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My research interests are in numerical analysis, numerical linera algebra, applied and computational mathematics. Below there is a description of problems I have been working on.

Cluster QC
Espanol, Kochmann, Conti and Ortiz (in preparation)
  • Computational Solid Mechanics

    • My current work focuses on the mathematical and numerical analysis of models arising in solid mechanics. In particular, I have been looking at atomistic-to-continuum models. 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. My current work consists of analyzing the QC method by means of Gamma-convergence analysis.

  • 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. I have developed multilevel methods for discrete ill-posed problems arising from the discretization of Fredholm integral equations of the first kind. In particular, I 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.

    Some of my work in image deblurring appears in the following papers:

    -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).

    -Multilevel Approach For Signal Restoration Problems With Toeplitz Matrices (with M. E. Kilmer). SIAM J. Scientific Computing 32(1): 229-319 (2010).

    Supervisees

  • Hyun Ji Jane Bae, SURF 2011
  • Ka Kin Kenneth Hung, SURF 2011
  • Andre Pradhana, SURF 2010, master student at the University of Cambridge, UK
  • Stephanie Tsuei, SURF 2010, Project Website

    • Current and past collaborators

  • Julianne Chung, University of Texas at Arlington
  • Sergio Conti, University of Bonn, Germany
  • Per Chirstian Hansen, Technical University of Denmark, Denmark
  • Misha Kilmer, Tufts University
  • Dennis Kochmann, Caltech
  • Michael Ortiz, Caltech

    • Current and 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)