arXiv subjects

Geoffrey McGregor

Publications and source records attributed to Geoffrey McGregor.

6 recordsLinked to original sources

arXiv2022

Improving sampling efficacy on high dimensional distributions with thin high density regions using Conservative Hamiltonian Monte Carlo

Hamiltonian Monte Carlo is a prominent Markov Chain Monte Carlo algorithm, which employs symplectic integrators to sample from high dimensional target distributions in many applications, such as statistical mechanics, Bayesian statistics and generative models. However, such distributions tend to have thin high density regions, posing a significant challenge for symplectic integrators to maintain the small energy errors needed for a high acceptance probability. Instead, we propose a variant called Conservative Hamiltonian Monte Carlo, using $R$--reversible energy-preserving integrators to retain a high acceptance probability. We show our algorithm can achieve approximate stationarity with an error determined by the Jacobian approximation of the energy-preserving proposal map. Numerical evidence shows improved convergence and robustness over integration parameters on target distributions with thin high density regions and in high dimensions. Moreover, a version of our algorithm can also be applied to target distributions without gradient information.

math.NA↗

arXiv2021

Comparing regional and provincial-wide COVID-19 models with physical distancing in British Columbia

We study the effects of physical distancing measures for the spread of COVID-19 in regional areas within British Columbia, using the reported cases of the five provincial Health Authorities. Building on the Bayesian epidemiological model of Anderson et al. (2020), we propose a hierarchical regional Bayesian model with time-varying regional parameters between March to December of 2020. In the absence of COVID-19 variants and vaccinations during this period, we examine the regionalized basic reproduction number, modelled prevalence, relative reduction in contact due to physical distancing, and proportion of anticipated cases that have been tested and reported. We observe significant differences between the regional and provincial-wide models and demonstrate the hierarchical regional model can better estimate regional prevalence, especially in rural regions. These results indicate that it can be useful to apply similar regional models to other parts of Canada or other countries.

stat.AP↗

arXiv2019

Parametric Interpolation Framework for 1-D Scalar Conservation Laws with Non-Convex Flux Functions

In this paper we present a novel framework for obtaining high order numerical methods for 1-D scalar conservation laws with non-convex flux functions. When solving Riemann problems, the Oleinik entropy condition, [16], is satisfied when the resulting shocks and rarefactions correspond to correct portions of the appropriate (upper or lower) convex envelope of the flux function. We show that the standard equal-area principle fails to select these solutions in general, and therefore we introduce a generalized equal-area principle which always selects the weak solution corresponding to the correct convex envelope. The resulting numerical scheme presented here relies on the area-preserving parametric interpolation framework introduced in [14] and locates shock position to fifth order in space, conserves area exactly and admits weak solutions which satisfy the Oleinik entropy condition numerically regardless of the initial states.

math.NA↗

arXiv2019

Parametric Interpolation Framework for Scalar Conservation Laws

In this paper we present a novel framework for obtaining high-order numerical methods for scalar conservation laws in one-space dimension for both the homogeneous and non-homogeneous case. The numerical schemes for these two settings are somewhat different in the presence of shocks, however at their core they both rely heavily on the solution curve being represented parametrically. By utilizing high-order parametric interpolation techniques we succeed to obtain fifth order accuracy ( in space ) everywhere in the computation domain, including the shock location itself. In the presence of source terms a slight modification is required, yet the spatial order is maintained but with an additional temporal error appearing. We provide a detailed discussion of a sample scheme for non-homogeneous problems which obtains fifth order in space and fourth order in time even in the presence of shocks.

math.NA↗

arXiv2018

Area-Preserving Geometric Hermite Interpolation

In this paper we establish a framework for planar geometric interpolation with exact area preservation using cubic B\'ezier polynomials. We show there exists a family of such curves which are $5^{th}$ order accurate, one order higher than standard geometric cubic Hermite interpolation. We prove this result is valid when the curvature at the endpoints does not vanish, and in the case of vanishing curvature, the interpolation is $4^{th}$ order accurate. The method is computationally efficient and prescribes the parametrization speed at endpoints through an explicit formula based on the given data. Additional accuracy (i.e. same order but lower error constant) may be obtained through an iterative process to find optimal parametrization speeds which further reduces the error while still preserving the prescribed area exactly.

math.NA↗

arXiv2017

A Parametric Interpolation Framework for 1D Scalar Conservation Laws using the Equal Area Principle

In this paper we develop a novel framework for numerically solving scalar conservation laws in one space dimension. Utilizing the method of characteristics in conjunction with the equal area principle we develop an approach where the weak solution is obtained purely as the solution of a parametric interpolation problem. As this framework hinges on the validity of the equal area principle, we provide a rigorous discussion of the equal area principle and show that, indeed, the equal area principle is equivalent to the Rankine-Hugoniot condition, within the specific context studied in this paper. Combining these results with properties of the characteristic equations yields the desired setting to define the equivalent parametric interpolation problem. We conclude by applying this framework to Burgers' equation and show how one obtains machine precision in the shock position when the initial condition can be represented exactly in the chosen space of parametric polynomials.

math.NA↗