Make method truly linear-time about spacetime

[Jannertje]

First 3-point hierarchical basis introduces some sorting steps, as well as calling O(N) list.index() methods.

I optimized IndexSet a little; see fbe3328.

[Jannertje]

Implementation works but the dot-product seems to become the bottleneck over time. If we track the computation time of apply() for the 3-point mass matrix on uniform grids with max level $l$, for $l = 1$ to $13$, then we get the following profiling results. Moreover, making a plot shows that for larger $l$, the total computation time becomes quadratic.. Hmm

ncalls	tottime	percall	cumtime	percall	filename:lineno(function)
114882	10.004	0.000	10.580	0.000	indexed_vector.py:60(dot)
16369	6.500	0.000	6.526	0.000	three_point_basis.py:138(col)
32829	4.617	0.000	4.680	0.000	three_point_basis.py:115(row)
392843	0.655	0.000	1.395	0.000	index_set.py:43(neighbours)
392843	0.525	0.000	0.525	0.000	{built-in method _bisect.bisect_left}
180748	0.428	0.000	1.442	0.000	three_point_basis.py:179(scaling_support)
65476	0.288	0.000	1.220	0.000	three_point_basis.py:189(wavelet_support)
360651	0.285	0.000	0.285	0.000	indexed_vector.py:62()
490394	0.235	0.000	0.235	0.000	three_point_basis.py:22(position_ss)
114882	0.223	0.000	0.508	0.000	{built-in method builtins.sum}
393051	0.209	0.000	0.255	0.000	three_point_basis.py:257(scaling_indices_on_level)
246588	0.163	0.000	0.163	0.000	interval.py:4(init)
312	0.152	0.000	15.392	0.049	linear_operator.py:25()

[Jannertje]

The problem seems to be in that while the matrices P and Q have a bounded number of nonzero elements per, the sets $\Pi_B$ and $\Lambda_l$ (the sets we restrict vectors to before applying P and Q) grow. As a result, the inner-product cost sum(self.vector[labda] * other[labda] for labda in index_mask & other.keys()) grows linearly in the size of other..

[Jannertje]

More careful phrasing (sum(self.vector[labda] * other[labda] for labda in other.keys() if labda in index_mask)) made a big difference. It's now linear and runs a lot faster :-)

[Jannertje]

The biggest hurdly to cross to making this truly linear time will be the computation of $\Pi_B$ and $\underline \Pi$. Maybe Rob can help shed some light on this.

[Jannertje]

With the tree-implementation, this is no longer really a current topic. Closing for staleness.