Brain Science and Pop Code Seminar
Recent papers of Jon's
Touryan J. and Dan Y. (2001) Analysis of sensory coding with
Touryan J., Lau B., Dan Y. (2002) Isolation of relevant visual |
Scientific Computing Seminar
Abstract: The construction of a bifurcation diagram requires the computation of steady states and of their stability. The most effective methods for carrying out these calculations are Newton's method and the inverse power/Arnoldi method, respectively, both of which require solving linear systems involving the Jacobian matrix. For physical phenomena described by elliptic or parabolic partial differential equations in two or three spatial dimensions, the Jacobian matrix is both too large to be inverted directly and too poorly conditioned to be inverted iteratively. The poor conditioning, due to the Laplacian operator, is manifested as stiffness when integrating the evolution and has led to the widespread use of implicit time-stepping methods. This suggests using the Laplacian inversion already present in time-stepping codes (as Poisson, Stokes, or Helmholtz solvers) as an effective preconditioner. We incorporate preconditioning by the inverse Laplacian into Newton's method and the inverse power/Arnoldi method, using BiCGSTAB to solve the resulting linear systems. Existing time-stepping codes can be easily modified to carry out either of these tasks. We describe two applications: spherical Couette flow described by the Navier-Stokes equations and Bose-Einstein condensation described by the nonlinear Schrodinger equation.
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