@article{ly_tran_2002, title={Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor}, volume={60}, ISSN={["1552-4485"]}, DOI={10.1090/qam/1939004}, abstractNote={Proper orthogonal decomposition (which is also known as the Karhunen-Loève decomposition) is a reduction method that is used to obtain low-dimensional dynamic models of distributed parameter systems. Roughly speaking, proper orthogonal decomposition (POD) is an optimal technique of finding a basis that spans an ensemble of data, collected from an experiment or a numerical simulation of a dynamical system, in the sense that when these basis functions are used in a Galerkin procedure, they will yield a finite-dimensional system with the smallest possible degrees of freedom. Thus, the technique is well suited to treat optimal control and parameter estimation of distributed parameter systems. In this paper, the method is applied to analyze the complex flow phenomenon in a horizontal chemical vapor deposition (CVD) reactor. In particular, we show that POD can be used to efficiently approximate solutions to the compressible viscous flows coupled with the energy and the species equations. In addition, we also examine the feasibility and efficiency of the POD method in the optimal control of the source vapors to obtain the most uniform deposition profile at the maximum growth rate. Finally, issues concerning the implementation of the method and numerical calculations are discussed.}, number={4}, journal={QUARTERLY OF APPLIED MATHEMATICS}, author={Ly, HV and Tran, HT}, year={2002}, month={Dec}, pages={631–656} }
 @article{ly_ito_banks_jolly_reitich_2001, title={Dynamic simulation of the temporal response of microstructure formation in magnetorheological fluids}, volume={15}, ISSN={["0217-9792"]}, DOI={10.1142/s0217979201005416}, abstractNote={ Efficient numerical simulations of microstructure development in magnetorheological (MR) fluids are conducted. The simulations, which are based upon a fast multipole algorithm, treat the magnetic inclusions as two-dimensional continuum magnetic entities. The development of microstructure is quantified by computing and recording the time evolution of the effective permeability of the composite fluid. Such a principle has been previously exploited for the experimental measurements of microstructure development [Jolly, Bender and Mathers, ERMR'97, Yonezawa, Japan 1997]. As was observed experimentally, numerical simulations reveal the evolution of microstructure to be multimodal in nature. Unlike the experiments, the numerical simulations afford us the ability to observe the physical mechanisms associated with various modes. }, number={6-7}, journal={INTERNATIONAL JOURNAL OF MODERN PHYSICS B}, author={Ly, HV and Ito, K and Banks, HT and Jolly, MR and Reitich, F}, year={2001}, month={Mar}, pages={894–903} }
 @article{ly_tran_2001, title={Modeling and control of physical processes using proper orthogonal decomposition}, volume={33}, ISSN={["1872-9479"]}, DOI={10.1016/S0895-7177(00)00240-5}, abstractNote={The proper orthogonal decomposition (POD) technique (or the Karhunan Loève procedure) has been used to obtain low-dimensional dynamical models of many applications in engineering and science. In principle, the idea is to start with an ensemble of data, called snapshots , collected from an experiment or a numerical procedure of a physical system. The POD technique is then used to produce a set of basis functions which spans the snapshot collection. When these basis functions are used in a Galerkin procedure, they yield a finite-dimensional dynamical system with the smallest possible degrees of freedom. In this context, it is assumed that the physical system has a mathematical model, which may not be available for many physical and/or industrial applications. In this paper, we consider the steady-state Rayleigh-Bénard convection whose mathematical model is assumed to be unknown, but numerical data are available. The aim of the paper is to show that, using the obtained ensemble of data, POD can be used to model accurately the natural convection. Furthermore, this approach is very efficient in the sense that it uses the smallest possible number of parameters, and thus, is suited for process control. Particularly, we consider two boundary control problems 1. (a) tracking problem, and 2. (b) avoiding hot spot in a certain region of the domain.}, number={1-3}, journal={MATHEMATICAL AND COMPUTER MODELLING}, author={Ly, HV and Tran, HT}, year={2001}, pages={223–236} }
 @article{ly_titi_1999, title={Global gevrey regularity for the Benard convection in a porous medium with zero Darcy-Prandtl number}, volume={9}, ISSN={["0938-8974"]}, DOI={10.1007/s003329900073}, number={3}, journal={JOURNAL OF NONLINEAR SCIENCE}, author={Ly, HV and Titi, ES}, year={1999}, pages={333–362} }
 @article{ly_reitich_jolly_banks_ito_1999, title={Simulations of particle dynamics in magnetorheological fluids}, volume={155}, ISSN={["1090-2716"]}, DOI={10.1006/jcph.1999.6335}, abstractNote={We present particle dynamics simulations for the response of magnetorheological (MR) fluids upon application of a magnetic field. The particles motion is considered to be governed by magnetic, hydrodynamic, and repulsive interactions. Fluid-particle interactions are accounted for via Stokes' drag while inter-particle repulsions are modeled through approximate hard-sphere rejections. In accordance with their greater significance, on the other hand (linear) magnetic interactions are fully simulated. The time evolution is considered to be magnetically quasi-static and magnetostatic forces are derived from the solution of (steady) Maxwell's equations, recomputed at each instant in time. For this we use a potential theoretic formulation where the boundary integral equations are solved with a fast multipole approach. We show that the resulting numerical codes can be effectively used to study a number of experimental observables such as effective magnetic permeabilities and response time-scales which are of crucial importance in the design of MR fluids.}, number={1}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Ly, HV and Reitich, E and Jolly, MR and Banks, HT and Ito, K}, year={1999}, month={Oct}, pages={160–177} }