Cache Efficient Data Structures and Algorithms for d -Dimensional Problems

transcript

Garching – 01.06.05 SCCS-Kolloquium

Cache Efficient Data Structures and Algorithms for

d-Dimensional Problems

Judith HartmannTU München, Institut für Informatik V

hartmanj@in.tum.de

SCCS 01.06.05 Cache Efficient Data Structures and Algorithms for d-Dimensional Problems

Overview

• cache hierarchies and data locality

• discretization by space-filling curve

• system of stacks

• measured results

• concluding remarks

Cache Hierarchy

100000

1980 1984 1988 1992 1996 2000 2004

Performance gap (Hennesy/Patterson)

CPU Memory

L2 cache

registers

L3 cache

memory

L1 data L1 inst

Cache Efficiency as Main Objective

• algorithm for solving PDEs

– modern concepts (finite elements, multigrid, etc.)

– cache-efficiency

• road to cache-efficiency

– spacefilling curve for discretization of the domain

– data transport via stack system

Iterative Construction of the Peano Curve

• Leitmotiv L

• for each dimension: decomposition of all diagonals

Important Properties for the Construction of a Stack System

• projection property

• palindrome property

• recursivity in dimension

Organization of a d-Dimensional Iteration

(d-1)-dim. process (d-1)-dim. process

(d-1)-dim. system of

stacks

d-dim. process

d-dim. input stack

d-dim. output stack

9• induced discretization

of the unit square and order for the grid cells

• processing of data

Necessary Properties of a Useful System of Stacks

• constant number of stacks

• determinsitic data access

• dimensional recursivity

• self similarity

Number of Stacks

type of stack 0D 1D 2D 3D 4D

d=1 2 2 4

d=2 4 4 2 10

d=3 8 12 6 2 28

d=4 16 32 24 8 2 28

• number of stacks for several dimensionalities

• number of i-dimensional stacks for a d-dimensional iteration

• total number of lower dimensional stacks: 3d-1

Stacks and Geometrical Boundary Elements

• matching of i-dimensional stack and i-dimensional boundary element of the unit cube

• ternary encoding of geometrical boundary elements

000 001

Data Access

• parameters for each grid point

– status: determination of the sort of the stack

• basic stack numbers

• mechanism of inheritance

– stack numbers: which stack of this sort

Necessary Properties of a Useful System of Stacks

• constant number of stacks

• determinsitic data access

• dimensional recursivity

• self similarity

Hardware PerformancePoisson Equation on the Unit Cube

• cache-hit-rates > 99,9 %

resolution d=2 d=3 d=4 d=5 d=6

9 x ··· x 9 99.06% 99.93% 99.95% 99.99% 99.81%

27 x ··· x 27 99.85% 99.99% 99.98% 99.97% -

81 x ··· x 81 99.98% 99.98% 99.96% - -

243 x ··· x 243 99.99% 99.97% - - -

• processing time per degree of freedom

– decreases with increasing grid size

– is augmnted by a factor ~2.7 for each dimension

• unpenalized disc-usage possible

resolution d=2 d=3 d=4 d=5

9 x ··· x 9 5.45·10-6 1.74·10-5 4.77·10-5 1.47·10-4

27 x ··· x 27 4.98·10-6 1.35·10-5 3.66·10-5 1.05·10-4

81 x ··· x 81 4.41·10-6 1.25·10-5 3.32·10-5 -

243 x ··· x 243 4.32·10-6 1.22·10-5 - -

• processing time per degree of freedom

AdaptivityPoisson Equation on a Sphere

• no loss of cache-efficiency:cache-hit-rates > 99,9 %

• locally refined grids:

– computing time improves (if appropriately approximated)

– no loss of cache-efficiency

dim. domain resolution time per dof L2 hit rate

unit cube 243 x ··· x 243 3.53·10-6 99.96%

sphere243 x ··· x 243(regular grid)

5.66·10-6 99.98%

sphere243 x ··· x 243

(only on Ω)3.90·10-6 99.97%

unit cube 81 x ··· x 81 2.64·10-5 99.96%

sphere81 x ··· x 81

(regular grid)1.24·10-4 99.96%

sphere81 x ··· x 81 (only on Ω)

3.83·10-5 99.96%

Conclusion & Further Developments

• method for solving d-dimensional PDEs

– modern numerical methods

– high cache-efficiency

– complicated geometrics

• further developements:

– potential for parallelization

– general system of PDEs, Stokes, Navier-Stokes, etc.

– potential for solving time-dependent PDEs in 3 space dimensions

Cache Efficient Data Structures and Algorithms for d -Dimensional Problems

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