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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

With my collaborators Dmitry Golobaty and Pat Wilber - Oct. 2016

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.

With my collaborators and friends Alicia Prieto Langarica and Julianne Chung - Feb. 2014

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:

  • Jonathan Wittmer, Spring 2018-Present. Undergraduate student in Applied Math.
  • Alexander Alberts, Fall 2017-Present. Master student in Applied Math co-advised with D. Golovaty and P. Wilber.
  • Emmanuel Rivera, Fall 2017-Present. Master student in Applied Math co-advised with D. Golovaty and P. Wilber.
  • Daniel Rhoads, Fall 2015-Present. PhD student co-advised with D. Golovaty and P. Wilber.

  • Previous:

  • Marissa Gross, Spring 2017. Research Assistant, (Co-advised with D. Golovaty and P. Wilber).
  • Alexander Alberts, Spring 2017. Research Assistant, (Co-advised with D. Golovaty and P. Wilber).
  • Emmanuel Rivera, Spring 2017. Research Assistant, (Co-advised with D. Golovaty and P. Wilber).
  • Lucas Stanek, MSc in Applied Math at UA, 2017 (Co-advised with D. Golovaty and P. Wilber). Current PhD student in the Computational Mathematics, Science and Engineering Program at Michigan State University.
  • Mackenzie Jones, Spring 2016. Research Project.
  • Oliver Evans, Research Assistant, Spring 2016.
  • Amirreza Hashemi, MSc in Applied Math at UA, 2016. Current PhD student in the Computational Modeling and Simulation Program at UPitt.
  • Michael Wransky, MSc in Applied Math at UA, 2015. Currently working as an AI Research Engineer at Barnstorm Research Corporation.
  • Rachel Richards, MSc in Applied Math at UA, 2015. Currently working as a Service Desk Analyst at IntegraTouch.
  • Daniel Rhoads, MSc in Applied Math at UA, 2015 (Co-advised with D. Golovaty and P. Wilber). Current PhD math student at UA.
  • Mona Matar, MSc in Applied Math at UA, 2014 (Co-advised with D. Golovaty and P. Wilber). Current PhD math student at Kent State University.
  • Tim Nixdorf, MSc in Applied Math at UA, 2014 (Co-advised with D. Golovaty and P. Wilber).
  • Michael Wransky, Spring 2013-Fall 2014. Applied Math Honors Thesis.
  • Christopher Brandt, Summer 2013 at UA.
  • Hannah Lebo, Summer 2013. Current applied math master student at UA.
  • Ting Gao, MSc in Applied Math at UA, 2013 (Co-advised with D. Golovaty and P. Wilber).
  • Arturo J. Mateos, MURF 2012 (Co-advised with M. Ortiz). Current aerospace PhD student at Caltech.
  • Hyun Ji Jane Bae, SURF 2011 (Co-advised with M. Ortiz). Current applied math PhD student at Stanford.
  • Ka Kin Kenneth Hung, SURF 2011 (Co-advised with M. Ortiz). Current PhD math student at UC Berkeley.
  • Andre Pradhana, SURF 2010 (Co-advised with M. Ortiz). Current applied math PhD student at UCLA.
  • Stephanie Tsuei, SURF 2010 (Co-advised with M. Ortiz). Currently working at Northrop Grumman Corporation.
  • Current and past collaborators

  • Julianne Chung, Virginia Tech
  • Sergio Conti, University of Bonn, Germany
  • Dmitry Golovaty, The University of Akron
  • Per Chirstian Hansen, Technical University of Denmark, Denmark
  • Pamela E. Harris, Williams College
  • Misha Kilmer, Tufts University
  • Dennis Kochmann, Caltech
  • Bryn Martin, University of Idaho
  • Michael Ortiz, Caltech
  • Alicia Prieto Langarica, Youngstown State University
  • Horacio Rotstein, New Jersey Institute of Technology
  • Aintzane Urbizu Serrano, Duke University
  • Patrick Wilber, The University of Akron
  • Former supervisors

  • Gabriel Acosta, Universidad de Buenos Aires (Licenciatura thesis advisor)
  • Penny Anderson, Mathworks (Summer intership 2006)
  • Misha Kilmer, Tufts University (Phd advisor)
  • Gabriel Kreiman, Harvard University (Summer research 2007)
  • Michael Ortiz, Caltech (Postoc supervisor)
  • Pat Quillen, Mathworks (Summer internship 2006)
  • Horacio Rotstein, New Jersey Institute of Technology (Licenciatura thesis co-advisor)