I am a researcher in 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. Each of these few words has a mathematical equivalent:
- after: boundary conditions in time. These select the physically relevan solution from the set of all solutions.
- a wave: the unperturbed wave is a solution to a partial differential equation describing the behaviour in the background material.
- hits: conditions connecting the unperturbed wave to the scattered wave.
- an obstacle: modelled by a second differential equation, often a perturbation of the first one.
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).
List of Publications PDF
- Recovering piecewise constant refractive indices by a single far-field pattern.
- Uniqueness for the inverse boundary value problem with singular potentials in 2D.
- On corners scattering stably and stable shape determination by a single far-field pattern.
Accepted in Peer-Reviewed Journals
- Well-posedness of the Goursat problem and stability for point source inverse backscattering, Inverse Problems, 33, 12, (2017) 125003.
- On vanishing near corners of transmission eigenfunctions, Journal of Functional Analysis, 273, 11 (2017), 3616–3632.
- On vanishing and localizing of transmission eigenfunctions near singular points: A numerical study, Inverse Problems, 33, 10 (2017), 105001.
- Translation-Invariant Estimates for Operators with Simple Characteristics, Journal of Differential Equations, 263, 9 (2017), 5656–5695.
- Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials, Inverse Problems and Imaging, 9, 3 (2015), 709–723.
- Corners Always Scatter, Communications in Mathematical Physics, 331, 2 (2014), 725–753.
- Completeness of generalized transmission eigenstates, Inverse Problems, 29, 10 (2013), 104002.
- On the Gel’Fand-Calderón inverse problem in two dimensions, Doctoral thesis, University of Helsinki, Faculty of Science, Department of Mathematics and Statistics, 2013.
- 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.
- Distribuutioteorian alkeet ja sovellutuksia, Master’s thesis, University of Helsinki, Faculty of Science, Department of Mathematics and Statistics, 2008.
- Addendum to: "On vanishing near corners of transmission eigenfunctions", arXiv: 1710.08089.
- Matematiikkaa Venäjällä, Solmu: matematiikkalehti, 3 (2007).
Description of projects
- 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.
- 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.
- 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.
- 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.
- 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?
- 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.
- 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.
- 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.
- 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 .
- Completeness of generalized transmission eigenstates
This article answers a question posed by Cakoni, Gintides and Haddar :
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.
- 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.
- 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.
- 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
- Applications of corner scattering: intrinsic properties of transmission eigenfunctions and single wave probing, School of Mathematical Sciences, Fudan University, China, 5 December 2017.
- 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.
- Inverse backscattering with point-source waves, Inverse Problems Seminar, Department of Mathematics and Statistics, University of Helsinki, Finland, 17 August 2017.
- Corners always scatter — quantitative results, Applied Inverse Problems Conference 2017, Zhejiang University, Hangzhou, China, 31 May 2017.
- Transmission eigenfunction localization, Annual meeting of the Hong Kong Mathematical Society, The Hong Kong University of Science and Technology, Hong Kong, 20 May 2017.
- 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.
- 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.
- 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.
- Non-scattering energies and interior transmission eigenvalues, Workshop on Inverse Problems and Related Topics, Zhejiang University, Hangzhou, China, 9 December 2015.
- 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.
- 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.
- 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.
- Completeness of the generalized transmission eigenstates, International Conference on Novel Directions in Inverse Scattering, University of Delaware, Delaware, USA, 29 July 2013.