Inverse scattering for a random potential

In this paper we consider an inverse problem for the $n$-dimensional random Schr\"{o}dinger equation $(\Delta-q+k^2)u = 0$. We study the scattering of plane waves in the presence of a potential $q$ which is assumed to be a Gaussian random function such that its covariance is described by a pseudodifferential operator. Our main result is as follows: given the backscattered far field, obtained from a single realization of the random potential $q$, we uniquely determine the principal symbol of the covariance operator of $q$. Especially, for $n=3$ this result is obtained for the full non-linear inverse backscattering problem. Finally, we present a physical scaling regime where the method is of practical importance.Comment: Previous version 48 pages; Current version 51 pages, 3 figures, several references have been adde

An adaptive finite element method in reconstruction of coefficients in Maxwell's equations from limited observations

We propose an adaptive finite element method for the solution of a coefficient inverse problem of simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions in the Maxwell's system using limited boundary observations of the electric field in 3D. We derive a posteriori error estimates in the Tikhonov functional to be minimized and in the regularized solution of this functional, as well as formulate corresponding adaptive algorithm. Our numerical experiments justify the efficiency of our a posteriori estimates and show significant improvement of the reconstructions obtained on locally adaptively refined meshes.Comment: Corrected typo

Induced magnetization in La0.7_{0.7}Sr0.3_{0.3}MnO3_3/BiFeO3_3 superlattices

Using polarized neutron reflectometry (PNR), we observe an induced magnetization of 75$\pm$ 25 kA/m at 10 K in a La$_{0.7}$Sr$_{0.3}$MnO$_3$ (LSMO)/BiFeO$_3$ superlattice extending from the interface through several atomic layers of the BiFeO$_3$ (BFO). The induced magnetization in BFO is explained by density functional theory, where the size of bandgap of BFO plays an important role. Considering a classical exchange field between the LSMO and BFO layers, we further show that magnetization is expected to extend throughout the BFO, which provides a theoretical explanation for the results of the neutron scattering experiment.Comment: 5 pages, 4 figures, with Supplemental Materials. To appear in Physical Review Letter

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

Depth Fields: Extending Light Field Techniques to Time-of-Flight Imaging

A variety of techniques such as light field, structured illumination, and time-of-flight (TOF) are commonly used for depth acquisition in consumer imaging, robotics and many other applications. Unfortunately, each technique suffers from its individual limitations preventing robust depth sensing. In this paper, we explore the strengths and weaknesses of combining light field and time-of-flight imaging, particularly the feasibility of an on-chip implementation as a single hybrid depth sensor. We refer to this combination as depth field imaging. Depth fields combine light field advantages such as synthetic aperture refocusing with TOF imaging advantages such as high depth resolution and coded signal processing to resolve multipath interference. We show applications including synthesizing virtual apertures for TOF imaging, improved depth mapping through partial and scattering occluders, and single frequency TOF phase unwrapping. Utilizing space, angle, and temporal coding, depth fields can improve depth sensing in the wild and generate new insights into the dimensions of light's plenoptic function.Comment: 9 pages, 8 figures, Accepted to 3DV 201

Fixed angle scattering: Recovery of singularities and its limitations

We prove that in dimension n ≥ 2 the main singularities of a complex potential q having a certain a priori regularity are contained in the Born approximation qθ constructed from fixed angle scattering data. Moreover, q-qθ can be up to one derivative more regular than q in the Sobolev scale. In fact, this result is optimal. We construct a family of compactly supported and radial potentials for which it is not possible to have more than one derivative gain. Also, these functions show that for n > 3, the maximum derivative gain can be very small for potentials in the Sobolev scale not having a certain a priori level of regularity which grows with the dimension.The author was supported by Spanish government predoctoral grant BES-2015- 074055 (project MTM2014-57769-C3-1-P