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The Foundations of Infinite-Dimensional Spectral Computations

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CATW03 - Computational complex analysis

Spectral computations in infinite dimensions are ubiquitous in the sciences and the problem of computing spectra is one of the most studied 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 in the self-adjoint case is spectral pollution in gaps of the essential spectrum).

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

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

1) Computing spectra of closed operators in the Attouch-Wets topology (local uniform convergence of closed sets). This algorithm uses estimates of the norm of the resolvent operator and a local minimisation scheme. As 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) Computing (projection-valued) spectral measures of self-adjoint operators as given 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 discuss local convergence properties and extensions to computing spectral decompositions (pure point, absolutely continuous and singular continuous parts).

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 chemistry and physics.

This talk is part of the Isaac Newton Institute Seminar Series series.

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