The model-based image reconstruction techniques for photoacoustic (PA) tomography require an explicit regularization. An error estimate (?2) minimization-based approach was proposed and developed for the determination of a regularization parameter for PA imaging. The regularization was used within Lanczos bidiagonalization framework, which provides the advantage of dimensionality reduction for ...

Communication – the movement of data between levels of memory hierarchy or between processors over a network – is the most expensive operation in terms of both time and energy at all scales of computing. Achieving scalable performance in terms of time and energy thus requires a dramatic shift in the field of algorithmic design. Solvers for sparse linear algebra problems, ubiquitous throughout s...

Given a large square matrix A and a sufficiently regular function f so that f(A) is well defined, we are interested in the approximation of the leading singular values and corresponding singular vectors of f(A), and in particular of ‖f(A)‖, where ‖ · ‖ is the matrix norm induced by the Euclidean vector norm. Since neither f(A) nor f(A)v can be computed exactly, we introduce and analyze an inexa...

In the robot navigation problem, noisy sensor data must be ltered to obtain the best estimate of the robot position. The discrete Kalman lter, which usually is used for prediction and detection of signal in communication and control problems has become a commonly used method to reduce the e ect of uncertainty from the sensor data. However, due to the special domain of robot navigation, the Kalm...

Tikhonov regularization of linear discrete ill-posed problems often is applied with a finite difference regularization operator that approximates a low-order derivative. These operators generally are represented by banded rectangular matrices with fewer rows than columns. They therefore cannot be applied in iterative methods that are based on the Arnoldi process, which requires the regularizati...

The total least squares (TLS) method is a successful method for noise reduction in linear least squares problems in a number of applications. The TLS method is suited to problems in which both the coefficient matrix and the right-hand side are not precisely known. This paper focuses on the use of TLS for solving problems with very ill-conditioned coefficient matrices whose singular values decay...

It is well-known that many Krylov solvers for linear systems, eigenvalue problems, and singular value decomposition problems have very simple and elegant formulas for residual norms. These formulas not only allow us to further understand the methods theoretically but also can be used as cheap stopping criteria without forming approximate solutions and residuals at each step before convergence t...

We describe the development of a method for the efficient computation of the smallest singular values and corresponding vectors for large sparse matrices [4]. The method combines state-of-the-art techniques that make it a useful computational tool appropriate for large scale computations. The method relies upon Lanczos bidiagonalization (LBD) with partial reorthogonalization [6], enhanced with ...