# A Top-Down Interactive Visual Analysis Approach for Physical Simulation Ensembles at Different Aggregation Levels

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## Abstract

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## 1. Introduction

## 2. Related Work

## 3. Visual Analysis of Multi-Run Simulation Data

- Overview analysis of field range distribution. In a first stage, one is interested in getting an overview of the ensemble, which can be achieved by investigating the range of the considered data field and the distribution of field values within the simulation runs. Respective histograms allow for first conclusions and to narrow down the field range for subsequent analysis stages (see Section 3.1).
- Analysis of field distribution over time. In this stage, one would like to investigate the change within the simulation runs over time, which supports multiple important tasks. First, one can detect features and the time intervals they occur, which narrows down the time interval for further analysis steps. Second, one can identify individual field values of interest, which can be further examined, e.g., by choosing them as isovalues. Third, one can detect overall patterns in the ensemble as well as outliers. A run identified to be of interest can also be observed individually as well as in further analyses with physical domain visualizations. Finally, one can also compare and correlate different fields of a multi-field data set at this level (see Section 3.2).
- Comparative analysis of individual runs. While the second stage was operating on an aggregation over multiple runs, this stage shall allow for a detailed understanding of the behavior of individual runs in a comparative view. Making appropriate selections in the preceding stage (i.e., identifying time interval and field value of interest) allows for an accurate and efficient analysis approach (see Section 3.3).

#### 3.1. Field Distribution Histogram

#### 3.2. Function Plot

#### 3.3. Similarity Plot

#### 3.4. Top-Down Analysis

## 4. Implementation Features

#### 4.1. Computation of Field Distribution Histogram for Interactive Use

#### 4.2. Computation of Function Plot for Interactive Use

#### 4.3. MDS for Similarity Plot Generation

- Set up matrix of squared distances $\mathbf{P}=[{d}_{i,j}^{2}]$.
- Apply double centering: $\mathbf{B}=-\frac{1}{2}\xb7\mathbf{J}\xb7\mathbf{P}\xb7\mathbf{J}$, where $\mathbf{J}=\mathbf{I}-{n}^{-1}\xb7\mathbf{E}$, $\mathbf{I}$ is the identity matrix, $\mathbf{E}$ is the matrix with all entries being 1, and n is the number of samples.
- Extract the m largest positive eigenvalues ${\lambda}_{1},\dots ,{\lambda}_{m}$ of $\mathbf{B}$ and the corresponding m eigenvectors ${v}_{1},\dots ,{v}_{m}$.
- An m-dimensional spatial configuration of the n objects is derived from the coordinate matrix $\mathbf{X}={\mathbf{V}}_{\mathbf{m}}\xb7{\mathsf{\Lambda}}_{\mathbf{m}}^{\mathbf{1}/\mathbf{2}}$, where ${\mathbf{V}}_{\mathbf{m}}$ is the matrix of the m eigenvectors and ${\mathsf{\Lambda}}_{\mathbf{m}}$ is the diagonal matrix of the m eigenvalues of $\mathbf{B}$.

## 5. Case Studies

#### 5.1. Astrophysical Simulations

**Stage 1—Field Distribution Histogram.**We start our analysis by computing the field distribution histogram for the scalar field of Internal Energy as shown in Figure 5. It is a simple plot, but nevertheless allows for some first interesting observations: (1) The distribution is skewed towards the lower values. In fact, only very few values are populating the upper half of the histogram. The respective simulation runs can immediately be identified as outliers by selecting the respective regions in the histogram. (2) After having identified the outliers, further analysis steps shall be applied to a narrowed (more saturated) field interval that excludes the outliers. This will make the automatic application of the transfer function in Stage 2 more effective. (3) Higher values do not occur in all simulation runs (red). The intersection areas (green and blue) are rather small. Still, due to the smooth transition, the entire range up to the dashed line seems to be of interest.

**Stage 2—Function Plot.**In the second stage, we operate on the function plots. Figure 6 shows the function plot for a single simulation run that was identified as an outlier in Stage 1. In this simulation run, both stars have the same mass. To observe the outlier values, we did not apply the narrowing of the field range from Stage 1. We can observe that there are very few field values with an internal energy greater than $3.0$ and that they occur around time step 300. Selecting those outliers and investigating them in a coordinated physical space visualization, one can observe that they belong to particles that transition from one star to the other. When hitting the other star the internal energy of these particles suddenly rises to high values, but also very quickly drops down again.

**Stage 3—Similarity Plot.**Having cropped the time intervals in Stage 2, which significantly reduces the amount of data to be handled, and having identified a representative field value, we can make use of that field value as the selected isovalue for the isosurface similarity computation. Having computed the isosurface similarity matrix, we generate the similarity plot. For the given application, we decided to generate a 3D similarity plot (see Figure 12a), which can be visually inspected using rotation and zooming. It shows all 45 astrophysical simulations. The polylines are color-coded using a continuous transfer function that maps the simulation parameter of the star’s mass to the hue of the color. Increasing ratio between the stars’ masses leads to changing the color towards yellow. We can observe a clear structure in the 3D similarity plot. Figure 12a confirms the finding from Stage 2 that we have three simulation phases. When selecting a point in the plot, the field of the respective time step of the respective run is displayed in the physical space visualization. Figure 12b–d show the physical space visualization of the selections made in Figure 12a. The linked views represent the initial phase (b), the merging phase (c), and the final phase (d). Moreover, when looking at the projection, we can see that beside of the main behavior pattern there is another repeating pattern, which produces a rotational structure in the upper part of the projection. Investigating this feature using linked views for different projection points it becomes clear, that this pattern is due to the rotation of the stars around their center of masses during the simulation.

**Domain Expert Feedback.**We discussed the MultiVisA tool and its components with the domain expert who generated the data set. We asked for advantages and limitations of our approach and to comment on the effectiveness or usefulness of our approach. The main findings were:

- To identify simulation features within a whole ensemble, researchers are using their own scripts and subroutines, as even advanced applications such as SPLASH [21] do not provide enough functionality. With our tool a multi-run simulation analysis becomes easy to visualize and it allows for a faster data investigation.
- The task of time alignment is one of the most time consuming for the researchers. From expert’s experience to perform the alignment on an ensemble of 250 runs one needs to spend couple of weeks, while with our tool one can do it by a single click.
- Correct and precise time definition of the analyzed features leads to increased accuracy of the analysis steps. For example, one can significantly increase the quality of the MDS projection when narrowing down the time interval. Figure 13d shows the same similarity plot as in (c), where in (c) one could use all time steps within the shorter time interval, while in (d) one could only use every other time step of the full time series.
- The domain expert has been working on this data set for a long time and knows it very well. Using our tools he was able to recognize most of the known data features in one session. Moreover, he even identified some additional features for further investigation.

#### Investigating Projection Dimensions

#### 5.2. Global Climate Simulations

- Visualization of the entire ensemble at once allows for estimating the diversity of the simulations’ behavior and identifying patterns and outliers.
- A strong advantage is the option to easily estimate activities of subregions. Usually one would need to look at some physical domain visualization for some selected time steps. Our tools leads to increased accuracy of feature detection.
- Estimating the influence of initial conditions to the simulation result is usually performed in sensitivity studies. A large number of statistical descriptors needs to be used. While it is complicated to capture the behavioral differences with a single value descriptor, our approach captures them in a multidimensional fashion and allows for interaction and navigation.

#### 5.3. Local Climate Simulations

#### 5.4. Computation Times

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) MultiVisA: Interactive visual analysis system for physical simulations: (

**left**) interaction panel for options and data settings; (

**top middle**) transfer function used for function plots; (

**top right**) plot view used for field distribution histograms or function plots; (

**bottom right**) similarity plot; (

**bottom middle**) domain visualization. (

**b**) System structure: orange arrows show analysis pipeline, black arrows show possible interactions between different components.

**Figure 2.**Schematic illustration of the similarity plot idea. Curves represent simulation states over time, where distances between points on the curves represent dissimilarity of corresponding simulation states.

**Figure 3.**Illustration of a re-scaling for the field distribution histogram for minimal resolution ${R}_{hist}=4$ and bin size changing from ${B}_{hist}=0.5$ to ${B}_{hist}=1.0$.

**Figure 4.**Illustration of hierarchical LOD scheme for function plots with resolutions ${L}_{0}=4$, ${L}_{1}=2\xb7{L}_{0}-1=7$, and ${L}_{2}=2\xb7{L}_{1}-1=13$.

**Figure 5.**Field distribution histogram (for astrophysical simulation). Field values from the intersection of all time steps’ ranges of all runs are colored in green, from the intersection of all simulations’ ranges (but not for all time steps) in blue, otherwise red.

**Figure 6.**Function plot of the simulation of two stars both with masses equal to 1.05 of the solar mass. Time steps around 300 contain outliers in field values and exhibit a significant change in the simulation structure, while before and after this change almost steady patterns can be observed.

**Figure 7.**Function plot of the simulation of two stars with different masses equal to 0.65 and 1.05 of the solar mass. Same three phases can be distinguished as in Figure 6, but additional feature can be observed for a hot matter. Interactive selection (shown in green) for coordinated view to the linked physical domain visualization.

**Figure 8.**Linked views of selections in Figure 7 in physical domain. Selection (

**a**) represents two separated stars. Selection (

**b**) shows that the shell of the core of one star is in the same condition, while selection (

**c**) shows that the matter of the other star is absorbed by the first one. Selection (

**d**) shows the merged structure. Please note that the representation of the heavier star seems smaller, as it represents data points with higher internal energy and therefore is only a core.

**Figure 9.**Function plot aggregated from all 45 astrophysical multi-run simulations without synchronization. General structure cannot be recognized. Vertical discontinuities indicate ends of simulations.

**Figure 10.**Result of automatic function plot synchronization for the astrophysical multi-run simulation data. As opposed to the unsynchronized representation in Figure 9, details of the general structure (three phases) can be observed.

**Figure 11.**Function plot representing standard deviation from Figure 10. Despite of similar structure of the field distribution in all simulation runs in the ensemble, this plot shows high deviations for lower field values. Green horizontal line indicates selection of representative field value used for isocontour similarity computation.

**Figure 12.**3D similarity plot (

**a**) with selected keyframes displayed in linked views to the domain visualization (

**b**–

**d**).

**Figure 13.**(

**a**) Plotting first principal component of projection over time. (

**b**) 2D similarity plot for one selected time step with embedded physical space visualizations. (

**c**) Similarity plot of one selected simulation run after cropping time interval. (

**d**) Similarity plot of same selected simulation run for the full time series but skipping every other time step, which leads to down-sampling artifacts.

**Figure 14.**Similarity plot matrix of 8-dimensional MDS projection for the astrophysical simulation ensemble (45 simulations differently color coded) with 28 different combinations of 8 dominant eigenvectors.

**Figure 15.**(

**a**) Field distribution histogram for sea snow thickness in global climate simulation. (

**b**) Interactive selection of a field value and linked visualization (

**c**) of its distribution of appearance. (

**d**) Interactive selection of a domain area and linked visualization (

**e**) of its field histogram.

**Figure 16.**(

**a**) Function plot for snow thickness aggregated over all climate simulation runs exhibits annual patterns of 3 years, which are selected as shown in green. (

**b**,

**c**) Function plots for snow thickness when filtering the trajectories according to selections a and b. (

**d**) Function plots for ice thickness (as in Figure 1) when filtering the trajectories according to selection a.

**Figure 17.**Coordinated views to physical space visualizations of selections in Figure 16 exhibit that selections correspond to arctic region (

**a**) and antarctic region (

**b**).

**Figure 18.**(

**top**) Function plot for sea surface temperature aggregated over all climate simulation runs exhibits annual patterns of 3 years. (

**middle**) Interactive selection of spatial region of interest (red) and display of similarly behaving spatial regions (green) using information from the function plot. (

**bottom**) Function plot for sea surface temperature when filtering the trajectories according to selected area.

**Figure 19.**Plotting principal component of projection (vertical axis) over time (horizontal axis) for all 11 simulations when selecting arctic (

**a**) and antarctic (

**b**) region separately. For isocontour similarity, we considered isovalue $0.25$ of sea snow thickness. We observe no activity during summer months in (

**a**), but activity throughout the year in (

**b**). Please note that the metric for the distance computation returns absolute values such that both plots are oriented the same way.

**Figure 20.**(

**a**) Field distribution histogram for water temperature t in the local climate simulation. (

**b**,

**c**) Linked visualizations of the spatial distribution of appearance for the selected temperatures in (

**a**) using transfer function (

**d**).

**Figure 21.**(

**a**) Function plot for water temperature aggregated over all simulation years. (

**b**) Function plot for water temperature when filtering the trajectories according to the small area highlighted in Figure 20b,c.

**Figure 22.**(

**a**) Field distribution histogram for a water temperature in the local climate simulation. (

**b**,

**c**) Linked visualizations of the distribution of appearance for the selected in (

**a**) field values.

**Table 1.**Pre-computation times (in seconds) for the visualization approaches used in the examples above.

Data | FH | FP | DM | MDS |
---|---|---|---|---|

Astrophysical data | 8743 | 7014 | 3107 | 679 |

Global climate data | 1061 | 1037 | 214 | 70 |

Local climate data | 117 | 100 | 103 | 189 |

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## Share and Cite

**MDPI and ACS Style**

Fofonov, A.; Linsen, L.
A Top-Down Interactive Visual Analysis Approach for Physical Simulation Ensembles at Different Aggregation Levels. *Information* **2018**, *9*, 163.
https://doi.org/10.3390/info9070163

**AMA Style**

Fofonov A, Linsen L.
A Top-Down Interactive Visual Analysis Approach for Physical Simulation Ensembles at Different Aggregation Levels. *Information*. 2018; 9(7):163.
https://doi.org/10.3390/info9070163

**Chicago/Turabian Style**

Fofonov, Alexey, and Lars Linsen.
2018. "A Top-Down Interactive Visual Analysis Approach for Physical Simulation Ensembles at Different Aggregation Levels" *Information* 9, no. 7: 163.
https://doi.org/10.3390/info9070163