Isaac Harris, Ph.D.
Isaac Harris, Ph.D.
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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
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
Data-driven PDE-based inverse problem, in theory and practice at UW Madison, August 2024
Mathematics of Computation, Application and Data Seminar at Simon Fraser University, September 2024
Henes Workshop on Computational Mathematics at Michigan Technological University, September 2024
Computational and Applied Mathematics Seminar at NC State University, September 2024
Co-Organizing a Mini-Symposium at the Conference on Mathematics of Wave Phenomena, February 2025
AMS Spring Central Sectional, University of Kansas Lawrence, March 2025
Colloquium Speaker at U of Dayton, April 2025
Applied Inverse Problems conference, Rio Brazil, July/August 2025
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
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 2025
Teaching Spring 2025
MA 52000-001 Boundary Value Problems Of Differential Equations
Class Description: Separation of variables; Fourier series; boundary value problems; Fourier transforms; Bessel functions; Legendre polynomials.
Textbook: Fourier Analysis and Its Applications–Gerald B. Folland.
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MA 34100-070 Foundations Of Analysis
Class Description: An introductory course in rigorous analysis, covering real numbers, sequences, series, continuous functions, differentiation, and Riemann integration. This is a proof-based course and will be writing intensive.
Textbook: Introduction to Real Analysis–Robert G. Bartle, Donald R. Sherbert – 4th Edition.
Research
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: Victor Hughes ; General Obiora Ozochiawaeze(co-advised w/ Peijun Li)
Former Ph.D. Students: Rafael Ceja Ayala Ph.D. 2024 ; Govanni Granados Ph.D. 2024
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:
Refereed Publications:
Submitted
Submitted
G. Granados, I. Harris and A. Kleefeld "Direct and inverse scattering for an isotropic media with a second-order transmission condition". Submitted (arXiv:2506.11818)
R. Ceja Ayala, I. Harris and T. Sánchez-Vizuet"Well--Posedness for the biharmonic scattering problem for a penetrable obstacle”. Submitted (arXiv:2506.10176)
I. Harris, A. Kleefeld and H. Lee "Sampling methods for recovering buried corroded boundaries from partial electrostatic Cauchy data". Submitted (arXiv:2409.02761)
Published
Published
I. Harris, A. Kleefeld and H. Lee "Sampling methods for recovering buried corroded boundaries from partial electrostatic Cauchy data". (Under Review) Inverse Problems and Imaging (arXiv:2409.02761)
I. Harris, V. Hughes and A. Kleefeld "Two direct sampling methods for an anisotropic scatterer with a conductive boundary". Applicable Analysis DOI: 10.1080/00036811.2025.2504034 (2025) (arXiv:2412.16605)
V. Hughes, I. Harris, and J. Sun ''The anisotropic transmission eigenvalue problem with a conductive boundary''. Comm. on Analysis & Computation 4 60-82 (2025) (arXiv:2311.00526)
I. Harris, H. Lee, and P. Li "Direct sampling for recovering a clamped cavity from Biharmonic far field data". Inverse Problems 41 035013 (2025) (arXiv:2409.04870)
I. Harris, T. Le and D.-L. Nguyen "A direct reconstruction method for radiating sources in Maxwell's equations with single-frequency data". Inverse Problems 41 015003 (2025) (arXiv:2408.03826)
I. Harris, V. Hughes and H. Lee "Analysis of the monotonicity method for an anisotropic scatterer with a conductive boundary". Inverse Problems 40 105003 (2024) (arXiv:2403.18644)
R. Ceja Ayala, I. Harris, and A. Kleefeld "Inverse parameter and shape problem for an isotropic scatterer with two conductivity coefficients". Analysis and Mathematical Physics, 14 No. 90 (2024) (arXiv:2402.07880)
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 18 No. 3 708-729 (2024) (arXiv:2305.15310)
G. Granados, I. Harris and H. Lee ''Reconstruction of extended regions in EIT with a generalized Robin transmission condition''. Communications on Analysis & Computation 1 No. 4 347-368 (2023) (arXiv:2310.08223)
G. Granados and I. Harris ''Reciprocity gap functional methods for potentials/sources with small volume support for two elliptic equations''. Applicable Analysis Volume 103 No. 11, 2015-2037 (2024) (arXiv:2302.05212)
I. Harris ''Regularized Factorization Method for a perturbed positive compact operator applied to scattering''. Inverse Problems, 39 115007 (2023) (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 Volume 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''. Statistica Sinica 34 (4), DOI: 10.5705/ss.202021.0027 (2024) (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, Volume 424: 114977 (2023).
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 Volume 16 No. 5, 1137-1162 (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 Volume 102No. 5, 1511-1529 (2023) (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 Volume 101 No. 8, 3065-3081 (2022) (arXiv:1908.05801)
I. Harris and A. Kleefeld "Analysis and computation of the transmission eigenvalues with a conductive boundary condition". Applicable Analysis Volume 101 No. 6, 1880-1895 (2022) (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, Volume 100 No. 9, 1996-2019 (2021) (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:
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:
Papers in Preparation:
I. Harris, P. Li and G. Ozochiawaeze "A Linear Sampling Method for the Inverse Cavity Scattering Problem of Biharmonic Waves". (Under Preparation-Working Title)
I. Harris and H. Lee "Existence and computation of the clamped transmission eigenvalues”. (Under Preparation-Working Title)
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