### Luca Rondi thesis topics

I am available as advisor for **Master** or **Ph.D. theses** in **Mathematical Analysis**. The topics of the thesis will be closely related to my research activity.

#### My research

My field of research is the study of **partial differential
equations**, with the main focus on **inverse problems** associated to them. Inverse problems, in particular for what concerns reconstruction methods, have strong connections to **shape optimization** and **imaging** problems. Since I like variational approaches to reconstruction, I often borrow techniques from the **Calculus of Variations**.

I try to couple a rigorous analysis of inverse problems for pde's with the development and analysis of efficient and effective numerical methods of reconstruction.

For the **analysis of inverse problems**, I address uniqueness and stability issues. This line of research requires a careful study of fine properties of solutions of the corresponding underlying pde's, such as unique continuation or properties of level sets and critical points. I am particularly interested in inverse problems whose unknowns are as little regular as possible, an extremely challenging case.

About the **numerical reconstruction for inverse problems**, my aim is to devise new methods or algorithms of reconstruction and to provide a rigorous analytical justification of these new, or of other already existing, reconstruction procedures.
My favourite approach is to use variational methods and regularization.

My research interests range from classical inverse problems, such as **Electrical Impedance Tomography** or **Inverse Acoustic or Electromagnetic Scattering**, to more applied problems arising from industrial applications such as the inverse photolithography problem.

If you are interested in pde's or inverse problems and you want to know more about these research themes, contact me
(e-mail to `luca.rondi at unimi.it` or drop by my office in room 2051), check some of my publications or read on for a brief explanation of what inverse problems are!

#### Inverse problems

In inverse problems there is a physical system, governed for example by differential equations, in which some data are not known and may not be directly measured. The aim is to determine such unknown data by indirect measurements. That is, we probe the system in a suitable way and we measure its reaction. From the additional information we gather through these indirect measurements, we aim to reconstruct the unknown data. Typically, the unknown is a coefficient of the differential equation or its domain of validity and we measure some properties of the solutions to suitable boundary value problems. The boundary value problem involved is usually called the direct problem.

The main issues are **uniqueness**, **stability** and **reconstruction**. The measurements performed, besides being feasible in
practice, need to guarantee uniqueness, that is, they must determine uniquely the unknown data. Measurements are subject to
errors, thus it is important to understand their effect on the reconstruction, that is, to study the stability of the inverse problem. Unfortunately, often there is no stability, unless
we impose suitable a-priori hypotheses on the unknown data. Even in this case, the stability may be very weak, like of
logarithmic type. This **severe ill-posedness** and the **nonlinearity** (the measurements depend nonlinearly on the unknown data
even if the direct problem is linear) are the main difficulties for developing effective numerical algorithms of reconstruction.

Inverse problems arise in many applications such as
**nondestructive evaluation** in engineering, **medical imaging** (CT scans and ultrasound tomography, for example), **sonar** or **radar** applications, **geophysical prospections**.

Two classical inverse
problems, that are prototypes for several others, are the **Electrical Impedance Tomography (EIT)** and **Inverse Scattering Problems**.

In **EIT** the aim is to recover the internal structure of a conducting body by nondestructive and noninvasive techniques, namely by electrostatic
measurements of current and voltage type on the boundary (we prescribe the voltage on the boundary and we measure the corresponding
current density, again on the boundary). The most studied problem is the **Calderón** or **inverse
conductivity problem** where the conductivity is to be determined by performing all possible measurements, that is, by
measuring the so-called **Dirichlet-to-Neumann map**. Here the conductivity is modeled by the coefficient of an **elliptic equation in divergence form**. EIT has applications in the nondestructive evaluation or in medical imaging for the detection of tumors, since one can exploit the fact that tumoral tissues have a much higher conductivity with respect to healthy tissues

The **Inverse Acoustic or Electromagnetic Scattering Problem** is the determination of obstacles or of the properties of a medium by **scattering
measurements**, that is, by sending (time-harmonic acoustic or electromagnetic) waves and measuring, at infinity, the reflected or scattered waves. For the determination of an (impenetrable) **obstacle**, the unknown is the domain of validity of the underlying pde (the **Helmholtz equation** in the acoustic case or the **Maxwell system** in the electromagnetic one).
These problems model sonar or radar applications and present many similarities with ultrasound tomography.