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Numerical Linear Algebra and Solvability of PDEs


Speaker: Maciej Zworski, UC Berkeley
Location: 693 Kerr
Start time: Mon, Apr 23 2001, 4:10PM

One of the goals of Numerical Linear Algebra is the accurate computation of eigenvalues of matrices. This is quite successful in the symmetric case but the situation is different for non-self adjoint matrices. These arise in the discretisation of engineering or physical problems involving friction or dissipation, and it is an important problem to compute their eigenvalues. It has been observed long time ago that the obstruction to accurately computing eigenvalues of nonselfadjoint matrices is inherent in the problem, and cannot be circumvented by using more powerful computers. The basic idea is that any algorithm for locating the eigenvalues will also find some `false eigenvalues'. These false eigenvalues also explain one of the most surprising phenomena in linear PDEs, namely the fact (discovered by Hans Lewy in 1957, in Berkeley) that one cannot always locally solve the PDE $ P u = f $. Local solvability is always possible if $ P $ is self-adjoint or has constant coefficients, but non-self-adjointness can destroy that property: Lewy's example was a simple vector-field with complex, non-constant coefficients arising in the study of several complex variables. Almost immediately after that discovery, H\"ormander provided an explanation of Lewy's example showing that {\em almost all} non-self-adjoint operators are not locally solvable. That was done by considering the essentially dual problem of existence of {\em non-propagating singularities}. I will give an elementary quantum mechanical interpretation of these issues of local solvability and non-propagation of singularities in terms of creation and annihilation operators. Finally, I will explain how the non-propagating singularities are the source of (at least some of) the computational problems in finding eigenvalues of non-self adjoint matrices arising in discretisation of high energy or semi-classical operators.