Kody Law

Kody Law
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Kody Law

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Statistics - Computation (11)
Mathematics - Numerical Analysis (7)
Mathematics - Probability (7)
Statistics - Methodology (4)
Physics - Data Analysis; Statistics and Probability (2)
Nonlinear Sciences - Pattern Formation and Solitons (2)
Mathematics - Optimization and Control (2)
Physics - Atomic Physics (1)
Physics - Optics (1)
Mathematics - Statistics (1)
Statistics - Applications (1)
Statistics - Theory (1)
Mathematics - Dynamical Systems (1)
Physics - Computational Physics (1)

Publications Authored By Kody Law

This article reviews the application of advanced Monte Carlo techniques in the context of Multilevel Monte Carlo (MLMC). MLMC is a strategy employed to compute expectations which can be biased in some sense, for instance, by using the discretization of a associated probability law. The MLMC approach works with a hierarchy of biased approximations which become progressively more accurate and more expensive. Read More

In this article we develop a new sequential Monte Carlo (SMC) method for multilevel (ML) Monte Carlo estimation. In particular, the method can be used to estimate expectations with respect to a target probability distribution over an infinite-dimensional and non-compact space as given, for example, by a Bayesian inverse problem with Gaussian random field prior. Under suitable assumptions the MLSMC method has the optimal $O(\epsilon^{-2})$ bound on the cost to obtain a mean-square error of $O(\epsilon^2)$. Read More

In this article we consider static Bayesian parameter estimation for partially observed diffusions that are discretely observed. We work under the assumption that one must resort to discretizing the underlying diffusion process, for instance using the Euler-Maruyama method. Given this assumption, we show how one can use Markov chain Monte Carlo (MCMC) and particularly particle MCMC [Andrieu, C. Read More

This work embeds a multilevel Monte Carlo (MLMC) sampling strategy into the Monte Carlo step of the ensemble Kalman filter (EnKF), thereby yielding a multilevel ensemble Kalman filter (MLEnKF) which has provably superior asymptotic cost to a given accuracy level. The development of MLEnKF for finite-dimensional state-spaces in the work [20] is here extended to models with infinite-dimensional state- spaces in the form of spatial fields. A concrete example is given to illustrate the results. Read More

This paper considers uncertainty quantification for an elliptic nonlocal equation. In particular, it is assumed that the parameters which define the kernel in the nonlocal operator are uncertain and a priori distributed according to a probability measure. It is shown that the induced probability measure on some quantities of interest arising from functionals of the solution to the equation with random inputs is well-defined; as is the posterior distribution on parameters given observations. Read More

This article considers the sequential Monte Carlo (SMC) approximation of ratios of normalizing constants associated to posterior distributions which in principle rely on continuum models. Therefore, the Monte Carlo estimation error and the discrete approximation error must be balanced. A multilevel strategy is utilized to substantially reduce the cost to obtain a given error level in the approximation as compared to standard estimators. Read More

In this paper the filtering of partially observed diffusions, with discrete-time observations, is considered. It is assumed that only biased approximations of the diffusion can be obtained, for choice of an accuracy parameter indexed by $l$. A multilevel estimator is proposed, consisting of a telescopic sum of increment estimators associated to the successive levels. Read More

This work considers black-box Bayesian inference over high-dimensional parameter spaces. The well-known adaptive Metropolis (AM) algorithm of (Haario etal. 2001) is extended herein to scale asymptotically uniformly with respect to the underlying parameter dimension for Gaussian targets, by respecting the variance of the target. Read More

This work embeds a multilevel Monte Carlo sampling strategy into the Monte Carlo step of the ensemble Kalman filter (EnKF) in the setting of finite dimensional signal evolution and noisy discrete-time observations. The signal dynamics is assumed to be governed by a stochastic differential equation (SDE), and a hierarchy of time grids is introduced for multilevel numerical integration of that SDE. The resulting multilevel EnKF is proved to asymptotically outperform EnKF in terms of computational cost versus approximation accuracy. Read More

Many Bayesian inference problems require exploring the posterior distribution of high-dimensional parameters that represent the discretization of an underlying function. This work introduces a family of Markov chain Monte Carlo (MCMC) samplers that can adapt to the particular structure of a posterior distribution over functions. Two distinct lines of research intersect in the methods developed here. Read More

The proof of convergence of the standard ensemble Kalman filter (EnKF) from Legland etal. (2011) is extended to non-Gaussian state space models. A density-based deterministic approximation of the mean-field limit EnKF (DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Read More

The Bayesian approach to inverse problems is of paramount importance in quantifying uncertainty about the input to and the state of a system of interest given noisy observations. Herein we consider the forward problem of the forced 2D Navier Stokes equation. The inverse problem is inference of the forcing, and possibly the initial condition, given noisy observations of the velocity field. Read More

We consider the inverse problem of estimating an unknown function $u$ from noisy measurements $y$ of a known, possibly nonlinear, map $\mathcal{G}$ applied to $u$. We adopt a Bayesian approach to the problem and work in a setting where the prior measure is specified as a Gaussian random field $\mu_0$. We work under a natural set of conditions on the likelihood which imply the existence of a well-posed posterior measure, $\mu^y$. Read More

Inverse problems lend themselves naturally to a Bayesian formulation, in which the quantity of interest is a posterior distribution of state and/or parameters given some uncertain observations. For the common case in which the forward operator is smoothing, then the inverse problem is ill-posed. Well-posedness is imposed via regularisation in the form of a prior, which is often Gaussian. Read More

In this paper we propose to numerically assess the performance of standard Gaussian approximations to probe the posterior distribution that arises from Bayesian data assimilation in petroleum reservoirs. In particular we assess the performance of (i) the linearization around the maximum a posterior estimate, (ii) the randomized maximum likelihood and (iii) standard ensemble Kalman filter-type methods. In order to fully resolve the posterior distribution we implement a state-of-the art MCMC method that scales well with respect to the dimension of the parameter space. Read More

The Ensemble Kalman filter (EnKF) was introduced by Evensen in 1994 [10] as a novel method for data assimilation: state estimation for noisily observed time-dependent problems. Since that time it has had enormous impact in many application domains because of its robustness and ease of implementation, and numerical evidence of its accuracy. In this paper we propose the application of an iterative ensemble Kalman method for the solution of a wide class of inverse problems. Read More

We study the structure, stability, and dynamics of dark solitary waves in parabolically trapped, collisionally inhomogeneous Bose-Einstein condensates (BECs) with spatially periodic variations of the scattering length. This collisional inhomogeneity yields a nonlinear lattice, which we tune from a small-amplitude, approximately sinusoidal structure to a periodic sequence of densely spaced spikes. We start by investigating time-independent inhomogeneities, and we subsequently examine the dynamical response when one starts with a collisionally homogeneous BEC and then switches on an inhomogeneity either adiabatically or nonadiabatically. Read More

We highlight an interesting mapping between the moving breather solutions of the generalized Nonlinear Schrodinger (NLS) equations and the static solutions of neutral scalar field theories. Using this connection, we then obtain several new moving breather solutions of the cubic-quintic NLS equation both with and without uniform phase in space. The stability of some stationary solutions is investigated numerically and the results confirmed via dynamical evolution. Read More

We show that double-charge discrete optical vortices may be completely stable in hexagonal photonic lattices where single-charge vortices always exhibit dynamical instabilities. Even when unstable the double-charge vortices typically have a much weaker instability than the single-charge vortices, and thus their breakup occurs at longer propagation distances. Read More