@article{nardini_bortz_2019, title={The influence of numerical error on parameter estimation and uncertainty quantification for advective PDE models}, volume={35}, ISSN={["1361-6420"]}, DOI={10.1088/1361-6420/ab10bb}, abstractNote={Advective partial differential equations can be used to describe many scientific processes. Two significant sources of error that can cause difficulties in inferring parameters from experimental data on these processes include (i) noise from the measurement and collection of experimental data and (ii) numerical error in approximating the forward solution to the advection equation. How this second source of error alters parameter estimation and uncertainty quantification during an inverse problem methodology is not well understood. As a step towards a better understanding of this problem, we present both analytical and computational results concerning how a least squares cost function and parameter estimator behave in the presence of numerical error in approximating solutions to the underlying advection equation. We investigate residual patterns to derive an autocorrelative statistical model that can improve parameter estimation and confidence interval computation for first order methods. Building on our results and their general nature, we provide guidelines for practitioners to determine when numerical or experimental error is the main source of error in their inference, along with suggestions of how to efficiently improve their results.}, number={6}, journal={INVERSE PROBLEMS}, author={Nardini, John T. and Bortz, D. M.}, year={2019}, month={Jun} }
 @article{banks_bortz_holte_2003, title={Incorporation of variability into the modeling of viral delays in HIV infection dynamics}, volume={183}, ISSN={["1879-3134"]}, DOI={10.1016/S0025-5564(02)00218-3}, abstractNote={We consider classes of functional differential equation models which arise in attempts to describe temporal delays in HIV pathogenesis. In particular, we develop methods for incorporating arbitrary variability (i.e., general probability distributions) for these delays into systems that cannot readily be reduced to a finite number of coupled ordinary differential equations (as is done in the method of stages). We discuss modeling from first principles, introduce several classes of non-linear models (including discrete and distributed delays) and present a discussion of theoretical and computational approaches. We then use the resulting methodology to carry out simulations and perform parameter estimation calculations, fitting the models to a set of experimental data. Results obtained confirm the statistical significance of the presence of delays and the importance of including delays in validating mathematical models with experimental data. We also show that the models are quite sensitive to the mean of the distribution which describes the delay in viral production, whereas the variance of this distribution has relatively little impact.}, number={1}, journal={MATHEMATICAL BIOSCIENCES}, author={Banks, HT and Bortz, DM and Holte, SE}, year={2003}, month={May}, pages={63–91} }