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SUMMARY:Inverse problems for non-linear partial differential equations and
  applications in tomography - Matti Lassas (University of Helsinki)
DTSTART:20230519T080000Z
DTEND:20230519T090000Z
UID:TALK198139@talks.cam.ac.uk
DESCRIPTION:In the talk we give an overview on how inverse problems can be
  used solved using non-linear interaction of the solutions. This method ca
 n be used for several different inverse problems for non-linear hyperbolic
  or elliptic equations and for Boltzmann equations. In this approach one d
 oes not consider the non-linearity as a troublesome perturbation term\, bu
 t as an effect that aids in solving the problem. Using it\, one can solve 
 inverse problems for non-linear equations for which the corresponding prob
 lem for linear equations is still unsolved.\nFor the hyperbolic equations\
 , we consider the non-linear wave equation $\\square_g u+u^m=f$ on a Loren
 tzian manifold $M\\times R$ and the source-to-solution map $\\Lambda_V:f\\
 to u|_V$ that maps a source $f$\, supported in an open domain $V\\subset M
 \\times R$\, to the restriction of $u$ in $V$. Under suitable conditions\,
  we show that the observations in $V$\, that is\, the map $\\Lambda_V$\, d
 etermine the metric $g$ in a larger domain which is the maximal domain whe
 re signals sent from $V$ can propagate and return back to $V$.\nWe apply n
 on-linear interaction of solutions of the linearized equation also to stud
 y non-linear elliptic equations. For example\, we consider $\\Delta_g u+qu
 ^m=0$ in $\\Omega\\subset R^n$ with the boundary condition $u|_{\\partial 
 \\Omega}=f$. For this equation we define the Dirichlet-to-Neumann map $\\L
 ambda_{\\partial \\Omega}:f\\to u|_V$. Using the high-order interaction of
  the solutions\, we consider various inverse problems for the metric $g$ a
 nd the potential $q$.&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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