Navneet Pratap Singh

Navneet Pratap Singh, Kapil Ahuja, Heike Fassbender

Recently a new algorithm for model reduction of second order linear dynamical systems with proportional damping, the Adaptive Iterative Rational Global Arnoldi (AIRGA) algorithm, has been proposed. The main computational cost of the AIRGA algorithm is in solving a sequence of linear systems. These linear systems do change only slightly from one iteration step to the next. Here we focus on efficiently solving these systems by iterative methods and the choice of an appropriate preconditioner. We propose the use of relevant iterative algorithm and the Sparse Approximate Inverse (SPAI) preconditioner. A technique to cheaply update the SPAI preconditioner in each iteration step of the model order reduction process is given. Moreover, it is shown that under certain conditions the AIRGA algorithm is stable with respect to the error introduced by iterative methods. Our theory is illustrated by experiments. It is demonstrated that SPAI preconditioned Conjugate Gradient (CG) works well for model reduction of a one dimensional beam model with AIRGA algorithm. Moreover, the computation time of preconditioner with update is on an average 2/3 rd of the computation time of preconditioner without update. With average timings running into hours for very large systems, such savings are substantial.

Seemandhar Jain, Aditya A. Shastri, Kapil Ahuja et al.

Clustering large amount of data is becoming increasingly important in the current times. Due to the large sizes of data, clustering algorithm often take too much time. Sampling this data before clustering is commonly used to reduce this time. In this work, we propose a probabilistic sampling technique called cube sampling along with K-Prototype clustering. Cube sampling is used because of its accurate sample selection. K-Prototype is most frequently used clustering algorithm when the data is numerical as well as categorical (very common in today's time). The novelty of this work is in obtaining the crucial inclusion probabilities for cube sampling using Principal Component Analysis (PCA). Experiments on multiple datasets from the UCI repository demonstrate that cube sampled K-Prototype algorithm gives the best clustering accuracy among similarly sampled other popular clustering algorithms (K-Means, Hierarchical Clustering (HC), Spectral Clustering (SC)). When compared with unsampled K-Prototype, K-Means, HC and SC, it still has the best accuracy with the added advantage of reduced computational complexity (due to reduced data size).

Navneet Pratap Singh, Kapil Ahuja

The main computational cost of algorithms for computing reduced-order models of parametric dynamical systems is in solving sequences of very large and sparse linear systems. We focus on efficiently solving these linear systems, arising while reducing second-order linear dynamical systems, by iterative methods with appropriate preconditioners. We propose that the choice of underlying iterative solver is problem dependent. We propose the use of block variant of the underlying iterative method because often all right-hand-side are available together. Since, Sparse Approximate Inverse (SPAI) preconditioner is a general preconditioner that can be naturally parallelized, we propose its use. Our most novel contribution is a technique to cheaply update the SPAI preconditioner, while solving the parametrically changing linear systems. We support our proposed theory by numerical experiments where we first show benefit of 80% in time by using a block iterative method, and a benefit of 70% in time by using SPAI updates.