Lecturer: Jed Brown, PhD student, VAW, ETH Zurich

PDF version. Updates will be posted here.

Update 2009-01-27: slides available for lecture 2

Update 2009-01-29: slides available for lecture 3

Update 2009-01-29: References

Update 2009-02-05: slides available for lecture 5

Time and place

• Five lectures: Tuesday and Thursday, 13:00 to 14:00
• Chapman building, room 106

What is a scalable solver?

• A scalable solver can solve N equations in N unknowns with O(N) work.
• Newton-Krylov methods offer:
  • Quadratic convergence on the nonlinearity
  • Parallel scalability and mesh-independence for the linear solve

Is it time to look at your solver?

• Is your solver using significantly more time or memory than the physics?
• Is your time stepping limited by stability?
• Are you putting loops around the analysis?

Proposed schedule

• Jan 22: Nonlinear systems

  • Motivation for coupled implicit methods
  • Scalability
  • Newton's method for large systems
  • Globalization

• Jan 27: Linear solvers slides (pdf)

  • Limitations of direct methods
  • Krylov methods
  • Representation of matrices and Jacobian-free methods
  • Preconditioners for simple definite problems
  • Parallel scalability

• Jan 29: Constraints and coupling slides (pdf)

  • Preconditioners for indefinite problems
  • Preconditioners for multi-physics

• Feb 3: Parallel software day: PETSc

  • Generic solver components
  • Physics-based preconditioners
  • Working with legacy code

• Feb 5: Higher order finite elements slides (pdf)

  • High-order elements at the cost of low-order elements
  • Exploiting the memory hierarchy and tensor-product operations

Schedule and content are flexible, let me know if you have requests.

Have a look at my AGU poster for a bit of a preview.

References

• Kelley. Solving nonlinear equations with Newton's method, 2003.
• Benzi, Golub, Liesen. Numerical solution of saddle point problems, Acta Numerica, 2005.
• Elman et. al. A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations, JCP, 2008.
• Knoll, Keyes. Jacobian-free Newton-Krylov methods: a survey of approaches and applications, JCP, 2004.

The Knoll and Keyes paper is especially recommended. Please email me if you need a copy of these or anything else.