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My research interests are in numerical analysis, numerical linear algebra, applied and computational mathematics. My research consists of designing and analyzing numerical methods for problems arising in solid mechanics, materials science, and image processing.

If you are an undergraduate or graduate student interested in research in any of these or related areas, please send me an email at mespanol@uakron.edu. I'll be happy to meet with you to discuss research opportunities.

Most of my research projects include interdisciplinary collaboration with scientists from various departments such as mathematics, computer science, and engineering. Please contact me if you are interested in potential collaborations.

For a list of publications see my CV.

**Research Projects**

__Carbon Nanostructures__

A graphene sheet is one-atom thick layer of carbon atoms arranged in a 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. 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. In particular, we work on the modeling and simulations of different carbon nanostructures such as multi-walled nanotubes and nanoscrolls.

This work is supported by NSF-DMS 1615952__The Quasicontinuum Method__

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. Current project consists of analyzing the QC method by means of Gamma-convergence.

__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. Current project consists of developing wavelet-based multilevel methods to find the ground state energy, that is, the smallest eigenvalue.

__Inverse Problems__

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 and Bayesian-based multilevel methods.

__Detection of Chiari Malformations__

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 study 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.

**Supervisees**

**Current**:

**Previous**:

**Current and past collaborators**

**Former supervisors**