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The document discusses quasi-Newton methods for solving nonlinear systems of equations and for unconstrained optimization problems. Quasi-Newton methods approximate Newton's method by iteratively updating an estimate of the inverse Hessian matrix without having to explicitly compute second derivatives. This avoids expensive evaluations of the Hessian and allows the methods to converge superlinearly. Examples of quasi-Newton methods discussed include Broyden's method and the BFGS method.
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Qualifier
- 1. Qualifier Exam in HPC February 10 th , 2010
- 2. Quasi-Newton methods Alexandru Cioaca
Editor's Notes
- Ball and beam, Navier-Stokes, stability of an airplane wrt to pilot commands. nonlinear systems, number of variables = number of equations The solution means determining some values of variables/parameters that strictly satisfy the given relationships (equations). Impossible to solve them by hand even if they are homogeneous.
- Direct solve – reuse the factorization (quasi-implicit Newton, chord method)
- Computational fluid dynamics – millions of variables (chesapeake bay)
- Examples of cost functions Talk about differentiability
- Convergence rates - LINEAR We want Bk to mimic the behavior of newton => symmetric positive definite (at least in a neighborhood)
- Trust regions determine a region around the current iterate where the approximate model of the objective function is accurate. The step is taken to be the minimizer on the trust region.
- Explain symmetry, positive definiteness Limited memory quasi-newton
- Z_k can be taken to be the