Isaac Harris, Ph.D.

Assistant Professor

Department of Mathematics

Purdue University

150 N. University Street, West Lafayette, IN 47907

Office: MATH 612

Email: iharris1107[at]gmail.com - or - harri814[at]purdue.edu

I graduated from The University of Delaware with a Ph.D. in Applied Mathematics under the advisement of Fioralba Cakoni in 2015. In August of 2018, I joined the Department of Mathematics at Purdue University as an Assistant Professor. From Summer 2015-2018 I was at Texas A&M University as a Visiting Assistant Professor continuing my research in Inverse Problems and Scattering Theory with my mentor Bill Rundell.

Here are links to my CV and Google Scholar page.

Some Updates

Education

Research

My research interests are in direct and inverse problems for PDEs, especially those arising in acoustic and electromagnetic scattering. I have been working on the inverse scattering problem of non-destructive testing for defects in complex media and parameter identification for periodic media. My research employs techniques from a multitude of areas in mathematical analysis, such as Functional analysis, Analysis of PDEs, Numerical and Asymptotic methods.

This figure is the cover image for Inverse Problems Volume 31 Number 2.

Reconstructions of circular cavities in an Anisotropic material via the factorization method.


Research Interests: Direct and Inverse Scattering, Transmission Eigenvalue Problems, Inverse Problems for (Fractional) Diffusion, Qualitative Reconstruction Methods, Electrical Impedance Tomography, Inverse Spectral Problems, Homogenization, and Asymptotic Analysis

Refereed Research Papers:

  • I.Harris ''Analysis of two transmission eigenvalue problems with a coated boundary condition ". Applicable Analysis - Under Revision (arXiv:1904.12908)
  • I. Harris "Detecting an inclusion with a generalized impedance condition from electrostatic data via sampling". Mathematical Methods in the Applied Sciences (Accepted) (arXiv:1708.03203)
  • L. Hoeltgen, A. Kleefeld, I. Harris and M. Breuss ''Theoretical Foundation of the Weighted Laplace Inpainting Problem" Applications of Mathematics, Volume 64, No. 3, 2019 281-300 (arXiv:1801.09068)
  • I. Harris and A. Kleefeld ''Analysis of new direct sampling indicators for Far-field measurements''. Inverse Problems 35 054002 (2019) (arXiv:1901.02351)
  • I. Harris and A. Kleefeld "The inverse scattering problem for a conductive boundary condition and transmission eigenvalues". Applicable Analysis, DOI: 10.1080/00036811.2018.1504028 (arXiv:1608.07560)
  • F. Cakoni, I. Harris and S. Moskow "The Imaging of Small Perturbations in an Anisotropic Media". Computers and Mathematics with Applications Volume 74, Issue 11, 2017 2769-2783 (arXiv:1702.06058)
  • L. Hoeltgen, I. Harris, A. Kleefeld and M. Breuss "Analytic Existence and Uniqueness Results for PDE-Based Image Reconstruction with the Laplacian" Scale Space and Variational Methods in Computer Vision (2017), 66-79. Lecture Notes in Computer Science Volume 10302, Springer
  • I. Harris and S. Rome "Near field imaging of small isotropic and extended anisotropic scatterers". Applicable Analysis Volume 96, No. 10, 2017 1713-1736 (arXiv:1601.02993)
  • O. Bondarenko I. Harris, and A. Kleefeld "The interior transmission eigenvalue problem for an inhomogeneous media with a conductive boundary". Applicable Analysis Volume 96, No. 1, 2017 2-22 (arXiv:1510.01762)
  • F. Cakoni, H. Haddar, and I. Harris "Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem". Inverse Problems and Imaging Volume 9, No. 4, 2015, 1025–1049 (arXiv:1410.3729)
  • F. Cakoni and I. Harris "The factorization method for a defective region in an anisotropic material". Inverse Problems, 31 025002 (2015) (arXiv:1410.3737)
  • I. Harris, F. Cakoni, and J. Sun "Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids". Inverse Problems, 30 035016 (2014) (Preprint)

Other Publications:

  • I. Harris ''Parameter identification for complex materials using Transmission Eigenvalues''. Oberwolfach Report, No. 45 (2016) 46-48. Extended Abstract for the workshop: Theory and Numerics of Inverse Scattering Problems
  • I. Harris "Non-Destructive Testing of Anisotropic Materials". The University of Delaware, Ph.D. Thesis (2015)

Papers in Preparation:

  • I. Harris and W. Rundell "A direct method for reconstructing inclusions and boundary conditions from electrostatic data" (Under Preparation) (arXiv:1704.07479)
  • I. Harris and J. Rezac ''A Sparsity-Constrained Qualitative Method for Parameter Estimation''. (Under Preparation)
  • I. Harris D.-L. Nguyen, J. Sands, T. Truong ''On Scattering from Anisotropic Periodic Layers and Transmission Eigenvalue''. (Under Preparation-Working Title)
  • I.Harris ''The direct and inverse problem for the fractional heat equation with an impedance subregion". (Under Preparation-Working Title)
  • I.Harris ''Numerical approximation for the inverse transmission eigenvalue problem with a conductive boundary". (Under Preparation-Working Title)

Teaching Summer 2019


This summer I am working with three Undergraduates, where they are studying the Analytical and Computational aspects of (Non-)Linear Spring Mass Systems as well as Electrical Impedance Tomography. The goal is for the students to understand the difficulties in solving some simple Inverse Problems that arise in the aforementioned areas. Some of the things that the students will explore are: Laplace Transform, Integral Equations, Fourier Series, Polynomial Interpolation and Regularization Techniques.