In this article, 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.

Balanced truncation of descriptor systems requires computation of spectral projectors and solution of the generalized projected Lyapunov equations, both of which have significant complexity. Frequency-weighted balancing methods are more efficient if the response over a specific frequency range is desired. However, a direct extension of these methods to descriptor systems requires the spectral projectors. In this paper, we propose an efficient frequency-weighted balanced truncation algorithm without finding the spectral projectors. Samples of the frequency-domain solution to the system are used to get an accurate estimate of the improper Gramians. The proper Gramians are computed after adjusting for the contribution of the improper subsystem. Low rank factors of these Gramians are used to obtain a basis that includes the contribution of both the proper and improper subsystems. Congruence transform is used to ensure passivity of RLC interconnect models. Results for standard benchmarks show that the method is accurate and efficient.

Vortex methods offer a grid-free alternative to simulating incompressible, viscous, fluid flows. They require the use of fairly sophisticated algorithms and can be complicated to implement for general flows. This article describes an object-oriented design used to implement a vortex particle based flow solver in two dimensions. We provide an overview of the various abstractions that arose as a result of this design. Several of the algorithms have common components that may be abstracted and reused. We demonstrate how the design allowed us to derive the traditional benefits of OOD. In addition, we show how the design directly suggested elegant generalizations of existing algorithms. Finally, we show the benefits of using software testing techniques and building a powerful scripting layer for the library.

A time-domain algorithm for computation of the noise power spectral density (PSD) is proposed in [17]. When applied to periodically varying circuits, this formulation requires efficient methods to obtain the quasi-periodic steady-state solution of a set of linear time-varying ordinary differential equations. We use both the mixed frequency-time technique(MFT) and the shooting Newton method for this computation. We show that, at each frequency for which the spectral density is sought, MFT requires only two integrations over a duration of the clock period. The results are compared with published experimental and computed data.