Browsing by Author "Gockenbach, Mark S."
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Item An Abstract Analysis of Differential Semblance Optimization(1994-04) Gockenbach, Mark S.Differential Semblance Optimization (DSO) is a novel way of approaching a class of inverse problems arising in exploration seismology. The promising feature of the DSO method is that it replaces a nonsmooth, highly nonconvex cost function (the Output Least-Squares (OLS) objective function) with a smooth cost function that is amenable to standard (local) optimization algorithms. The OLS problem can be written abstractly as a partially linear least-squares problem with linear constraints. The DSO objective function is derived from the associated quadratic penalty function. It is shown that one way to view the DSO objective function is as a regularization of a function that is dual (in a certain sense) to the OLS objective function. By viewing the DSO problem as a perturbation of this dual problem, this method can be shown to be effective. In particular, it is demonstrated that, under suitable assumptions, the DSO method defines a parameterized path of minimizers converging to the desired solution, and that for certain values of the parameter, standard optimization techniques can be used to find a point on the path. The predictions of the theory are motivated and illustrated on two simple model problems for seismic velocity inversion, the plane wave detection problem and the "layer-over-half-space" problem. It is shown that the theory presented in this thesis extends the existing theory for the plane wave detection problem.Item An Infeasible Point Method for Minimizing the Lennard-Jones Potential(1993-10) Gockenbach, Mark S.; Kearsley, Anthony J.; Symes, William W.Minimizing the Lennard-Jones potential, the most-studied problem for molecular conformation, is an unconstrained global optimization problem. In this paper, the problem is reformulated as an equality constrained nonlinear program in such a way that the likelihood of finding a global minimizer is increased. Implementation of an algorithm for solving this nonlinear program is discussed, and results of numerical tests are presented.Item An Overview of HCL 1.0(1999-10) Gockenbach, Mark S.; Symes, William W.The Hilbert Class Library (HCL) is a collection of C++ classes which apply object-oriented programming principles to implement mathematical objects such as vectors, linear and nonlinear operators, and functions. HCL provides a convenient environment for implementing algorithms for optimization and linear algebra at a natural, abstract level, without reference to the implementations of data structures, simulators, and other complex, application-specific details. Because coordinate representations, data storage formats, and other domain-specific idiosyncrasies are not entangled in these implementations, the resulting code is reusable across applications of widely varying size and structure. The design of HCL also results in several very important capabilities, such as the ability to treat very large out-of-core data sets as vector objects, and to manipulate linear operators not defined explicitly by matrices, which distinguish HCL from other object oriented numerics libraries.Item Coherent Noise Suppression in Velocity Inversion(1999) Gockenbach, Mark S.; Symes, William W.Data components with well-defined moveout other than primary reflections are sometimes called coherent noise. Coherent noise makes velocity analysis ambiguous, since no single velocity function explains incompatible moveouts simultaneously. Contemporary data processing treats the control of coherent noise influence on velocity as an interpretive step. Dual regularization theory suggests an alternative, automatic inversion algorithm for suppression of coherent noise when primary reflection phases dominate the data. Experiments with marine data illustrate the robustness and effectiveness of the algorithm.Item Comparing Objective Functions for Velocity Inversion(1993-10) Gockenbach, Mark S.; Symes, William W.The success of automatic velocity inversion is highly dependent on the numerical tractability of the optimization problem which defines the solution. The purpose of this paper is to compare the objective (cost) functions arising from two different formulations of the problem. It is shown by example that the output least-squares approach defines an objective function which can be highly nonconvex and which can have local, nonglobal minima. In contrast, the method of Differential Semblance Optimization defines a cost function which has a unique minimizer and which appears to be nearly convex. These two approaches are applied to the simple problem of determining the depth of a horizontal reflector and the velocity of the layer above it.Item Implementing Functionals in HCL(1999-10) Gockenbach, Mark S.Item Implementing Nonlinear Operators in HCL(1999-09) Gockenbach, Mark S.Item Optimal Signal Sets for Non-Gaussian Detectors(1995-05) Gockenbach, Mark S.; Kearsley, Anthony J.Identifying a maximally-separated set of signals is important in the design of modems. The notion of optimality is dependent on the model chosen to describe noise in the measurements; while some analytic results can be derived under the assumption of Gaussian noise, no such techniques are known for choosing signal acts in the non-Gaussian case. To obtain numerical solutions for non-Gaussian detectors, minimax problems are transformed into nonlinear programs,resulting in a novel formulation yielding problems with relatively few variables and many inequality constraints. Using sequential quadratic programming, optimal signal sets are obtained for a variety of noise distributions.Item Understanding code generated by TAMC(2000-06) Gockenbach, Mark S.The Tangent linear and Adjoint Model Compiler (TAMC) is an automatic differentiation package designed and implemented by Ralf Giering. The code generated by TAMC can be understood in terms of a careful mathematical model for a subroutine implementing an operator.