M. Ramakrishna

V. Charumathi, M. Ramakrishna, Vinita Vasudevan

In this paper, we propose a computationally efficient iterative algorithm for proper orthogonal decomposition (POD) using random sampling based techniques. In this algorithm, additional rows and columns are sampled and a merging technique is used to update the dominant POD modes in each iteration. We derive bounds for the spectral norm of the error introduced by a series of merging operations. We use an existing theorem to get an approximate measure of the quality of subspaces obtained on convergence of the iteration. Results on various datasets indicate that the POD modes and/or the subspaces are approximated with excellent accuracy with a significant runtime improvement over computing the truncated SVD. We also propose a method to compute the POD modes of large matrices that do not fit in the RAM using this iterative sampling and merging algorithms.

Vinita Vasudevan, M. Ramakrishna

Singular value decomposition (SVD) is a widely used technique for dimensionality reduction and computation of basis vectors. In many applications, especially in fluid mechanics and image processing the matrices are dense, but low-rank matrices. In these cases, a truncated SVD corresponding to the most significant singular values is sufficient. In this paper, we propose a tree based merge-and-truncate algorithm to obtain an approximate truncated SVD of the matrix. Unlike previous methods, our technique is not limited to "tall and skinny" or "short and fat" matrices and it can be used for matrices of arbitrary size. The matrix is partitioned into blocks and the truncated SVDs of blocks are merged to obtain the final SVD. If the matrices are low rank, this algorithm gives significant speedup over finding the truncated SVD, even when run on a single core. The error is typically less than 3\%.