1. Ph.D. Thesis Numerical Solution of PDEs and Their Object-oriented Parallel Implementations
Xing Cai
October 26, 1998

2. Overview 10 publications covering: Numerical solution of PDEs
Object orientation & parallel computing
Additional work
October 26, 1998

3. Nonlinear water waves Free water surface Dynamic solution domain
3D velocity potential field
Divergence free, irrotational
Efficient numerical simulation
October 26, 1998

4. Mathematical model
October 26, 1998

5. Domain transformation
October 26, 1998

6. Solving Laplace’s equation
Variable coefficient after transformation
Domain imbedding
Finite element discretization
Fast Poisson solver as preconditioner
Fixed computational domain
Simple shape
October 26, 1998

7. An optimal preconditioner
Pressure equation
Low permeable zone
Preconditioner
Eigenvalue analysis
Two-level preconditioning scheme
October 26, 1998

8. A question Starting point: sequential PDE simulators.
How to do the parallelization?
Resulting parallel simulators should have
Good parallel performance
Good overall numerical performance
A relative simple parallelization process
We need
a good parallelization strategy
a good implementation of the strategy
October 26, 1998

9. 3 key words Parallel Computing faster solution, larger simulation
Domain Decomposition (additive Schwarz method)
good algorithmic efficiency
mathematical foundation of parallelization
Object-Oriented Programming
extensible sequential simulator
flexible implementation framework for parallelization
October 26, 1998

10. A simulator-parallel model
Each processor hosts an arbitrary number of subdomains
One subdomain is assigned with a sequential simulator
Flexibility - different linear system solvers, preconditioners, convergence monitors etc. can easily be chosen for different subproblems
Domain decomposition at the level of subdomain simulators!
October 26, 1998

11. Advantages & disadvantages
Reuse of existing sequential simulators
Data distribution is implied
No need for global data
Needs additional functionalities for exchanging nodal values inside the overlapping region
Needs some global administration
October 26, 1998

12. A generic framework Object-oriented programming
An add-on library (SPMD model)
Flexibility and portability
Simplified parallelization process for end-user
October 26, 1998

13. Administrator
Parameter Interface
solution method or preconditioner, max iterations, stopping criterion etc
DD algorithm Interface
access to predefined numerical algorithm e.g. CG
Operation Interface (standard codes & UDC)
access to subdomain simulators, matrix-vector product, inner product etc

14. O-O implementation ParaPDESolver SPAdmUDC BasicDDSolver KrylovDDSolver
ConjGradDD
BiCGStabDD
October 26, 1998

15. Subdomain Simulator Subdomain Simulator -- a generic representation
C++ class hierarchy
Standard interface of generic member functions
October 26, 1998

16. Coding example Existing sequential simulator PoissonSolver
New subdomain solver
SubdomainFESolver
SubdomainFEMSolver
PoissonSolver
SubdomainFESolver
October 26, 1998

17. P: number of processors.
2D Poisson’s equation
Fixed M=32 subdomains based on a 481 x 481 global grid.
Straightforward parallelization of an existing simulator.
Subdomain solves use CG+FFT
P: number of processors.
October 26, 1998

18. 2-phase porous media flow
October 26, 1998

19. Adaptivity
October 26, 1998

20. Multigrid 1 2 3 4
Common
October 26, 1998

21. DD approach Schwarz two level method Multigrid for coarse grid
Multigrid for local sub-problems
October 26, 1998

22. Grid generation/partition
Non-overlapping grid partition on a single processor (metis)
Create overlap
Pick subgrid for each processor
Uniform refinement of subgrid
Communication pattern determination
October 26, 1998

23. Partition example
October 26, 1998

24. Solving 2D Lapalce’s equation
October 26, 1998

25. CPU-measurements
October 26, 1998

26. Scalability
Measurements obtained on 16 processors
October 26, 1998

27. Scientific visualization
October 26, 1998

28. Acknowledgements Sincere thanks to my supervisors: Aslak Tveito
Even Mehlum
Hans Petter Langtangen
October 26, 1998