Research

My research area is analysis of partial differential equations, with a specialization in inverse boundary value problems. Inverse problems arise naturally in many physical contexts, where the goal is to reconstruct material properties of an object by nonintrusive measurements taken on the surface of, or at a distance from, the body; examples include medical imaging, nondestructive testing of materials, and geoprospection. My supervisor is Prof. Ting Zhou.

My doctoral research focused on inverse boundary value problems for Maxwell’s equations in electromagnetism. The accessible data in this case are boundary measurements of the electric and magnetic fields, and one wishes to recover the conductivity, electric permittivity, and magnetic permeability of the object under consideration. I studied the unique solvability of such problems in different physical scenarios including bounded and unbounded domains, assuming either full access to boundary measurements or access to measurements on only a part of the boundary – the latter is a particularly practically relevant scenario, since in practice, not all of the surface of the object may be accessible, or it may simply to be too costly to take measurements on the whole surface.

My most recent project involved partial data inverse problems on a slab, an infinite domain bounded by two parallel planes. This geometry is interesting in view of applications in modeling waveguides and in medical imaging, among others. I have been able to show that the electromagnetic material properties are uniquely determined if boundary measurements of the electric and magnetic fields are available on (subsets of) the same hyperplanes, or on opposite hyperplanes. The preprint can be found on arXiv.

 

I previously worked as a graduate teaching assistant at the State University of New York at Buffalo, where I conducted research on the Inverse Scattering Transform, a method to find solutions to certain classes of partial differential equations, under the supervision of Prof. Gino Biondini. I used this method to construct novel solutions to the focusing nonlinear Schrödinger equation.

 

I received my Bachelor’s and Master’s degrees in Technical Mathematics at the Vienna University of Technology. My Bachelor thesis is titled “The quantum-hydrodynamic model for semiconductors in thermal equilibrium” and was supervised by Ansgar Jüngel. My Master thesis “Boundary behavior of singular integrals” was supervised by Harald Woracek.

 

Preprints

[1] Inverse problems for Maxwell’s equations in a slab with partial boundary data. arXiv

 

Publications

[1] M.P., Gino Biondini; On the focusing non-linear Schrödinger equation with non-zero boundary conditions and double poles. IMA J Appl Math 2017; 82 (1): 131-151.  journal

[2] An inverse problem for Maxwell’s equations with Lipschitz parameters, Inverse Problems 2018, 34(2):025006. journal / arXiv

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Recent talks and presentations

Two partial data inverse problems for Maxwell’s equations in a slab. AMS Fall Central Sectional Meeting, University of Michigan, Ann Arbor, MI, October 2018

The Inverse Problem for Maxwell’s Equations on a Bounded Lipschitz Domain with Lipschitz Parameters. Poster Flash and Poster Session, Heidelberg Laureate Forum, Heidelberg, Germany, September 2018. video

Inverse Problems for Maxwell’s Equations in a Slab with Partial Boundary Data. SIAM Conference on Mathematics of Planet Earth (MPE18), Philadelphia PA, September 2018. slides

An inverse boundary value problem for Maxwell’s equations. Boston Graduate Math Colloquium, Harvard University, Cambridge MA, April 28, 2018.

An inverse problem for Maxwell’s equations with Lipschitz parameters. AMS Special Session on New Developments in Inverse Problems and Imaging, Northeastern University, Boston MA, April 22, 2018. slides

An inverse boundary value problem for Maxwell’s equations.AMS Graduate Student Conference at Brown, Brown University, Providence RI, February 22, 2018.