My research interests are in the areas of Applied and Computational Mathematics. I am interested in the development, analysis, and application of mathematical models and numerical methods for solving problems arising in science and engineering. My work has been focused on problems related to materials science, image processing, and medical diagnostics. More specifically, my research has been at the intersection of many different mathematical areas including numerical analysis, numerical linear algebra, calculus of variations, and machine learning.
If you are an undergraduate or graduate student interested in doing research in any of these or related areas, please send me an email at email@example.com. I'd be happy to meet with you to discuss research opportunities.
Most of my research projects include interdisciplinary collaboration with scientists, engineers, and medical doctors. Please contact me if you are interested in potential collaborations.
For a list of publications see my CV.
Research Projects2D Materials
A graphene sheet is a one-atom thick layer of carbon atoms arranged in a honeycomb hexagonal lattice. Graphene continues to attract strong interest within the scientific community because it is a particularly exceptional material that promises a wide range of new applications and yields insights on fundamental physics of 2D materials. In this project, we focus on the developing of rigorous atomistic-to-continuum procedures that upscales the energy of a discrete system of atoms to a continuum energy. We are interested in the modeling and simulations of van der Waals heterostructures. More specifically, we are working on the modeling of moire patterns that appear in a suspended graphene sheet.This work is supported by NSF-DMS 1615952
-Discrete-to-Continuum Modeling of Weakly Interacting Incommensurate Two-Dimensional Lattices (with D. Golovaty and J. P. Wilber). Proceedings of the Royal Society A 474: 20170612 (2018).
-Euler Elastica as a Gamma-Limit of Discrete Bending Energies of One-Dimensional Chains of Atoms (D. Golovaty and J. P. Wilber). Mathematics and Mechanics of Solids 23 (7): 1104-1116 (2018).
-Discrete-to-Continuum Modeling of Weakly Interacting Incommensurate Chains (D. Golovaty and J. P. Wilber). Physical Reviews E 96, 033003 (2017).Chromonic Liquid Crystals
Chromonic liquid crystals are materials that move like a liquid, but have a crystal-like structure formed by flat molecules. Chromonic liquid crystals include drugs, dyes, and DNA nucleotides. In this project, we are working in the modeling of a colorant called ‘sunset yellow' that is used in food, cosmetics, and drugs. We are particularly interested in understanding the role that the level of concentration of sunset yellow plays in its crystal-like structure. To do so, we solve free boundary problems to determine the shape of chromonic aggregates for a fixed volume of material and compare them to experimental observations. This project started in May 2018 at the Women in Mathematics of Materials Workshop at the Michigan Center for Applied and Interdisciplinary Mathematics (University of Michigan). My collaborators in this project are Maria-Carme Calderer (U. Minnesota), Lidia Mrad (U. Arizona), Eleni Panagiotou (U. Tennessee), Robin Selinger (Kent Sate), Ling Xu (U. Michigan), and Longhua Zhao (Case Western).Electronic Structure of Materials
The Schrodinger equation is the foundation of quantum mechanics. It describes the behavior of small particles, such as atoms and electrons, at the molecular level. Since all solids are made up of many small particles, quantum mechanics can describe the electronic structure and behavior of materials under any conditions and stresses that are applied. The Schrodinger equation and its associated Kohn-Sham equation are linear and nonlinear eigenvalue problems respectively. We have developed wavelet-based multilevel methods to find the ground state energy, that is, the smallest eigenvalue.
Continuum mechanics models of solids have certain limitations as the length scale of interest approaches the atomistic scale, for instance when studying defects. A possible solution in such situations is to use a pure atomistic model. However, this approach could be computational prohibited as we are dealing with million of atoms. The Quasicontinuum (QC) method is a computational technique that reduces the atomic degrees of freedom. We have analyzed the QC method by means of Gamma-convergence bringing new insights on similar type numerical schemes.
-A Gamma-convergence Analysis of the Quasicontinuum Method (with D. Kochmann, S. Conti, and M. Ortiz), Multiscale Modeling and Simulation 11(3): 766-794 (2013).
Inverse problems are situations where hidden information is computed from external observations. For instance in image deblurring one wants to recover an image from one that is blurred and noisy. We have developed wavelet-based multilevel methods for signal and image restoration problems as well as for blind deconvolution problems. In these methods, an orthogonal wavelet transform is used to define restriction and prolongation operators within a multigrid-type iteration. The choice of the Haar wavelet operator has the advantage of preserving matrix structure, such as Toeplitz, between grids, which can be exploited to obtain faster solvers on each level where an edge-preserving Tikhonov regularization is applied. Current projects include the development of multilevel methods for the design of optimal regularization operators, Bayesian-based multilevel methods, and extensions of these methods to light-field cameras.
-Learning Regularization Parameters for General-Form Tikhonov (with J. M. Chung). Inverse Problems 33 074004 (2017).
-A Wavelet-Based Multilevel Approach for Blind Deconvolution Problems (with M. E. Kilmer). SIAM J. Scientific Computing 36(4): A1432-A1450 (2014).
-Multilevel Approach for Signal Restoration Problems with Toeplitz Matrices (with M. E. Kilmer). SIAM J. Scientific Computing 32(1): 229-319 (2010).
-A Projection-Based Approach to General-Form Tikhonov Regularization (with M. E. Kilmer and P. C. Hansen). SIAM J. Scientific Computing 29(1): 315-330 (2007).
Chiari malformation (CM) is a serious neurological disorder where the bottom part of the brain, the cerebellum, descends out of the skull and crowds the spinal cord, putting pressure on both the brain and spine and causing many symptoms. Magnetic resonance imaging (MRI) is currently an indispensable diagnostic imaging technique in the detection of CM. In this project, we studied MRI-based classifiers to detect CM. For more information on CM and other projects on CM at The University of Akron, visit the Conquer Chiari Research Center, or watch this promotional video.This work was funded by the C&S Patient Education Foundation and The University of Akron Faculty Research Committee Summer Grant.
-Machine learning applied to neuroimaging for diagnosis of adult classic Chiari malformation: role of the basion as a key morphometric indicator (with A. Urbizu, B. Martin, D. Moncho, A. Rovira, M. A. Poca. J. Sahuquillo, and A. Macaya). Journal of Neurosurgery 129(3): 779-791 (2018).