## Line Chart: Estimator Performance Comparison ### Overview The image is a line chart comparing the performance of different estimators (PDE, PIKL, and Sobolev) against the norm of f* and the PDE error. The x-axis represents an unspecified parameter, ranging from 1 to 5. The y-axis represents the estimator values, ranging from -3 to 2. The chart includes shaded regions around the estimator lines, likely representing confidence intervals or standard deviations. ### Components/Axes * **X-axis:** Ranges from 1 to 5, incrementing by 1. No explicit label is provided. * **Y-axis:** Ranges from -3 to 2, incrementing by 1. No explicit label is provided. * **Legend (bottom-left):** * Black dashed line: "Norm of f*" * Black dotted line: "PDE error" * Light pink line: "PDE estimator" * Blue line: "PIKL estimator" * Orange line: "Sobolev estimator" ### Detailed Analysis * **Norm of f* (Black dashed line):** A horizontal line at approximately y = 1.3. * **PDE error (Black dotted line):** A horizontal line at approximately y = -0.9. * **PDE estimator (Light pink line):** * Starts at approximately (1, 1). * Decreases to approximately (3, -1). * Remains relatively stable around -1 after x=3. * Has a shaded region around the line, indicating variability. * **PIKL estimator (Blue line):** * Starts at approximately (1, 0.8). * Decreases steadily to approximately (5, -3). * Has a shaded region around the line, indicating variability. * **Sobolev estimator (Orange line):** * Starts at approximately (1, 1.2). * Decreases steadily to approximately (5, -3). * Has a shaded region around the line, indicating variability. ### Key Observations * The "Norm of f*" serves as a benchmark, remaining constant across the x-axis. * The "PDE error" also remains constant, indicating a baseline error level. * The PIKL and Sobolev estimators show a similar downward trend, suggesting comparable performance as x increases. * The PDE estimator initially performs well but plateaus around the PDE error level. ### Interpretation The chart illustrates the performance of three different estimators in relation to a target norm and a baseline error. The PIKL and Sobolev estimators converge towards lower values as the x-axis parameter increases, potentially indicating improved accuracy or stability. The PDE estimator, while initially close to the target norm, plateaus at the PDE error level, suggesting a limitation in its ability to reduce error beyond a certain point. The shaded regions around the estimator lines indicate the variability or uncertainty associated with each estimator's performance. The data suggests that for higher values of the x-axis parameter, PIKL and Sobolev estimators may be preferable to the PDE estimator.

\n ## Line Chart: Error Estimation Comparison ### Overview The image presents a line chart comparing the performance of different error estimators (PDE estimator, PIKL estimator, and Sobolev estimator) against the norm of f* and the PDE error. The chart displays the estimators' values across a range of values from approximately 1 to 5 on the x-axis. Shaded regions around each line represent uncertainty or variance in the estimator's performance. ### Components/Axes * **X-axis:** Labeled with numerical values ranging from 1 to 5, with tick marks at integer values. * **Y-axis:** Ranges from approximately -3 to 2, with tick marks at integer values. * **Lines:** Four lines are present, each representing a different estimator or error metric. * Black dashed line: "Norm of f*" * Red dotted line: "PDE error" * Pink line with shaded region: "PDE estimator" * Blue line with shaded region: "PIKL estimator" * Orange line with shaded region: "Sobolev estimator" * **Legend:** Located in the top-left corner, associating colors with each estimator/error metric. ### Detailed Analysis The chart shows the following trends and approximate data points: * **Norm of f* (Black Dashed Line):** This line is approximately horizontal, remaining relatively constant at a value of around 1.2 across the entire x-axis range. * **PDE Error (Red Dotted Line):** This line is also approximately horizontal, positioned around -1.1. * **PDE Estimator (Pink Line):** This line starts at approximately 1.1 at x=1 and slopes downward, reaching approximately -0.8 at x=5. The shaded region indicates a relatively wide range of uncertainty, particularly between x=1 and x=3. * **PIKL Estimator (Blue Line):** This line begins at approximately 0.8 at x=1 and exhibits a steeper downward slope than the PDE estimator, reaching approximately -2.5 at x=5. The shaded region is narrower than that of the PDE estimator, suggesting lower variance. * **Sobolev Estimator (Orange Line):** This line starts at approximately 0.9 at x=1 and slopes downward, similar to the PDE estimator, but with a slightly more pronounced decline. It reaches approximately -3.2 at x=5. The shaded region is comparable in width to that of the PDE estimator. ### Key Observations * All three estimators (PDE, PIKL, and Sobolev) show a decreasing trend as the x-value increases, indicating that the estimated error generally decreases with increasing values on the x-axis. * The PIKL estimator consistently estimates a lower error than the other two estimators across the entire range. * The Sobolev estimator generally estimates a lower error than the PDE estimator. * The uncertainty (as indicated by the shaded regions) is most significant for the PDE estimator. * The "Norm of f*" and "PDE error" lines provide reference levels for evaluating the performance of the estimators. ### Interpretation The chart demonstrates a comparison of different error estimation techniques. The downward trends of the estimators suggest that the error is reduced as the parameter (represented by the x-axis) increases. The PIKL estimator appears to be the most accurate, consistently providing the lowest error estimates. The wider uncertainty range for the PDE estimator suggests that it is less reliable than the PIKL and Sobolev estimators. The horizontal lines representing the "Norm of f*" and "PDE error" serve as benchmarks for assessing the quality of the error estimations. The fact that the estimators generally fall below the "PDE error" line indicates that they are successfully identifying and quantifying the error in the system. The differences between the estimators could be attributed to their underlying methodologies and assumptions. The chart suggests that the PIKL estimator is a robust and reliable method for error estimation in this context.

## Line Chart: Comparison of Estimators Against Norm and PDE Error ### Overview The image displays a line chart comparing the performance of three different estimators (PDE, PIKL, Sobolev) against two reference benchmarks (Norm of f*, PDE error) over a range of values on the x-axis (from 1 to 5). The chart includes shaded regions representing confidence intervals or variance for the estimator lines. ### Components/Axes * **X-Axis:** The horizontal axis is labeled with integer markers: `1`, `2`, `3`, `4`, `5`. There is no explicit axis title. * **Y-Axis:** The vertical axis is labeled with numerical markers: `-3`, `-2`, `-1`, `0`, `1`, `2`. There is no explicit axis title. * **Legend:** Located in the bottom-left quadrant of the chart area. It contains five entries: 1. `Norm of f*` (represented by a black dashed line `--`) 2. `PDE error` (represented by a black dotted line `...`) 3. `PDE estimator` (represented by a pink solid line with a light pink shaded confidence band) 4. `PIKL estimator` (represented by a blue solid line with a light blue shaded confidence band) 5. `Sobolev estimator` (represented by an orange solid line with a light orange shaded confidence band) ### Detailed Analysis **Trend Verification & Data Points:** 1. **Norm of f* (Black Dashed Line):** * **Trend:** Perfectly horizontal, constant value. * **Value:** Approximately `y = 1.4` across the entire x-axis range (1 to 5). 2. **PDE error (Black Dotted Line):** * **Trend:** Perfectly horizontal, constant value. * **Value:** Approximately `y = -1.0` across the entire x-axis range (1 to 5). 3. **PDE estimator (Pink Line & Shading):** * **Trend:** Starts high, decreases rapidly between x=1 and x=2, then flattens out, asymptotically approaching the `PDE error` line from above. * **Approximate Points:** * At x=1: y ≈ 1.5 (with a wide confidence interval from ~1.0 to ~2.0) * At x=2: y ≈ -0.5 * At x=3: y ≈ -0.9 * At x=4: y ≈ -1.0 * At x=5: y ≈ -1.0 (very close to the `PDE error` line) 4. **PIKL estimator (Blue Line & Shading):** * **Trend:** Shows a consistent, nearly linear downward slope across the entire range. It crosses below the `PDE error` line between x=2 and x=3. * **Approximate Points:** * At x=1: y ≈ 1.0 * At x=2: y ≈ -0.2 * At x=3: y ≈ -1.5 * At x=4: y ≈ -2.4 * At x=5: y ≈ -3.2 5. **Sobolev estimator (Orange Line & Shading):** * **Trend:** Also shows a consistent downward slope, but less steep than the PIKL estimator after x=3. It remains above the PIKL line for the entire visible range. * **Approximate Points:** * At x=1: y ≈ 1.2 * At x=2: y ≈ 0.5 * At x=3: y ≈ -0.8 * At x=4: y ≈ -1.8 * At x=5: y ≈ -2.8 ### Key Observations * **Convergence:** The `PDE estimator` (pink) clearly converges to the `PDE error` benchmark (dotted line) as the x-value increases. * **Divergence:** Both the `PIKL estimator` (blue) and `Sobolev estimator` (orange) diverge downward from the `PDE error` benchmark, showing no sign of leveling off within the plotted range. * **Relative Performance:** The `PIKL estimator` decreases at the fastest rate, followed by the `Sobolev estimator`. The `PDE estimator` stabilizes. * **Uncertainty:** The shaded confidence bands are widest at low x-values (especially for the PDE estimator at x=1) and narrow as x increases, suggesting reduced variance or increased certainty in the estimates for larger x. * **Crossing Point:** The `PIKL estimator` line crosses below the `PDE error` line at approximately x=2.5. The `Sobolev estimator` crosses it at approximately x=2.8. ### Interpretation This chart likely illustrates the behavior of different numerical methods or estimators for a problem involving a function `f*` and a Partial Differential Equation (PDE). The `Norm of f*` serves as a baseline magnitude. The `PDE error` represents a target or known error level. The key insight is the starkly different asymptotic behavior: * The **PDE estimator** appears to be *consistent* with the PDE error model, as it converges to that value. This suggests it correctly captures the underlying PDE structure as the parameter (x-axis, possibly related to model complexity, data size, or iteration count) increases. * The **PIKL and Sobolev estimators** do not converge to the PDE error. Their continuous decrease suggests they might be optimizing a different objective or are subject to a different error mechanism that does not align with the PDE error benchmark in this regime. The PIKL estimator's steeper decline indicates it is more sensitive to the changing x-parameter. The narrowing confidence intervals imply that all estimators become more precise (less variable) as x increases, even if their accuracy (closeness to the PDE error) differs. The chart effectively demonstrates that choice of estimator critically impacts long-term behavior and convergence properties relative to the PDE error standard.

## Line Graph: Estimator Performance vs. x-axis ### Overview The image is a line graph comparing the performance of four estimators (PDE, PIKL, Sobolev) and two reference lines (Norm of f*, PDE error) across an x-axis range of 1 to 5. The y-axis spans from -3 to 2, with key reference lines at y=1 (dashed) and y=-1 (dotted). The graph includes shaded regions around the Sobolev and PIKL estimators, suggesting uncertainty or confidence intervals. ### Components/Axes - **X-axis**: Labeled "x" with integer ticks at 1, 2, 3, 4, 5. - **Y-axis**: Labeled "y" with integer ticks at -3, -2, -1, 0, 1, 2. - **Legend**: Located at the bottom-left corner, with the following entries: - Dashed black: Norm of f* - Dotted black: PDE error - Pink: PDE estimator - Blue: PIKL estimator - Orange: Sobolev estimator - **Reference Lines**: - Dashed black horizontal line at y=1. - Dotted black horizontal line at y=-1. ### Detailed Analysis 1. **Norm of f* (Dashed Black Line)**: - Constant at y=1 across all x-values. - No variation observed. 2. **PDE Error (Dotted Black Line)**: - Constant at y=-1 across all x-values. - No variation observed. 3. **PDE Estimator (Pink Line)**: - Starts at approximately y=1.5 at x=1. - Gradually decreases to y=-1.0 at x=5. - Slope: Approximately -0.5 per unit x (calculated as (1.5 - (-1.0)) / (5 - 1) = 2.5/4 ≈ 0.625, but visually less steep due to curvature). 4. **PIKL Estimator (Blue Line)**: - Starts at approximately y=1.0 at x=1. - Decreases to y=-3.0 at x=5. - Slope: Approximately -0.8 per unit x (calculated as (1.0 - (-3.0)) / (5 - 1) = 4/4 = 1.0, but visually slightly less steep due to curvature). 5. **Sobolev Estimator (Orange Line)**: - Starts at approximately y=1.2 at x=1. - Decreases to y=-2.8 at x=5. - Slope: Approximately -0.8 per unit x (calculated as (1.2 - (-2.8)) / (5 - 1) = 4/4 = 1.0, but visually slightly less steep due to curvature). - Shaded region (confidence interval) widens slightly as x increases, peaking around x=2.5. ### Key Observations - The Sobolev and PIKL estimators exhibit nearly identical slopes but differ in starting values and endpoint magnitudes. - The PDE estimator shows a less steep decline compared to the PIKL and Sobolev estimators. - The Sobolev estimator’s shaded region indicates increasing uncertainty as x increases. - The Norm of f* and PDE error lines remain constant, suggesting they are independent of x. ### Interpretation The graph demonstrates that the Sobolev and PIKL estimators perform similarly in terms of slope but differ in baseline values. The PDE estimator’s shallower decline suggests it may be less sensitive to changes in x. The constant Norm of f* and PDE error lines imply these metrics are invariant to x, possibly indicating a fixed reference or error threshold. The widening confidence interval for the Sobolev estimator hints at increasing variability in its predictions as x grows. This could reflect model instability or higher sensitivity to input perturbations in the Sobolev framework compared to others.