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Abstract
We study two inverse problems on a globally hyperbolic Lorentzian manifold (M, g). The problems are:
Passive observations in spacetime: consider observations in an open set . The light observation set corresponding to a point source at is the intersection of V and the lightcone emanating from the point q.
Let be an unknown open, relatively compact set. We show that under natural causality conditions, the family of light observation sets corresponding to point sources at points determine uniquely the conformal type of W.
Active measurements in spacetime: we develop a new method for inverse problems for nonlinear hyperbolic equations that utilizes the nonlinearity as a tool. This enables us to solve inverse problems for nonlinear equations for which the corresponding problems for linear equations are still unsolved. To illustrate this method, we solve an inverse problem for semilinear wave equations with quadratic nonlinearities. We assume that we are given the neighborhood V of the timelike path and the sourcetosolution operator that maps the source supported on V to the restriction of the solution of the wave equation to V. When M is 4dimensional, we show that these data determine the topological, differentiable, and conformal structures of the spacetime in the maximal set where waves can propagate from and return back to mu.
Original language  English 

Journal  Inventiones Mathematicae 
Volume  212 
Issue number  3 
Pages (fromto)  781857 
Number of pages  77 
ISSN  00209910 
DOIs  
Publication status  Published  Jun 2018 
MoE publication type  A1 Journal articlerefereed 
Fields of Science
 CUSP
 ELASTOGRAPHY
 OPERATORS
 PROGRESSING WAVES
 SCATTERING
 SEMILINEAR WAVEEQUATIONS
 SINGULAR SYMBOLS
 SPACETIMES
 TIME
 UNIQUE CONTINUATION
 111 Mathematics
 112 Statistics and probability
Projects
 1 Active

Centre of Excellence in Inverse Modelling and Imaging
Lassas, M., Siltanen, S., Piiroinen, P., Bubba, T., Balehowsky, T., Meaney, A., LatvaÄijö, S., Kujanpää, A., Liimatainen, T. J., Ylinen, L., Korpela, J., Helin, T., Santacesaria, M., Kyriakopoulou, D., Mach, M., Zhang, Z., Murthy, R., Dunlop, M., Rautio, S. I., Purisha, Z., Juvonen, M., Hyttinen, V., Enwald, J., Virtanen, H. J., SuomenrinneNordvik, A., Lehtonen, J., Ola, P., Ratti, L., Partio, H., LatvaÄijö, S., Enwald, J., Tyni, T. K., Agnelli, J. P., Silva de Moura, F., Oksanen, L., Heikkilä, T., Blåsten, E. & Lu, J.
01/01/2018 → 31/12/2025
Project: Research project