Matthew O. Williams

Matthew O. Williams
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Matthew O. Williams

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Pub Categories

Mathematics - Dynamical Systems (4)
Physics - Fluid Dynamics (3)
Computer Science - Computers and Society (2)
Physics - Physics and Society (2)
Nonlinear Sciences - Adaptation and Self-Organizing Systems (1)
Physics - Optics (1)
Computer Science - Cryptography and Security (1)
Physics - Biological Physics (1)
Mathematics - Number Theory (1)
Mathematics - Numerical Analysis (1)
Physics - Soft Condensed Matter (1)

Publications Authored By Matthew O. Williams

For positive integers m and n, denote S(m,n) as the associated Stirling number of the second kind and let z be a complex variable. In this paper, we introduce the Stirling functions S(m,n,z) which satisfy S(m,n,z) = S(m,n) for any z which lies in the zero set of a certain polynomial P(m,n,z). For all real z, the solutions of S(m,n,z) = S(m,n) are computed and all real roots of the polynomial P(m,n,z) are shown to be simple. Read More

In many commercial and academic settings, numerical solvers fail to achieve their theoretical performance levels due to issues in the system definition, parameterization, and even implementation. We propose a pair of methods for detecting and localizing these convergence rate issues in applications that can be treated as homotopy problems including numerical continuation and the evolution of differential algebraic equations. Both approaches are rooted in dynamical systems theory, in particular, the numerical techniques used to perform bifurcation studies on "black-box" systems, and can be applied across a range of numerical solvers and systems without significant modification. Read More

By means of extensive replica-exchange simulations of generic coarse-grained models for helical polymers, we systematically investigate the structural transitions into all possible helical phases for flexible and semiflexible elastic polymers with self-interaction under the influence of torsion barriers. The competing interactions lead to a variety of conformational phases including disordered helical arrangements, single helices, and ordered, tertiary helix bundles. Most remarkably, we find that a bending restraint entails a clear separation and stabilization of the helical phases. Read More

Dynamic mode decomposition (DMD) provides a practical means of extracting insightful dynamical information from fluids datasets. Like any data processing technique, DMD's usefulness is limited by its ability to extract real and accurate dynamical features from noise-corrupted data. Here we show analytically that DMD is biased to sensor noise, and quantify how this bias depends on the size and noise level of the data. Read More

Recent advances in spatial and temporal networks have enabled researchers to more-accurately describe many real-world systems such as urban transport networks. In this paper, we study the response of real-world spatio-temporal networks to random error and systematic attack, taking a unified view of their spatial and temporal performance. We propose a model of spatio-temporal paths in time-varying spatially embedded networks which captures the property that, as in many real-world systems, interaction between nodes is non-instantaneous and governed by the space in which they are embedded. Read More

The election forecasting 'industry' is a growing one, both in the volume of scholars producing forecasts and methodological diversity. In recent years a new approach has emerged that relies on social media and particularly Twitter data to predict election outcomes. While some studies have shown the method to hold a surprising degree of accuracy there has been criticism over the lack of consistency and clarity in the methods used, along with inevitable problems of population bias. Read More

With the advent of GPS enabled smartphones, an increasing number of users is actively sharing their location through a variety of applications and services. Along with the continuing growth of Location-Based Social Networks (LBSNs), security experts have increasingly warned the public of the dangers of exposing sensitive information such as personal location data. Most importantly, in addition to the geographical coordinates of the user's location, LBSNs allow easy access to an additional set of characteristics of that location, such as the venue type or popularity. Read More

A data-driven procedure for identifying the dominant transport barriers in a time-varying flow from limited quantities of Lagrangian data is presented. Our approach partitions state space into pairs of coherent sets, which are sets of initial conditions chosen to minimize the number of trajectories that "leak" from one set to the other under the influence of a stochastic flow field during a pre-specified interval in time. In practice, this partition is computed by posing an optimization problem, which once solved, yields a pair of functions whose signs determine set membership. Read More

We demonstrate that numerically computed approximations of Koopman eigenfunctions and eigenvalues create a natural framework for data fusion in applications governed by nonlinear evolution laws. This is possible because the eigenvalues of the Koopman operator are invariant to invertible transformations of the system state, so that the values of the Koopman eigenfunctions serve as a set of intrinsic coordinates that can be used to map between different observations (e.g. Read More

A data driven, kernel-based method for approximating the leading Koopman eigenvalues, eigenfunctions, and modes in problems with high dimensional state spaces is presented. This approach approximates the Koopman operator using a set of scalar observables, which are functions defined on state space, that is determined {\em implicitly} by the choice of a kernel. This circumvents the computational issues that arise due to the number of basis functions required to span a "sufficiently rich" subspace of the space of scalar observables in these problems. Read More

We demonstrate the utility of the equation free methodology developed by one of the authors (I.G.K) for the study of scalar conservation laws with disordered initial conditions. Read More

The Koopman operator is a linear but infinite dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system, and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a "black box" integrator. Read More

We formulate a low-storage method for performing dynamic mode decomposition that can be updated inexpensively as new data become available; this formulation allows dynamical information to be extracted from large datasets and data streams. We present two algorithms: the first is mathematically equivalent to a standard "batch-processed" formulation; the second introduces a compression step that maintains computational efficiency, while enhancing the ability to isolate pertinent dynamical information from noisy measurements. Both algorithms reliably capture dominant fluid dynamic behaviors, as demonstrated on cylinder wake data collected from both direct numerical simulations and particle image velocimetry experiments Read More

We demonstrate theoretically that robust mode-locking can be achieved on a semiconductor chip with a waveguide array architecture. The waveguide arrays are used as an ideal saturable absorption mechanism for initial noise start-up as well as pulse shaping and stabilization. The cavity gain is provided by an injection current and forward biasing of the semiconductor material. Read More