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Scalable Applications with Dense Linear Algebra and Sparse Solvers

Explore impactful applications like genome research, heart simulations, and Poisson solvers using dense linear algebra and sparse solvers. Learn about memory parameters, communication overhead, and key algorithms in high-performance computing.

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Scalable Applications with Dense Linear Algebra and Sparse Solvers

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  1. Global Address Space Applications Kathy Yelick NERSC/LBNL and U.C. Berkeley

  2. Algorithm Space Search Two-sided dense linear algebra Grobner Basis (“Symbolic LU”) FFTs Sorting Reuse Sparse iterative solvers Asynchronous discrete even simulation Sparse direct solvers One-sided dense linear algebra Regularity

  3. Scaling Applications • Machine Parameters • Floating point performance • Application dependent, not theoretical peak • Amount of memory per processor • Use 1/10th for algorithm data • Communication Overhead • Time processor is busy sending a message • Cannot be overlapped • Communication Latency • Time across the network (can be overlapped) • Communication Bandwidth • Single node and bisection • Back-of-the envelope calculations !

  4. Running Sparse MVM on a Pflop • 1 GHz * 8 pipes * 8 ALUs/Pipe = 64 GFLOPS/node peak • 8 Address generators limit performance to 16 Gflops • 500ns latency, 1 cycle put/get overhead, 100 cycle MP overhead • Programmability differences too: packing vs. global address space

  5. Effect of Memory Size • Low overhead is important for • Small memory nodes or smaller problem sizes • Programmability

  6. Parallel Applications in Titanium • Genome Application • Heart simulation • AMR elliptic and hyperbolic solvers • Scalable Poisson for infinite domains • Genome application • Several smaller benchmarks: EM3D, MatMul, LU, FFT, Join

  7. MOOSE Application • Problem: Microarray construction • Used for genome experiments • Possible medical applications long-term • Microarray Optimal Oligo Selection Engine (MOOSE) • A parallel engine for selecting the best oligonucleotide sequences for genetic microarray testing • Uses dynamic load balancing within Titanium

  8. Heart Simulation • Problem: compute blood flow in the heart • Model as elastic structure in incompressible fluid. • “Immersed Boundary Method” [Peskin and McQueen] • Particle/Mesh method stress communication performance • 20 years of development in model • Many other applications: blood clotting, inner ear, insect flight, embryo growth,… • Can be used for design of prosthetics • Artificial heart valves • Cochlear implants

  9. -6.47x10-9 0 1.31x10-9 Scalable Poisson Solver • MLC for Finite-Differences by Balls and Colella • Poisson equation with infinite boundaries • arise in astrophysics, some biological systems, etc. • Method is scalable • Low communication • Performance on • SP2 (shown) and t3e • scaled speedups • nearly ideal (flat) • Currently 2D and non-adaptive • Point charge example shown • Rings & star charges • Relative error shown

  10. AMR Gas Dynamics • Developed by McCorquodale and Colella • 2D Example (3D supported) • Mach-10 shock on solid surface at oblique angle • Future: Self-gravitating gas dynamics package

  11. UPC Application Investigations • Pyramid • 3D Mesh generation [Shewchuk] • 2D version (triangle) critical in Quake project • Written in C, challenge to parallelize • SuperLU • Sparse direct solver [Li,Demmel] • Written in C+MPI or threads • UPC may enable new algorithmic techniques • N-Body simulation • “Simulating the Universe”

  12. Summary • UPC Killer App should • Leverage programmability: hard in MPI • Use fine-grained, irregular, asynchronous communication • Libraries • Must allow for interface to libraries • MPI libraries, multithreaded libraries, serial libraries • Compilation needs • High performance on at least one machine • Portability across many machines

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