Knob sweep — how sensitive is the grid to the thresholds?

The adaptive loop’s behaviour on the one-parameter diffusion benchmark depends on a handful of tolerance values. This page is a self-contained sensitivity study: it walks each of those thresholds across a range and records how the terminal refinement level and the per-level verdict counts respond. The goal is to show that the loop’s output is sensitive to the specific tolerance values rather than fixed — small changes move the grid — and to locate which knobs matter and which do not.

As a fixed yardstick, each cell is also scored against a reference verdict-count target: a per-level tuple (n_points, n_subintervals, n_wrong, n_uncertified) that a well-tuned run is expected to produce. The study reports how many of the four target rows each cell reproduces exactly. No single cell reproduces all four.

Model

Same one-parameter 2D diffusion setup as the one-parameter sweep — only the algorithm thresholds vary.

Sweep design

Six threshold axes are walked, all anchored to a single baseline (t_\pi=0.21, t_\lambda=10^{-3}, cluster tolerance 10^{-4}, n_{\text{eigs}}=8, max levels =4):

Lift the artificial L=4 level cap to 8.
t_\lambda \in \{0.03, 0.085, 0.15\} (the a-priori relative-gap threshold).
t_\pi \in \{0.30, 0.43, 0.57\} (the a-posteriori pruning threshold).
n_{\text{eigs}} \in \{4, 6\} (bands tracked per point).
Combined relaxations of (t_\pi, t_\lambda) toward looser values, with and without n_{\text{eigs}}=6.
Fine-tune around the best-scoring (0.43, 0.085, 6) cell.

The reference verdict-count target, per refinement level, is:

L	reference target
0	`(3, 2, 2, 2)`
1	`(5, 4, 3, 2)`
2	`(7, 4, 0, 1)`
3	`(8, 2, 0, 0)`

Each sweep cell’s L0..L3_exact_matches column counts how many of those four rows it reproduces verbatim. The table below ranks the cells by how close they come.

Summary figure

Per-knob termination depth (top) and number of reference rows reproduced exactly (bottom). Bars at `4` in the lower panel would mean perfect reproduction of the reference target; none reach it.

Full sweep table

label	t_pi	t_lambda	cluster_tol	n_eigs	max_levels	terminal_level	ran_out	L0..L3_exact_matches	L0	L1	L2	L3	L4
`baseline`	`0.21`	`0.001`	`0.0001`	`8`	`4`	`3`	`False`	`1`	`(3,2,2,2)`	`(5,4,3,3)`	`(8,6,5,1)`	`(9,2,6,0)`	\| \| `max_levels=8 only` \| `0.21` \| `0.001` \| `0.0001` \| `8` \| `8` \| `3` \| `False` \| `1` \| `(3,2,2,2)` \| `(5,4,3,3)` \| `(8,6,5,1)` \| `(9,2,6,0)` \|
`t_lambda=0.03`	`0.21`	`0.03`	`0.0001`	`8`	`8`	`3`	`False`	`1`	`(3,2,2,2)`	`(5,4,3,3)`	`(8,6,5,1)`	`(9,2,6,0)`	\| \| `t_lambda=0.085` \| `0.21` \| `0.085` \| `0.0001` \| `8` \| `8` \| `3` \| `False` \| `1` \| `(3,2,2,2)` \| `(5,4,3,3)` \| `(8,6,5,1)` \| `(9,2,6,0)` \|
`t_lambda=0.15`	`0.21`	`0.15`	`0.0001`	`8`	`8`	`2`	`False`	`0`	`(3,2,1,2)`	`(5,4,1,1)`	`(6,2,2,0)`	`\|`
`t_pi=0.3`	`0.3`	`0.001`	`0.0001`	`8`	`8`	`3`	`False`	`1`	`(3,2,2,2)`	`(5,4,3,3)`	`(8,6,5,1)`	`(9,2,6,0)`	\| \| `t_pi=0.43` \| `0.43` \| `0.001` \| `0.0001` \| `8` \| `8` \| `3` \| `False` \| `1` \| `(3,2,2,2)` \| `(5,4,3,3)` \| `(8,6,5,2)` \| `(10,4,6,0)` \|
`t_pi=0.57`	`0.57`	`0.001`	`0.0001`	`8`	`8`	`4`	`False`	`1`	`(3,2,2,2)`	`(5,4,3,4)`	`(9,8,6,4)`	`(13,8,9,1)`	`(14,2,10,0)`
`n_eigs=4`	`0.21`	`0.001`	`0.0001`	`4`	`8`	`1`	`False`	`0`	`(3,2,2,1)`	`(4,2,2,0)`	`\|`	\| \| `n_eigs=6` \| `0.21` \| `0.001` \| `0.0001` \| `6` \| `8` \| `1` \| `False` \| `0` \| `(3,2,2,1)` \| `(4,2,2,0)` \|	`\|`
`t_pi=0.43, t_lambda=0.085`	`0.43`	`0.085`	`0.0001`	`8`	`8`	`3`	`False`	`1`	`(3,2,2,2)`	`(5,4,3,3)`	`(8,6,5,2)`	`(10,4,6,0)`	\| \| `t_pi=0.57, t_lambda=0.085` \| `0.57` \| `0.085` \| `0.0001` \| `8` \| `8` \| `3` \| `False` \| `1` \| `(3,2,2,2)` \| `(5,4,3,4)` \| `(9,8,6,4)` \| `(13,8,9,0)` \|
`t_pi=0.43, t_lambda=0.085, n_eigs=6`	`0.43`	`0.085`	`0.0001`	`6`	`8`	`2`	`False`	`0`	`(3,2,1,2)`	`(5,4,2,1)`	`(6,2,2,0)`	`\|`
`tight t_pi=0.21, t_lambda=0.085, n_eigs=6`	`0.21`	`0.085`	`0.0001`	`6`	`8`	`1`	`False`	`0`	`(3,2,2,1)`	`(4,2,2,0)`	`\|`	\| \| `tight t_pi=0.3, t_lambda=0.085, n_eigs=6` \| `0.3` \| `0.085` \| `0.0001` \| `6` \| `8` \| `2` \| `False` \| `0` \| `(3,2,1,1)` \| `(4,2,1,1)` \| `(5,2,1,0)` \|	\| \| `tight t_pi=0.35, t_lambda=0.085, n_eigs=6` \| `0.35` \| `0.085` \| `0.0001` \| `6` \| `8` \| `2` \| `False` \| `0` \| `(3,2,1,1)` \| `(4,2,1,1)` \| `(5,2,1,0)` \|

The takeaway is twofold: the terminal level and verdict counts shift visibly as t_\pi and n_{\text{eigs}} move (the grid is genuinely sensitive to them), while t_\lambda barely registers in this regime. That the threshold choice is load-bearing is the same lesson the tolerance-perturbation page draws from the downstream ROM error.

--- title: "Knob sweep — how sensitive is the grid to the thresholds?" --- The adaptive loop's behaviour on the one-parameter diffusion benchmark depends on a handful of tolerance values. This page is a self-contained sensitivity study: it walks each of those thresholds across a range and records how the terminal refinement level and the per-level verdict counts respond. The goal is to show that the loop's output is **sensitive to the specific tolerance values** rather than fixed — small changes move the grid — and to locate which knobs matter and which do not. As a fixed yardstick, each cell is also scored against a **reference verdict-count target**: a per-level tuple `(n_points, n_subintervals, n_wrong, n_uncertified)` that a well-tuned run is expected to produce. The study reports how many of the four target rows each cell reproduces exactly. No single cell reproduces all four. ## Model Same one-parameter 2D diffusion setup as [the one-parameter sweep](diffusion-3-3-2.qmd) — only the algorithm thresholds vary. ## Sweep design Six threshold axes are walked, all anchored to a single baseline ($t_\pi=0.21$, $t_\lambda=10^{-3}$, cluster tolerance $10^{-4}$, $n_{\text{eigs}}=8$, max levels $=4$): 1. Lift the artificial $L=4$ level cap to $8$. 2. $t_\lambda \in \{0.03, 0.085, 0.15\}$ (the a-priori relative-gap threshold). 3. $t_\pi \in \{0.30, 0.43, 0.57\}$ (the a-posteriori pruning threshold). 4. $n_{\text{eigs}} \in \{4, 6\}$ (bands tracked per point). 5. Combined relaxations of $(t_\pi, t_\lambda)$ toward looser values, with and without $n_{\text{eigs}}=6$. 6. Fine-tune around the best-scoring $(0.43, 0.085, 6)$ cell. The reference verdict-count target, per refinement level, is: | $L$ | reference target | |---|---| | 0 | `(3, 2, 2, 2)` | | 1 | `(5, 4, 3, 2)` | | 2 | `(7, 4, 0, 1)` | | 3 | `(8, 2, 0, 0)` | Each sweep cell's `L0..L3_exact_matches` column counts how many of those four rows it reproduces verbatim. The table below ranks the cells by how close they come. ## Summary figure ![Per-knob termination depth (top) and number of reference rows reproduced exactly (bottom). Bars at `4` in the lower panel would mean perfect reproduction of the reference target; none reach it.](../figures/knob-sweep.png) ## Full sweep table {{< include ../figures/knob-sweep.md >}} The takeaway is twofold: the terminal level and verdict counts shift visibly as $t_\pi$ and $n_{\text{eigs}}$ move (the grid is genuinely sensitive to them), while $t_\lambda$ barely registers in this regime. That the threshold choice is load-bearing is the same lesson the [tolerance-perturbation page](tolerance-perturbation.qmd) draws from the downstream ROM error.