Sarah Day
College of William & Mary
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Publication
Featured researches published by Sarah Day.
SIAM Journal on Numerical Analysis | 2007
Sarah Day; Jean-Philippe Lessard; Konstantin Mischaikow
One of the most efficient methods for determining the equilibria of a continuous parameterized family of differential equations is to use predictor-corrector continuation techniques. In the case of partial differential equations this procedure must be applied to some finite-dimensional approximation, which of course raises the question of the validity of the output. We introduce a new technique that combines the information obtained from the predictor-corrector steps with ideas from rigorous computations and verifies that the numerically produced equilibrium for the finite-dimensional system can be used to explicitly define a set which contains a unique equilibrium for the infinite-dimensional partial differential equation. Using the Cahn-Hilliard and Swift-Hohenberg equations as models we demonstrate that the cost of this new validated continuation is less than twice the cost of the standard continuation method alone.
Siam Journal on Applied Dynamical Systems | 2004
Sarah Day; Oliver Junge; Konstantin Mischaikow
We present a numerical method to prove certain statements about the global dynamics of infinite-dimensional maps. The method combines set-oriented numerical tools for the computation of invariant sets and isolating neighborhoods, the Conley index theory, and analytic considerations. It not only allows for the detection of a certain dynamical behavior, but also for a precise computation of the corresponding invariant sets in phase space. As an example computation we show the existence of period points, connecting orbits, and chaotic dynamics in the Kot--Schaffer growth-dispersal model for plants.
Siam Journal on Applied Dynamical Systems | 2005
Sarah Day; Yasuaki Hiraoka; Konstantin Mischaikow; Toshiyuki Ogawa
This paper presents a rigorous numerical method for the study and verification of global dynamics. In particular, this method produces a conjugacy or semiconjugacy between an attractor for the Swift--Hohenberg equation and a model system. The procedure involved relies on first verifying bifurcation diagrams produced via continuation methods, including proving the existence and uniqueness of computed branches as well as showing the nonexistence of additional stationary solutions. Topological information in the form of the Conley index, also computed during this verification procedure, is then used to build a model for the attractor consisting of stationary solutions and connecting orbits.
Siam Journal on Applied Dynamical Systems | 2008
Sarah Day; Rafael M. Frongillo; Rodrigo Treviño
The aim of this paper is to introduce a method for computing rigorous lower bounds for topological entropy. The topological entropy of a dynamical system measures the number of trajectories that separate in finite time and quantifies the complexity of the system. Our method relies on extending existing computational Conley index techniques for constructing semiconjugate symbolic dynamical systems. Besides offering a description of the dynamics, the constructed symbol system allows for the computation of a lower bound for the topological entropy of the original system. Our overall goal is to construct symbolic dynamics that yield a high lower bound for entropy. The method described in this paper is algorithmic and, although it is computational, yields mathematically rigorous results. For illustration, we apply the method to the Henon map, where we compute a rigorous lower bound of 0.4320 for topological entropy.
Multiscale Modeling & Simulation | 2009
Sarah Day; William D. Kalies; Thomas Wanner
Homology has long been accepted as an important computational tool for quantifying complex structures. In many applications these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study based on suitable discretizations. Such an approach immediately raises the question of how accurately the resulting homology can be computed. In this paper we present an algorithm for correctly computing the homology of one- and two-dimensional nodal domains. The approach relies on constructing an appropriate cubical approximation for the nodal domain based on the behavior of the defining function at the vertices of a fixed grid. Betti numbers for these cubical sets are readily computable using [T. Kaczynski, K. Mischaikow, and M. Mrozek, Computational Homology, Springer-Verlag, New York, 2004; W. Kalies, M. Mrozek, and P. Pilarczyk, Computational Homology Project, http://chomp.rutgers.edu/ (2006)]. Here, we present a technique to verify that the cubical representatio...
SIAM Journal on Numerical Analysis | 2013
Sarah Day; William D. Kalies
Topological tools, such as Conley index theory, have inspired rigorous computational methods for studying dynamics. These methods rely on the construction of an outer approximation, a combinatorial representation of the system that incorporates small, bounded error. In this work, we present an automated approach to constructing outer approximations for systems in a class of integrodifference operators with smooth nonlinearities. Chebyshev interpolants and Galerkin projections form the basis for the construction, while analysis and interval arithmetic are used to incorporate explicit error bounds. This represents a significant advance to the approach given by Day, Junge, and Mischaikow [SIAM J. Appl. Dyn. Syst., 3 (2004), pp. 117--160], extending the nonlinearities that may be studied from low degree polynomials to smooth functions and the studied portion of phase space from a simulated attracting region to the global maximal invariant set. As a demonstration of the techniques, a Morse decomposition of the...
Nonlinearity | 2008
Sarah Day; Hiroshi Kokubu; Stefano Luzzatto; Konstantin Mischaikow; Hiroe Oka; P Pilarczyk
We develop a rigorous computational method for estimating the Lyapunov exponents in uniformly expanding regions of the phase space for one-dimensional maps. Our method uses rigorous numerics and graph algorithms to provide results that are mathematically meaningful and can be achieved in an efficient way.
Proceedings of the International Conference on Differential Equations | 2005
Sarah Day; Oliver Junge; Konstantin Mischaikow
We describe a set of algorithms (and corresponding codes) that serve as a basis for an automated proof of existence of a certain dynamical behaviour in a given dynamical system. In particular it is in principle possible to automatically verify the presence of complicated dynamics using these tools. The Hénon map is used to illustrate these techniques.
Electronic Research Announcements of The American Mathematical Society | 2007
Sarah Day; William D. Kalies; Konstantin Mischaikow; Thomas Wanner
Homology has long been accepted as an important computable tool for quantifying complex structures. In many applications these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study, based on suitable discretizations. Such an approach immediately raises the question of how accurate the resulting homology computations are. In this paper we present a probabilistic approach to quantifying the validity of homology computations for nodal domains of random Fourier series in one and two space dimensions, which furnishes explicit probabilistic apriori bounds for the suitability of certain discretization sizes. In addition, we introduce a numerical method for verifying the homology computation using interval arithmetic.
Journal of Mathematical Imaging and Vision | 2018
Yu-Min Chung; Sarah Day
We develop a method based on persistent homology to analyze topological structure in noisy digital images. The method returns threshold(s) for image segmentation to represent inherent topological structure as well as estimates of topological quantities in the form of Betti numbers. Two motivating data sets are scans of binary alloys and firn, the intermediate stage between snow and ice.