Department of Mathematics
http://hdl.handle.net/1957/13737
Thu, 03 Sep 2015 07:14:03 GMT2015-09-03T07:14:03ZSymmetry breaking and uniqueness for the incompressible Navier-Stokes equations
http://hdl.handle.net/1957/57110
Symmetry breaking and uniqueness for the incompressible Navier-Stokes equations
Dascaliuc, Radu; Michalowski, Nicholas; Thomann, Enrique; Waymire, Edward C.
The present article establishes connections between the structure of the deterministic Navier-Stokes equations and the structure of (similarity) equations that govern self-similar solutions as expected values of certain naturally associated stochastic cascades. A principle result is that explosion criteria for the stochastic cascades involved in the probabilistic representations of solutions to the respective equations coincide. While the uniqueness problem itself remains unresolved, these connections provide interesting problems and possible methods for investigating symmetry breaking and the uniqueness problem for Navier-Stokes equations. In particular, new branching Markov chains, including a dilogarithmic branching random walk on the multiplicative group (0, ∞), naturally arise as a result of this investigation.
Copyright 2015 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.; The following article appeared in Chaos: An Interdisciplinary Journal of Nonlinear Science and may be found at http://scitation.aip.org/content/aip/journal/chaos
Wed, 01 Jul 2015 00:00:00 GMThttp://hdl.handle.net/1957/571102015-07-01T00:00:00ZJoint optimization of well placement and control for nonconventional well types
http://hdl.handle.net/1957/56985
Joint optimization of well placement and control for nonconventional well types
Humphries, T. D.; Haynes, R. D.
Optimal well placement and optimal well control are two important areas of study in
oilfield development. Although the two problems differ in several respects, both are important
considerations in optimizing total oilfield production, and so recent work in the field
has considered the problem of addressing both problems jointly. Two general approaches
to addressing the joint problem are a simultaneous approach, where all parameters are optimized
at the same time, or a sequential approach, where a distinction between placement
and control parameters is maintained by separating the optimization problem into two (or
more) stages, some of which consider only a subset of the total number of variables. This
latter approach divides the problem into smaller ones which are easier to solve, but may not
explore search space as fully as a simultaneous approach.
In this paper we combine a stochastic global algorithm (Particle Swarm Optimization)
and a local search (Mesh Adaptive Direct Search) to compare several simultaneous and sequential
approaches to the joint placement and control problem. In particular, we study
how increasing the complexity of well models (requiring more variables to describe the well’s
location and path) affects the respective performances of the two approaches. The results of
several experiments with synthetic reservoir models suggest that the sequential approaches
are better able to deal with increasingly complex well parameterizations than the simultaneous
approaches.
This is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Elsevier and can be found at: http://www.journals.elsevier.com/journal-of-petroleum-science-and-engineering/
Sun, 01 Feb 2015 00:00:00 GMThttp://hdl.handle.net/1957/569852015-02-01T00:00:00ZTechnical Note: Convergence analysis of a polyenergetic SART algorithm
http://hdl.handle.net/1957/56984
Technical Note: Convergence analysis of a polyenergetic SART algorithm
Humphries, T.
PURPOSE: We analyze a recently proposed polyenergetic version of the simultaneous algebraic
reconstruction technique (SART). This algorithm, denoted pSART, replaces the monoenergetic
forward projection operation used by SART with a post-log, polyenergetic forward projection, while
leaving the rest of the algorithm unchanged. While the proposed algorithm provides good results
empirically, convergence of the algorithm was not established mathematically in the original paper.
METHODS: We analyze pSART as a nonlinear fixed point iteration by explicitly computing the Jacobian of the iteration. A necessary condition for convergence is that the spectral radius of the Jacobian, evaluated at the fixed point, is less than one. A short proof of convergence for SART is also provided as a basis for comparison..
RESULTS: We show that the pSART algorithm is not guaranteed to converge, in general. The Jacobian of the iteration depends on several factors, including the system matrix and how one models the energy dependence of the linear attenuation coefficient. The authors provide a simple numerical example that shows that the spectral radius of the Jacobian matrix is not guaranteed to be less than one. A second set of numerical experiments using realistic CT system matrices, however, indicates that conditions for convergence are likely to be satisfied in practice
CONCLUSION: Although pSART is not mathematically guaranteed to converge, our numerical
experiments indicate that it will tend to converge at roughly the same rate as SART for system
matrices of the type encountered in CT imaging. Thus we conclude that the algorithm is still a
useful method for reconstruction of polyenergetic CT data.
This is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by American Association of Physicists in Medicine and can be found at: http://www.medphys.org/
Wed, 01 Jul 2015 00:00:00 GMThttp://hdl.handle.net/1957/569842015-07-01T00:00:00ZThe Variable Speed Wave Equation and Perfectly Matched Layers
http://hdl.handle.net/1957/56308
The Variable Speed Wave Equation and Perfectly Matched Layers
Kim, Dojin
A perfectly matched layer (PML) is widely used to model many different types of wave propagation in different media. It has been found that a PML is often very effective and also easy to set, but still many questions remain.
We introduce a new formulation from regularizing the classical Un-Split PML of the acoustic wave equation and show the well-posedness and numerical efficiency. A PML is designed to absorb incident waves traveling perpendicular to the PML, but there is no effective absorption of waves traveling with large incident angles. We suggest one method to deal with this problem and show well-posedness of the system, and some numerical experiments. For the 1-d wave equation with a constant speed equipped a PML, stability and the exponential decay rate of energy has been proved, but the question for variable sound speed equation remained open. We show that the energy decays exponentially in the 1-d PML wave equation with variable sound speed.
Most PML wave equations appear as a first-order hyperbolic system with as a zero-order perturbation. We introduce a general formulation and show well-posedness and stability of the system. Furthermore we develop a discontinuous Galerkin method and analyze both the semi-discrete and fully discretized system and provide a priori error estimations.
Graduation date: 2016
Thu, 04 Jun 2015 00:00:00 GMThttp://hdl.handle.net/1957/563082015-06-04T00:00:00Z