### 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.

#### 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)

#### 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.

#### Introducing special speakers

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• Gitta Kutyniok from TU Berlin is an expert on "sparsity-promoting" reconstruction methods. Inverse problems are about recovering objects based on measurement data which is insufficient. The data needs to be complemented with extra information about the object, such as sparsity. Sparsity means representing the object using building blocks specifically chosen so that only very few of them are needed. Professor Kutyniok often uses "shearlets" for representing images. Shearlets are versatile building blocks adapting to image details of any scale and representing edges with a variety of orientations.

In the attached picture she applies shear let reconstruction to an inverse scattering problem, resulting in a result much improved over a traditional method. In her plenary talk at the AIP2015 conference, Professor Kutyniok gives an introduction to the theory and computational use of the shearlet transform.

• Peter Markowich from KAUST is an expert of partial differential equations which arise from systems depending on many variables and involving change. Due to the generality of mathematics, such models apply to wildly different areas of application.

In his Special Keynote Address, Professor Markowich discusses biological transportation networks, price formation in economic markets and fluid flow in porous matter. The picture shows models for a large crowd of people in three groups exiting a building as fast as possible. Different models of human behaviour lead to different dynamics. This is a joint work with Martin Burger, Marco Di Francesco and Marie-Therese Wolfram.

• Peijun Li from Purdue University studies direct and inverse scattering problems. One of the central contributions in his work is the design of imaging methods accepting realistic near-field measurements (as opposed to mathematically ideal far-field patterns). In the picture is shown reconstructions of a two-dimensional shape. Here the unknown shape is probed with acoustic waves send from different directions. Various datasets are considered with limited angles of view. Observe that the "dark side" of the shape is more difficult to recover. This work is joint between Peijun Li and Yuliang Wang.

In his plenary talk at AIP, Peijun Li will describe his recent work on achieving sub-wavelength resolution for inverse surface scattering problems.

• Hongyu Liu from Hong Kong Baptist University knows how to recover objects from remote measurements. Below is an example of sending elastic vibrations through an unknown body, and recovering inhomogeneities (red) inside. This 2013 result is a joint work between four authors: Guanghui Hu, Jingzhi Li, Hongyu Liu and Hongpeng Sun.

At AIP, Professor Liu will explain how to hide objects from remote sensing. Such cloaking techniques are already used widely in fiction: think Harry Potter and his invisibility cloak.

• Xiaoqun Zhang from Shanghai Jiao Tong University is an expert in inverse problems related to image processing. Here is an example of her work (this one done jointly with Tony Chan). On the left is the original "Barbara" image. Second image from left shows many missing pixels that should be filled back in using so-called "inpainting." Third image from left shows the result of a standard baseline technique, whereas the rightmost picture shows the excellent inpainting result using a nonlocal method developed by Zhang & Chan in 2010.

• Recent work of Thomas Schuster from Saarland University, Germany, (joint with Arne Wöstehoff) paves the way to self-diagnosing airplanes. The idea is to equip the aircraft with vibration sources and sensors. Cracks and other defects can be detected by sending vibrations along the plane, and measuring the response at the sensors.

Prof. Schuster's plenary talk at AIP will be about vector tomography, which allows new imaging techniques in the fields of medicine, industry, oceanography, plasma physics, polarization tomography and electron microscopy.

• Katya Krupchyk from University of California at Irvine, USA. Professor Krupchyk is an expert on mathematical models of a range of indirect physical measurements. In one of her works, joint with Matti Lassas and Samuli Siltanen, she studied an extension of the imaging method called electrical impedance tomography.

In this work, electrical voltage-to-current measurements are preformed on the boundary of a physical body. The resulting currents flowing inside the body produce heat. The surface of the body is covered with heat flow sensors (interlaced with electrodes used for electrical measurements), providing extra information. Now the electrical and thermal measurements can be combined to yield improved information about the internal structure of the body.

• Takashi Kako from University of Electro-Communications, Chofu-Tokyo, Japan, is an expert on resonances, and he will talk about their role in the formation of vowels in human speech. The related inverse problem is quite tricky: given a recording of a vowel sound, recover the shape of the vocal tract and the excitation signal arising from the vocal folds flapping against each other.

Pictured are simplified vocal tract models for the five Japanese vowels: /a/, /i/, /u/, /e/, /o/.

• Eero Saksman, University of Helsinki: Adaptive Markov chain Monte Carlo (MCMC) methods (joint with Johanna Tamminen and Heikki Haario). In Bayesian inversion, one often needs to compute high dimensional integrals (posterior mean). Due to the "curse of dimensionality" it is not a good idea to use a quadrature method.

Instead, MCMC shoots plenty of points in the space, distributed according to the posterior probability. The average of the points is close to the integral. Now if the posterior probability has a weird shape, regular MCMC may not visit all corners of positive probability. Adaptive MCMC monitors the chain and modifies the search strategy on the fly, guiding the process to all relevant areas.