Xing Cai University of Oslo Numerical Simulation of 3D Fully Nonlinear Waters Waves on Parallel Computers Xing Cai University of Oslo

Outline of the Talk Mathematical model Numerical scheme (sequential) Parallelization strategy (domain decomposition) Object-oriented implementation Numerical experiment PARA'98

Mathematical Model Fully nonlinear 3D water waves Primary unknowns: PARA'98

Numerical Scheme Physical domain: Transformation: (a fixed domain) PARA'98

Numerical Scheme Operator splitting At each time level: PARA'98 FDM for updating free surface conditions FEM solution of an elliptic boundary value problem in PARA'98

Preconditioning Computational cost Elliptic boundary value problem - most CPU intensive Resulting system of linear equations Preconditiong Computational cost PARA'98 N- number of unknowns

The Question Starting point: an o-o water wave simulator (built in Diffpack: C++ environment for scientific computing) How to do the parallelization? Different approaches on different levels: Automatic parallelization Parallelization on the low matrix-vector level Parallelization on the level of simulators PARA'98

Parallelization Strategy Domain Decomposition Divide and conquer Solution of the original large problem through iteratively solving many smaller subproblems -- solution method or preconditioner Flexible -- localized treatment of irregular geometries, singularities etc Very efficient numerical methods -- even on sequential computers Suitable for coarse grained parallelization PARA'98

Overlapping Domain Decomposition Alternating Schwarz method for two subdomains Example: solving an elliptic boundary value problem in A sequence of approximations where PARA'98

Additive Schwarz Method Numerical Foundation Additive Schwarz Method Subproblems are of the same form as the original large problem, with possibly different boundary conditions on artificial boundaries. Subproblems can be solved in parallel. PARA'98

Convergence of the Solution Example: Solving the Poisson problem on the unit square PARA'98

Coarse Grid Correction Numerical Foundation Coarse Grid Correction Important for good DD convergence Run on each processor, shared with subdomain simulators on the same processor PARA'98

Some Observations Parallel Computing Domain Decomposition efficiency relies on the parallelization Domain Decomposition suits well for parallel computing a good parallelization strategy Object-Oriented Programming Technique flexible and efficient sequential simulators can be used in subdomain solves -- main ingredient of DD PARA'98

A simulator-parallel model New Programming Model A simulator-parallel model Each processor hosts an arbitrary number of subdomains balance between numerical efficiency and load balancing One subdomain is assigned a sequential simulator Flexibility -- different types of grids, linear system solvers, preconditioners, convergence monitors etc. are allowed for different subproblems Domain decomposition on the level of subdomain simulators! PARA'98

Simulator-Parallel 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 PARA'98

A Generic Programming Framework An add-on library (SPMD model) Use of object-oriented programming technique Flexibility and portability Simplified parallelization process for end-user PARA'98

The Administrator Parameter Interface solution method or preconditioner, max iterations, stopping criterion etc DD algorithm Interface access to predifined numerical algorithm e.g. CG Operation Interface (standard codes & UDC) access to subdomain simulators, matrix-vector product, inner product etc PARA'98

The Subdomain Simulator Subdomain Simulator -- a generic representation C++ class hierarchy Interface of generic member functions PARA'98

Adaptation of Sequential Simulator Class SubdomainSimulator - generic representation of a sequential simulator. Class SubdomainFEMSolver - generic representation of a sequential simulator using FEM. A new sequential wave simulator that fits in the framework is readily extended from the existing sequential simulator, also being a subclass of SubdomainFEMSolver. SubdomainSimulator PARA'98 SubdomainFEMSolver WaveSimulator NewWSimulator

Algorithmic efficiency Performance Algorithmic efficiency efficiency of original sequential simulator(s) efficiency of domain decomposition method Parallel efficiency communication overhead (low) coarse grid correction overhead (normally low) synchronization overhead load balancing subproblem size work on subdomain solves PARA'98

Parallel Simulation of Waves

Parallel Efficiency Fixed number of subdomains M=16. Subdomain grids from partition of a global 41x41x41 grid. Simulation over 32 time steps. DD as preconditioner of CG for the Laplace eq. Multigrid V-cycle as subdomain solver. PARA'98

Overall Efficiency Number of subdomains equal to number of processors PARA'98 *For P=2 parallel BiCGStab is used.

Summary Efficient solution of elliptic boundary value problems Parallelization based on DD Introduction of a simulator-parallel model A generic framework for implementation http:www.nobjects.com PARA'98