### Summer school on inverse problems

There will be a summer school on inverse problems aimed at graduate students and postdocs. The time of the summer school is May 18-22, 2015, which is the week before the AIP conference. The summer school will be organized at the Department of Mathematics and Statistics of University of Helsinki in the Exactum building at the Kumpula Campus. The visiting address of the department is Gustaf Hällströmin katu 2b. The lectures will take place at the auditorium CK112.Summer School Poster (pdf)

Schedule of the summer school (pdf)

How to get to the Kumpula Campus (pdf)

List of summer school participants

Excursion

Dinner

Scientific committee:

**Miren Zubeldia** (chair), BCAM, Spain & University of Helsinki, Finland

**Roberta Bosi**, University of Helsinki, Finland

**Pedro Caro**, ICMAT, Spain

**Nuutti Hyvönen**, Aalto University, Finland

**Matti Lassas**, University of Helsinki, Finland

**Samuli Siltanen**, University of Helsinki

There are four minicourses in the summer school given by

**Maarten de Hoop**, Purdue University, USA.

**Tanja Tarvainen**, University of Eastern Finland.

**Eric Bonnetier**, Laboratoire Jean Kuntzmann, Université de Grenoble-Alpes, France.

**Gabriel Paternain**, University of Cambridge, UK.

**Keynote Talks:**

#### Ill-posedness and regularisation: the case of X-ray tomography

**Samuli Siltanen**, University of Helsinki, Finland, Slides (pdf)

#### Harry Potter's Cloak via Transformation Optics

**Gunther Uhlmann**, University of Helsinki, Finland, and University of Washington, USA, Slides (pdf)

**Minicourses:**

#### How to obtain large gradients in composite media and the Neumann-Poincaré operator

**Eric Bonnetier**, Laboratoire Jean Kuntzmann, Université de Grenoble-Alpes, France.

Abstract (pdf)

Slides part 1 (pdf)

In this course, I will present work that has been developed over the last decade concerning large gradients in composite media made of inclusions with smooth boundaries embedded in a matrix phase. The problems originates from a question of Ivo Babuŝka, who was concerned with the possible onset of cracks in the narrow channels between close-to-touching inclusions in such composites. If large values of the gradients can be detrimental in mechanics, they may be beneficial in other contexts, such as plasmonics. There, large gradients may be the signature of localized resonant modes, which can be used to identify single molecules or to destroy cancerous cells. This resonant mechanism is also key in a form of cloaking known as claoking via anomalous localized resonance, hence the connection with inverse problems.

We will first discuss the case of the conduction equation in a medium made of 2 discs of conductivity $k$ embedded in a matrix phase of conductivity 1, and separated by a distance $\delta$. We will see how the parameters $\delta$ and $k$ govern the pointwise behavior of the gradient of the voltage potential. I will survey the results concerning pointwise bounds on the gradient, that have been obtained by several groups, when $0 < k < \infty$, but also in the degenerate case when $k = \infty$.

Next, we will study the Neumann-Poincaré
operator $K^*$ associated to a system of 2 inclusions. This integral operator
naturally appears when one seeks a representation of the voltage
potential in terms of layer potentials. After recalling the basic theory
concerning layer potentials, I will derive the system of integral equations

associated to a system of 2 close-to-touching discs. I will show how the spectral properties of this operator are related to the possible blow up of the gradient of the voltage potential. Finally, I will describe how cloacking via anomalous localized resonance is related to the Neumann-Poincar\'e operator and to creating large gradients in composite media.

#### The geodesic ray transform in 2D

**Gabriel Paternain**, University of Cambridge, UK.

This mini-course will focus on the geodesic ray transform in two dimensions and related geometric inverse problems. The transform is obtained by integrating a function or a tensor along geodesics of a Riemannian manifold and is a fundamental object of study in the field. Our goal would be to understand the kernel of the transform when acting on symmetric tensors of degree two and how it relates to other geometric inverse problems like boundary rigidity and spectral rigidity.

#### Computational inverse problems with applications in optical tomography

**Tanja Tarvainen**, University of Eastern Finland

course_notes (pdf)

Imaging biological tissues using visible light has been in the interest of researchers for many decades. The interest to optical imaging modalities comes from the capability of optical methods to provide information on the internal properties of tissues based on endogenous (e.g. haemoglobin) or exogenous (e.g. dyes) contrast. Many of the optical imaging problems are ill-posed, and therefore they need to be approached in the framework of inverse problems. In this lecture, optical imaging of biological tissues is discussed with emphasis on mathematical modelling and computational inverse problems.

#### The seismic inverse problem, stability and reconstruction via hierarchical compression

**Maarten de Hoop**, Purdue University, USA

We consider the inverse boundary value problems for the acoustic and elastic wave equations with the Dirichlet-to-Neumann map as the data. The seismic inverse problem can be formulated in this way via some relations between this map and field data acquisition: For example, in the elastic case, vibroseis generated data which include surface waves provide the multi-frequency Neumann-to-Dirichlet map while, in the acoustic case, `simultaneous source' marine acquisition can be identified with the Dirichlet-to-Neumann map via a single-layer potential operator.

An optimization approach, also referred to as full waveform inversion, to try and estimate from these data the coefficients of the equations was introduced in exploration and global seismology in the 1980s and has gained significant interest in recent years, primarily due to the increase in available computation power. We address the connection between this approach with the analysis of inverse problems, accommodating partial boundary data, and iterative reconstruction. We apply a time-Fourier transform and study the corresponding multi-frequency inverse boundary value problems, in particular, conditional Lipschitz stability leading to the introduction of coefficients with structure associated with domain partitions. Estimates of the stability constants play a key role in the development of multi-level schemes. Underlying these is the notion of bounded frequency data.

We conclude with a concise introduction to the system of equations describing Earth's free oscillations, discuss how the above mentioned inverse problems can be extracted and various open problems.