BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:Reduced order models for spectral domain inversion: embedding into the continuous problem and generation of internal data.* - Shari Moskow ( Drexel University) DTSTART:20191212T160000Z DTEND:20191212T163000Z UID:TALK135625@talks.cam.ac.uk CONTACT:INI IT DESCRIPTION:We generate data-driven reduced order models (ROMs) for invers ion of the one and two dimensional Schr\\"odinger equation in the spectral domain given boundary data at a few frequencies. The ROM is the Galerkin projection of the Schr\\"odinger operator onto the space spanned by soluti ons at these sample frequencies\, and corresponds to a rational interpolan t of the Neumann to Dirichlet map. The ROM matrix is in general full\, an d not good for extracting the potential. However\, using an orthogonal cha nge of basis via Lanczos iteration\, we can transform the ROM to a block t riadiagonal form from which it is easier to extract the unknown coefficien t. In one dimension\, the tridiagonal matrix corresponds to a three-point staggered finite-difference system for the Schr\\"odinger operator discret ized on a so-called spectrally matched grid which is almost independent o f the medium. In higher dimensions\, the orthogonalized basis functions pl ay the role of the grid steps. The orthogonalized basis functions are loca lized and also depend only very weakly on the medium\, and thus by embeddi ng into the continuous problem\, the reduced order model yields highly acc urate internal solutions. That is to say\, we can obtain\, just from bound ary data\, very good approximations of the solution of the Schr\\"odinger equation in the whole domain for a spectral interval that includes the sam ple frequencies. We present inversion experiments based on the internal so lutions in one and two dimensions.

* joint with L. Borcea\, V. D ruskin\, A. Mamonov\, M. Zaslavsky LOCATION:Seminar Room 1\, Newton Institute END:VEVENT END:VCALENDAR