Bayesian synthetic likelihood

Having the ability to work with complex models can be highly beneficial, but the computational cost of doing so is often large. Complex models often have intractable likelihoods, so methods that directly use the likelihood function are infeasible. In these situations, the benefits of working with likelihood-free methods become apparent. Likelihood-free methods, such as parametric Bayesian indirect likelihood that uses the likelihood of an alternative parametric auxiliary model, have been explored throughout the literature as a good alternative when the model of interest is complex. One of these methods is called the synthetic likelihood (SL), which assumes a multivariate normal approximation to the likelihood of a summary statistic of interest. This paper explores the accuracy and computational efficiency of the Bayesian version of the synthetic likelihood (BSL) approach in comparison to a competitor known as approximate Bayesian computation (ABC) and its sensitivity to its tuning parameters and assumptions. We relate BSL to pseudo-marginal methods and propose to use an alternative SL that uses an unbiased estimator of the exact working normal likelihood when the summary statistic has a multivariate normal distribution. Several applications of varying complexity are considered to illustrate the findings of this paper.

Mathematical modelling of human corneal oxygenation and cell kinetics in corneal epithelial wound healing

Healthy transparent cornea depends upon the regulation of fluid, nutrient and oxygen transport through the tissue to sustain cell metabolism and other critical processes for normal functioning. This research considers the corneal geometry and investigates oxygen distribution using a two-dimensional Monod kinetic model, showing that previous studies make assumptions that lead to predictions of near-anoxic levels of oxygen tension in the limbal regions of the cornea. It also considers the comparison of experimental spatial and temporal data with the predictions of novel mathematical models with respect to distributed mitotic rates during corneal epithelial wound healing.

Open-pit block sequencing optimization: A mathematical model and solution technique

This study presents a comprehensive mathematical formulation model for a short-term open-pit mine block sequencing problem, which considers nearly all relevant technical aspects in open-pit mining. The proposed model aims to obtain the optimum extraction sequences of the original-size (smallest) blocks over short time intervals and in the presence of real-life constraints, including precedence relationship, machine capacity, grade requirements, processing demands and stockpile management. A hybrid branch-and-bound and simulated annealing algorithm is developed to solve the problem. Computational experiments show that the proposed methodology is a promising way to provide quantitative recommendations for mine planning and scheduling engineers.

Mathematical modelling and numerical simulation of phosphine flow during grain fumigation in leaky cylindrical silos

The phosphine distribution in a cylindrical silo containing grain is predicted. A three-dimensional mathematical model, which accounts for multicomponent gas phase transport and the sorption of phosphine into the grain kernel is developed. In addition, a simple model is presented to describe the death of insects within the grain as a function of their exposure to phosphine gas. The proposed model is solved using the commercially available computational fluid dynamics (CFD) software, FLUENT, together with our own C code to customize the solver in order to incorporate the models for sorption and insect extinction. Two types of fumigation delivery are studied, namely, fan- forced from the base of the silo and tablet from the top of the silo. An analysis of the predicted phosphine distribution shows that during fan forced fumigation, the position of the leaky area is very important to the development of the gas flow field and the phosphine distribution in the silo. If the leak is in the lower section of the silo, insects that exist near the top of the silo may not be eradicated. However, the position of a leak does not affect phosphine distribution during tablet fumigation. For such fumigation in a typical silo configuration, phosphine concentrations remain low near the base of the silo. Furthermore, we find that half-life pressure test readings are not an indicator of phosphine distribution during tablet fumigation.

Characterization of anomalous relaxation using the time-fractional Bloch equation and multiple echo T2 *-weighted magnetic resonance imaging at 7T

PURPOSE To study the utility of fractional calculus in modeling gradient-recalled echo MRI signal decay in the normal human brain. METHODS We solved analytically the extended time-fractional Bloch equations resulting in five model parameters, namely, the amplitude, relaxation rate, order of the time-fractional derivative, frequency shift, and constant offset. Voxel-level temporal fitting of the MRI signal was performed using the classical monoexponential model, a previously developed anomalous relaxation model, and using our extended time-fractional relaxation model. Nine brain regions segmented from multiple echo gradient-recalled echo 7 Tesla MRI data acquired from five participants were then used to investigate the characteristics of the extended time-fractional model parameters. RESULTS We found that the extended time-fractional model is able to fit the experimental data with smaller mean squared error than the classical monoexponential relaxation model and the anomalous relaxation model, which do not account for frequency shift. CONCLUSIONS We were able to fit multiple echo time MRI data with high accuracy using the developed model. Parameters of the model likely capture information on microstructural and susceptibility-induced changes in the human brain.

Mathematical modelling of rock-bed solar thermal storage tank

An important application of thermal storage is solar energy for power generation or process heating. Low temperature thermal storage in a packed rock bed is considered best option for thermal storage for solar drying applications. In this paper, mathematical formulations for conical and cylindrical rock bed storage tanks have been developed. The model equations are solved numerically for charging/discharging cycles. From the simulated results, it was observed that for the same aspect ratio between the diameter and the length of the thermal storages, the conical thermal storage had better performance. The temperature distribution was found to be more uniform in the truncated conical shape rock bed storage. Also, the pressure drop over long period of time in the conical thermal storage was lower than that of the cylindrical thermal storage. Hence, the amount of power required from a centrifugal fan was lower.

Human health risk assessment of heavy metals in urban stormwater

Toxic chemical pollutants such as heavy metals (HMs) are commonly present in urban stormwater. These pollutants can pose a significant risk to human health and hence a significant barrier for urban stormwater reuse. The primary aim of this study was to develop an approach for quantitatively assessing the risk to human health due to the presence of HMs in stormwater. This approach will lead to informed decision making in relation to risk management of urban stormwater reuse, enabling efficient implementation of appropriate treatment strategies. In this study, risks to human health from heavy metals were assessed as hazard index (HI) and quantified as a function of traffic and land use related parameters. Traffic and land use are the primary factors influencing heavy metal loads in the urban environment. The risks posed by heavy metals associated with total solids and fine solids (<150µm) were considered to represent the maximum and minimum risk levels, respectively. The study outcomes confirmed that Cr, Mn and Pb pose the highest risks, although these elements are generally present in low concentrations. The study also found that even though the presence of a single heavy metal does not pose a significant risk, the presence of multiple heavy metals could be detrimental to human health. These findings suggest that stormwater guidelines should consider the combined risk from multiple heavy metals rather than the threshold concentration of an individual species. Furthermore, it was found that risk to human health from heavy metals in stormwater is significantly influenced by traffic volume and the risk associated with stormwater from industrial areas is generally higher than that from commercial and residential areas.

Ship collision avoidance and COLREGS compliance using simulation-based control behavior selection with predictive hazard assessment

This paper describes a concept for a collision avoidance system for ships, which is based on model predictive control. A finite set of alternative control behaviors are generated by varying two parameters: offsets to the guidance course angle commanded to the autopilot and changes to the propulsion command ranging from nominal speed to full reverse. Using simulated predictions of the trajectories of the obstacles and ship, compliance with the Convention on the International Regulations for Preventing Collisions at Sea and collision hazards associated with each of the alternative control behaviors are evaluated on a finite prediction horizon, and the optimal control behavior is selected. Robustness to sensing error, predicted obstacle behavior, and environmental conditions can be ensured by evaluating multiple scenarios for each control behavior. The method is conceptually and computationally simple and yet quite versatile as it can account for the dynamics of the ship, the dynamics of the steering and propulsion system, forces due to wind and ocean current, and any number of obstacles. Simulations show that the method is effective and can manage complex scenarios with multiple dynamic obstacles and uncertainty associated with sensors and predictions.