arXiv subjects

Sivaram Ambikasaran

Publications and source records attributed to Sivaram Ambikasaran.

23 records · Page 2Linked to original sources

arXiv2015

A Fast Block Low-Rank Dense Solver with Applications to Finite-Element Matrices

This article presents a fast solver for the dense "frontal" matrices that arise from the multifrontal sparse elimination process of 3D elliptic PDEs. The solver relies on the fact that these matrices can be efficiently represented as a hierarchically off-diagonal low-rank (HODLR) matrix. To construct the low-rank approximation of the off-diagonal blocks, we propose a new pseudo-skeleton scheme, the boundary distance low-rank approximation, that picks rows and columns based on the location of their corresponding vertices in the sparse matrix graph. We compare this new low-rank approximation method to the adaptive cross approximation (ACA) algorithm and show that it achieves betters speedup specially for unstructured meshes. Using the HODLR direct solver as a preconditioner (with a low tolerance) to the GMRES iterative scheme, we can reach machine accuracy much faster than a conventional LU solver. Numerical benchmarks are provided for frontal matrices arising from 3D finite element problems corresponding to a wide range of applications.

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arXiv2014

The Inverse Fast Multipole Method

This article introduces a new fast direct solver for linear systems arising out of wide range of applications, integral equations, multivariate statistics, radial basis interpolation, etc., to name a few. \emph{The highlight of this new fast direct solver is that the solver scales linearly in the number of unknowns in all dimensions.} The solver, termed as Inverse Fast Multipole Method (abbreviated as IFMM), works on the same data-structure as the Fast Multipole Method (abbreviated as FMM). More generally, the solver can be immediately extended to the class of hierarchical matrices, denoted as $\mathcal{H}^2$ matrices with strong admissibility criteria (weak low-rank structure), i.e., \emph{the interaction between neighboring cluster of particles is full-rank whereas the interaction between particles corresponding to well-separated clusters can be efficiently represented as a low-rank matrix}. The algorithm departs from existing approaches in the fact that throughout the algorithm the interaction corresponding to neighboring clusters are always treated as full-rank interactions. Our approach relies on two major ideas: (i) The $N \times N$ matrix arising out of FMM (from now on termed as FMM matrix) can be represented as an extended sparser matrix of size $M \times M$, where $M \approx 3N$. (ii) While solving the larger extended sparser matrix, \emph{the fill-in's that arise in the matrix blocks corresponding to well-separated clusters are hierarchically compressed}. The ordering of the equations and the unknowns in the extended sparser matrix is strongly related to the local and multipole coefficients in the FMM~\cite{greengard1987fast} and \emph{the order of elimination is different from the usual nested dissection approach}. Numerical benchmarks on $2$D manifold confirm the linear scaling of the algorithm.

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arXiv2014

A Kalman filter powered by $\mathcal{H}^2$-matrices for quasi-continuous data assimilation problems

Continuously tracking the movement of a fluid or a plume in the subsurface is a challenge that is often encountered in applications, such as tracking a plume of injected CO$_2$ or of a hazardous substance. Advances in monitoring techniques have made it possible to collect measurements at a high frequency while the plume moves, which has the potential advantage of providing continuous high-resolution images of fluid flow with the aid of data processing. However, the applicability of this approach is limited by the high computational cost associated with having to analyze large data sets within the time constraints imposed by real-time monitoring. Existing data assimilation methods have computational requirements that increase super-linearly with the size of the unknowns $m$. In this paper, we present the HiKF, a new Kalman filter (KF) variant powered by the hierarchical matrix approach that dramatically reduces the computational and storage cost of the standard KF from $\mathcal{O}(m^2)$ to $\mathcal{O}(m)$, while producing practically the same results. The version of HiKF that is presented here takes advantage of the so-called random walk dynamical model, which is tailored to a class of data assimilation problems in which measurements are collected quasi-continuously. The proposed method has been applied to a realistic CO$_2$ injection model and compared with the ensemble Kalman filter (EnKF). Numerical results show that HiKF can provide estimates that are more accurate than EnKF, and also demonstrate the usefulness of modeling the system dynamics as a random walk in this context.

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arXiv2014

A fast direct solver for high frequency scattering from a large cavity in two dimensions

We present a fast direct solver for the simulation of electromagnetic scattering from an arbitrarily-shaped, large, empty cavity embedded in an infinite perfectly conducting half space. The governing Maxwell equations are reformulated as a well-conditioned second kind integral equation and the resulting linear system is solved in nearly linear time using a hierarchical matrix factorization technique. We illustrate the performance of the scheme with several numerical examples for complex cavity shapes over a wide range of frequencies.

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arXiv2013

A simple way to speedup Gauss Elimination

This article looks at a simple modification to speedup the conventional Gauss Elimination. The proposed modification speeds up the conventional Gauss Elimination by a factor of nearly 9/7 (in the asymptotic limit).

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