1,254 research outputs found
A weakly stable algorithm for general Toeplitz systems
We show that a fast algorithm for the QR factorization of a Toeplitz or
Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A.
Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx =
A^Tb, we obtain a weakly stable method for the solution of a nonsingular
Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the
solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further
details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm
A Fast Algorithm for the Inversion of Quasiseparable Vandermonde-like Matrices
The results on Vandermonde-like matrices were introduced as a generalization
of polynomial Vandermonde matrices, and the displacement structure of these
matrices was used to derive an inversion formula. In this paper we first
present a fast Gaussian elimination algorithm for the polynomial
Vandermonde-like matrices. Later we use the said algorithm to derive fast
inversion algorithms for quasiseparable, semiseparable and well-free
Vandermonde-like matrices having complexity. To do so we
identify structures of displacement operators in terms of generators and the
recurrence relations(2-term and 3-term) between the columns of the basis
transformation matrices for quasiseparable, semiseparable and well-free
polynomials. Finally we present an algorithm to compute the
inversion of quasiseparable Vandermonde-like matrices
Multigrid waveform relaxation for the time-fractional heat equation
In this work, we propose an efficient and robust multigrid method for solving
the time-fractional heat equation. Due to the nonlocal property of fractional
differential operators, numerical methods usually generate systems of equations
for which the coefficient matrix is dense. Therefore, the design of efficient
solvers for the numerical simulation of these problems is a difficult task. We
develop a parallel-in-time multigrid algorithm based on the waveform relaxation
approach, whose application to time-fractional problems seems very natural due
to the fact that the fractional derivative at each spatial point depends on the
values of the function at this point at all earlier times. Exploiting the
Toeplitz-like structure of the coefficient matrix, the proposed multigrid
waveform relaxation method has a computational cost of
operations, where is the number of time steps and is the number of
spatial grid points. A semi-algebraic mode analysis is also developed to
theoretically confirm the good results obtained. Several numerical experiments,
including examples with non-smooth solutions and a nonlinear problem with
applications in porous media, are presented
Laplace deconvolution on the basis of time domain data and its application to Dynamic Contrast Enhanced imaging
In the present paper we consider the problem of Laplace deconvolution with
noisy discrete non-equally spaced observations on a finite time interval. We
propose a new method for Laplace deconvolution which is based on expansions of
the convolution kernel, the unknown function and the observed signal over
Laguerre functions basis (which acts as a surrogate eigenfunction basis of the
Laplace convolution operator) using regression setting. The expansion results
in a small system of linear equations with the matrix of the system being
triangular and Toeplitz. Due to this triangular structure, there is a common
number of terms in the function expansions to control, which is realized
via complexity penalty. The advantage of this methodology is that it leads to
very fast computations, produces no boundary effects due to extension at zero
and cut-off at and provides an estimator with the risk within a logarithmic
factor of the oracle risk. We emphasize that, in the present paper, we consider
the true observational model with possibly nonequispaced observations which are
available on a finite interval of length which appears in many different
contexts, and account for the bias associated with this model (which is not
present when ). The study is motivated by perfusion imaging
using a short injection of contrast agent, a procedure which is applied for
medical assessment of micro-circulation within tissues such as cancerous
tumors. Presence of a tuning parameter allows to choose the most
advantageous time units, so that both the kernel and the unknown right hand
side of the equation are well represented for the deconvolution. The
methodology is illustrated by an extensive simulation study and a real data
example which confirms that the proposed technique is fast, efficient,
accurate, usable from a practical point of view and very competitive.Comment: 36 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:1207.223
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