 # Mathematics

## Research topics

I deal with mathematical inverse problems and scattering theory. The latter is a subfield of partial differential equations where the purpose is to study phenomena which occur after a wave hits an obstacle. The field of inverse problems is related to mathematical modeling. In traditional or direct modeling, the goal is to predict the effects from the causes. On the other hand, in inverse problems, one is interested in finding the model, or causes, when a set of observations is given. Famous examples include

• electrical impedance tomography: find the electrical conductivity at various points in the interior of an object by doing voltage–current measurements on its surface,
• travel-time tomography: calculate the density of the Earth by observing the travel times of earthquakes,
• 3D X-ray tomography: determine the 3D structure of a body by taking X-ray pictures from various directions around it (CT-scan).
All of these examples involve tomography, which means imaging an object by sending waves into it. This is related to non-destructive testing by not having to cut the object open.

## List of Publications PDF

### Submitted

1. E. Blåsten, H. Liu, J. Xiao, On an electromagnetic problem in a corner and its applications.
2. E. Blåsten, H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems.
3. E. Blåsten, H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern.

### Accepted in Peer-Reviewed Journals

1. E. Blåsten, E. Vesalainen, Non-Scattering Energies and Transmission Eigenvalues in Hn. Annales Academiæ Scientiarum Fennicæ Mathematica, 45, 1 (2020) 547–576.
2. E. Blåsten, H. Li, H. Liu, Y. Wang, Localization and geometrization in plasmon resonances and geometric structures of Neumann-Poincaré eigenfunctions. ESAIM: Mathematical Modelling and Numerical Analysis, accepted 2019
3. E. Blåsten, L. Tzou, J. Wang, Uniqueness for the inverse boundary value problem with singular potentials in 2D. Mathematische Zeitschrift, in press 2019.
4. F. Zouari, M. Louati, S. Meniconi, E. Blåsten, M. S. Ghidaoui and B. Brunone, Experimental verification of the accuracy and robustness of Area Reconstruction Method for a Pressurized Water Pipe System. Journal of Hydraulic Engineering, ASCE, 146, 3 (2020).
5. E. Blåsten, F. Zouari, M. Louati and M. S. Ghidaoui, Blockage detection in networks: the area reconstruction method. Mathematics in Engineering, 1, 4 (2019).
6. E. Blåsten, H. Liu, On corners scattering stably and stable shape determination by a single far-field pattern. Indiana University Mathematics Journal, accepted 2019.
7. F. Zouari, E. Blåsten, M. Louati, and M. S. Ghidaoui, Internal pipe area reconstruction as a tool for blockage detection. Journal of Hydraulic Engineering, ASCE, 145, 6 (2019)
8. E. Blåsten, Y.-H. Lin Radiating and non-radiating sources in elasticity. Inverse Problems, 35, 1 (2019) 015005.
9. E. Blåsten, Nonradiating sources and transmission eigenfunctions vanish at corners and edges. SIAM Journal on Mathematical Analysis, 50, 6 (2018) 6255–6270.
10. E. Blåsten, Well-posedness of the Goursat problem and stability for point source inverse backscattering, Inverse Problems, 33, 12 (2017) 125003.
11. E. Blåsten, H. Liu, On vanishing near corners of transmission eigenfunctions, Journal of Functional Analysis, 273, 11 (2017), 3616–3632.
12. E. Blåsten, X. Li, H. Liu, Y. Wang, On vanishing and localizing of transmission eigenfunctions near singular points: A numerical study, Inverse Problems, 33, 10 (2017), 105001.
13. E. Blåsten, J. Sylvester, Translation-Invariant Estimates for Operators with Simple Characteristics, Journal of Differential Equations, 263, 9 (2017), 5656–5695.
14. E. Blåsten, O. Yu. Imanuvilov, M. Yamamoto, Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials, Inverse Problems and Imaging, 9, 3 (2015), 709–723.
15. E. Blåsten, L. Päivärinta, J. Sylvester, Corners Always Scatter, Communications in Mathematical Physics, 331, 2 (2014), 725–753.
16. E. Blåsten, L. Päivärinta, Completeness of generalized transmission eigenstates, Inverse Problems, 29, 10 (2013), 104002.

### Conference Proceedings

1. F. Zouari, M. Louati, E. Blåsten, and M. S. Ghidaoui, Multiple defects detection and characterization in pipes. 13th International Conference on Pressure Surges, Bordeaux, France, 14th–16th November 2018.

### Theses

1. E. Blåsten, On the Gel’Fand-Calderón inverse problem in two dimensions, Doctoral thesis, University of Helsinki, Faculty of Science, Department of Mathematics and Statistics, 2013.
2. E. Blåsten, The Inverse Problem of the Schrödinger Equation in the Plane: A Dissection of Bukhgeim's Result, Licentiate thesis, University of Helsinki, Faculty of Science, Department of Mathematics and Statistics, 2010.
3. E. Blåsten, Distribuutioteorian alkeet ja sovellutuksia, Master’s thesis, University of Helsinki, Faculty of Science, Department of Mathematics and Statistics, 2008.

### Other publications

1. E. Blåsten, H. Liu, Addendum to: "On vanishing near corners of transmission eigenfunctions", arXiv: 1710.08089.
2. E. Blåsten, Matematiikkaa Venäjällä, Solmu: matematiikkalehti, 3 (2007).

## Description of projects

E. Blåsten, H. Liu, J. Xiao, On an electromagnetic problem in a corner and its applications.

We consider the topic of scattering produced by electromagnetic sources having a corner, and was done by Blåsten  for acoustic source scattering. This also includes studying the corresponding electromagnetic transmission eigenvalue problem.

The conclusions are consistent with those for the acoustic case: namely that corners always radiate, and consequently that a source's shape can be determined by a single far-field measurement if it is known to be in the convex polyhedral class. This gives even more evidence for corners behaving universally in a very unique way, as unlike in the previous studies, the Maxwell equations are not elliptic.

E. Blåsten, E. Vesalainen, Non-Scattering Energies and Transmission Eigenvalues in Hn.

We extend the study of Blåsten, Päivärinta, Sylvester  from the Euclidean space to any comformally equivaĺent geometric space, for example the hyperbolic plane in two dimensions.

We conclude that potentials which have a hyperbolic corner or a corner made by a horocycle and a line (or higher dimensional equivalents) would always produce non-zero scattering given any non-zero incident wave. Se also show the existence of transmission eigenvalues in these geometries, so we can again conclude that non-scattering energies and transmission eigenvalues are different despite the change in the underlying spatial geometry.

E. Blåsten, F. Zouari, M. Louati, and M. S. Ghidaoui, Blockage detection in networks: the area reconstruction method.

The study is about developing and using an inversion algorithm for the wave equation with the goal of determining the cross-sectional area inside a network, e.g. the water supply network. The large picture idea come from Zouari, Blåsten, Louati, Ghidaoui , i.e. it is based on a time-domain boundary control method.

We show that the internal cross-sectional area of a tree network is determined by boundary measurements done at all network boundary ends except one. We present a proof of that, and also explicit numerical code for reconstructing the area given the network geometry.

F. Zouari, M. Louati, S. Menicone, E. Blåsten, M. S. Ghidaoui and B. Brunone, Experimental Verification of the Accuracy and Robustness of Area Reconstruction Method for a Pressurized Water Pipe System.

This is an experimental verification of the method introduced by Zouari, Blåsten, Louati, Ghidaoui  We conclude that the method is stable enough for use with experimentally measured noisy data.

F. Zouari, M. Louati, E. Blåsten, and M. S. Ghidaoui, Multiple defects detection and characterization in pipes.

This is a continuation and generalization of the work done for pipe area reconstruction. We extend that method to the detection of leaks and discrete blockages. Because of a mathematical non-uniqueness to this one-dimensional inverse problem, certain configurations of defects cannot be uniquely identified from certain others. We propose a solution to this problem: measure the impulse-response function from both ends of the pipe.

E. Blåsten, H. Li, H. Liu, Y. Wang Localization and geometrization in plasmon resonances and geometric structures of Neumann-Poincaré eigenfunctions

This article provides a new point of view to plasmon resonance. The motivation comes from high curvature scattering by myself and Liu . Usually plasmon resonance occurs around particles that are smaller than the wavelength of the incident light. Previously we showed that small scatterers always scatter, but also that the same phenomenon occurs near certain high curvature points of large scatterers. In this manuscript we study several theoretical and numerical examples, and conclude that the same is true of plasmon resonance. In other words, plasmon resonance occurs not only for small particles, but also near high curvature points on the boundary of large particles!

E. Blåsten, H. Liu Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems

We study how high curvature affects wave scattering. Our previous work focused on having a corner point in the boundary of a penetrable scatterer, and this had various consequences: corners always scatter, the shape of polyhedral scatterers can be determined by a single measurement, and smooth enough transmission eigenfunctions vanish at convex corners. In essence these properties stay true with slight modifications if the corner is replaced by high curvature point.

Our research shows a somewhat surprising result: an active source or inactive scatterer cannot produce a zero far-field if it is small (as measured by the wavenumber), or if its boundary has an admissible high curvature point. This fact is counter-intuitive. Surely small scatterers are barely seen, however the mathematical model implies that it cannot be completely invisible.

An additional consequence of this study is that a single measurement of the complete far-field of a scattered wave determines the rough location and exact number of small well-separated scatterers. This gives a theoretical proof for the results by Griesmaier, Hanke, Raasch 2013a and 2013b.

We consider the inverse source problem for the Navier system, in other words linear isotropic elasticity. We consider a homogeneous background, and waves created by a force applied to part of the domain. The force can be inhomogeneous and anisotropic.

We show two things. First of all the literature seemed to lack an example of a non-radiating source for elasticity. We show that a constant force applied on a disc causes no scattered elastic waves beyond the disc at a fixed frequency, if the radius of the disc, frequency and elastic parameters are in a given algebraic relation.

Secondly, we show that if an inhomogeneous force is applied on a region which has a corner on its exterior boundary, and if the magnitude of the force is nonzero there, then no matter how the rest of region, or the rest of the force looks like, this force would always cause scattered waves that can be detected in the far-field.

E. Blåsten, Nonradiating sources and transmission eigenfunctions vanish at corners and edges

This paper considers the inverse source problem, namely we do not send an incident wave, and are instead using data provided by an unknown source itself. We show that so called nonradiating sources cannot have sharp corner or edge jumps on their boundary. This also implies a simpler proof for the result of Blåsten, Liu 2017a assuming that the transmission eigenfunctions are smooth enough. Furthermore it gives uniqueness for the shape determination of convex polyhedral sources.

The method has some similarities with Ikehata's enclosure method. A major difference is that we use a more complicated probing function which allows us to consider non-convex geometry. This function is the exponential of a complex square root. The branch is chosen such that the modulus of this function decays in all directions except the branch cut.

F. Zouari, E. Blåsten, M. Louati, and M. S. Ghidaoui, Internal pipe area reconstruction as a tool for blockage detection

Water supply systems around the cities of the world suffer from various defects occasionally: the pipes can deteriorate, start leaking or become blocked by mineral deposits. We introduce a new method for detecting the location and shape of such blockages from an impulse-response measurement at the pipe end. Such non-invasive methods avoid the expenses of having to dig out and open the pipe just to inspect it. The method is based on determining the internal cross-sectional area of the pipe at any location inside it.

The area reconstruction method is introduced, both from an engineering point of view and a mathematical one. It is furthermore compared numerically to other state of the art blockage detection algorithms. We conclude that in noiseless numerical experiments this method is superior to others, especially when the number of blockages is unknown, or the blockages have irregular shape.

Blåsten, Well-posedness of the Goursat problem and stability for point source inverse backscattering

The topic of this paper is on the inverse backscattering method by Rakesh and Uhlmann, which they use both for the ordinary and point-source backscattering. Their method lives in the time-domain and works for potentials that are angularly controlled. Our primary goal is to prove stability for their uniqueness result of point-source inverse backscattering.

A difficulty arising in the proof relates to required norm-estimates. It is very difficult to find suitable estimates for the direct point-source problem, or a related problem which is called the Goursat problem.

The Goursat problem is the wave equation with boundary data given on a characteristic cone. Some call it the characteristic initial value problem. We prove well-posedness for it. That is, we show that given the initial data and coefficients, the solution exists, is unique (in a chosen function space), and depends continuously on the inputs. This then leads to the well-posedness of the point-source problem for the wave equation.

Blåsten, Liu, Recovering piecewise constant refractive indices by a single far-field pattern

We improve past corner scattering results in two ways. Compared to the earlier papers Hu, Salo, Vesalainen 2015 and Blåsten, Liu 2016 the results now apply to arbritrary three-dimensional convex polyhedra instead of just rectangular boxes. A second improvement is more useful: we show that a single far-field pattern not only determines the shape of a convex polyhedral scatterer, as known before, but also the values of the potential function at the vertices.

Using the corner value recovery and Holmgren's uniqueness theorem we are able to show the following under certain geometric conditions: given a single incident wave, two polyhedral piecewise constant potentials produce the same far-field pattern only if the potentials are the same.

Depending on application, the geometric a-priori conditions might be bothersome. However if the potentials are assumed to be piecewise constant on a square lattice, no matter how fine the mesh is, then our results apply. We hope that this would give ideas for novel reconstruction methods in engineering.

Blåsten, Tzou, Wang, Uniqueness for the inverse boundary value problem with singular potentials in 2D

In this paper we improve the integrability condition used in Blåsten, Imanuvilov, Yamamoto 2015. We show that the Dirichlet-Neumann map of a potential defined on a two-dimensional bounded domain determines uniquely the potential. Previously this could be done for almost square-integrable potentials. In this paper we improve that bound up to almost 4/3-integrable potentials.

Our improvement is based on two techniques. Firstly we use the fact that if two potentials have the same boundary data, then their difference is smoother (and hence more integrable) than either of the original potentials. Secondly we prove a more refined estimate for the Cauchy operators conjugated by a complex Gaussian function.

Blåsten, Li, Liu, Wang, On vanishing and localizing of transmission eigenfunctions near singular points: A numerical study

We study numerically the observations proved in Blåsten, Liu 2017 in a more general setting. Namely, we study where the energy of interior transmission eigenfunctions are located in domains with corners, edges or cusps.

We find the following: transmission eigenfunctions indeed do vanish on corners and edges of various domain in two and three dimensions. The order of vanishing also seems to depend inversely on the magnitude of the angle. If the angle is less than π then the eigenfunction decays when approacing the corner. If the angle is larger than π then the eigenfunction blows up at the corner. This is a completely new finding in the study of the interior transmission problem.

The method used for calculating the transmission eigenfunctions is by first transforming the interior transmission problem into an integral formulation, and then using a finite element approximation. The resulting sparse non-Hermitean discrete eigenvalue problem is solved by the `sptarn`-function in MATLAB.

Blåsten, Liu, On vanishing near corners of transmission eigenfunctions

We prove that transmission eigenfunctions carry geometrical information about their domain. Namely, that under some conditions the eigenfunctions vanish at corner points. To achieve this we also show a lower bound for the energy of the far-field pattern of a Herglotz wave scattered by penetrable polygonal scatterer. We have written an addendum arXiv:1710.08089 that relaxes the assumptions of the theorem.

This is a first result showing intrinsic properties of the transmission eigenfunctions. Just as in spectral theory, the transmission eigenvalues have been studied in great detail previously. Big challenges have prevented touching the eigenfunctions until now.

A related numerical work, which is still in progress, raises many interesting theoretical questions: what's the effect on the angle of the corner? what about complex transmission eigenvalues? what about localization?

Blåsten, Liu, On corners scattering stably and stable shape determination by a single far-field pattern

In this work we prove stability estimates for penetrable potential support recovery and also corner scattering by incident plane-waves. The corresponding uniqueness results are in [BPS2014], [PSV] and [HSV2016].

A new key tool we show is a quantitative Rellich's theorem for penetrable scatterers. This means that if the scattered waves created by two potentials are close to each other at infinity, then the values of the these waves are close to each other on the boundary of the obstacle.

The stability of corner scattering means that given any incident plane-wave of unit modulus, and a penetrable potential having a sharp corner, the energy of the far-field pattern has a uniform lower bound. This raises questions on whether interior transmission eigenfunctions could in fact be approximated by the more general Herglotz incident waves as is commonly believed.

Blåsten, Sylvester, Translation-Invariant Estimates for Operators with Simple Characteristics

The goal of this work is to prove new estimates for constant coefficient partial differential equations. The estimates have a simple formulation (they are seminorm estimates) and are invariant under geometric transformations of the Euclidean space. Once a solution to an inhomogeneous PDE satisfies our estimate the Agmon-Hörmander estimates from scattering theory are a trivial corollary. We prove the estimates for a wide class of PDE's, including all second order operators with real coefficients, and give some examples showing that the method can still be generalized, for example to systems of PDE's.

Blåsten, Imanuvilov, Yamamoto, Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials

This article combines methods from my PhD thesis  and a manuscript by Imanuvilov and Yamamoto  to solve the Gel'Fand-Calderón problem. The new unified method allowed for a simpler proof of uniqueness for potentials which are just integrable and stability for potentials having a fractional derivative of any positive order.

The change from Bukhgeim's method  is a smarter choice for the first two terms in the Neumann series of the complex geometric optics solutions.

Blåsten, Päivärinta, Sylvester, Corners Always Scatter

In this paper we proved that in single-frequency Helmholtz scattering, no matter which incident wave (Herglotz wave) and no matter at which frequency, a potential with a jump shaped like a 90° sharp corner will always produce a non-trivial scattered wave. This implies that transmission eigenvalues are not the same as non-scattering energies, i.e. energies at which the the relative scattering operator (which maps the asymptotics of the incident wave to the asymptotics of the scattered wave) has a non-trivial kernel. This is in contrast to the radially symmetric case where these two concepts were known to be the same.

A key-ingredient for the proof is a new resolvent-type estimate that uses norms from the Fourier transforms of Besov spaces. These give a better local integrability than past estimates by Agmon and Hörmander  or Sylvester and Uhlmann .

Blåsten, Päivärinta, Completeness of generalized transmission eigenstates

Although the results of this paper provide an important step forward in understanding the spectral properties of the interior transmission problem, many questions still remain. We think that some interesting open problems in this direction are ... and the completeness of the eigensystem of the interior transmission problem.

We proved that the eigensystems associated with all the transmission eigenvalues form a complete set. We used the consept of parameter-ellipticity by Agranovich and Vishik, analytic Fredholm theory, and Nevanlinna theory for operator-valued meromorphic functions. These imply that a fourth-order operator related to the transmission eigenvalue problem satisfies three assumptions that are stated in the language of meromorphic functions. These then imply the completeness of the eigensystem.

Blåsten, On the Gel’Fand-Calderón inverse problem in two dimensions

This is my PhD thesis, succesfully defended at the University of Helsinki in 2013. In it I prove stability for the Gel'Fand-Calderón problem with rough, or non-smooth potential, using the methods of Bukhgeim .

The problem of uniqueness for the Gel'Fand-Calderón problem had been open in two dimensions for at least twenty years and Bukhgeim was the first one to manage to solve it for a general potential. His arguments work when the potential has one derivative. Using function theory and interpolation theory, I managed to prove stability in addition to uniqueness, and for potentials that have any positive fractional derivative.

Blåsten, The Inverse Problem of the Schrödinger Equation in the Plane: A Dissection of Bukhgeim's Result

In this licentiate thesis I studied and explained Bukhgeim's solution to the Gel'Fand-Calderón problem in two dimensions .

The Finnish licentiate degree is an elective intermediate degree between master's degree, which is five years of university-level study, and the doctor's degree, which totals nine to ten years of study.

Blåsten, Distribuutioteorian alkeet ja sovellutuksia

An English translation of the title would be An introduction to the theory of distributions and its applications. This is my master's thesis, written in Finnish. In it I state and prove elementary results from distribution theory. The more advanced topics include the Fourier transform, how to define the division of arbitrary polynomials in one dimension and the division by homogeneous polynomials in arbitrary dimensions.

## Conference and Seminar Talks

1. Detecting blockages in water supply networks using boundary control, Inverse Days 2019; University of Jyväskylä, Finland, 17 December 2019.
2. Inverse problems with one measurement, Inverse Days 2018; Aalto University, Finland, 12 December 2018.
3. Inverse problems with one measurement, Inverse problems, PDE and geometry; University of Jyväskylä, Finland, 22 August 2018.
4. Inverse problems with one measurement, The 9th International Conference on Inverse Problems and Related Topics; National University of Singapore, Singapore, 15 August 2018.
5. Applications of corner scattering: intrinsic properties of transmission eigenfunctions and single wave probing, School of Mathematical Sciences, Fudan University, China, 5 December 2017.
6. Planar inverse boundary value problem for Lp potentials with p>4/3, Analysis Seminar, Department of Mathematics and Statistics; University of Jyväskylä, Finland, 23 August 2017.
7. Inverse backscattering with point-source waves, Inverse Problems Seminar, Department of Mathematics and Statistics; University of Helsinki, Finland, 17 August 2017.
8. Corners always scatter — quantitative results, Applied Inverse Problems Conference 2017; Zhejiang University, Hangzhou, China, 31 May 2017.
9. Transmission eigenfunction localization, Annual meeting of the Hong Kong Mathematical Society; The Hong Kong University of Science and Technology, Hong Kong, 20 May 2017.
10. Topics in Corner Scattering: Non-Scattering Waves, Potential Probing with a Single Incident Wave, and the Interior Transmission Problem, NCTS PDE and Analysis Seminar; National Center for Theoretical Sciences, National Taiwan University, Taipei, Taiwan, 9 Marh 2017.
11. Inverse scattering using a single incident wave, 2nd East Asia Section of IPIA, Young Scholars Symposium; National Center for Theoretical Sciences, National Taiwan University, Taipei, Taiwan, 5 November 2016.
12. Non-scattering energies, new resolvent estimates and other projects, 1st East Asia Symposium of IPIA; South University of Science and Technology, Shenzhen, China, 29 February 2016.
13. Non-scattering energies and interior transmission eigenvalues, Workshop on Inverse Problems and Related Topics; Zhejiang University, Hangzhou, China, 9 December 2015.
14. A new viewpoint to scattering theory à la Hörmander, Spectral and Analytic Inverse Problems, Thematic Programme on Inverse Problems; Institut Henri Poincaré, Paris, France, 4 May 2015.
15. Solving the Inverse Problem for the 2D Schrödinger Equation with Lp-potential, 17th Annual Workshop on Applications and generalizations of complex analysis; University of Aveiro, Aveiro, Portugal, 21 March 2015.
16. Solving the Inverse Problem for the 2D Schrödinger Equation with Lp-potential, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications; Instituto de Ciencias Matemáticas (ICMAT) and the Universidad Autónoma de Madrid (UAM), Madrid, Spain, 9 July 2014.
17. Completeness of the generalized transmission eigenstates, International Conference on Novel Directions in Inverse Scattering; University of Delaware, Delaware, USA, 29 July 2013.