@article{bokil_banks_cioranescu_griso_2018, title={A MULTISCALE METHOD FOR COMPUTING EFFECTIVE PARAMETERS OF COMPOSITE ELECTROMAGNETIC MATERIALS WITH MEMORY EFFECTS}, volume={76}, ISSN={["1552-4485"]}, DOI={10.1090/qam/1503}, abstractNote={We consider the problem of computing (macroscopic) effective properties of composite materials that are mixtures of complex dispersive dielectrics described by polarization and magnetization laws. We assume that the micro-structure of the composite material is described by spatially periodic and deterministic parameters. Mathematically, the problem is to homogenize Maxwell’s equations along with constitutive laws that describe the material response of the micro-structure comprising the mixture, to obtain an equivalent effective model for the composite material with constant effective parameters. The novel contribution of this paper is the homogenization of a hybrid model consisting of the Maxwell partial differential equations along with ordinary (auxiliary) differential equations modeling the evolution of the polarization and magnetization, as a model for the complex dielectric material. This is in contrast to our previous work [3] in which we employed a convolution in time of a susceptibility kernel with the electric field to model the delayed polarization effects in the dispersive material. In this paper, we describe the auxiliary differential equation approach to modeling material responses in the composite material and use the periodic unfolding method to construct a homogenized model.}, number={4}, journal={QUARTERLY OF APPLIED MATHEMATICS}, author={Bokil, V. A. and Banks, H. T. and Cioranescu, D. and Griso, G.}, year={2018}, month={Dec}, pages={713–738} }
 @article{banks_bokil_hu_2007, title={Monotone approximation for a nonlinear size and class age structured epidemic model}, volume={8}, ISSN={["1468-1218"]}, DOI={10.1016/j.nonrwa.2006.03.008}, abstractNote={In this paper, we study a nonautonomous size and class age structured epidemic model with nonlinear and nonlocal boundary conditions. We establish a comparison principle and construct convergent monotone sequences to prove the existence of solutions. Uniqueness of solutions is also established.}, number={3}, journal={NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS}, author={Banks, H. T. and Bokil, V. A. and Hu, Shuhua}, year={2007}, month={Jul}, pages={834–852} }
 @article{banks_bokil_cioranescu_gibson_griso_miara_2006, title={Homogenization of periodically varying coefficients in electromagnetic materials}, volume={28}, ISSN={["1573-7691"]}, DOI={10.1007/s10915-006-9091-y}, abstractNote={In this paper, we employ the periodic unfolding method for simulating the electromagnetic field in a composite material exhibiting heterogeneous microstructures which are described by spatially periodic parameters. We consider cell problems to calculate the effective parameters for a Debye dielectric medium in the case of a circular microstructure in two dimensions. We assume that the composite materials are quasi-static in nature, i.e., the wavelength of the electromagnetic field is much larger than the relevant dimensions of the microstructure.}, number={2-3}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Banks, H. T. and Bokil, V. A. and Cioranescu, D. and Gibson, N. L. and Griso, G. and Miara, B.}, year={2006}, month={Sep}, pages={191–221} }
 @article{banks_bokil_2005, title={A computational and statistical framework for multidimensional domain acoustooptic material interrogation}, volume={63}, ISSN={["0033-569X"]}, DOI={10.1090/S0033-569X-05-00949-0}, abstractNote={Abstract : We consider an electromagnetic interrogation technique in two and three dimensions for identifying the dielectric parameters (including the permittivity, the conductivity and the relaxation time) of a Debye medium. In this technique a travelling acoustic pressure wave in the Debye medium is used as a virtual reflector for an interrogating microwave electromagnetic pulse that is generated in free space. The reflections of the microwave pulse from the air-Debye interface and from the acoustic pressure wave are recorded at a remote antenna. the data is used in an inverse problem to estimate the locally pressure dependent dielectric parameters of the Debye medium. We present a time domain formulation that is solved using finite differences (FDTD) in time and in space. Perfectly matched layer (PML) absorbing boundary conditions are used to absorb outgoing waves at the finite boundaries of the computational domain, preventing spurious reflections from reentering the domain. Using the method of least squares for the parameter identification problem, we compare two different algorithms (the gradient based Levenberg-Marquardt method, and the gradient free, simplex based Nelder-Mead method) in solving an inverse problem to calculate estimates for two or more dielectric parameters. Finally we use statistical error analysis to construct confidence intervals for all the presented estimates, thereby providing a probabilistic statement about the computational procedure with uncertainty aspects of estimates.}, number={1}, journal={QUARTERLY OF APPLIED MATHEMATICS}, author={Banks, HT and Bokil, VA}, year={2005}, month={Mar}, pages={156–200} }
 @article{bokil_glowinski_2005, title={An operator splitting scheme with a distributed Lagrange multiplier based fictitious domain method for wave propagation problems}, volume={205}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2004.10.040}, abstractNote={We propose a novel fictitious domain method based on a distributed Lagrange multiplier technique for the solution of the time-dependent problem of scattering by an obstacle. We study discretizations that include a fully conforming approach as well as mixed finite element formulations utilizing the lowest order Nedelec edge elements (in 2D) on rectangular grids. We also present a symmetrized operator splitting scheme for the scattering problem, which decouples the operator that propagates the wave from the operator that enforces the Dirichlet condition on the boundary of an obstacle. A new perfectly matched layer (PML) model is developed to model the unbounded problem of interest. This model is based on a formulation of the wave equation as a system of first-order equations and uses a change of variables approach that has been developed to construct PML's for Maxwell's equations. We present an analysis of our fictitious domain approach for a one-dimensional wave problem. Based on calculations of reflection coefficients, we demonstrate the advantages of our fictitious domain approach over the staircase approximation of the finite difference method. We also demonstrate some important properties of the distributed multiplier approach that are not shared by a boundary multiplier fictitious domain approach for the same problem. Numerical results for two-dimensional wave problems that validate the effectiveness of the different methods are presented.}, number={1}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Bokil, VA and Glowinski, R}, year={2005}, month={May}, pages={242–268} }