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    <div class="paper">
      <section class="content">

        <h1>Mathematics</h1>

        <section>
          <a class="anchor" id="topics"></a>
	  <h2>Research topics</h2>
          <p>
            I deal with
            mathematical <a href="https://en.wikipedia.org/wiki/Inverse_problem">inverse
            problems</a>
            and <a href="https://en.wikipedia.org/wiki/Scattering_theory">scattering
            theory</a>. The latter is a subfield of partial
            differential equations where the purpose is to study
            phenomena which occur <em>after a wave hits an
            obstacle</em>. The field of inverse problems is related to
            mathematical modeling. In traditional or <em>direct</em>
            modeling, the goal is to predict the effects from the
            causes. On the other hand, in <em>inverse problems</em>,
            one is interested in finding the model, or causes, when a
            set of observations is given. Famous examples include
            
            <ul>
              <li><em>electrical impedance tomography</em>: find the
                electrical conductivity at various points in the
                interior of an object by doing voltage&ndash;current
                measurements on its surface,</li>
              
              <li><em>travel-time tomography</em>: calculate the
                density of the Earth by observing the travel times of
                earthquakes,</li>
              
              <li><em>3D X-ray tomography</em>: determine the 3D
                structure of a body by taking X-ray pictures from
                various directions around it (CT-scan).</li>
            </ul>
            
            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.
          </p>
        </section>


        <section>
	  <a class="anchor" id="publications"></a>
          <h2>List of
            Publications <a href="pdf/blasten_publications.pdf">PDF</a></h2>
          <ul class="navsubbar">
            <li><a href="#pubs_submitted">Submitted</a></li>
            <li><a href="#pubs_accepted">Accepted</a></li>
            <li><a href="#pubs_proceedings">Proceedings</a></li>
            <li><a href="#pubs_theses">Theses</a></li>
            <li><a href="#pubs_others">Others</a></li>
          </ul>

          <p style="clear:left">
	    <a class="anchor" id="pubs_submitted"></a>
            <h3>Submitted</h3>
            <ol reversed>
              <li>
		<a class="anchor" id="Blasten-Li-Liu-Wang-2018"></a>
                <span class="ref-authors">E. Blåsten, H. Li, H. Liu, Y. Wang,</span>
                <span class="ref-title">Localization and
		  geometrization in plasmon resonances and geometric
		  structures of Neumann-Poincaré
		  eigenfunctions.</span>
                <div class="ref-links">
                  <a href="#desc_Blasten-Li-Liu-Wang-2018">Description</a>
                  <a href="https://arxiv.org/abs/1809.08533">Preprint</a>
		</div>
	      </li>
              <li>
		<a class="anchor" id="Blasten-Vesalainen-2018"></a>
                <span class="ref-authors">E. Blåsten, E. V. Vesalainen,</span>
                <span class="ref-title">Non-Scattering Energies and Transmission Eigenvalues in
		  <i>H<sup>n</sup></i>.</span>
                <div class="ref-links">
                  <a href="#desc_Blasten-Vesalainen-2018">Description</a>
                  <a href="http://arxiv.org/abs/1809.04426/">Preprint</a>
		</div>
	      </li>
              <li>
		<a class="anchor" id="Blasten-Liu-2018"></a>
                <span class="ref-authors">E. Blåsten, H. Liu,</span>
                <span class="ref-title">Scattering by curvatures,
                radiationless sources, transmission eigenfunctions and
                inverse scattering problems.</span>
                <div class="ref-links">
                  <a href="#desc_Blasten-Liu-2018">Description</a>
                  <a href="https://arxiv.org/abs/1808.01425">Preprint</a>
                </div>
              </li>
              <li>
		<a class="anchor" id="BlastenLiu2017b"></a>
                <span class="ref-authors">E. Blåsten, H. Liu,</span>
                <span class="ref-title">Recovering piecewise constant
                  refractive indices by a single far-field
                  pattern.</span>
                <div class="ref-links">
                  <a href="#desc_BlastenLiu2017b">Description</a>
                  <a href="https://arxiv.org/abs/1705.00815">Preprint</a>
                </div>
              </li>
              <li>
		<a class="anchor" id="BlastenTzouWang2017"></a>
                <span class="ref-authors">E. Blåsten, L. Tzou,
                  J. Wang,</span>
                <span class="ref-title">Uniqueness for the inverse
                  boundary value problem with singular potentials in
                  2D.</span>
                <div class="ref-links">
                  <a href="#desc_BlastenTzouWang2017">Description</a>
                  <a href="https://arxiv.org/abs/1704.06397">Preprint</a>
                </div>
              </li>
              <li>
		<a class="anchor" id="BlastenLiu2016"></a>
                <span class="ref-authors">E. Blåsten, H. Liu,</span>
                <span class="ref-title"> On corners scattering stably
                  and stable shape determination by a single far-field
                  pattern.</span>
                <div class="ref-links">
                  <a href="#desc_BlastenLiu2016">Description</a>
                  <a href="https://arxiv.org/abs/1611.03647">Preprint</a>
                </div>
              </li>
            </ol>
          </p>

          <p>
	    <a class="anchor" id="pubs_accepted"></a>
            <h3>Accepted in Peer-Reviewed Journals</h3>
            <ol reversed>
              <li>
		<a class="anchor" id="Blasten2018"></a>
                <span class="ref-authors">E. Blåsten,</span>
                <span class="ref-title">Nonradiating sources and
		  transmission eigenfunctions vanish at corners and
		  edges.</span>
                <div class="ref-links">
                  <a href="#desc_Blasten2018">Description</a>
		  <a href="https://doi.org/10.1137/18M1182048">Publication</a>
                  <a href="http://arxiv.org/abs/1803.10917/">Preprint</a>
                </div>
              </li>
              <li>
		<a class="anchor" id="Pipes1"></a>
                <span class="ref-authors">F. Zouari, E. Blåsten,
                  M. Louati, and M.&nbsp;S. Ghidaoui,</span>
                <span class="ref-title">Internal pipe area
                  reconstruction as a tool for blockage
                  detection.</span>
		<span class="ref-journal">Journal of Hydraulic
		  Engineering, ASCE,</span>
		<span class="ref-year">accepted 2018.</span>
                <div class="ref-links">
                  <a href="#desc_Pipes1">Description</a>
                </div>
              </li>
              <li>
		<a class="anchor" id="BlastenLin2018"></a>
                <span class="ref-authors">E. Blåsten, Y.-H. Lin</span>
                <span class="ref-title">Radiating and non-radiating
                  sources in elasticity.</span>
		<span class="ref-journal">Inverse Problems,</span>
		<span class="ref-year">accepted (2018).</span>
                <div class="ref-links">
                  <a href="#desc_BlastenLin2018">Description</a>
		  <a href="https://doi.org/10.1088/1361-6420/aae99e">Publication</a>
                  <a href="http://arxiv.org/abs/1807.07225/">Preprint</a>
                </div>
              </li>
              <li>
		<a class="anchor" id="Blasten2017"></a>
                <span class="ref-authors">E. Blåsten,</span>
                <span class="ref-title">Well-posedness of the Goursat
                  problem and stability for point source inverse
                  backscattering,</span>
                <span class="ref-journal">Inverse Problems,</span>
                <span class="ref-volume">33,</span>
                <span class="ref-issue">12,</span>
                <span class="ref-year">(2017)</span>
                <span class="ref-pages">125003.</span>
                <div class="ref-links">
                  <a href="#desc_Blasten2017">Description</a>
                  <a href="https://doi.org/10.1088/1361-6420/aa941f">Publication</a>
                  <a href="https://arxiv.org/abs/1705.09442">Preprint</a>
                </div>
              </li>
              <li>
		<a class="anchor" id="BlastenLiu2017a"></a>
                <span class="ref-authors">E. Blåsten, H. Liu,</span>
                <span class="ref-title">On vanishing near corners of
                  transmission eigenfunctions,</span>
                <span class="ref-journal">Journal of Functional
                  Analysis,</span>
                <span class="ref-volume">273,</span>
                <span class="ref-issue">11</span>
                <span class="ref-year">(2017),</span>
                <span class="ref-pages">3616&ndash;3632.</span>
                <div class="ref-links">
                  <a href="#desc_BlastenLiu2017a">Description</a>
                  <a href="https://doi.org/10.1016/j.jfa.2017.08.023">Publication</a>
                  <a href="https://arxiv.org/abs/1701.07957">Preprint</a>
                </div>
              </li>
              <li>
		<a class="anchor" id="BlastenLiLiuWang2017"></a>
                <span class="ref-authors">E. Blåsten, X. Li, H. Liu,
                  Y. Wang,</span>
                <span class="ref-title"> On vanishing and localizing
                  of transmission eigenfunctions near singular points:
                  A numerical study,</span>
                <span class="ref-journal">Inverse Problems,</span>
                <span class="ref-volume">33,</span>
                <span class="ref-issue">10</span>
                <span class="ref-year">(2017),</span>
                <span class="ref-pages">105001.</span>
                <div class="ref-links">
                  <a href="#desc_BlastenLiLiuWang2017">Description</a>
                  <a href="https://doi.org/10.1088/1361-6420/aa8826">Publication</a>
                  <a href="https://arxiv.org/abs/1704.01885">Preprint</a>
                </div>
              </li>
              <li>
		<a class="anchor" id="BlastenSylvester2016"></a>
                <span class="ref-authors">E. Blåsten, J. Sylvester,</span>
                <span class="ref-title">Translation-Invariant
                  Estimates for Operators with Simple
                  Characteristics,</span>
                <span class="ref-journal">Journal of Differential
                  Equations,</span>
                <span class="ref-volume">263,</span>
                <span class="ref-issue">9</span>
                <span class="ref-year">(2017),</span>
                <span class="ref-pages">5656&ndash;5695.</span>
                <div class="ref-links">
                  <a href="#desc_BlastenSylvester2016">Description</a>
                  <a href="https://doi.org/10.1016/j.jde.2017.06.033">Publication</a>
                  <a href="http://arxiv.org/abs/1607.06214">Preprint</a>
                </div>
              </li>
              <li>
		<a class="anchor" id="BlastenImanuvilovYamamoto2015"></a>
                <span class="ref-authors">E. Blåsten,
                  O. Yu. Imanuvilov, M. Yamamoto,</span>
                <span class="ref-title">Stability and uniqueness for a
                  two-dimensional inverse boundary value problem for
                  less regular potentials,</span>
                <span class="ref-journal">Inverse Problems and
                  Imaging,</span>
                <span class="ref-volume">9,</span>
                <span class="ref-issue">3</span>
                <span class="ref-year">(2015),</span>
                <span class="ref-pages">709&ndash;723.</span>
                <div class="ref-links">
                  <a href="#desc_BlastenImanuvilovYamamoto2015">Description</a>
                  <a href="http://dx.doi.org/10.3934/ipi.2015.9.709">Publication</a>
                  <a href="http://arxiv.org/abs/1504.02207">Preprint</a>
                </div>
              </li>
              <li>
		<a class="anchor" id="BlastenPaivarintaSylvester2014"></a>
                <span class="ref-authors">E. Blåsten, L. Päivärinta,
                  J. Sylvester,</span>
                <span class="ref-title">Corners Always Scatter,</span>
                <span class="ref-journal">Communications in
                  Mathematical Physics,</span>
                <span class="ref-volume">331,</span>
                <span class="ref-issue">2</span>
                <span class="ref-year">(2014),</span>
                <span class="PUGpages">725&ndash;753.</span>
                <div class="ref-links">
                  <a href="#desc_BlastenPaivarintaSylvester2014">Description</a>
                  <a href="http://dx.doi.org/10.1007/s00220-014-2030-0">Publication</a>
                  <a href="http://arxiv.org/abs/1211.1848">Preprint</a>
                </div>
              </li>
              <li>
		<a class="anchor" id="BlastenPaivarinta2013"></a>
                <span class="ref-authors">E. Blåsten,
                  L. Päivärinta,</span>
                <span class="ref-title">Completeness of generalized
                  transmission eigenstates,</span>
                <span class="ref-journal">Inverse Problems,</span>
                <span class="ref-volume">29,</span>
                <span class="ref-issue">10</span>
                <span class="ref-year">(2013),</span>
                <span class="ref-pages">104002.</span>
                <div class="ref-links">
                  <a href="#desc_BlastenPaivarinta2013">Description</a>
                  <a href="http://dx.doi.org/10.1088/0266-5611/29/10/104002">Publication</a>
                  <a href="http://arxiv.org/abs/1307.4863">Preprint</a>
                </div>
              </li>
            </ol>
          </p>

          <p>
	    <a class="anchor" id="pubs_proceedings"></a>
            <h3>Conference Proceedings</h3>
            <ol reversed>
              <li>
		<a class="anchor" id="PipesProceedings1"></a>
                <span class="ref-authors">F. Zouari, M. Louati,
                  E. Blåsten, and M.&nbsp;S. Ghidaoui,</span>
                <span class="ref-title">Multiple defects detection and
                  characterization in pipes.</span>
		<span class="ref-journal">13<sup>th</sup> International Conference on
		  <span style="font-variant: small-caps">Pressure
		  Surges</span>,</span>
		<span class="ref-year">Bordeaux, France,
		  14<sup>th</sup>&ndash;16<sup>th</sup> November
		  2018.</span>
                <div class="ref-links">
                  <a href="#desc_PipesProceedings1">Description</a>
                </div>
              </li>
            </ol>
          </p>

          <p>
	    <a class="anchor" id="pubs_theses"></a>
            <h3>Theses</h3>
            <ol reversed>
              <li>
		<a class="anchor" id="Blasten2013"></a>
                <span class="ref-authors">E. Blåsten,</span>
                <span class="ref-title">On the Gel’Fand-Calderón
                  inverse problem in two dimensions,</span>
                <span class="ref-journal">Doctoral thesis, University
                  of Helsinki, Faculty of Science, Department of
                  Mathematics and Statistics,</span>
                <span class="ref-year">2013.</span>
                <div class="ref-links">
                  <a href="#desc_Blasten2013">Description</a>
                  <a href="http://urn.fi/URN:ISBN:978-952-10-8699-1">Publication</a>
                  <a href="http://arxiv.org/abs/1307.4870">Preprint</a>
                </div>
              </li>
              <li>
		<a class="anchor" id="Blasten2010"></a>
                <span class="ref-authors">E. Blåsten,</span>
                <span class="ref-title">The Inverse Problem of the
                  Schrödinger Equation in the Plane: A Dissection of
                  Bukhgeim's Result,</span>
                <span class="ref-journal">Licentiate thesis,
                  University of Helsinki, Faculty of Science,
                  Department of Mathematics and Statistics,</span>
                <span class="ref-year">2010.</span>
                <div class="ref-links">
                  <a href="#desc_Blasten2010">Description</a>
                  <a href="http://arxiv.org/abs/1103.6200">arXiv version</a>
                </div>
              </li>
              <li>
		<a class="anchor" id="Blasten2008"></a>
                <span class="ref-authors">E. Blåsten,</span>
                <span class="ref-title">Distribuutioteorian alkeet ja
                  sovellutuksia,</span>
                <span class="ref-journal">Master’s thesis, University
                  of Helsinki, Faculty of Science, Department of
                  Mathematics and Statistics,</span>
                <span class="ref-year">2008.</span>
                <div class="ref-links">
                  <a href="#desc_Blasten2008">Description</a>
                  <a href="pdf/blasten_master.pdf">PDF (in Finnish)</a>
                </div>
              </li>
            </ol>
          </p>

          <p>
	    <a class="anchor" id="pubs_others"></a>
            <h3>Other publications</h3>
            <ol reversed>
              <li>
                <span class="ref-authors">E. Blåsten, H. Liu,</span>
                <span class="ref-title">Addendum to: "On vanishing
                  near corners of transmission eigenfunctions",</span>
                <span class="ref-journal">arXiv:</span>
                <span class="ref-pages">1710.08089.</span>
                <div class="ref-links">
                  <a href="https://arxiv.org/abs/1710.08089">Addendum</a>
                </div>
              </li>
              <li>
                <span class="ref-authors">E. Blåsten,</span>
                <span class="ref-title">Matematiikkaa
                  Venäjällä,</span>
                <span class="ref-journal">Solmu:
                  matematiikkalehti,</span>
                <span class="ref-issue">3</span>
                <span class="ref-year">(2007).</span>
                <div class="ref-links">
                  <a href="http://matematiikkalehtisolmu.fi/2007/3/Dubna.pdf">Article</a>
                </div>
              </li>
            </ol>
          </p>

        </section>


        <section>
	  <a class="anchor" id="descriptions"></a>
          <h2>Description of projects</h2>
	  <dl>
            <dt>
	      <a class="anchor" id="desc_PipesProceedings1"></a>
              <a href="#PipesProceedings1">
                <span class="ref-authors">F. Zouari, M. Louati,
                  E. Blåsten, and M.&nbsp;S. Ghidaoui,</span>
                <span class="ref-title">Multiple defects detection and
                  characterization in pipes.</span>
              </a>
            </dt>
            <dd>
              <p>
		This is a continuation and generalization of the work
		done for pipe <a href="#Pipes1">area
		reconstruction</a>. 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.
	      </p>
            </dd>

	    <dt>
	      <a class="anchor" id="desc_Blasten-Li-Liu-Wang-2018"></a>
	      <a href="#Blasten-Li-Liu-Wang-2018">
		<span class="ref-authors">E. Blåsten, H. Li, H. Liu, Y. Wang</span>
		<span class="ref-title">Localization and
		  geometrization in plasmon resonances and geometric
		  structures of Neumann-Poincaré eigenfunctions</span>
	      </a>
	    </dt>
	    <dd>
	      <p>
		This article provides a new point of view to plasmon
		resonance. The motivation comes from high curvature
		scattering by myself and
		Liu <a href="#Blasten-Liu-2018">[2018]</a>. 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!
	      </p>
	    </dd>
	    <dt>
	      <a class="anchor" id="desc_Blasten-Vesalainen-2018"></a>
	      <a href="#Blasten-Vesalainen-2018">
		<span class="ref-authors">E. Blåsten, E. V. Vesalainen</span>
		<span class="ref-title">Non-Scattering Energies and
		Transmission Eigenvalues
		in <i>H<sup>n</sup></i></span>
	      </a>
	    </dt>
	    <dd>
	      <p>
		The article concerns scattering theory for the
		Helmholtz and Schrödinger equations in hyperbolic
		spaces. We show that corners always scatter there too
		despite there being so-called transmission
		eigenvalues, which we show also. This brings the
		results of myself, Päivärinta and
		Sylvester <a href="#BlastenPaivarintaSylvester2014">[2014]</a>,
		and Hu, Salo,
		Vesalainen <a href="http://dx.doi.org/10.1137/15M1032958">[2015]</a>
		into the setting of hyperbolic manifolds.
	      </p>
	    </dd>
	    <dt>
	      <a class="anchor" id="desc_Blasten-Liu-2018"></a>
	      <a href="#Blasten-Liu-2018">
		<span class="ref-authors">E. Blåsten, H. Liu</span>
		<span class="ref-title">Scattering by curvatures,
		radiationless sources, transmission eigenfunctions and
		inverse scattering problems</span>
	      </a>
	    </dt>
	    <dd>
	      <p>
		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.
	      </p>
	      <p>
		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.
	      </p>
	      <p>
		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 <a href="https://doi.org/10.1137/110855880">2013a</a>
		and <a href="https://doi.org/10.1137/130908658">2013b</a>.
	      </p>
	    </dd>
	    <dt>
	      <a class="anchor" id="desc_BlastenLin2018"></a>
	      <a href="#BlastenLin2018">
		<span class="ref-authors">E. Blåsten, Y.-H. Lin</span>
		<span class="ref-title">Radiating and non-radiating
		  sources in elasticity</span>
	      </a>
	    </dt>
	    <dd>
	      <p>
		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.
	      </p>
	      <p>
		We show two things. First of all the literature seemed
		to lack an example of a <em>non-radiating source</em>
		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.
	      </p>
	      <p>
		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.
	      </p>
	    </dd>
	    <dt>
	      <a class="anchor" id="desc_Blasten2018"></a>
	      <a href="#Blasten2018">
		<span class="ref-authors">E. Blåsten,</span>
		<span class="ref-title">Nonradiating sources and
		  transmission eigenfunctions vanish at corners and
		  edges</span>
	      </a>
	    </dt>
	    <dd>
	      <p>
		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 <em>nonradiating
		sources</em> cannot have sharp corner or edge jumps on
		their boundary. This also implies a simpler proof for
		the result of <a href="#BlastenLiu2017a">Blåsten, Liu
		2017a</a> assuming that the transmission
		eigenfunctions are smooth enough. Furthermore it gives
		uniqueness for the shape determination of convex
		polyhedral sources.
	      </p>
	      <p>
		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.
	      </p>
	    </dd>
            <dt>
	      <a class="anchor" id="desc_Pipes1"></a>
              <a href="#Pipes1">
                <span class="ref-authors">F. Zouari, E. Blåsten,
                  M. Louati, and M.&nbsp;S. Ghidaoui,</span>
                <span class="ref-title">Internal pipe area
                  reconstruction as a tool for blockage
                  detection</span>
              </a>
            </dt>
            <dd>
              <p>
		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 <em>non-invasive</em> 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.
	      </p>
	      <p>
		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.
              </p>
            </dd>
            <dt>
	      <a class="anchor" id="desc_Blasten2017"></a>
	      <a href="#Blasten2017">
                <span class="ref-authors">Blåsten,</span>
                <span class="ref-title">Well-posedness of the Goursat
                  problem and stability for point source inverse
                  backscattering</span>
	      </a>
            </dt>
            <dd>
	      <p>
                The topic of this paper is on the inverse
                backscattering method by Rakesh and Uhlmann, which
                they use both for
                the <a href="https://doi.org/10.1088/0266-5611/30/6/065005">ordinary</a>
                and <a href="http://www.ams.org/books/conm/644/">point-source</a>
                backscattering. Their method lives in the time-domain
                and works for potentials that are <em>angularly
                controlled</em>. Our primary goal is to prove
                stability for their uniqueness result of point-source
                inverse backscattering.
	      </p>
	      <p>
                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.
	      </p>
	      <p>
                The Goursat problem is the wave equation with boundary
                data given on a characteristic cone. Some call it
                the <em>characteristic initial value problem</em>. 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.
	      </p>
            </dd>
            
            <dt>
	      <a class="anchor" id="desc_BlastenLiu2017b"></a>
              <a href="#BlastenLiu2017b">
                <span class="ref-authors">Blåsten, Liu,</span>
                <span class="ref-title">Recovering piecewise constant
                  refractive indices by a single far-field
                  pattern</span>
              </a>
            </dt>
            <dd>
              <p>
                We improve past corner scattering results in two
                ways. Compared to the earlier
                papers <a href="http://dx.doi.org/10.1137/15M1032958">Hu,
                Salo, Vesalainen 2015</a>
                and <a href="#BlastenLiu2016">Blåsten, Liu 2016</a>
                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.
              </p>
              <p>
                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.
              </p>
              <p>
                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.
              </p>
            </dd>
            
            <dt>
	      <a class="anchor" id="desc_BlastenTzouWang2017"></a>
              <a href="#BlastenTzouWang2017">
                <span class="ref-authors">Blåsten, Tzou, Wang,</span>
                <span class="ref-title">Uniqueness for the inverse
                  boundary value problem with singular potentials in
                  2D</span>
              </a>
            </dt>
            <dd>
              <p>
                In this paper we improve the integrability condition
                used
                in <a href="#BlastenImanuvilovYamamoto2015">Blåsten,
                Imanuvilov, Yamamoto 2015</a>. 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.
              </p>
              <p>
                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.
              </p>
            </dd>
            
            <dt>
	      <a class="anchor" id="desc_BlastenLiLiuWang2017"></a>
              <a href="#BlastenLiLiuWang2017">
                <span class="ref-authors">Blåsten, Li, Liu, Wang,</span>
                <span class="ref-title"> On vanishing and localizing
                  of transmission eigenfunctions near singular points:
                  A numerical study</span>
              </a>
            </dt>
            <dd>
              <p>
                We study numerically the observations proved
                in <a href="#BlastenLiu2017a">Blåsten, Liu 2017</a> in
                a more general setting. Namely, we study where the
                energy of interior transmission eigenfunctions are
                located in domains with corners, edges or cusps.
              </p>
              <p>
                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 &pi;
                then the eigenfunction decays when approacing the
                corner. If the angle is larger than &pi; then the
                eigenfunction blows up at the corner. This is a
                completely new finding in the study of the interior
                transmission problem.
              </p>
              <p>
                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 <code>sptarn</code>-function
                in MATLAB.
              </p>
            </dd>
            
            <dt>
	      <a class="anchor" id="desc_BlastenLiu2017a"></a>
              <a href="#BlastenLiu2017a">
                <span class="ref-authors">Blåsten, Liu,</span>
                <span class="ref-title"> On vanishing near corners of
                  transmission eigenfunctions</span>
              </a>
            </dt>
            <dd>
              <p>
                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 <a href="https://arxiv.org/abs/1710.08089">arXiv:1710.08089</a>
                that relaxes the assumptions of the theorem.
              </p>
              <p>
                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.
              </p>
              <p>
                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?
              </p>
            </dd>
            
            <dt>
	      <a class="anchor" id="desc_BlastenLiu2016"></a>
              <a href="#BlastenLiu2016">
                <span class="ref-authors">Blåsten, Liu,</span>
                <span class="ref-title">On corners scattering stably
                  and stable shape determination by a single far-field
                  pattern</span>
              </a>
            </dt>
            <dd>
              <p>
                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 <a href="#BlastenPaivarintaSylvester2014">[BPS2014]</a>,
                <a href="https://arxiv.org/abs/1404.2513">[PSV]</a>
                and <a href="http://dx.doi.org/10.1137/15M1032958">[HSV2016]</a>.
              </p>
              <p>
                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.
              </p>
              <p>
                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.
              </p>
            </dd>
            
            <dt>
	      <a class="anchor" id="desc_BlastenSylvester2016"></a>
              <a href="#BlastenSylvester2016">
                <span class="ref-authors">Blåsten, Sylvester,</span>
                <span class="ref-title">Translation-Invariant
                  Estimates for Operators with Simple
                  Characteristics</span>
              </a>
            </dt>
            <dd>
              <p>
                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.
              </p>
            </dd>
            
            <dt>
	      <a class="anchor" id="desc_BlastenImanuvilovYamamoto2015"></a>
              <a href="#BlastenImanuvilovYamamoto2015">
                <span class="ref-authors">Blåsten, Imanuvilov, Yamamoto,</span>
                <span class="ref-title">Stability and uniqueness for a
                  two-dimensional inverse boundary value problem for
                  less regular potentials</span>
              </a>
            </dt>
            <dd>
              <p>
                This article combines methods from my PhD
                thesis <a class="referenceNbr"
                href="#Blasten2013">[2013]</a>  and a manuscript by
                Imanuvilov and Yamamoto <a class="referenceNbr"
                href="http://arxiv.org/abs/1208.3775">[2012]</a> 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.
              </p>
              <p>
                The change from Bukhgeim's
                method <a class="referenceNbr"
                href="http://dx.doi.org/10.1515/jiip.2008.002">[2008]</a>
                is a smarter choice for the first two terms in the
                Neumann series of the complex geometric optics
                solutions.
              </p>
            </dd>
            
            <dt>
	      <a class="anchor" id="desc_BlastenPaivarintaSylvester2014"></a>
              <a href="#BlastenPaivarintaSylvester2014">
                <span class="ref-authors">Blåsten, Päivärinta,
                  Sylvester,</span>
                <span class="ref-title">Corners Always Scatter</span>
              </a>
            </dt>
            <dd>
              <p>
                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&deg; sharp
                corner will always produce a non-trivial scattered
                wave. This implies that transmission eigenvalues are
                not the same as <em>non-scattering energies</em>,
                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.
              </p>
              <p>
                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 <a class="referenceNbr"
                href="http://dx.doi.org/10.1007/BF02786703">[1976]</a>
                or Sylvester and Uhlmann <a class="referenceNbr"
                href="http://dx.doi.org/10.2307/1971291">[1987]</a>.
              </p>
            </dd>
            
            <dt>
	      <a class="anchor" id="desc_BlastenPaivarinta2013"></a>
              <a href="#BlastenPaivarinta2013">
                <span class="ref-authors">Blåsten, Päivärinta,</span>
                <span class="ref-title">Completeness of generalized
                  transmission eigenstates</span>
              </a>
            </dt>
            <dd>
              <p>
                This article answers a question posed by Cakoni,
                Gintides and
                Haddar <a href="http://dx.doi.org/10.1137/090769338"
                class="referenceNbr">[2010]</a>:
                <blockquote>
                  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.
                </blockquote>
              </p>
              <p>
                We proved that the eigensystems associated with all
                the transmission eigenvalues form a complete set. We
                used the consept of <em>parameter-ellipticity</em> 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.
              </p>
            </dd>
            
            <dt>
	      <a class="anchor" id="desc_Blasten2013"></a>
              <a href="#Blasten2013">
                <span class="ref-authors">Blåsten,</span>
                <span class="ref-title">On the Gel’Fand-Calderón
                  inverse problem in two dimensions</span>
              </a>
            </dt>
            <dd>
              <p>
                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 <a class="referenceNbr"
                href="http://dx.doi.org/10.1515/jiip.2008.002">[2008]</a>.
              </p>
              <p>
                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.
              </p>
            </dd>
            
            <dt>
	      <a class="anchor" id="desc_Blasten2010"></a>
              <a href="#Blasten2010">
                <span class="ref-authors">Blåsten,</span>
                <span class="ref-title">The Inverse Problem of the
                  Schrödinger Equation in the Plane: A Dissection of
                  Bukhgeim's Result</span>
              </a>
            </dt>
            <dd>
              <p>
                In this licentiate thesis I studied and explained
                Bukhgeim's solution to the Gel'Fand-Calderón problem
                in two dimensions <a class="referenceNbr"
                href="http://dx.doi.org/10.1515/jiip.2008.002">[2008]</a>.
              </p>
              <p>
                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.
              </p>
            </dd>
            
            <dt>
	      <a class="anchor" id="desc_Blasten2008"></a>
              <a href="#Blasten2008">
                <span class="ref-authors">Blåsten,</span>
                <span class="ref-title">Distribuutioteorian alkeet ja
                  sovellutuksia</span>
              </a>
            </dt>
            <dd>
              <p>
                An English translation of the title would be <q>An
                introduction to the theory of distributions and its
                applications</q>. 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.
              </p>
            </dd>
          </dl>
        </section>


        <section>
	  <a class="anchor" id="talks"></a>
          <p>
            <h2>Conference and Seminar Talks</h2>
            <ol reversed>
              <li>
                <span class="ref-title">Inverse problems with one
                measurement,</span>
                <span class="ref-journal">Inverse Days 2018; Aalto
                University, Finland,</span>
                <span class="ref-year">12 December 2018.</span>
                <div class="ref-links">
                  <a href="https://math.aalto.fi/en/research/applied/inverseproblems/inverse_days-2018/">Event</a>
                  <a href="pdf/blasten_talk_2018_helsinki.pdf">Talk</a>
                </div>
              </li>
              <li>
                <span class="ref-title">Inverse problems with one
                measurement,</span>
                <span class="ref-journal">Inverse problems, PDE and
                geometry; University of Jyväskylä, Finland,</span>
                <span class="ref-year">22 August 2018.</span>
                <div class="ref-links">
                  <a href="https://www.jyu.fi/science/en/maths/research/inverse-problems/inverse-problems-pde-and-geometry-2018">Event</a>
                  <a href="pdf/blasten_talk_2018_jyvaskyla.pdf">Talk</a>
                </div>
              </li>
              <li>
                <span class="ref-title">Inverse problems with one
		  measurement,</span>
                <span class="ref-journal">The 9th International
                  Conference on Inverse Problems and Related Topics;
                  National University of Singapore, Singapore,</span>
                <span class="ref-year">15 August 2018.</span>
                <div class="ref-links">
                  <a href="https://www.ece.nus.edu.sg/stfpage/elechenx/Conference/index.htm">Event</a>
                  <a href="pdf/blasten_talk_2018_singapore.pdf">Talk</a>
                </div>
              </li>
              <li>
                <span class="ref-title">Applications of corner
                  scattering: intrinsic properties of transmission
                  eigenfunctions and single wave probing,</span>
                <span class="ref-journal">School of Mathematical
                  Sciences, Fudan University, China,</span>
                <span class="ref-year">5 December 2017.</span>
                <div class="ref-links">
                  <a href="http://math.fudan.edu.cn/show.aspx?info_lb=765&flag=527&info_id=5565">Event</a>
                  <a href="pdf/blasten_talk_2017_shanghai.pdf">Talk</a>
                </div>
              </li>
              <li>
                <span class="ref-title">Planar inverse boundary value
                  problem for <i>L<sup>p</sup></i> potentials
                  with <i>p&gt;4/3</i>,</span>
                <span class="ref-journal">Analysis Seminar, Department
                  of Mathematics and Statistics; University of
                  Jyväskylä, Finland,</span>
                <span class="ref-year">23 August 2017.</span>
              </li>
              <li>
                <span class="ref-title">Inverse backscattering with
                  point-source waves,</span>
                <span class="ref-journal">Inverse Problems Seminar,
                  Department of Mathematics and Statistics; University
                  of Helsinki, Finland,</span>
                <span class="ref-year">17 August 2017.</span>
                <div class="ref-links">
                  <a href="pdf/blasten_talk_2017_UH.pdf">Notes</a>
                </div>
              </li>
              <li>
                <span class="ref-title">Corners always scatter &mdash;
                  quantitative results,</span>
                <span class="ref-journal">Applied Inverse Problems
                  Conference 2017; Zhejiang University, Hangzhou,
                  China,</span>
                <span class="ref-year">31 May 2017.</span>
                <div class="ref-links">
                  <a href="http://aip2017.csp.escience.cn/dct/page/70004">Programme</a>
                  <a href="http://aip2017.csp.escience.cn/dct/page/1">Event</a>
                  <a href="pdf/blasten_talk_2017_AIP.pdf">Talk</a>
                </div>
              </li>
              <li>
                <span class="ref-title">Transmission eigenfunction
                  localization,</span>
                <span class="ref-journal">Annual meeting of the Hong
                  Kong Mathematical Society; The Hong Kong University
                  of Science and Technology, Hong Kong,</span>
                <span class="ref-year">20 May 2017.</span>
                <div class="ref-links">
                  <a href="pdf/blasten_talk_2017_HKMS.pdf">Talk</a>
                </div>
              </li>
              <li>
                <span class="ref-title">Topics in Corner Scattering:
                  Non-Scattering Waves, Potential Probing with a
                  Single Incident Wave, and the Interior Transmission
                  Problem,</span>
                <span class="ref-journal">NCTS PDE and Analysis
                  Seminar; National Center for Theoretical Sciences,
                  National Taiwan University, Taipei, Taiwan,</span>
                <span class="ref-year">9 Marh 2017.</span>
                <div class="ref-links">
                  <a href="pdf/blasten_talk_2017_taipei.pdf">Talk</a>
                </div>
              </li>
              <li>
                <span class="ref-title">Inverse scattering using a
                  single incident wave,</span>
                <span class="ref-journal">2<sup>nd</sup> East Asia
                  Section of IPIA, Young Scholars Symposium; National
                  Center for Theoretical Sciences, National Taiwan
                  University, Taipei, Taiwan,</span>
                <span class="ref-year">5 November 2016.</span>
                <div class="ref-links">
                  <a href="pdf/taipei2016_programme.pdf">Programme</a>
                  <a href="pdf/blasten_talk_2016_taipei.pdf">Talk</a>
                </div>
              </li>
              <li>
                <span class="ref-title">Non-scattering energies, new
                  resolvent estimates and other projects,</span>
                <span class="ref-journal">1<sup>st</sup> East Asia
                  Symposium of IPIA; South University of Science and
                  Technology, Shenzhen, China,</span>
                <span class="ref-year">29 February 2016.</span>
                <div class="ref-links">
                  <a href="pdf/shenzhen2016_programme.pdf">Programme</a>
                  <a href="pdf/blasten_talk_2016_shenzhen.pdf">Talk</a>
                </div>
              </li>
              <li>
                <span class="ref-title">Non-scattering energies and
                  interior transmission eigenvalues,</span>
                <span class="ref-journal">Workshop on Inverse Problems
                  and Related Topics; Zhejiang University, Hangzhou,
                  China,</span>
                <span class="ref-year">9 December 2015.</span>
                <div class="ref-links">
                  <a href="pdf/hangzhou2015_programme.pdf">Programme</a>
                  <a href="pdf/blasten_talk_2015_hangzhou.pdf">Talk</a>
                </div>
              </li>
              <li>
                <span class="ref-title">A new viewpoint to scattering
                  theory à la Hörmander,</span>
                <span class="ref-journal">Spectral and Analytic
                  Inverse Problems, Thematic Programme on Inverse
                  Problems; Institut Henri Poincaré, Paris,
                  France,</span>
                <span class="ref-year">4 May 2015.</span>
                <div class="ref-links">
                  <a href="http://www.ihp.fr/en/CEB/T2-2015">Programme</a>
                  <a href="http://www.ihp.fr/en/CEB/T2-2015/workshop1">Event</a>
                  <a href="pdf/blasten_talk_2015_paris.pdf">Talk</a>
                </div>
              </li>
              <li>
                <span class="ref-title">Solving the Inverse Problem
                  for the 2D Schrödinger Equation with
                  Lp-potential,</span>
                <span class="ref-journal">17<sup>th</sup> Annual
                  Workshop on Applications and generalizations of
                  complex analysis; University of Aveiro, Aveiro,
                  Portugal,</span>
                <span class="ref-year">21 March 2015.</span>
                <div class="ref-links">
                  <a href="http://sweet.ua.pt/pceres/complex2015/Webpage/Workshop.html">Event</a>
                  <a href="pdf/blasten_talk_2015_aveiro.pdf">Talk</a>
                </div>
              </li>
              <li>
                <span class="ref-title">Solving the Inverse Problem
                  for the 2D Schrödinger Equation with
                  Lp-potential,</span>
                <span class="ref-journal">The 10<sup>th</sup> AIMS
                  Conference on Dynamical Systems, Differential
                  Equations and Applications; Instituto de Ciencias
                  Matemáticas (ICMAT) and the Universidad Autónoma de
                  Madrid (UAM), Madrid, Spain,</span>
                <span class="ref-year">9 July 2014.</span>
                <div class="ref-links">
                  <a href="http://aimsciences.org/conferences/2014/">Event</a>
                  <a href="pdf/blasten_talk_2014_madrid.pdf">Talk</a>
                </div>
              </li>
              <li>
                <span class="ref-title">Completeness of the
                  generalized transmission eigenstates,</span>
                <span class="ref-journal">International Conference on
                  Novel Directions in Inverse Scattering; University
                  of Delaware, Delaware, USA,</span>
                <span class="ref-year">29 July 2013.</span>
                <div class="ref-links">
                  <a href="http://www.cmap.polytechnique.fr/~colton/">Event</a>
                  <a href="pdf/blasten_talk_2013_delaware.pdf">Talk</a>
                </div>
              </li>
            </ol>
          </p>
        </section>


      </section>
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