A Levinson-Galerkin algorithm for regularized trigonometric approximation

Trigonometric polynomials are widely used for the approximation of a smooth function $f$ from a set of nonuniformly spaced samples $\{f(x_j)\}_{j=0}^{N-1}$. If the samples are perturbed by noise, controlling the smoothness of the trigonometric approximation becomes an essential issue to avoid overfitting and underfitting of the data. Using the polynomial degree as regularization parameter we derive a multi-level algorithm that iteratively adapts to the least squares solution of optimal smoothness. The proposed algorithm computes the solution in at most $\cal{O}(NM + M^2)$ operations ($M$ being the polynomial degree of the approximation) by solving a family of nested Toeplitz systems. It is shown how the presented method can be extended to multivariate trigonometric approximation. We demonstrate the performance of the algorithm by applying it in echocardiography to the recovery of the boundary of the Left Ventricle

Immittance- versus scattering-domain fast algorithms for non-Hermitian Toeplitz and quasi-Toeplitz matrices

AbstractThe classical algorithms of Schur and Levinson are efficient procedures to solve sets of Hermitian Toeplitz linear equations or to invert the corresponding coefficient matrices. They propagate pairs of variables that may describe incident and scattered waves in an associated cascade-of-layered-media model, and thus they can be viewed as scattering-domain algorithms. It was recently found that a certain transformation of these variables followed by a change from two-term to three-term recursions results in reduction in computational complexity in the abovementioned algorithms roughly by a factor of two. The ratio of such pairs of transformed variables can be interpreted in the above layered-media model as an impedance or admittance; hence the name immittance-domain variables. This paper provides extensions for previous immittance Schur and Levinson algorithms from Hermitian to non-Hermitian matrices. It considers both Toeplitz and quasi-Toeplitz matrices (matrices with certain “hidden” Toeplitz structure) and compares two- and three-term recursion algorithms in the two domains. The comparison reveals that for non-Hermitian matrices the algorithms are equally efficient in both domains. This observation adds new comprehension to the source and value of algorithms in the immittance domain. The immittance algorithms, like the scattering algorithms, exploit the (quasi-)Toeplitz structure to produce fast algorithms. However, unlike the scattering algorithms, they can respond also to symmetry of the underlying matrix when such extra structure is present, and yield algorithms with improved efficiency

Block pivoting implementation of a symmetric Toeplitz solver

Toeplitz matrices are characterized by a special structure that can be exploited in order to obtain fast linear system solvers. These solvers are difficult to parallelize due to their low computational cost and their closely coupled data operations. We propose to transform the Toeplitz system matrix into a Cauchy-like matrix since the latter can be divided into two independent matrices of half the size of the system matrix and each one of these smaller arising matrices can be factorized efficiently in multicore computers. We use OpenMP and store data in memory by blocks in consecutive positions yielding a simple and efficient algorithm. In addition, by exploiting the fact that diagonal pivoting does not destroy the special structure of Cauchy-like matrices, we introduce a local diagonal pivoting technique which improves the accuracy of the solution and the stability of the algorithm.This work was partially supported by the Spanish Ministerio de Ciencia e Innovacion (Project TIN2008-06570-C04-02 and TEC2009-13741), Vicerrectorado de Investigacion de la Universidad Politecnica de Valencia through PAID-05-10 (ref. 2705), and Generalitat Valenciana through project PROMETEO/2009/2013.Alonso-Jordá, P.; Dolz Zaragozá, MF.; Vidal Maciá, AM. (2014). Block pivoting implementation of a symmetric Toeplitz solver. Journal of Parallel and Distributed Computing. 74(5):2392-2399. https://doi.org/10.1016/j.jpdc.2014.02.003S2392239974

Preconditioned Lanczos Methods for the Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrix

In this paper, we apply the preconditioned Lanczos (PL) method to compute the minimum eigenvalue of a symmetric positive definite Toeplitz matrix. The sine transform-based preconditioner is used to speed up the convergence rate of the PL method. The resulting method involves only Toeplitz and sine transform matrix-vector multiplications and hence can be computed efficiently by fast transform algorithms. We show that if the symmetric Toeplitz matrix is generated by a positive $2 \pi$-periodic even continuous function, then the PL method will converge sufficiently fast. Numerical results including Toeplitz and non-Toeplitz matrices are reported to illustrate the effectiveness of the method.published_or_final_versio

Linear predictive modelling of speech : constraints and line spectrum pair decomposition

In an exploration of the spectral modelling of speech, this thesis presents theory and applications of constrained linear predictive (LP) models. Spectral models are essential in many applications of speech technology, such as speech coding, synthesis and recognition. At present, the prevailing approach in speech spectral modelling is linear prediction. In speech coding, spectral models obtained by LP are typically quantised using a polynomial transform called the Line Spectrum Pair (LSP) decomposition. An inherent drawback of conventional LP is its inability to include speech specific a priori information in the modelling process. This thesis, in contrast, presents different constraints applied to LP models, which are then shown to have relevant properties with respect to root loci of the model in its all-pole form. Namely, we show that LSP polynomials correspond to time domain constraints that force the roots of the model to the unit circle. Furthermore, this result is used in the development of advanced spectral models of speech that are represented by stable all-pole filters. Moreover, the theoretical results also include a generic framework for constrained linear predictive models in matrix notation. For these models, we derive sufficient criteria for stability of their all-pole form. Such models can be used to include a priori information in the generation of any application specific, linear predictive model. As a side result, we present a matrix decomposition rule for Toeplitz and Hankel matrices.reviewe

Discrete Gel'fand-Levitan and Marchenko matrix equations and layer stripping algorithms for the discrete two-dimensional Schrödinger equation inverse scattering problem with a nonlocal potential

We develop discrete counterparts to the Gel'fand-Levitan and Marchenko integral equations for the two-dimensional (2D) discrete inverse scattering problem in polar coordinates with a nonlocal potential. We also develop fast layer stripping algorithms that solve these systems of equations exactly. The significance of these results is: (1) they are the first numerical implementation of Newton's multidimensional inverse scattering theory; (2) they show that the result will almost always be a nonlocal potential, unless the data are `miraculous'; (3) they show that layer stripping algorithms implement fast `split' signal processing fast algorithms; (4) they link 2D discrete inverse scattering with 2D discrete random field linear least-squares estimation; and (5) they formulate and solve 2D discrete Schrödinger equation inverse scattering problems in polar coordinates.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/49103/2/ip8322.pd

Inverse scattering for time-varying one-dimensional layered media: algorithms and applications

We formulate and present algorithms and applications for the inverse scattering problem for a one-dimensional layered medium whose reflection coefficients vary with time. We show that given the reflection responses to impulsive plane waves incident on the medium at different times, this problem can be solved either by solving a series of nested non-Toeplitz systems of equations, or by propagating a set of coupled layer-stripping algorithms with reflection coefficients varying in time. All multiple scattering effects are included here. We show that our results reduce to previous results in the special case of time-invariant media. Applications include the two-dimensional (2D) inverse resistivity problem of reconstructing the 2D resistivity of a 2D medium from surface measurements, and the reconstruction of rapidly curing layered media. We also present a time-varying discrete Miura transform linking the time-varying discrete wavesystem inverse scattering problem to a time-varying discrete Schrödinger equation, and a new important feasibility condition on the reflection responses, which seems to be new in the context of the 2D inverse resistivity problem. A numerical example is provided.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/49101/2/ip7318.pd