Shiv Chandrasekaran

An eigenspace update algorithm for image analysis

During the past few years several interesting applications of eigenspace representation of images have been proposed. These include face recognition, video coding, pose estimation, etc. However, the vision research community has largely overlooked parallel developments in signal processing and numerical linear algebra concerning efficient eigenspace updating algorithms. These new developments are significant for two reasons: adopting them makes some of the current vision algorithms more robust and efficient. More important is the fact that incremental updating of eigenspace representations opens up new and interesting research applications in vision such as active recognition and learning. The main objective of the paper is to put these in perspective and discuss a recently introduced updating scheme that has been shown to be numerically stable and optimal. We provide an example of one particular application to 3D object representation projections and give an error analysis of the algorithm. Preliminary experimental results are shown.

A Fast

We consider an algebraic representation that is useful for matrices with off-diagonal blocks of low numerical rank. A fast and stable solver for linear systems of equations in which the coefficient matrix has this representation is presented. We also present a fast algorithm to construct the hierarchically semiseparable representation in the general case.

ULV

In this paper we present a fast direct solver for certain classes of dense structured linear systems that works by first converting the given dense system to a larger system of block sparse equations and then uses standard sparse direct solvers. The kind of matrix structures that we consider are induced by numerical low rank in the off-diagonal blocks of the matrix and are related to the structures exploited by the fast multipole method (FMM) of Greengard and Rokhlin. The special structure that we exploit in this paper is captured by what we term the hierarchically semiseparable (HSS) representation of a matrix. Numerical experiments indicate that the method is probably backward stable.

Decomposition Solver for Hierarchically Semiseparable Representations

In this paper we develop a new superfast solver for Toeplitz systems of linear equations. To solve Toeplitz systems many people use displacement equation methods. With displacement structures, Toeplitz matrices can be transformed into Cauchy-like matrices using the FFT or other trigonometric transformations. These Cauchy-like matrices have a special property, that is, their off-diagonal blocks have small numerical ranks. This low-rank property plays a central role in our superfast Toeplitz solver. It enables us to quickly approximate the Cauchy-like matrices by structured matrices called sequentially semiseparable (SSS) matrices. The major work of the constructions of these SSS forms can be done in precomputations (independent of the Toeplitz matrix entries). These SSS representations are compact because of the low-rank property. The SSS Cauchy-like systems can be solved in linear time with linear storage. Excluding precomputations the main operations are the FFT and SSS system solve, which are both very efficient. Our new Toeplitz solver is stable in practice. Numerical examples are presented to illustrate the efficiency and the practical stability.

A Fast Solver for HSS Representations via Sparse Matrices

It is shown that the numerical rank of the off-diagonal blocks of certain Schur complements of matrices that arise from the finite-difference discretization of constant coefficient, elliptic PDEs in two spatial dimensions is bounded by a constant independent of the grid size. Moreover, in three-dimensional problems the Schur complements are shown to have off-diagonal blocks whose numerical rank is a slowly growing function.

A Superfast Algorithm for Toeplitz Systems of Linear Equations

We present fast and numerically stable algorithms for the solution of linear systems of equations, where the coefficient matrix can be written in the form of a banded plus semiseparable matrix. Such matrices include banded matrices, banded bordered matrices, semiseparable matrices, and block-diagonal plus semiseparable matrices as special cases. Our algorithms are based on novel matrix factorizations developed specifically for matrices with such structures. We also present interesting numerical results with these algorithms.

On the Numerical Rank of the Off-Diagonal Blocks of Schur Complements of Discretized Elliptic PDEs

Abstract This paper presents a method of producing higher order discretization weights for linear differential and integral operators using the Minimum Sobolev Norm idea[1][2] in arbitrary geometry and grid configurations. The weight computation involves solving a severely ill-conditioned weighted least-squares system. A method of solving this system to very high-accuracy is also presented, based on the theory of Vavasis et al [3]. An end-to-end planar partial differential equation solver is developed based on the described method and results are presented. Results presented include the solution error, discretization error, condition number and time taken to solve several classes of equations on various geometries. These results are then compared with those obtained using the Matlabs FEM based PDE Solver as well as Dealii[4].

Fast and Stable Algorithms for Banded Plus Semiseparable Systems of Linear Equations

In this paper, a method for modeling the deformation behavior of organs and soft tissue is presented. The purpose is to predict the global deformation effect that arbitrary, time-varying external perturbations have on an organ. The perturbation might be caused by an instrument (e.g., through the surgeonpsilas grasping and pinching actions), or it might be from organ-organ, organ-body-wall collisions in a bodily cavity. A methodology, employing (1) a surface representation based on the Boundary-Element Method-or BEM, of the deformation equations and (2) recently developed linear-algebra techniques (known as the ldquoHierarchical Semi-Separablerdquo matrix representation-or HSS), is proposed. We demonstrate that the proposed framework achieves an almost linear time complexity of O(n1.14), a significant speed up comparing to the traditional O(n3) schemes employing brute-force linear-algebra solution methods based on Finite-Element Method (FEM) formulations. Furthermore, unlike some previous approach, no restriction is placed on the external perturbation pattern and how it can change over time.