Bayesian Estimation for Transport Equations for Nanocapacitors

We use and evaluate different Bayesian estimation methods to quantify uncertainties in model parameters in nanocapacitors. Here, randomness arises due to process variations; also, parameters that cannot be measured directly are to be determined. The methods include the direct approach, the Markov-chain Monte-Carlo (MCMC) method, and an iterative version of the latter that we have developed, where we use the calculated posterior distribution as the prior distribution for a new MCMC analysis. We investigate the influence of the number of samples in each Markov chain and the number of iterations on the total computational work and the error achieved. In addition, we discuss the methods for estimating the posterior distribution based on samples provided by the MCMC analysis. We apply our algorithms to the Poisson-Boltzmann and Poisson-Nernst-Planck equations which arise from modeling nanoelectrode biosensors, which have recently been used to detect minute concentrations of target particles. This technology has many applications in precision medicine. Numerical examples show the estimation of parameters such as ionic concentration, size of Stern layer, and the sizes of multiple electrodes (multilevel Bayesian estimation) of sensors for which experimental data are available.