BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:The Foundations of Infinite-Dimensional Spectral Computations - Ma tthew Colbrook (University of Cambridge) DTSTART:20191209T150000Z DTEND:20191209T153000Z UID:TALK135448@talks.cam.ac.uk CONTACT:INI IT DESCRIPTION:Spectral computations in infinite dimensions are ubiquitous in the sciences and the problem of computing spectra is one of the most stud ied areas of computational mathematics over the last half-century. However \, such computations are infamously difficult\, since standard approaches do not\, in general\, produce correct solutions (the most famous problem i n the self-adjoint case is spectral pollution in gaps of the essential spe ctrum).

The goal of this talk is to introduce classes of resolven t based algorithms that compute spectral properties of operators on separa ble Hilbert spaces. As well as solving computational problems for the firs t time\, these algorithms are proven to be optimal\, and the computational problems themselves can be classed in a hierarchy (the SCI hierarchy) wit h ramifications beyond spectral theory.

For concreteness\, I shall focus on two problems for a very general class of operators on $l^2(\\mat hbb{N})$\, where algorithms access the matrix elements of the operator:

1) Computing spectra of closed operators in the Attouch-Wets topolog y (local uniform convergence of closed sets). This algorithm uses estimate s of the norm of the resolvent operator and a local minimisation scheme. A s well as solving the long-standing computational spectral problem\, this algorithm computes spectra with error control. It can also be extended to partial differential operators with coefficients of locally bounded total variation with algorithms point sampling the coefficients.

2) Comp uting (projection-valued) spectral measures of self-adjoint operators as g iven by the spectral theorem. This algorithm uses computation of the full resolvent operator (with asymptotic error control) to compute convolutions of rational kernels with the measure before taking a limit. I shall discu ss local convergence properties and extensions to computing spectral decom positions (pure point\, absolutely continuous and singular continuous part s).

Finally\, these algorithms are embarrassingly parallelisable. Numerical examples will be given\, demonstrating efficiency\, and tackling difficult problems taken from mathematics and other fields such as chemis try and physics. LOCATION:Seminar Room 1\, Newton Institute END:VEVENT END:VCALENDAR