Underdetermined System Of Linear Equations Research Articles

A regularizing L-curve Lanczos method for underdetermined linear systems

Many real applications give rise to the solution of underdetermined linear systems of equations with a very ill conditioned matrix A, whose dimensions are so large as to make solution by direct methods impractical or infeasible. Image reconstruction from projections is a well-known example of such systems. In order to facilitate the computation of a meaningful approximate solution, we regularize the linear system, i.e., we replace it by a nearby system that is better conditioned. The amount of regularization is determined by a regularization parameter. Its optimal value is, in most applications, not known a priori. A well-known method to determine it is given by the L-curve approach. We present an iterative method based on the Lanczos algorithm for inexpensively evaluating an approximation of the points on the L-curve and then determine the value of the optimal regularization parameter which lets us compute an approximate solution of the regularized system of equations.

The Meridional Oceanic Transports of Heat and Nutrients in the South Atlantic

Meridional transports of mass, heat, nutrients, and carbon across coast-to-coast WOCE and pre-WOCE sections between 11°S and 45°S in the South Atlantic are calculated using an inverse model. Usually salt preservation is used as a condition in the inverse model, and only in the case of heat transport the condition of zero total mass transport is taken instead. Other constraints include silica conservation, prescribed southward fluxes of salt and phosphate, and transports in the southward Brazil Current and in the northward Antarctic Bottom Water flow obtained from WOCE moored current meter arrays. The constraints set the underdetermined system of linear equations of the inverse model whose solutions depend on weights, scales, and matrix ranks. The discussion emphasizes the sensitivity of the fluxes to changes in the model input. The transports given in the following are obtained as the means of “reasonable” solutions at 30°S. The error numbers in parentheses include uncertainties due to wind stress and temporal variability, the numbers without parentheses do not contain these terms:0.53 ± 0.03 (0.09) Tg s−1 mass to the south, 0.29 ± 0.05 (0.24) PW heat to the north, 15 ± 120 (500) kmol s−1 oxygen to the south, 121 ± 22 (75) kmol s−1 nitrate to the south, 64 ± 110 (300) silica to the north, and 1997 ± 215 (600) kmol s−1 dissolved inorganic carbon to the south. The above errors in transports are obviously dominated by uncertainties in wind stress and temporal variability. The divergence in meridional heat and mass transport is consistent with integral surface flux changes between corresponding zonal bands. The mass compensation of southward flowing North Atlantic Deep Water occurs to a greater extent in the warm surface waters than in the Antarctic Intermediate Water below. If one follows the arguments of earlier authors on the relation between meridional fluxes and the significance of the two possible pathways for the global thermohaline circulation, the warm water path south of Africa seems to be somewhat more important than the cold water path through Drake Passage.

A regularizing Lanczos iteration method for underdetermined linear systems

This paper is concerned with the solution of underdetermined linear systems of equations with a very ill-conditioned matrix A, whose dimensions are so large to make solution by direct methods impractical or infeasible. Image reconstruction from projections often gives rise to such systems. In order to facilitate the computation of a meaningful approximate solution, we regularize the linear system, i.e., we replace it by a nearby system that is better conditioned. The amount of regularization is determined by a regularization parameter. Its optimal value is, in most applications, not known a priori. We present a new iterative method based on the Lanczos algorithm for determining a suitable value of the regularization parameter by the discrepancy principle and an approximate solution of the regularized system of equations.

Minimization of Power Losses in Cooperating Manipulators

The control of multiple manipulators handling a common object entails the solution of an underdetermined system of linear equations which represents the system’s dynamics. In order to choose an optimal solution to this problem, various approaches have been proposed: minimum internal force and minimum power, among others. The present work investigates an approach for minimizing the power losses in these systems. It is shown that the power imparted to the manipulator/payload system cannot be optimized once the system’s motion is prescribed. However, assuming certain loss characteristics for the dc servomotors commonly used on robotic manipulators, it is shown that the minimization of power losses can be cast as a linear-quadratic optimization problem. Local and global performance indices are introduced to allow comparison of the minimum power loss and the minimum internal force approaches. An example of two Puma 560 robots handling a common payload is shown to demonstrate the proposed technique.

High-Resolution Signal and Noise Field Estimation Using the<tex>L</tex>1 (Least Absolute Values) Norm

In this paper a new method for obtaining a quantitative estimate of an acoustic field consisting of a set of discrete sources and background noise is described. The method is based on the L 1 (least absolute values) norm solution to an underdetermined system of linear equations defining the Fourier transform of the signal series. Implementations of the method with either equality or inequality constraints are presented and discussed. The much faster and more compact equality constraint version with a provision for modeling the noise field is recommended in practice. Experience with real data has shown the necessity of correcting for an observed Gaussian decay on the covariances. A simple means of estimating this effect and taking it into account in the signal estimation procedure is discussed, and the implications of this effect in high-resolution beamforming are considered. The effectiveness and versatility of the L 1 method indicate that it has a useful role in high-resolution signal estimation.

Error analysis of an algorithm for solving an underdetermined linear system

The accumulation of rounding errors in a method used to compute the solution of an underdetermined system of linear equations at the least distance from a given point is being studied. The method used is based on orthogonal matrix decompositions.

Linear inversion of gravity data using the spectral expansion method

Inversion of gravity anomalies in terms of an anomalous mass distribution with irregular outline but uniform density leads to a nonlinear inverse problem. An alternative approach based on the thin‐sheet approximation can, however, be formulated as a linear inverse problem provided the structures are two‐dimensional. The anomalous mass is represented by a thin sheet, which is located at a depth [Formula: see text] and is divided into M strips with N < M data points. Thus, an underdetermined system of linear equations is obtained which is solved by the spectral expansion method for the surface density distribution of each segment. This set of parameters is then transformed into a prismatic structure with variable depth but uniform density. The modeling procedure involves a noniterative method. A gravity problem is investigated, and the solution obtained compares well with previous interpretations.

Solution of Sparse Underdetermined Systems of Linear Equations

The problem of computing the minimum-norm solution to a sparse, underdetermined system of linear equations is considered. An algorithm is developed which is based on orthogonal factorization. The algorithm is generalized to cope with systems which are rank deficient, inconsistent, or require column updating.

On the use of singular value decomposition and decimation in discrete-time band-limited signal extrapolation

The problem of extrapolating a band-limited signal in discrete time is viewed as one of solving an underdetermined system of linear equations. Choosing the minimum norm least-squares (MNLS) solution is one criterion for singling out an extrapolation from all the possible solutions to the linear system. Use of the Moore-Penrose inverse yields the MNLS solution, and singular value decomposition (SVD) provides a means for implementing the Moore-Penrose inverse. An expression for the mean-square error incurred in solving a linear system via SVD is derived. This can be used to estimate the number of singular values needed to form the inverse. The error expression also indicates that decimation can be applied in the extrapolation problem to reduce the high computational cost of SVD without degrading the extrapolation. The results developed for the one-dimensional case are extended to higher dimensions. Examples of the SVD approach to extrapolation are given, along with examples using other extrapolation techniques for comparison. The SVD approach compares favorably with known MNLS extrapolation methods.

Screening Sensitivity Theory

A comprehensive rigorous theory is developed for screening sensitivity coefficients in largescale modeling applications. The theory uses Bayesian inference and group theory to establish a probabilistic framework for solving an underdetermined system of linear equations. The underdetermined problem is directly related to statistical screening sensitivity theory as developed in recent years. Several examples of the new approach to screening are worked out in detail and comparisons are made with statistical approaches to the problem. The drawbacks of these latter methods are discussed at some length.

Minimum norm solutions to linear elastic analysis problems

AbstractA basic problem in the linear elastic analysis is that of finding the vectors of stresses and strains, given a finite element model of a structure and a set of external loads. One purpose of this paper is to show that the problem is a special case of the minimum norm problem for underdetermined systems of linear equations. In this regard, the three conventional structural analysis approaches, i.e. the displacement method, the natural factor formulation and the force method, are unified and interpreted in the framework of the minimum norm problem, which is divided into two approaches—the primal formulation and the dual formulation. Numerical comparisons of several computational procedures capable of solving the minimum norm problem are given from computational efficiency and accuracy points of view. Included in the comparisons are the three conventional structural analysis approaches mentioned above, and several alternative approaches.

Estimation of ocean-surface directional-frequency spectra: The inverse scattering problem

This paper demonstrates, by theoretical means, that measured acoustic scattering data can be inverted to yield estimates of the ocean-surface directional-frequency spectrum. The inverse scattering technique developed is based on least-squares inversion through the generalized inverse for underdetermined linear systems of equations. The procedure is based on a recently developed acoustic scattering model which is capable of sampling all frequencies and directions of the gravity-wave field in a localized area of ocean surface. The technique is numerically tested on simulated data samples, constructed with the scattering model using pseudorandom numbers, for a Gaussian ocean surface. The estimation procedure is independent of the assumed form of the simulated directional-frequency spectrum, and the results obtained show that estimation agrees with that spectrum.

Estimating Ventilation/Perfusion Distributions from Inert Gas Data: A Bayesian Approach

In recent years, experimental techniques using inert gas studies have been developed for determining essentially continuous distributions of the ventilation/perfusion ratio over a lung. Mathematically, the technique requires the recovery of 'smooth' solutions of grossly underdetermined systems of linear equations. This paper investigates the use of Bayesian smoothing techniques and shows that, provided the true underlying distribution of ventilation/perfusion ratios is relatively smooth and well-behaved, it is possible to recover reliable estimates of such distributions.

Minimum L∞ solution of underdetermined systems of linear equations

The problem of obtaining a minimum L∞ solution of an underdetermined system of consistent linear equations is reduced to a linear programming problem. A modified simplex algorithm is then described. In this algorithm no conditions are imposed on the coefficient matrix, minimum computer storage is required and no artificial variables are needed. The algorithm is a simple and fast one. Numerical results are given.

Neutron Spectroscopy by Foil Activation in Radiation Effects Studies

A new technique is presented for determining neutron spectra from foil activation data. The set of activation integral equations are replaced by an underdetermined system of linear equations whose solutions are piecewise linear representations of the differential flux. The technique then selects that solution which is closest to a trial solution in a least squares sense. Examples are presented showing various neutron spectra determined from activation data and certain radiation effects computed with these resultant spectra. A comparison of these "calculated" effects with the "actual" effects as computed with the actual spectra shows excellent agreement.