Isaac Harris, Ph.D.
Barbara A. Kunze New Frontiers Associate Professor
Purdue University
150 N. University Street, West Lafayette, IN 47907
Office: MATH 612
Email: 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 with my mentor Bill Rundell. I am Co-Organizing the Spectral and Scattering Theory Seminar here in the Math Department at Purdue. For anyone interested in speaking at our Spectral and Scattering Theory Seminar please contact Kiril Datchev or myself.
Here are links to my CV, ORCID profile, and Google Scholar page.
Some Updates
SPIRA Award for Excellence in Graduate Teaching and Mentoring 2023-2024
My research is partially supported by the NSF DMS Grant 2107891 and 2208256
In January 2023 I received the Barbara A. Kunze New Frontiers Named Professorship
Promoted to Associate Professor w/ Tenure starting Aug 2023
Mathematically Gifted and Black featured mathematician (link) February 2022
Recent Advances in Nonlinear Partial Differential Equations and the Applications Special Session, JMM Boston MA 2023 (Invited)
Rice University Computational Applied Mathematics & Operations Research Colloquium February 2023
SIAM CSE Special Session on Novel Methods for Inverse Scattering, Amsterdam February-March 2023
AMS Sectional Meeting, University of Cincinnati, April 2023 (Invited)
New Trends in Computational Wave Propagation and Imaging, NJIT, May 2023 (Invited)
AIMS 13th Conference on Dynamical Systems, Differential Equations and Applications, Wilmington NC, May-June 2023 (Invited)
Applied Inverse Problems Conference, Goettingen Germany September 2023 (2 Invited Talks)
Spectral Theory and Applications, Texas A&M University, October 2023
SIAM NNP Annual Meeting, NJIT, October 2023
Advanced Research Initiative Colloquium, Marshall University November 2023
AMS Sectional Meeting, Howard University, April 2024 (Invited)
AMS Sectional Meeting, Florida State University, March 2024 (Invited)
CSULB Mathematics and Statistics Colloquium Series April 2024 (invited)
Plenary Talk on 'Transmission Eigenvalues for a Conductive Boundary' At the 24th Inverse Days Conference at Aalto University
Talk at the BIRS workshop on New Ideas in Computational Inverse Problems on a Regularization of the Factorization Method
Education
B.A. in Mathematics, Kean University, 2010
M.S. in Applied Mathematics, University of Delaware, 2012
Ph.D. in Applied Mathematics, University of Delaware, 2015
My Dissertation is titled "Non-Destructive Testing of Anisotropic Materials".
Teaching Spring 2024
MA 35100 Section 112-– Elementary Linear Algebra
Class: TTh at 4:30pm-5:45pm in SCHM 123
Office Hrs: TF at 11am -12pm
Class Description: Systems of linear equations, finite dimensional vector spaces, matrices, determinants, eigenvalues, and eigenvector applications to analytical geometry. Not open to students with credit in MA 26500.
Reference Text:
Linear Algebra: Ideas and Applications by Richard C. Penney. (available via Purdue Libraries online)
Linear Algebra for students by James E McClure. (will be available on Brightspace)
Research
My research interests are in inverse problems and spectral methods 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, and Asymptotic methods.
My research is partially supported by the NSF DMS Grant 2107891 and 2208256.
Current Ph.D. Students: Rafael Ceja Ayala ; Govanni Granados ; Victor Hughes ; General Obiora Ozochiawaeze(co-advised w/ Peijun Li)
Current PostDoc: Heejin Lee
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, Qualitative Methods, Fractional Diffusion, Electrical Impedance Tomography, Inverse Spectral Problems, Homogenization, and Asymptotic Analysis.
Refereed Publications:
Submitted
V. Hughes, I. Harris, and J. Sun ''The anisotropic transmission eigenvalue problem with a conductive boundary''. Submitted (arXiv:2311.00526)
R. Ceja Ayala, I. Harris, and A. Kleefeld "Inverse parameter and shape problem for an isotropic scatterer with two conductivity coefficients". Submitted (arXiv:2402.07880)
I. Harris, V. Hughes and H. Lee "Analysis of the monotonicity method for an anisotropic scatterer with a conductive boundary". Submitted (arXiv:2403.18644)
Published
G. Granados, I. Harris and H. Lee ''Reconstruction of extended regions in EIT with a generalized Robin transmission condition''. Communications on Analysis & Computation DOI:10.3934/cac.2023017 (arXiv:2310.08223)
R. Ceja Ayala, I. Harris and A. Kleefeld ''Direct sampling method via Landweber iteration for an absorbing scatterer with a conductive boundary''. Inverse Problems and Imaging DOI:10.3934/ipi.2023051 (arXiv:2305.15310)
G. Granados and I. Harris ''Reciprocity gap functional methods for potentials/sources with small volume support for two elliptic equations''. Applicable Analysis DOI: 10.1080/00036811.2023.2279951 (2023) (arXiv:2302.05212)
I. Harris ''Regularized Factorization Method for a perturbed positive compact operator applied to scattering''. Inverse Problems DOI: 10.1088/1361-6420/acfd59 (arXiv:2304.01324)
I. Harris, T. Le and D.-L. Nguyen''On reconstruction of small sources from Cauchy data at a fixed frequency''. Applied Mathematics Letters, 145 108765 (2023) (arXiv:2305.03932)
R. Ceja Ayala, I. Harris, A. Kleefeld and N. Pallikarakis ''Analysis of the transmission eigenvalue problem with two conductivity parameters''. Applicable Analysis 103 No. 1: 211–239 (2024) (arXiv:2209.07247)
C.-L. Sung, W. Wang, F. Cakoni, I. Harris and Y. Hung ''Functional-Input Gaussian Processes with Applications to Inverse Scattering Problems''. (in-press)Statistica Sinica DOI: 10.5705/ss.202022.0180 (arXiv:2201.01682)
U. Kalachikova, M. Vasilyeva, I. Harris and E. Chung "Generalized Multiscale Finite Element Method for scattering problem in heterogeneous media". Journal of Computational and Applied Mathematics, DOI: 10.1016/j.cam.2022.114977 (2022).
G. Granados and I. Harris "Reconstruction of small and extended regions in EIT with a Robin transmission condition". Inverse Problems, 38 105009 (2022) (arXiv:2203.09551)
I. Harris "Regularization of the Factorization Method with Applications to Inverse Scattering". AMS Contemporary Mathematics 784 DOI: 10.1090/conm/784/15752 (2023) (arXiv:2202.13411)
I. Harris ''Dirichlet Spectral-Galerkin approximation method for simply supported plate eigenvalue problems''. Partial Differential Equations and Applications 3 No. 34 (2022) (arXiv:2005.03433)
I. Harris, D.-L. Nguyen and T.-P. Nguyen ''Direct sampling methods for isotropic and anisotropic scatterers with point source measurements''. Inverse Problems and Imaging DOI: 10.3934/ipi.2022015 (2022) (arXiv:2107.08138)
I. Harris and J. Rezac ''A sparsity-constrained sampling method with applications to communications and inverse scattering''. Journal of Computational Physics 451 110890 (2022) (arXiv:2010.08032)
I. Harris ''The direct and inverse problem for sub-diffusion equations with a generalized impedance subregion". Applications of Mathematics 67 1-20 (2022) (arXiv:2004.07212)
I. Harris "Direct methods for recovering sound soft scatterers from point source measurements". Computation 9 No. 11, 120 (2021) (arXiv:2110.10767)
I. Harris "Regularization of the Factorization Method applied to diffuse optical tomography". Inverse Problems 37 125010 (2021) (arXiv:2106.07743)
I. Harris "A direct method for reconstructing inclusions and boundary conditions from electrostatic data". Applicable Analysis DOI:10.1080/00036811.2021.1991326 (2021) (arXiv:1704.07479)
I. Harris ''Approximation of the inverse scattering Steklov eigenvalues and the inverse spectral problem''. Research in the Mathematical Sciences 8 No. 31 DOI: 10.1007/s40687-021-00268-1 (2021) (arXiv:2006.10567)
I. Harris, D.-L. Nguyen, J. Sands, and T. Truong ''On the inverse scattering from anisotropic periodic layers and transmission eigenvalues''. Applicable Analysis DOI:10.1080/00036811.2020.1836349 (2020) (arXiv:1908.05801)
I. Harris and A. Kleefeld "Analysis and computation of the transmission eigenvalues with a conductive boundary condition". Applicable Analysis DOI:10.1080/00036811.2020.1789598 (2020) (arXiv:2001.07188)
I. Harris and D.-L. Nguyen "Orthogonality Sampling Method for the Electromagnetic Inverse Scattering Problem". SIAM Journal on Scientific Computing 42 No. 3 B722–B737 (2020) (arXiv:1908.05800)
I. Harris ''Approximation of the zero-index transmission eigenvalues with conductive boundary and parameter estimation". Journal of Scientific Computing 82 No. 80 (2020) (arXiv:1910.01009)
U. Gavrilieva, M. Vasilyeva, I. Harris, E. Chung, and Y. Efendiev ''Multiscale Finite Element Method for scattering problem in heterogeneous domain''. Journal of Physics: Conference Series 1392 012067 (2019)
I.Harris ''Analysis of two transmission eigenvalue problems with a coated boundary condition ". Applicable Analysis, DOI: 10.1080/00036811.2019.1672869 (2019) (arXiv:1904.12908)
I. Harris "Detecting an inclusion with a generalized impedance condition from electrostatic data via sampling". Mathematical Methods in the Applied Sciences, Volume 49, No. 18, 6741-6756 (2019) (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, 281-300 (2019) (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, Volume 99 No. 3, 508–529 (2020) (arXiv:1608.07560)
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
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, 2769-2783 (2017) (arXiv:1702.06058)
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 ''Direct sampling algorithms based on the factorization method for inverse scattering". IEEE RAPID Conference (2021) DOI: 10.1109/RAPID51799.2021.9521406
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, A. Kleefeld and H. Lee "Sampling Method for recovering buried corroded type boundaries in EIT". (Under Preparation-Working Title)
I. Harris and H. Lee "Direct Sampling for recovering a clamped obstacle from Biharmonic far field data". (Under Preparation-Working Title)
I. Harris, T. Le and D.-L. Nguyen “A fast reconstruction method for point-like sources for Maxwell’s equations with single-frequency data”. (Under Preparation–Working Title)
I. Harris and H. Lee " On the transmission eigenvalues for scattering by a clamped planar region". (Under Preparation-Working Title)