Speaker
Description
For nonsymmetric operators, GMRES convergence is governed not only by the eigenvalues but also by the field of values and the resolvent norm. Non-normality amplifies the resolvent $||(zI - A)^{-1}||$ far from the spectrum, and the spectral geometry of the numerical range $W(A)$ determines how rapidly GMRES residual polynomials can decrease. The intrinsic information dimension $K_\infty$, a problem-dependent quantity measuring the spectral information a Krylov process can resolve above the floating-point noise floor, provides an effective subspace dimension for non-normal operators.
This talk presents an $s$-step Newton-Leja GMRES method that addresses non-normality directly. The Newton-Leja polynomial basis generates the Krylov subspace with conditioning controlled by the spectral geometry of $W(A)$, achieving constant memory through the s-step recurrence. NL--GMRES minimises the residual with one global reduction per MGS orthogonalisation step. A polar preconditioner based on the Newton-Schulz iteration applied per block to the basis vectors restores orthogonality without forming the Gram matrix, counteracting the basis degradation caused by non-normality. Together, these components achieve constant memory and bounded synchronisation cost, with convergence governed by the field of values and bounded by $K_\infty$.
Numerical experiments on non-normal convection-diffusion operators illustrate the role of spectral geometry in determining both convergence and basis conditioning.