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University of Colorado at Boulder
Department of Computer Science
ECOT 717 Engineering Center
Campus Box 430
Boulder, Colorado 803090430
(303) 4927514, FAX: (303) 4922844
CS COLLOQUIUM
Terascale Spectral Element Algorithms and Implementations
Dr. Henry Tufo
Argonne National Laboratory/University of Chicago
Monday, April 29, 2002
2:00 - 2:50 PM
Engineering Center Classroom 200
Refreshments will be served afterwards in ECOT 832

The spectral element method is a weighted residual approach that employs 
Nth-order tensor-product polynomial bases (typ., N = 5-15) within K 
deformable subdomains. This approach combines the accuracy and efficiency 
of spectral methods with the geometric flexibility of finite elements. 
Because of its low numerical dispersion and dissipation, the method is 
ideal for the simulation of transitional and weakly turbulent flows.

We begin with an overview of the method and then focus on the algorithms 
and implementations which have resulted in terascale performance (0.376 
Tflops on 2048 nodes of ASCI Option Red). The key ingredients to achieving 
this level of performance are the local tensor-product structure, a high 
degree of parallelism at the inter-element level, and scalable multilevel 
solvers.

Locally, matrix-vector products involving O(KN^3) degrees-of-freedom and 
full stiffness matrices can be recast as matrix-matrix products requiring 
only O(KN^4) work - a substantial savings over the O(KN^6) work and storage 
required without tensor-product forms.

Globally, parallelism derives from the usual SPMD model in which clusters 
of elements are distributed to separate processors.  Communication required 
to update interface vertices during the matrix-vector product phase is 
nearest neighbor and is handled asynchronously by our stand-alone 
gather-scatter package, GS.

Incompressibility is enforced through iterative solution of an elliptic 
problem at each time step. Efficiency of the preconditioned CG scheme rests 
upon a multi-level preconditioner that includes a coarse grid solver to 
rapidly effect the transfer of low-wave number information throughout the 
domain. Because of its sparsity and distributed nature (n/P = O(1)), the 
coarse grid solve is communication intensive and frequently a bottleneck 
for P ~> 64. We have developed a fast parallel coarse grid solver that will 
scale to tens of thousands of processors. The method is based on projection 
onto a sparse basis for R^n and, for appropriately ordered FEM or FD meshes 
with n degrees of freedom, leads to communication costs of O(n^{1/2} log P) 
for two-dimensional meshes and O(n^{2/3} log P) for three-dimensional 
meshes. This is a substantial improvement over previous coarse-grid solve 
strategies, which require O(n log P) or O(n) communication. This complexity 
analysis is supported by experiments for P=1-4096.

We close with results from a series of challenging problems in fluid
mechanics and heat transfer

Host:  Richard Byrd
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http://www.cs.colorado.edu/department/events/colloquia/colloquia.html

Sign Language Interpreters Available Upon Request.  Please contact 
Stephanie Morris, Department of
Computer Science, Engineering  ECOT 722, 303-492-6101, at least five days 
prior to the colloquium.

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