arXiv subjects

David Kelly

Publications and source records attributed to David Kelly.

13 recordsLinked to original sources

arXiv2023

Experimental study of developing free-falling annular flow in a large-scale vertical pipe

Annular flow is the primary characteristic of unsteady wastewater flow, which initiates entrained air and sets up the air pressure regime within the system - an important design consideration. This paper reports on an experimental investigation of free-falling annular flow in a vertical pipe with different inlets at extended flow ranges up to Re = 3 x 10e4, similar to those in Building Drainage Systems (BDS). In the experimental setup, a vertical pipe system (5 m) was used to record velocity profiles and film thickness in the developing region through Particle Image Velocimetry (PIV) measurements. Entrained droplets were collected through a separator, and the entrainment fraction was calculated at different flow conditions. The study reports on the development process of the film velocity and thickness along the vertical pipe, which agrees well with empirical predictions. The results of the droplet entrainment of a vertical annular flow show the development process to the steady state. Additionally, a Tee-junction inlet in drainage system generates a higher and different entrainment profile.

physics.flu-dyn↗

arXiv2019

SafeStrings: Representing Strings as Structured Data

Strings are ubiquitous in code. Not all strings are created equal, some contain structure that makes them incompatible with other strings. CSS units are an obvious example. Worse, type checkers cannot see this structure: this is the latent structure problem. We introduce SafeStrings to solve this problem and expose latent structure in strings. Once visible, operations can leverage this structure to efficiently manipulate it; further, SafeStrings permit the establishment of closure properties. SafeStringsharness the subtyping and inheritance mechanics of their host language to create a natural hierarchy of string subtypes. SafeStrings define an elegant programming model over strings: the front end use of a SafeString is clear and uncluttered, with complexity confined inside the definition of a particular SafeString. They are lightweight, language-agnostic and deployable, as we demonstrate by implementing SafeStrings in TypeScript. SafeStrings reduce the surface area for cross-site scripting, argument selection defects, and they can facilitate fuzzing and analysis.

cs.PL↗

arXiv2016

Ergodicity and Accuracy of Optimal Particle Filters for Bayesian Data Assimilation

For particle filters and ensemble Kalman filters it is of practical importance to understand how and why data assimilation methods can be effective when used with a fixed small number of particles, since for many large-scale applications it is not practical to deploy algorithms close to the large particle limit asymptotic. In this paper we address this question for particle filters and, in particular, study their accuracy (in the small noise limit) and ergodicity (for noisy signal and observation) without appealing to the large particle number limit. We first overview the accuracy and minorization properties for the true filtering distribution, working in the setting of conditional Gaussianity for the dynamics-observation model. We then show that these properties are inherited by optimal particle filters for any fixed number of particles, and use the minorization to establish ergodicity of the filters. For completeness we also prove large particle number consistency results for the optimal particle filters, by writing the update equations for the underlying distributions as recursions. In addition to looking at the optimal particle filter with standard resampling, we derive all the above results for (what we term) the Gaussianized optimal particle filter and show that the theoretical properties are favorable for this method, when compared to the standard optimal particle filter.

math.PR↗

arXiv2016

Fluctuations in the heterogeneous multiscale methods for fast-slow systems

How heterogeneous multiscale methods (HMM) handle fluctuations acting on the slow variables in fast-slow systems is investigated. In particular, it is shown via analysis of central limit theorems (CLT) and large deviation principles (LDP) that the standard version of HMM artificially amplifies these fluctuations. A simple modification of HMM, termed parallel HMM, is introduced and is shown to remedy this problem, capturing fluctuations correctly both at the level of the CLT and the LDP. Similar type of arguments can also be used to justify that the tau-leaping method used in the context of Gillespie's stochastic simulation algorithm for Markov jump processes also captures the right CLT and LDP for these processes.

math.PR↗

arXiv2015

Nonlinear stability of the ensemble Kalman filter with adaptive covariance inflation

The Ensemble Kalman filter and Ensemble square root filters are data assimilation methods used to combine high dimensional nonlinear models with observed data. These methods have proved to be indispensable tools in science and engineering as they allow computationally cheap, low dimensional ensemble state approximation for extremely high dimensional turbulent forecast models. From a theoretical perspective, these methods are poorly understood, with the exception of a recently established but still incomplete nonlinear stability theory. Moreover, recent numerical and theoretical studies of catastrophic filter divergence have indicated that stability is a genuine mathematical concern and can not be taken for granted in implementation. In this article we propose a simple modification of ensemble based methods which resolves these stability issues entirely. The method involves a new type of adaptive covariance inflation, which comes with minimal additional cost. We develop a complete nonlinear stability theory for the adaptive method, yielding Lyapunov functions and geometric ergodicity under weak assumptions. We present numerical evidence which suggests the adaptive methods have improved accuracy over standard methods and completely eliminate catastrophic filter divergence. This enhanced stability allows for the use of extremely cheap, unstable forecast integrators, which would otherwise lead to widespread filter malfunction.

math.PR↗

arXiv2014

Deterministic homogenization for fast-slow systems with chaotic noise

Consider a fast-slow system of ordinary differential equations of the form $\dot x=a(x,y)+\varepsilon^{-1}b(x,y)$, $\dot y=\varepsilon^{-2}g(y)$, where it is assumed that $b$ averages to zero under the fast flow generated by $g$. We give conditions under which solutions $x$ to the slow equations converge weakly to an It\^o diffusion $X$ as $\varepsilon\to0$. The drift and diffusion coefficients of the limiting stochastic differential equation satisfied by $X$ are given explicitly. Our theory applies when the fast flow is Anosov or Axiom A, as well as to a large class of nonuniformly hyperbolic fast flows (including the one defined by the well-known Lorenz equations), and our main results do not require any mixing assumptions on the fast flow.

math.PR↗

arXiv2014

Smooth approximation of stochastic differential equations

Consider an It\^{o} process $X$ satisfying the stochastic differential equation $dX=a(X)\,dt+b(X)\,dW$ where $a,b$ are smooth and $W$ is a multidimensional Brownian motion. Suppose that $W_n$ has smooth sample paths and that $W_n$ converges weakly to $W$. A central question in stochastic analysis is to understand the limiting behavior of solutions $X_n$ to the ordinary differential equation $dX_n=a(X_n)\,dt+b(X_n)\,dW_n$. The classical Wong--Zakai theorem gives sufficient conditions under which $X_n$ converges weakly to $X$ provided that the stochastic integral $\int b(X)\,dW$ is given the Stratonovich interpretation. The sufficient conditions are automatic in one dimension, but in higher dimensions the correct interpretation of $\int b(X)\,dW$ depends sensitively on how the smooth approximation $W_n$ is chosen. In applications, a natural class of smooth approximations arise by setting $W_n(t)=n^{-1/2}\int_0^{nt}v\circ\phi_s\,ds$ where $\phi_t$ is a flow (generated, e.g., by an ordinary differential equation) and $v$ is a mean zero observable. Under mild conditions on $\phi_t$, we give a definitive answer to the interpretation question for the stochastic integral $\int b(X)\,dW$. Our theory applies to Anosov or Axiom A flows $\phi_t$, as well as to a large class of nonuniformly hyperbolic flows (including the one defined by the well-known Lorenz equations) and our main results do not require any mixing assumptions on $\phi_t$. The methods used in this paper are a combination of rough path theory and smooth ergodic theory.

math.DS↗

arXiv2014

Rough path recursions and diffusion approximations

In this article, we consider diffusion approximations for a general class of stochastic recursions. Such recursions arise as models for population growth, genetics, financial securities, multiplicative time series, numerical schemes and MCMC algorithms. We make no particular probabilistic assumptions on the type of noise appearing in these recursions. Thus, our technique is well suited to recursions where the noise sequence is not a semi-martingale, even though the limiting noise may be. Our main theorem assumes a weak limit theorem on the noise process appearing in the random recursions and lifts it to diffusion approximation for the recursion itself. To achieve this, we approximate the recursion (pathwise) by the solution to a stochastic equation driven by piecewise smooth paths; this can be thought of as a pathwise version of backward error analysis for SDEs. We then identify the limit of this stochastic equation, and hence the original recursion, using tools from rough path theory. We provide several examples of diffusion approximations, both new and old, to illustrate this technique.

math.PR↗

arXiv2013

Reusability in Science: From Initial User Engagement to Dissemination of Results

Effective use of parallel and distributed computing in science depends upon multiple interdependent entities and activities that form an ecosystem. Active engagement between application users and technology catalysts is a crucial activity that forms an integral part of this ecosystem. Technology catalysts play a crucial role benefiting communities beyond a single user group. An effective user-engagement, use and reuse of tools and techniques has a broad impact on software sustainability. From our experience, we sketch a life-cycle for user-engagement activity in scientific computational environment and posit that application level reusability promotes software sustainability. We describe our experience in engaging two user groups from different scientific domains reusing a common software and configuration on different computational infrastructures.

cs.SE↗

arXiv2012

Geometric versus non-geometric rough paths

In this article we consider rough differential equations (RDEs) driven by non-geometric rough paths, using the concept of branched rough paths introduced in Gubinelli (2004). We first show that branched rough paths can equivalently be defined as $\gamma$-H\"older continuous paths in some Lie group, akin to geometric rough paths. We then show that every branched rough path can be encoded in a geometric rough path. More precisely, for every branched rough path $\mathbf{X}$ lying above a path $X$, there exists a geometric rough path $\bar{\mathbf{X}}$ lying above an extended path $\bar X$, such that $\bar{\mathbf{X}}$ contains all the information of $\mathbf{X}$. As a corollary of this result, we show that every RDE driven by a non-geometric rough path $\mathbf{X}$ can be rewritten as an extended RDE driven by a geometric rough path $\bar{\mathbf{X}}$. One could think of this as a generalisation of the It\^o-Stratonovich correction formula.

math.PR↗

arXiv2012

Stochastic PDEs with multiscale structure

We study the spatial homogenisation of parabolic linear stochastic PDEs exhibiting a two-scale structure both at the level of the linear operator and at the level of the Gaussian driving noise. We show that in some cases, in particular when the forcing is given by space-time white noise, it may happen that the homogenised SPDE is not what one would expect from existing results for PDEs with more regular forcing terms.

math.PR↗

arXiv2010

Inferring hidden Markov models from noisy time sequences: a method to alleviate degeneracy in molecular dynamics

We present a new method for inferring hidden Markov models from noisy time sequences without the necessity of assuming a model architecture, thus allowing for the detection of degenerate states. This is based on the statistical prediction techniques developed by Crutchfield et al., and generates so called causal state models, equivalent to hidden Markov models. This method is applicable to any continuous data which clusters around discrete values and exhibits multiple transitions between these values such as tethered particle motion data or Fluorescence Resonance Energy Transfer (FRET) spectra. The algorithms developed have been shown to perform well on simulated data, demonstrating the ability to recover the model used to generate the data under high noise, sparse data conditions and the ability to infer the existence of degenerate states. They have also been applied to new experimental FRET data of Holliday Junction dynamics, extracting the expected two state model and providing values for the transition rates in good agreement with previous results and with results obtained using existing maximum likelihood based methods.

q-bio.QM↗

arXiv2009

Groupoid identities common to four abelian group operations

We exhibit a finite basis M for a certain variety $\mathbf{V}$ of medial groupoids. The set M consists of the medial law (xy)(zt)=(xz)(yt) and five other identities involving four variables. The variety $\mathbf{V}$ is generated by the four groupoids $\pm x\pm y$ on the integers. Since $\mathbf{V}$ is a very natural variety, proving it to be finitely based should be of interest. In an earlier paper, we made a conjecture which implies that $\mathbf{V}$ is finitely based. In this paper, we show that $\mathbf{V}$ is finitely based by proving that M is a basis. Based on our proof, we think that our conjecture will be difficult to prove. We used four medial groupoids to define $\mathbf{V}$. We also present a finite basis for the variety generated by any proper subset of these four groupoids. In an earlier paper with R. Padmanabhan, we gave the corresponding finite bases when the constant zero is allowed.

math.RA↗