Space Filling CurvesHierarchical Basis

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2.d Usage in numerical simulations
• So far we’ve only spoken about the generation of space filling curves. But how can this be of advantage for the data transfer in numerical simulations? To answer this question let’s have a short look at a simple grid. Within a square we need the values at the corner points of that square. When we go through this square following e.g. Peano’s space filling curve we can build up inner lines of data points, which are processed in a forward or backward order. This scheme corresponds to a stack, on which we can only access the very most upper element (forward: writing to stack, backward: reading from stack). In our example we would need 2 stacks (blue and red). 0,0 1,1
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2.b Hilbert’s Space-Filling Curve
The German mathematician David Hilbert (1862-1943) was the first one to give the so far only algebraically described space-filling curves a geometric interpretation. In the 2 dimensional case he splat I into four subintervals and Ω into four sub squares. This splitting is recursively repeated for the emerging subspaces, leading to the below shown curve. An important fact is the so called full inclusion: whenever we further split a subinterval, the mapping to the corresponding finer sub squares stays in the former interval. Repeating the partitioning symmetrically (for all subspaces) again and again leads to 22n subintervals / sub squares in case of a 2-dimensional target space. (n: # of partitioning steps)
3. Hierarchical basis and generating systems
a. Standard nodal basis for FEM-analysis b. Hierarchical basis c. Generating systems
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1. Motivation
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2.c Peano’s Space-Filling Curve
The Italian mathematician Giuseppe Peano (1858-1932) was the first one who constructed (only algebraically) a space filling curve in 1890. According to him we split the starting domain Ω into 3m sub areas. The Peano curve goes always from 0 to 1 in all dimensions. The way we run through the target space is up to us.Two different ways are marked with blue / red numbers. Only the first and the last subspaces are mapped in a fixed way. Recursive partitioning leads to finer curves (compare with Hilbert’s curve). Symmetrical partitioning leads to 3mn sub areas.
Space-Filling Curves
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Use (basis) functions for FEM-analysis with which LSE solvers show faster convergence
Hierarchical Basis Generating System (for higher dimensional problems)
Ι =[0,1] →Ω =[0,1]2 Ι =[0,1] →Ω =[0,1]3 Ι =[0,1] →Ω =[0,1]n , n = 4,5,...
If the curve of this mapping passes through every point of the target space we call this a “Space-Filling Curve”. We are interested in a continuous mapping. This cannot be bijective, it’s only surjective.
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Now we begin to split this cube into finer sub cubes. We can select the dimension we want to begin with.
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– The latter can be optimised by arranging the data in an effective way to avoid cache misses and support optimisation techniques (e.g. pre-fetching). Standard organization in matrices / arrays leads to bad cache behaviour.
Time needed for simulating phenomena depends on both performing the corresponding calculation steps and transferring data to and from the processor(s).
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2.a Definition of Space-Filling Curves
We can construct a mapping from a 1-D interval to a n-D interval where n is finite, i.e. we can construct mappings as follow:
2.c Peano curve in 3-D (continued)
Now the emerging sub spaces are split in one on the two remaining dimensions.
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After splitting the sub spaces in the last dimension we end up with 27 equally sized cubes (symmetrical case, here: 7 cuboids.)
Space Filling Curves Hierarchical Basis
JAivation 2. Space filling curves
a. b. c. d. Definition Hilbert’s space filling curve Peano’s space filling curve Usage in numerical simulations
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