed improvements provide highly accurate truncated singular value decomposition. For precise discussion, both large-scale sparse matrices and large-scale dense matrices are included in the experiments.