Applied Mathematics Colloquium

Monday October 24, 2011 4:15 PM

Inverse Problems for Non-Self-Adjoint Operators

Speaker: Mark Embree, Computational and Applied Mathematics, Rice University
Location: Annenberg 105
Given a set of eigenvalues, can one accurately reconstruct a matrix or operator that produced them? When the eigenvalues are real and the operator is self-adjoint with fixed structure (Sturm-Liouville operators and their discretizations, Jacobi matrices), the answer is yes, and a variety of algorithms are available to effect the reconstruction.

Non-self-adjoint problems pose considerable practical and theoretical challenge, but are important for applications such as the design of damping for physical structures. Non-orthogonal eigenfunctions and consequent eigenvalue sensitivity complicate analysis, while damping can hamper eigenvalue measurement.

This talk will describe progress on two classes of inverse problems for non-self-adjoint operators: recovery of a viscous damping parameter in a vibrating string, and the inverse field of values problem for Rayleigh-Ritz eigenvalue estimates. The latter reveals that the Cauchy interlacing observed for Hermitian matrices has a natural analogue that holds even for non-diagonalizable matrices.

[Joint work with Russell Carden and Steve Cox]
Series Applied Mathematics Colloquium Series

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