Emulation-accelerated Hamiltonian Monte Carlo algorithms for parameter estimation and uncertainty quantification in differential equation models

We propose to accelerate Hamiltonian and Lagrangian Monte Carlo algorithms by coupling them with Gaussian processes for emulation of the log unnormalised posterior distribution. We provide proofs of detailed balance with respect to the exact posterior distribution for these algorithms, and validate the correctness of the samplers’ implementation by Geweke consistency tests. We implement these algorithms in a delayed acceptance (DA) framework, and investigate whether the DA scheme can offer computational gains over the standard algorithms. A comparative evaluation study is carried out to assess the performance of the methods on a series of models described by differential equations, including a real-world application of a 1D fluid-dynamics model of the pulmonary blood circulation. The aim is to identify the algorithm which gives the best trade-off between accuracy and computational efficiency, to be used in nonlinear DE models, which are computationally onerous due to repeated numerical integrations in a Bayesian analysis. Results showed no advantage of the DA scheme over the standard algorithms with respect to several efficiency measures based on the effective sample size for most methods and DE models considered. These gradient-driven algorithms register a high acceptance rate, thus the number of expensive forward model evaluations is not significantly reduced by the first emulator-based stage of DA. Additionally, the Lagrangian Dynamical Monte Carlo and Riemann Manifold Hamiltonian Monte Carlo tended to register the highest efficiency (in terms of effective sample size normalised by the number of forward model evaluations), followed by the Hamiltonian Monte Carlo, and the No U-turn sampler tended to be the least efficient.

Simu-D: A Simulator-Descriptor Suite for Polymer-Based Systems under Extreme Conditions

We present Simu-D, a software suite for the simulation and successive identification of local structures of atomistic systems, based on polymers, under extreme conditions, in the bulk, on surfaces, and at interfaces. The protocol is built around various types of Monte Carlo algorithms, which include localized, chain-connectivity-altering, identity-exchange, and cluster-based moves. The approach focuses on alleviating one of the main disadvantages of Monte Carlo algorithms, which is the general applicability under a wide range of conditions. Present applications include polymer-based nanocomposites with nanofillers in the form of cylinders and spheres of varied concentration and size, extremely confined and maximally packed assemblies in two and three dimensions, and terminally grafted macromolecules. The main simulator is accompanied by a descriptor that identifies the similarity of computer-generated configurations with respect to reference crystals in two or three dimensions. The Simu-D simulator-descriptor can be an especially useful tool in the modeling studies of the entropy- and energy-driven phase transition, adsorption, and self-organization of polymer-based systems under a variety of conditions.

Monte Carlo Algorithms for the Extracting of Electrical Capacitance

We present new Monte Carlo algorithms for extracting mutual capacitances for a system of conductors embedded in inhomogeneous isotropic dielectrics. We represent capacitances as functionals of the solution of the external Dirichlet problem for the Laplace equation. Unbiased and low-biased estimators for the capacitances are constructed on the trajectories of the Random Walk on Spheres or the Random Walk on Hemispheres. The calculation results show that the accuracy of these new algorithms does not exceed the statistical error of estimators, which is easily determined in the course of calculations. The algorithms are based on mean value formulas for harmonic functions in different domains and do not involve a transition to a difference problem. Hence, they do not need a lot of storage space.

Random walk on spheres algorithm for solving steady-state and transient diffusion-recombination problems

We further develop in this study the Random Walk on Spheres (RWS) stochastic algorithm for solving systems of coupled diffusion-recombination equations first suggested in our recent article [K. Sabelfeld, First passage Monte Carlo algorithms for solving coupled systems of diffusion–reaction equations, Appl. Math. Lett. 88 2019, 141–148]. The random walk on spheres process mimics the isotropic diffusion of two types of particles which may recombine to each other. Our motivation comes from the transport problems of free and bound exciton recombination. The algorithm is based on tracking the trajectories of the diffusing particles exactly in accordance with the probabilistic distributions derived from the explicit representation of the relevant Green functions for balls and spheres. Therefore, the method is mesh free both in space and time. In this paper we implement the RWS algorithm for solving the diffusion-recombination problems both in a steady-state and transient settings. Simulations are compared against the exact solutions. We show also how the RWS algorithm can be applied to calculate exciton flux to the boundary which provides the electron beam-induced current, the concentration of the survived excitons, and the cathodoluminescence intensity which are all integral characteristics of the solution to diffusion-recombination problem.

A diffusion Monte Carlo method for charge density on a conducting surface at non-constant potentials

We develop a last-passage Monte Carlo algorithm on a conducting surface at non-constant potentials. In the previous researches, last-passage Monte Carlo algorithms on conducting surfaces with a constant potential have been developed for charge density at a specific point or on a finite region and a hybrid BIE-WOS algorithm for charge density on a conducting surface at non-constant potentials. In the hybrid BIE-WOS algorithm, they used a deterministic method for the contribution from the lower non-constant potential surface. In this paper, we modify the hybrid BIE-WOS algorithm to a last-passage Monte Carlo algorithm on a conducting surface at non-constant potentials, where we can avoid the singularities on the non-constant potential surface very naturally. We demonstrate the last-passage Monte Carlo algorithm for charge densities on a circular disk and the four rectangle plates with a simple voltage distribution, and update the corner singularities on the unit square plate and cube.

A Parallel Dynamic Asynchronous Framework for Uncertainty Quantification by Hierarchical Monte Carlo Algorithms

The necessity of dealing with uncertainties is growing in many different fields of science and engineering. Due to the constant development of computational capabilities, current solvers must satisfy both statistical accuracy and computational efficiency. The aim of this work is to introduce an asynchronous framework for Monte Carlo and Multilevel Monte Carlo methods to achieve such a result. The proposed approach presents the same reliability of state of the art techniques, and aims at improving the computational efficiency by adding a new level of parallelism with respect to existing algorithms: between batches, where each batch owns its hierarchy and is independent from the others. Two different numerical problems are considered and solved in a supercomputer to show the behavior of the proposed approach.