Laplace Smoothing

Double X

We examine the most basic type of smoothing, Laplace smoothing, , without hierarchical control, with the Double X example.

../examples/unit_tests/double_x.png

Figure: The Double X two-element example.

Table. The neighborhoods table. A node, with its neighbors, is considered a single neighborhood. The table has twelve neighborhoods.

nodenode neighbors
12, 4, 7
21, 3, 5, 8
32, 6, 9
41, 5, 10
52, 4, 6, 11
63, 5, 12
71, 8, 10
82, 7, 9, 11
93, 8, 12
104, 7, 11
115, 8, 10, 12
126, 9, 11

Hierarchy

Following is a test where all nodes are BOUNDARY from the Hierarchy enum.

node_hierarchy: NodeHierarchy = (
    Hierarchy.BOUNDARY,
    Hierarchy.BOUNDARY,
    Hierarchy.BOUNDARY,
    Hierarchy.BOUNDARY,
    Hierarchy.BOUNDARY,
    Hierarchy.BOUNDARY,
    Hierarchy.BOUNDARY,
    Hierarchy.BOUNDARY,
    Hierarchy.BOUNDARY,
    Hierarchy.BOUNDARY,
    Hierarchy.BOUNDARY,
    Hierarchy.BOUNDARY,
)

Since there are no INTERIOR nodes nor PRESCRIBED nodes, the effect of hiearchical smoothing is nill, and the same effect would be observed were all nodes categorized as INTERIOR nodes.

Iteration 1

Table: The smoothed configuration (x, y, z) after one iteration of Laplace smoothing.

nodexyz
10.10.10.1
21.00.0750.075
31.90.10.1
40.10.90.1
51.00.9250.075
61.90.90.1
70.10.10.9
81.00.0750.925
91.90.10.9
100.10.90.9
111.00.9250.925
121.90.90.9

free_laplace_iter_1.png

Figure: Two element test problem (left) original configuration, (right) subject to two iterations of Laplace smoothing.

Iteration 2

nodexyz
10.190.17750.1775
21.00.14250.1425
31.810.17750.1775
40.190.82250.1775
51.00.85750.1425
61.810.82250.1775
70.190.17750.8225
81.00.14250.8575
91.810.17750.8225
100.190.82250.8225
111.00.85750.8575
121.810.82250.8225

free_laplace_iter_2.png

Figure: Two element test problem (left) original configuration, (right) subject to two iterations of Laplace smoothing.

Iteration 100

A known drawback of Laplace smoothing is that it can fail to preserve volumes. In the limit, volumes get reduced to a point, as illustrated in the figure below.

free_laplace_iter_100.gif

Figure: Two element test problem (left) original configuration, (right) subject to [1, 2, 3, 4, 5, 10, 20, 30, 100 iterations of Laplace smoothing. Animation created with Ezgif.