a parallelisation approach for multi-resolution grids based upon the peano space-filling curve...

A Parallelisation Approach A Parallelisation Approach for Multi-Resolution Grids for Multi-Resolution Grids

Based Upon the Peano Based Upon the Peano Space-Filling CurveSpace-Filling Curve

Student: Adriana BocoiAdvisor: Dipl.-Inf.Tobias Weinzierl

Adriana Bocoi - Peano space filling curveAdriana Bocoi - Peano space filling curve 2201.04.2007

Content

Domain discretisation

Trees [Computer representation]

Space-Filling Curves [Traversal]

Parallelisation


Solving Real Life Problems

Model the real life problem Usually we end up with PDE

(Partial Differential Equations) Consider the domain Discretise (grid) Solve the discrete problem


ParallelisationParallel computing is the division of one task into a set of subtasks

assigned to multiple processors in order to obtain results faster or to reduce memory requirements per processor.

Example: input: a={2,3,4}, b={5,6,1}result: c=a+b={7,9,5}

Serial Parallel (3 processors)


Domain Partitioning

Domain partitioning is the process of dividing a domain into two or more

regions.


Domain PartitioningWhich is better ?

hhh

hh

qualityPartition

hhh

hh

qualityPartition

volume

surface

surface

lengthdatalocal

dataedcommunicatqualityPartition

square

stripes

DD

8

22

222

10

4

42

32


Domain Partitioning

We have (regarding partition quality):

stripes < square

Which is optimal ?

Minimum length/maximum surface -> circle

thus: stripes < square < circle

hh

hqualityPartition circle

222


Domain Partitioning Types of structured grids:

Regular grids

Adaptive grids


Pro’s and Con’s

Regular Grid Adaptive Grid

Generation Easy Difficult

Storage Scheme Easy Complicated

Algorithm Implementation Easy Difficult

PrecisionGreat memory and

computational effort

Good


Conclusion Grids

If we can find a way to memorise and work with adaptive grids

efficiently, they can help us to make more precise

computations with less effort (memory and computation).


Content




Parallelisation


Data structure - Trees !

Recursivity

=>

Tree data structure


Data structure - Trees !

Terms that we encounter:

Quadtrees(2D)bipartition

Octrees (3D)

Space Trees Refinement trees – representation of the

refinement process that created the grid [1]


How to read from that structure We have two traversals:

BFS DFS (Breadth First Search) (Depth First Search)


Ordering

Deterministic algorithm(Where to go next ?)


Content




Parallelisation


We need an unique ordering

We have space filling curves to

define the order of the children

The Peano Space - Filling Curve


How does this actually help ?

We obtain a traversal of the tree given by the Peano

Space - Filling Curve.


What do we have till now ?

Adaptive grids Methods to handle them on the

computer

Data Structure: Tree Structure

Ordering: Peano Space-Filling Curve


Content




Parallelisation


Remember the domain partitioning

Adaptive grid

Regular grid


Parallelisation

What is wrong?

Adaptive grids - One processor has to do more work than the other three!

[Load imbalance]

Regular grids – nothing


We want a well-balanced workload distribution!

Load balancing is a technique to spread work between processors

in order to get better resource utilisation and, thus, decrease

computing time.


Partition strategy

split the grid <=> split the tree One strategy [1]:

p processors n nodes in the treewe get approx. [n/p] nodes for each

processor


How do we find it ? Example:


How good is the partition ?

The papers in this field [2] tell us that actually domain partitioning

using Peano Space-Filling Curves is besides a constant as good as an

optimal partitioning (circle).

rectangle < square < peano SFC < circle


Why Peano and not other SFC ?

Because of the 3D projection propriety:


Conclusion

Partitioned Domain+

Trees as data structure+

Peano space filling curve for ordering+

Splitting strategy=

Good parallelisation

Thank you for your attention !!!

Q & A

a parallelisation approach for multi-resolution grids based upon the peano space-filling curve...

Documents

adriana bocoiadvisor

computerdata structure

refinement process

tree structureordering

multiresolution grids

real life problemusually

memory requirements

effort memory