Information Geometry For Nonlinear Least-Squares Data Fitting And Calculation Of The Superconducting Superheating Field
This thesis consist of two parts, each of which consist of two chapters. First we explore the information geometric properties of least squares data fitting, particularly for so-called "sloppy" models. Second we describe a calculation of the superconducting superheating field, relevant for advancing gradients in particle accelerator resonance cavities. Parameter estimation by nonlinear least squares minimization is a ubiquitous problem that has an elegant geometric interpretation: all possible parameter values induce a manifold embedded within the space of data. The minimization problem is then to find the point on the manifold closest to the data. By interpreting nonlinear models as a generalized interpolation scheme, we find that the manifolds of many models, known as sloppy models, have boundaries and that their widths form a hierarchy. We describe this universal structure as a hyper-ribbon. The hyper-ribbon structure explains many of the difficulties associated with fitting nonlinear models and suggests improvements to standard algorithms. We add a "geodesic acceleration" correction to the standard Levenberg-Marquardt algorithm and observe a dramatic increase in success rate and convergence speed on many fitting problems. We study the superheating field of a bulk superconductor within the GinzburgLandau, which is valid only near Tc , and Eilenberger theory, which is valid at all temperatures. We calculate as functions of both the Ginzburg-Landau parameter [kappa] and reduced temperature t = T /Tc the superheating field Hsh and the critical momentum kc describing the wavelength of the unstable perturbations to flux penetration. By mapping the two-dimensional linear stability theory into a one-dimensional eigenfunction problem for a linear operator, we solve the problem numerically. Within the Ginzburg-Landau theory, We demonstrate agreement between the numerics and analytics, and show convergence to the known results at both small and large [kappa]. Within the Eilenberger theory we demonstrate agreement with the results of Ginzburg-Landau theory near Tc , but find discrepancies with the temperature-dependent results for large [kappa]. We speculate that this discrepancy is due to a lack of convergence at low temperatures due to small length scales of the perturbations analogous to the small length scales associated with the vortex cores of the mixed state.
nonlinear regression; superconductivity; information geometry
Sethna, James Patarasp
Hoffstaetter, Georg Heinz; Teukolsky, Saul A
Ph. D., Physics
Doctor of Philosophy
dissertation or thesis