Typical optimal design methods for inverse or parameter estimation problems are designed to choose optimal sampling distributions through minimization of a specific cost function related to the resulting error in parameter estimates. It is hoped that the inverse problem will produce parameter estimates with increased accuracy using data collected according to the optimal sampling distribution. Here we formulate the classical optimal design problem in the context of general optimization problems over distributions of sampling times. We present a new Prohorov metric based theoretical framework that permits one to treat succinctly and rigorously any optimal design criteria based on the Fisher Information Matrix (FIM). A fundamental approximation theory is also included in this framework. A new optimal design, SE-optimal design (standard error optimal design), is then introduced in the context of this framework. We compare this new design criteria with the more traditional D-optimal and E-optimal designs. The optimal sampling distributions from each design are used to compute and compare standard errors; the standard errors for parameters are computed using asymptotic theory or bootstrapping and the optimal mesh. We use three examples to illustrate ideas: the Verhulst-Pearl logistic population model [13], the standard harmonic oscillator model [13] and a popular glucose regulation model [16, 19, 29].

We computationally investigate two approaches for uncertainty quantification in inverse problems for nonlinear parameter dependent dynamical systems. We compare the bootstrapping and asymptotic theory approaches for problems involving data with several noise forms and levels. We consider both constant variance absolute error data and relative error which produces non-constant variance data in our parameter estimation formulations. We compare and contrast parameter estimates, standard errors, confidence intervals, and computational times for both bootstrapping and asymptotic theory methods.

Highly time-resolved air monitoring data are widely being collected over long time horizons in order to characterize ambient and near-source air quality trends. In many applications, it is desirable to split the time-resolved data into two or more components (e.g., local and regional) for apportionment and mitigation purposes. While there may be increased information content in highly time-resolved data, the temporal resolution may also increase entropic effects on the data, thereby dramatically clouding the very information sought in time-resolved data. Specialized methods such as filtering may be required to extract the underlying information content. Constrained and Adaptive Decomposition of Time Series (CADETS) is a new method that can help carve out components of time series based on the content of the frequencies present in the time series. CADETS is also a flexible approach that allows the user to choose the bifurcation point with minimal negative impacts. Using this algorithm, we demonstrate that a time series signal may be decomposed into two useful and interpretable signals that can help identify aspects that may otherwise be hidden or distorted. Using the output from the CADETS algorithm, we show that ultrafine particles (30-100 nm) collected near a major highway may be split into a 64:36 ratio of highly varying (local) and slowly varying (regional) components, meanwhile identical measurements at a background location were estimated to split into a 56:44 local versus regional ratio.

We present a preliminary first-pass dynamic model for delivery of drug compounds to the lungs and heart. We use a compartmental mass balance approach to develop a system of nonlinear differential equations for mass accumulated in the heart as a result of intravenous injection. We discuss sensitivity analysis as well as methodology for minimizing mass in the heart while maximizing mass delivered to the lungs on a first circulatory pass.