Crustal structure of the Borderland-Continent Transition Zone of southern California adjacent to Los Angeles

We use data from the onshore-offshore component of Los Angeles Region Seismic Experiment (LARSE) to model the broad-scale features of the midcrust to upper mantle beneath a north-south transect that spans the continental borderland in the Los Angeles, California, region. We have developed an analysis method for wide-angle seismic data that consists primarily of refractions, lacks near-offset recordings, and contains wide gaps in coverage. Although the data restrict the analysis to the modeling of broad-scale structure, the technique allows one to explore the limits of the data and determine the resolving power of the data set. The resulting composite velocity model constrains the crustal thickness and location and width of the continent-Borderland transition zone. We find that the mid to lower crust layer velocities of the Inner Borderland are slightly lower than the corresponding layers in the average southern California crust model, while the upper mantle velocity is significantly higher. The data require the Moho to deepen significantly to the north. We constrain the transition zone to initiate between the offshore slope and the southwest Los Angeles Basin. If the Inner Borderland crust is 22 km thick, then the transition zone is constrained to initiate within a 2 km wide region beneath the southwest Los Angeles Basin, and have a width of 20–25 km. The strong, coherent, and continuous Pn phase suggests the Moho is coherent and laterally continuous beneath the Inner Borderland and transition zone. The Inner California Borderland seems to be modified and thickened oceanic crust, with the oceanic upper mantle intact beneath it

An Adaptive Method for Minimizing a Sum of Squares of Nonlinear Functions

The Gauss-Newton and the Levenberg-Marquardt algorithms for solving nonlinear least squares problems, minimize F(x) = sum_i=1^m (f_i(x))^2 for x in R^n, are both based upon the premise that one term in the Hessian of F(x) dominates its other terms, and that the Hessian may be approximated by this dominant term J^T J, where J_ij = ( delta f_i / delta x_j ). We are motivated here by the need for an algorithm which works well when applied to problems for which this premise is substantially violated, and is yet able to take advantage of situations where the premise holds. We describe and justify a method for approximating the Hessian of F ( x ) which uses a convex combination of J^T J and a matrix obtained by making quasi-Newton updates. In order to evaluate the usefulness of this idea, we construct a nonlinear least squares algorithm which uses this Hessian approximation, and report test results obtained by applying it to a set of test problems. A merit of our approach is that it demonstrates how a single adaptive algorithm can be used to efficiently solve unconstrained nonlinear optimization problems (whose Hessians have no particular structure), small residual and large residual, nonlinear least squares problems. Our paper can also be looked upon as an investigation for one problem area, of the following more general question: how can one combine two different Hessian approximations (or model functions) which are simultaneously available? The technique suggested here may thus be more widely applicable and may be of use, for example, when minimizing functions which are only partly composed of sums of squares arising in penalty function methods

Analogues of Dixon's and Powell's Theorems for Unconstrained Minimization with Inexact Line Searches

By modifying the way in which search directions are defined, we show how to relax the restrictive assumption that line searches must be exact in the theorems of Dixon and Powell. We show also that the BFGS algorithm modified in this way is equivalent to the three-term-recurrence (TTR) method for quadratic functions

Design and Implementation of a Stochastic Programming Optimizer with Recourse and Tenders

This paper serves two purposes, to which we give equal emphasis. First, it describes an optimization system for solving large-scale stochastic linear programs with simple (i.e. decision-free in the second stage) recourse and stochastic right-hand-side elements. Second, it is a study of the means whereby large-scale Mathematical Programming Systems may be readily extended to handle certain forms of uncertainty, through post-optimal options akin to sensitivity on parametric analysis, which we term "recourse analysis". This latter theme (implicit throughout the paper) is explored in a proselytizing manner, in the concluding section

Variants on Dantzig-Wolfe Decomposition with Applications to Multistage Problems

The initial representation of an LP problem to which the Dantzig-Wolfe decomposition procedure is applied, is of the essence. We study this here, and, in particular, we consider two transformations of the problem, by introducing suitable linking rows and variables. We study the application of the Dantzig-Wolfe procedure to these new representations of the original problem and the relationship to previously proposed algorithms. Advantages and disadvantages from a computational viewpoint are discussed. Finally we develop a decomposition algorithm based upon these ideas for solving multistage staircase-structured LP problems

Algorithms Based upon Generalized Linear Programming for Stochastic Programs with Recourse

In this paper, the author discusses solution algorithms for a particular form of two-stage stochastic linear programs with recourse. The algorithms considered are based upon the generalized linear programming method of Wolfe. The author first gives an alternative formulation of the original problem and uses this to examine the relation between tenders and certainty equivalents. He then considers problems with simple recourse, discussing algorithms for two cases: (a) when the distribution is discrete and probabilities are known explicitly; (b) when the probability distribution is other than discrete or when it is only known implicitly through some simulation model. The latter case is especially useful because it makes possible the transition to general recourse. Some possible solution strategies based upon generalized programming for general recourse problems are then discussed. This paper is a product of the Adaptation and Optimization Project within the System and Decision Sciences Program

Implementation Aids for Optimization Algorithms that Solve Sequences of Linear Programs by the Revised Simplex Method

We describe a collection of subroutines designed a) to facilitate the implementation of algorithms that are based upon linear programming, b) to serve as a tutorial on the development of such implementations. We make this collection the basis for a discussion of some of the broader issues of software development