@article{adams_banks_davidian_kwon_tran_wynne_rosenberg_2005, title={HIV dynamics: Modeling, data analysis, and optimal treatment protocols}, volume={184}, ISSN={["1879-1778"]}, DOI={10.1016/j.cam.2005.02.004}, abstractNote={We present an overview of some concepts and methodologies we believe useful in modeling HIV pathogenesis. After a brief discussion of motivation for and previous efforts in the development of mathematical models for progression of HIV infection and treatment, we discuss mathematical and statistical ideas relevant to Structured Treatment Interruptions (STI). Among these are model development and validation procedures including parameter estimation, data reduction and representation, and optimal control relative to STI. Results from initial attempts in each of these areas by an interdisciplinary team of applied mathematicians, statisticians and clinicians are presented.}, number={1}, journal={JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS}, author={Adams, BM and Banks, HT and Davidian, M and Kwon, HD and Tran, HT and Wynne, SN and Rosenberg, ES}, year={2005}, month={Dec}, pages={10–49} }
@article{margolin_titi_wynne_2003, title={The postprocessing Galerkin and nonlinear Galerkin methods - A truncation analysis point of view}, volume={41}, ISSN={["1095-7170"]}, DOI={10.1137/S0036142901390500}, abstractNote={We revisit the postprocessing algorithm and give a justification from a classical truncation analysis point of view. We assume a perturbation expansion for the high frequency mode component of solutions to the underlying equation. Keeping terms to certain orders, we then generate approximate systems which correspond to numerical schemes. We show that the first two leading order methods are in fact the postprocessed Galerkin and postprocessed nonlinear Galerkin methods, respectively. Hence postprocessed Galerkin is a natural leading order method, more natural than the standard Galerkin method, for approximating solutions of parabolic dissipative PDEs. The analysis is presented in the framework of the two-dimensional Navier--Stokes equation (NSE); however, similar analysis may be done for any parabolic, dissipative nonlinear PDE. The truncation analysis is based on asymptotic estimates (in time) for the low and high mode components. We also introduce and investigate an alternative postprocessing scheme, which we call the dynamic postprocessing method, for the case in which the asymptotic estimates (in time) do not hold (i.e., in the situation of long transients, nonsmooth initial data, or highly oscillatory time-dependent solutions).}, number={2}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={Margolin, LG and Titi, ES and Wynne, S}, year={2003}, pages={695–714} }