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

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

Cache Hierarchy

1

10

100

1000

10000

100000

Pe

rfo

rma

nc

e (

19

80

= 1

)

1980 1984 1988 1992 1996 2000 2004

Performance gap (Hennesy/Patterson)

CPU Memory

L2 cache

registers

CPU

L3 cache

main

memory

L1 data L1 inst

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

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

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

Iterative Construction of the Peano Curve

• Leitmotiv L

• for each dimension: decomposition of all diagonals

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

Important Properties for the Construction of a Stack System

• projection property

• palindrome property

• recursivity in dimension

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

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

1

2

3

6

5

4

7

8

9• induced discretization

of the unit square and order for the grid cells

• processing of data

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

Necessary Properties of a Useful System of Stacks

• constant number of stacks

• determinsitic data access

• dimensional recursivity

• self similarity

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

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

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

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

011

101

111

100

110

002

201

021

211

120

112

102

200

121

202

221

122

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

Data Access

002

201

021

211

120

112

102

200

121

100

120

102

200

100

200

• parameters for each grid point

100

– status: determination of the sort of the stack

• basic stack numbers

• mechanism of inheritance

– stack numbers: which stack of this sort

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

Necessary Properties of a Useful System of Stacks

• constant number of stacks

• determinsitic data access

• dimensional recursivity

• self similarity

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

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

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

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

d=3

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%

d=4

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%

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

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