Recovery of a Smooth Metric via Wave Field and Coordinate Transformation Reconstruction

In this paper, we study the inverse boundary value problem for the wave equation with a view towards an explicit reconstruction procedure. We consider both the anisotropic problem where the unknown is a general Riemannian metric smoothly varying in a domain, and the isotropic problem where the metric is conformal to the Euclidean metric. Our objective in both cases is to construct the metric, using either the Neumann-to-Dirichlet (N-to-D) map or Dirichlet-to-Neumann (D-to-N) map as the data. In the anisotropic case we construct the metric in the boundary normal (or semi-geodesic) coordinates via reconstruction of the wave field in the interior of the domain. In the isotropic case we can go further and construct the wave speed in the Euclidean coordinates via reconstruction of the coordinate transformation from the boundary normal coordinates to the Euclidean coordinates. Both cases utilize a variant of the Boundary Control method, and work by probing the interior using special boundary sources. We provide a computational experiment to demonstrate our procedure in the isotropic case with N-to-D data.Comment: 24 pages, 6 figure

On the Construction of Virtual Interior Point Source Travel Time Distances from the Hyperbolic Neumann-to-Dirichlet Map

We introduce a new algorithm to construct travel time distances between a point in the interior of a Riemannian manifold and points on the boundary of the manifold, and describe a numerical implementation of the algorithm. It is known that the travel time distances for all interior points determine the Riemannian manifold in a stable manner. We do not assume that there are sources or receivers in the interior, and use the hyperbolic Neumann-to-Dirichlet map, or its restriction, as our data. Our algorithm is a variant of the Boundary Control method, and to our knowledge, this is the first numerical implementation of the method in a geometric setting

Techniques for Reconstructing a Riemannian Metric Via the Boundary Control Method

In this dissertation, we consider some new techniques related to the solution of the inverse boundary value problem for the wave equation with partial boundary data. Most results are formulated in a geometric setting, where waves propagate in the interior of a smooth manifold with smooth boundary M, and the wave speed is modelled by an unknown Riemannian metric g. For data, we focus mostly on using the Neumann-to-Dirichlet (N-to-D) map with sources and receivers restricted to a measurement set Γ ⊂ ∂M. The goal of the inverse problem, in this setting, is to use these wave boundary measurements to recover the geometry of (M, g) near the measurement set. We note that this geometric perspective accomodates, as special cases, both the scalar acoustic wave equation and elliptically anisotropic wave speeds. We consider three problems. In the first problem, we provide a technique to use the N-to-D map to construct the travel times between interior points with known semi-geodesic coordinates and boundary points belonging to Γ. Such travel times can be used to reconstruct the metric in semi-geodesic coordinates using one of several existing techniques, so this procedure can be viewed as providing a data processing step for a metric reconstruction procedure. In the second problem, we consider a redatuming procedure, where we use data on the boundary and known near-boundary geometry to synthesize wave measurements in this known near-boundary region. This allows us to construct a map which plays a similar role to the N-to-D map, but for interior sources and interior measurements. Our motivation for this procedure is that it can serve as a data propagation step for a layer stripping reconstruction method, in which one first reconstructs the metric near the boundary and then propagates data into this region to serve as data for an interior reconstruction step. In the third problem, we restrict attention to the case where M is a domain in Rn, and consider two related procedures to use the N-to-D map or Dirichlet-to-Neumann (D-to-N) map to directly reconstruct the metric. In the anisotropic case, we construct the metric in semi-geodesic coordinates via reconstruction of the wave field in the interior of the domain. In the isotropic case, we can go further and construct the wave speed in the Euclidean coordinates via reconstruction of the coordinate transformation from the boundary normal coordinates to the Euclidean coordinates. In addition to providing constructive procedures, we analyze the stability of some steps from these procedures. In particular we consider the stability of the redatuming procedure and the stability of the metric reconstruction procedure from internal data (for the third problem). Moreover, we provide computational experiments to demonstrate our three main procedures

Unique Determination of Sound Speeds for Coupled Systems of Semi-linear Wave Equations

We consider coupled systems of semi-linear wave equations with different sound speeds on a finite time interval $[0,T]$ and a bounded Lipschitz domain $\Omega$ in $\mathbb{R}^3$, with boundary $\partial\Omega$. We show the coupled systems are well posed for variable coefficient sounds speeds and short times. Under the assumption of small initial data, we prove the source to solutions map on $[0,T]\times\partial\Omega$ associated with the nonlinear problem is sufficient to determine the source-to-solution map for the linear problem. We can then reconstruct the sound speeds in $\Omega$ for the coupled nonlinear wave equations under certain geometric assumptions. In the case of the full source to solution map in $\Omega\times[0,T]$ this reconstruction could also be accomplished under fewer geometric assumptions.Comment: minor update

Inverse Problems with Microlocal Observations

Each of the articles of this dissertation is related to a specific measurement setup for determining the interior structure of a target object using indirect observations. We study microlocal singularities such as peaks and non-regularities, and use their dynamical properties to solve the associated inverse problems. The works combine methods of microlocal analysis, differential geometry and techniques of geometric inverse problems. The first article (I) is about the inverse problem of recovering the law of a random potential from empirical correlations in the peak scattering of several plane waves interacting simultaneously with the potential. Observing peaks in the total wave at a distance makes it possible to recover correlations between the X-ray transforms of the potential provided that it is almost surely H^2-valued random variable and supported in some fixed compact set. This is then applied to show that the law V^*P is uniquely determined by the measurements in a reasonable class of random potentials. In the second article (II), the relativistic Boltzmann equation with a source is studied. We investigate the inverse problem of recovering the corresponding Lorentzian tensor of a system behaving according to the Boltzmann equation by observing light in a confined, possibly small, area V in space and time. We show that it is possible to uniquely recover the underlying Lorentzian metric in causally attainable regions outside V from source-to-solution data defined on sources with support in V. The work is based on the microlocal techniques developed for non-linear waves by Kurylev, Lassas and Uhlmann. In comparison to their result, we were able to recover the actual metric instead of merely the conformal class. In the third article (III), Cherenkov radiation in anisotropic materials with scalar wave impedance is shown to have a microlocal description as a propagation of singularities which can be used as a method to recover the Riemannian metric determining the permittivity and permeability tensors, and hence the fundamental electromagnetic properties of the medium. The work is most likely the first time that Cherenkov scattering has been applied to solve a geometric inverse problem. It strongly suggests that the phenomenon can be used to develop new modalities for imaging.Tämä väitöskirja koskee matemaattisia malleja, jotka kuvaavat fysikaalisia mittausasetelmia, joissa tavoitteena on selvittää tarkasteltavan tutkimuskohteen sisäinen rakenne sen ulkopuolelta käsin. Kohteeseen kohdistetaan energiaa aaltojen tai hiukkasten muodossa samalla mitaten prosessissa muodostuvaa sirontaa. Väitöskirja koostuu johdanto-osion lisäksi kolmesta tieteellisestä julkaisusta. Näissä töissä sovelletaan mikrolokaalia analyysiä ja geometrisiä menetelmiä inversio ongelmiin. Keskiössä on ns. singulariteettien tutkiminen havaintoaineistona. Ensimmäinen julkaisu (I) liittyy fysikaaliseen asetelmaan, missä tuntematon, ergodisesti kehittyvä satunnaispotentiaali vuorovaikuttaa eri suunnista samanaikaisesti lähetettyjen tasoaaltojen kanssa. Mittauksena työssä käytetään sironneen kokonaisaallon pääsingulariteettien amplitudeja. Asetelma vastaa jokseenkin useiden röntgenkuvien ottamista samanaikaisesti eri kulmista. Tutkimme tällä tavoin kerätyn statistisen mittausaineiston mallia ja osoitamme, että suurelle luokalle satunnaispotentiaaleja niiden funktioarvoiset todennäköisyysjakaumat voidaan yksikäsitteisesti selvittää mittauksista. Toinen julkaisu (II) koskee kineettisen teorian epälineaarista mallia, jossa suuri määrä hiukkasia vuorovaikuttaa törmäilemällä toisiinsa yleisen suhteellisuusteorian kontekstissa. Matemaattisesti kyse on relativistisestä Boltzmannin yhtälöstä, jossa on lisäksi ylimääräinen, kontrolloitavissa oleva hiukkaslähde. Mallin epälineaarisuutta aktiivisesti hyödyntäen osoitamme, että hiukkasten törmäyksissä muodostuvia fotoneja tarkastelemalla on mahdollista selvittää globaalisti hyperbolisen aika-avaruuden rakenne alueissa, jotka ovat kausaalisen vuorovaikutuksen piirissä. Kolmas julkaisu (III) käsittelee matemaattisesti Cherenkovin säteilyä epähomogeenisessa, anisotrooppisessa väliaineessa. Kyseistä säteilyä muodostuu, kun varattu hiukkanen liikkuu aineessa nopeampaa kuin väliaineen valon nopeus. Ilmiö on ikään kuin sähkömagneettinen analogia äänivallin rikkoontumiselle. Työssä osoitetaan, että Cherenkovin säteilyä tarkastelemalla on tietyin edellytyksin mahdollista selvittää väliaineen sähkömagneettiset ominaisuudet. Tulos viittaa siihen, että Cherenkovin säteilyä voi käyttää anisotrooppisen aineen tarkkaan kuvantamiseen