Large Sparse Systems Of Equations Research Articles

Tomographic inversion for three‐dimensional velocity structure at Mount St. Helens using earthquake data

Tomographic inversion is applied to 17,659 P phase observations at 21 stations from 2023 earthquakes in the vicinity of Mount St. Helens to study the three‐dimensional velocity structure. Block size for the inversion is 2 km horizontally and 2 km or more vertically. Locations of hypocenters are assumed known and are based on a reference one‐dimensional, layered velocity structure. A conjugate gradient technique (LSQR) is used to invert the large sparse system of equations, augmented by regularization with a Laplacian roughening matrix. Resolution is estimated by computing the impulse response of the inversion for various critical locations, and uncertainties of the estimates are determined by a jackknife approach. The results of the inversion show a remarkable correlation with known geological and geophysical features. The Spirit Lake and Spud Mt. plutons are characterized by high‐velocity regions extending to approximately 9 km depth. The St. Helens seismic zone, a band of diffuse seismicity extending NNW from the volcano is evident as a prominent low‐velocity lineation. The change in character of the velocity anomalies south of St. Helens corresponds well with the near cessation of seismic activity there. A low‐velocity anomaly beneath the crater from 6 to 16 km depths may represent modern magma accumulations.

Some shared memory is desirable in parallel sparse matrix computation

Over the past few years a number of algorithms for solving large sparse systems of equations on distributed-memory multiprocessors have been developed. In this article the authors point out that the properties of sparse matrix problems generally, along with the characteristics of these parallel algorithms for solving them, lead to inefficient use of memory. An example is presented to show that a (relatively small) amount of shared memory on an otherwise pure distributed-memory multiprocessor is very desirable when it is being used to execute these parallel algorithms.

A Parallel Solution Method for Large Sparse Systems of Equations

This paper presents a new distributed multifrontal sparse matrix decomposition algorithm suitable for message passing parallel processors. The algorithm uses a nested dissection ordering and a multifrontal distribution of the matrix to minimize interprocessor data dependencies and overcome the communication bottleneck previously reported for sparse matrix decomposition [1]. Distributed multifrontal forward elimination and back substitution algorithms are also provided. Results of an implementation on the Intel iPSC are presented. Up to 16 processors are used to solve systems with as many as 7225 equations. With 16 processors, speedups of 10.2 are observed and the decomposition is shown to achieve 67 percent processor utilization. This work was motivated by the need to reduce the computational bottleneck in the Stanford PISCES [2] device simulator; however, it should be applicable to a wide range of scientific and engineering problems

Solving large sparse systems of equations in econometric models

AbstractComparative studies of Gauss‐Seidel and Newton‐type algorithms for solving large sparse systems of equations are reported by Népomiastchy and Ravelli (1978), Gabay et al. (1980) and Norman et al. (1983). The first two favour Newton's method, the third favours Gauss‐Seidel. Apart from working on different test models, their setups differ in the implementation of both schemes.This paper studies the performance of both methods on ten different econometric models of varying size and complexity. First the choice of implementation (equation reordering, updating rules for Newton's Jacobian) is studied on a relatively small model. Qualitative and quantitative feedback criteria are considered, and an efficient reordering algorithm is discussed. On the ten models considered, the selected Newton method is almost uniformly cheaper, generally reducing the number of iterations by more than 30 percent A final section draws attention to the possible extra gains of Newton's method in evaluating multipliers for policy analysis.

Large-scale geodetic least-squares adjustment by dissection and orthogonal decomposition

Very large-scale matrix problems currently arise in the context of accurately computing the coordinates of points on the surface of the earth. Here geodesists adjust the approximate values of these coordinates by computing least-squares solutions to large sparse systems of equations which result from relating the coordinates to certain observations such as distances or angles between points. The purpose of this paper is to suggest an alternative to the formation and solution of the normal equations for these least-squares adjustment problems. In particular, it is shown how a block-orthogonal decomposition method can be used in conjunction with a nested dissection scheme to produce an algorithm for solving such problems which combines efficient data management with numerical stability. The approach given here parallels somewhat the development of the natural factor formulation, by Argyris et al., for the use of orthogonal decomposition procedures in the finite-element analysis of structures. As an indication of the magnitude that these least-squares adjustment problems can sometimes attain, the forthcoming readjustment of the North American Datum in 1983 by the National Geodetic Survey is discussed. Here it becomes necessary to linearize and solve an overdetermined system of approximately 6,000,000 equations in 400,000 unknowns—a truly large scale matrix problem.