Search results for "inversio-ongelmat"
showing 6 items of 76 documents
Increasing stability in the linearized inverse Schrödinger potential problem with power type nonlinearities
2022
We consider increasing stability in the inverse Schr\"{o}dinger potential problem with power type nonlinearities at a large wavenumber. Two linearization approaches, with respect to small boundary data and small potential function, are proposed and their performance on the inverse Schr\"{o}dinger potential problem is investigated. It can be observed that higher order linearization for small boundary data can provide an increasing stability for an arbitrary power type nonlinearity term if the wavenumber is chosen large. Meanwhile, linearization with respect to the potential function leads to increasing stability for a quadratic nonlinearity term, which highlights the advantage of nonlinearit…
On some partial data Calderón type problems with mixed boundary conditions
2021
In this article we consider the simultaneous recovery of bulk and boundary potentials in (degenerate) elliptic equations modelling (degenerate) conducting media with inaccessible boundaries. This connects local and nonlocal Calderón type problems. We prove two main results on these type of problems: On the one hand, we derive simultaneous bulk and boundary Runge approximation results. Building on these, we deduce uniqueness for localized bulk and boundary potentials. On the other hand, we construct a family of CGO solutions associated with the corresponding equations. These allow us to deduce uniqueness results for arbitrary bounded, not necessarily localized bulk and boundary potentials. T…
Partial Data Problems and Unique Continuation in Scalar and Vector Field Tomography
2022
AbstractWe prove that if P(D) is some constant coefficient partial differential operator and f is a scalar field such that P(D)f vanishes in a given open set, then the integrals of f over all lines intersecting that open set determine the scalar field uniquely everywhere. This is done by proving a unique continuation property of fractional Laplacians which implies uniqueness for the partial data problem. We also apply our results to partial data problems of vector fields.
Applications of Microlocal Analysis in Inverse Problems
2020
This note reviews certain classical applications of microlocal analysis in inverse problems. The text is based on lecture notes for a postgraduate level minicourse on applications of microlocal analysis in inverse problems, given in Helsinki and Shanghai in June 2019.
Kinemaattinen inversio-ongelma pallosymmetrisellä monistolla
2017
Tutkielman pääaiheena on maanjäristysaaltoihin ja Maan sisärakenteen tutkimiseen liittyvä käänteinen kinemaattinen ongelma. Maapalloa mallinnetaan kolmiulotteisella kompaktilla reunallisella monistolla \(\bar{B}^3(0, R)\), jonka säde normitetaan ykköseksi \(R=1\). Aaltorintamat kulkevat pitkin geodeeseja, jotka sijaitsevat kokonaan avoimessa pallossa \(B^3(0, 1)\) lukuun ottamatta päätepisteitä, jotka ovat reunalla \(S^2(0, 1)\). Symmetrioiden nojalla tarkastelu voidaan siirtää tasoon \(\mathbb{R}^2\), jossa riittää tutkia kiekon \(\bar{B}^2(0, 1)\) geodeeseja. Äänennopeus \(v=v(r)\) oletetaan isotrooppiseksi ja aidosti positiiviseksi \(C^{1,1}([0, 1])\)-funktioksi, jolle \(v^{\prime}(0)=0\…