Causal discretizations of seismic imaging problems.
Alex Vladimirsky (Math, Cornell)
Seismic imaging involves solving many inverse problems. A common practical task is to reconstruct the subsurface structure based on the experimental data -- a signal/disturbance propagating from (one or several) source locations and the "times of signal arrival" observed at many receiver locations.
Algorithms for solving such inverse problems rely on repeatedly solving the corresponding "forward" problem (i.e., find the arrival times assuming the substructure and the slowness field are already known). The latter is accomplished by solving a PDE, and has to be done very efficiently. The exact type of that PDE depends on which arrivals need to be captured (the first vs. the most "energetic" vs. all of them) and whether we are computing the arrivals from all sources simultaneously (one-to-many vs. many-to-many problems).
Causality is the property needed to de-couple systems of non-linear equations, making it possible to solve them in a non-iterative (efficient) fashion. In this talk, I will informally describe the challenges of causal discretizations for several classes of PDEs arising in seismic imaging. No prior familiarity with this application area will be assumed. The discussed methods are a joint work with Sergey Fomel, Vladimir Bashkardin, and Siwei Li.