Joint Contour Net Analysis for Feature Detection in Lattice Quantum Chromodynamics Data
Dean P. Thomas, Rita Borgo, Robert S. Laramee, Simon J. Hands
JJoint Contour Net Analysis for Feature Detection inLattice Quantum Chromodynamics Data
Dean P.Thomas a,b , Rita Borgo c , Robert S.Laramee a , Simon J.Hands b a Department of Computer Science, Swansea University, Swansea, United Kingdom b Department of Physics, Swansea University, Swansea, United Kingdom c Department of Informatics, Kings College London, London, United Kingdom
Abstract
In this paper we demonstrate the use of multivariate topological algorithms toanalyse and interpret Lattice Quantum Chromodynamics (QCD) data. LatticeQCD is a long established field of theoretical physics research in the pursuit ofunderstanding the strong nuclear force. Complex computer simulations modelinteractions between quarks and gluons to test theories regarding the behaviourof matter in a range of extreme environments. Data sets are typically gener-ated using Monte Carlo methods, providing an ensemble of configurations, fromwhich observable averages must be computed. This presents issues with regardto visualisation and analysis of the data as a typical ensemble study can generatehundreds or thousands of unique configurations.We show how multivariate topological methods, such as the Joint ContourNet, can assist physicists in the detection and tracking of important featureswithin their data in a temporal setting. This enables them to focus upon thestructure and distribution of the core observables by identifying them withinthe surrounding data. These techniques also demonstrate how quantitativeapproaches can help understand the lifetime of objects in a dynamic system.
Keywords: multivariate, topology driven visualisation, temporal data,analysis, volume visualisation, lattice quantum chromodynamics
Email addresses: (Dean P.Thomas), [email protected] (Rita Borgo), [email protected] (Robert S.Laramee), [email protected] (Simon J.Hands)
Preprint submitted to Big Data Research April 2, 2019 a r X i v : . [ h e p - l a t ] M a r . Introduction Recent advances in multivariate topological visualisation have provided newapproaches for detecting interesting phenomena in scalar fields of more than onevariable. Much of this work builds upon existing techniques used to understandthe topology of scalar fields — where critical events, such as the creation andmerging of unique features, are captured using graph structures. The ability toexamine multiple fields in parallel also presents a method for tracking objectsin higher dimensional data sets.In this paper we use the joint contour net (JCN) algorithm [1] to trackobjects with a finite lifetime across multiple time steps. (Anti-)Instantons are4D “pseudo-particles” studied by domain scientists that are localised to specificlocations in 4D Euclidean space-time. Existing statistical physics methods areable to predict the existence of these objects; however, more complex propertiessuch as their structure and lifetime are more difficult to evaluate.To demonstrate how multivariate topological visualisation techniques canbenefit lattice QCD scientists we focus upon analysing a single instanton pre-identified by existing physics methods. Due to the Euclidean nature of latticeQCD, where space and time are treated equivalently, the techniques used in thispaper can also be used to scan volumes with a temporal component ( xyt , xzt , yzt ) along a spatial axis. Whilst not seen as a direct replacement for viewingthese fields in their native 4D embeddings, this technique presents an interestingapproach to pin-point critical events in the topology of a single evolving field.By carrying out this case study we intend to answer the following questions: • Can we use the JCN to track an instanton between multiple neighbouringtime slices? • Can extra properties about instantons be determined through multivariatepersistence measures? • Can the Reeb skeleton be used to simplify the field and summarise prop-erties of observables? 2he remainder of this paper begins with an overview of the necessary back-ground information in Section 2. Section 3 briefly introduces the applicationused to carry out the work in this paper. We then give a description of theobserved data in Section 4, this forms the basis of a quantitative approach toevaluating the topological structure of the data in Section 5. We then showan experimental use of the JCN to locate and study the structure of an (an-ti-)instanton within an entire 4D data set in Section 6. The paper is concludedin Section 7 where we summarise our findings from the case study.
2. Background
In this section we introduce the relevant background information required toread the rest of the paper. We begin by providing an overview of multivariatetopological visualisation algorithms, before introducing aspects of lattice QCDthat are relevant to this paper. Much of multivariate topology uses a generalisa-tion of univariate topology concepts where structures such as the Reeb graph [2]summarise the topology of a scalar field. For a more in depth overview of theuse of topology in visualisation we refer the reader to the survey paper by Heineet al. [3].
The
Reeb space is a generalisation of the Reeb graph enabling multivariate ortemporal data to analysed. The first discussion of using the Reeb space to com-pute topological structure of multiple functions is presented by Edelsbrunneret al. [4], where it is suggested that the Reeb space can be modelled mathe-matically in the form f : M (cid:55)→ R k , where M represents the domain and f theoutput of k scalar functions. For the simple case, where k = 1, this is directlycomparable to the Reeb graph. The Reeb space extends this formulation tosituations where k ≥
2. 3 .1.1. The joint contour net
Carr et al. [1] presented the first discrete representation of the Reeb space usingthe
Joint Contour Net (JCN). For functions of n variables defined in an R m dimensional space the algorithm approximates the Reeb space as a number ofmultivariate contours named joint contour slabs. These represent connectedregions of the domain with respect to the isovalue of multiple functions. Insituations where n ≥ m the JCN can still be computed; however, the output isnot an approximation of the Reeb space but instead a subdivision of the inputgeometry over n variables. The JCN captures the Reeb space as an undirectedgraph structure, where vertices represents slabs of n isovalue tuples, and edgesare used to show adjacency between regions. An example JCN of two scalarfunctions is presented in Figures 1 and 2. The JCN has previously been usedto study multivariate data originating from nuclear scission simulations [5, 6, 7]and hurricane measurements [8].The Reeb skeleton is a simplified graph structure that takes into account thesize of connected components, allowing measures of persistence to be assignedto its arcs to aid multivariate simplification [9]. Lip pruning techniques, similarto the leaf pruning method of simplification found in univariate topologicalstructures [10] can then be applied to progressively remove noisy features inthe multi-field. Example persistence measures applied to the JCN include theaccumulated volume of joint contour slabs in a connected region. Alternatively,the Reeb skeleton can be used to quantify regions of the multivariate topologyfor analysis. Kenneth Wilson was the first physicist to suggest that a discrete 4D latticecould be used to model properties of quark-gluon fields [11]. The lattice is ahyper-torus in Euclidean space-time, meaning that the three spatial dimensionsand the time dimension are treated as equal. Periodic boundary conditions,where the minima and maxima on each axis are connected, are used so that itis impossible to consider any position on the lattice to be on a boundary. The4
91 88 56743 46 7 8 93 2 1 3843 756 129
Image taken from Duke et al. [5]Figure 1: Two simple scalar functions defined on a simplicial grid (left) where the dotted linesrepresent the quantisation intervals. The quantised contour tree for each function (right) isshown mapped to scalar field in the centre. (3,1) (4,1)(3,2) (4,2)(4,3) (5,2) (6,2) (7,2)(4,4)(5,4)(6,5)(7,6)(8,7) (7,7)(6,6)(5,5)(6,7)(5,4)(6,4)(6,3) (7,4) (8,3) (9,3)(3,5)(1,8)(1,9)(4,9) (3,9) (2,9)(6,8) (3,6)
Image taken from Duke et al. [5]Figure 2: The JCN capturing the bivariate topology of the two simple functions shown inFig. 1. The bivariate field is decomposed by overlaying the quantisation intervals of the twoinput fields (dotted line). A vertex is placed at the barycentre of each region, or joint contourslab, and edges mark adjacency. − m), with state of the art simulations having a lattice spacing a of 0 .
02 to0 .
04 femtometres.Quarks are located on the lattice at positions with integer indices referred toas sites . From each lattice site four link variables are used to model the gluonpotential in the x, y, z and t directions between two sites. Each link variable isa member of the special unitary group of matrices, identified using the notation SU ( n ). The value of n represents the number of charge colours used in thegauge theory, with true QCD defined with n = 3. However, in this work weuse a simplified two colour model of the theory using SU (2) matrices. Colour isused in this context to parametrise the concept of colour neutrality in a formsimilar to that of positive and negative charge. One of the primary reasons forusing a simplified model is that it allows us the freedom to vary the chemicalpotential of the system [12]. Chemical potential represents the energy changeas either a quark is added or an anti-quark is removed from a system. Varyingthe parameter enables exotic forms of matter, such as neutron star cores, to besimulated.The discrete nature of the lattice means it is possible to calculate pathsaround sites in space and time. Certain configurations of closed loops on thelattice are used to generate the scalar field observables of lattice QCD. The mostbasic unit closed loop in any directions is commonly referred to as a plaquette ;computing the average plaquette in all 4 dimensions produces the Wilson action observable. In this work we mainly focus upon the topological charge densityfield, computed as a loop in all four space-time dimensions from each site [13].Regions of the lattice where the topological charge density reach a minima ormaxima indicate the presence of (anti-)instanton pseudo-particles — one of theprimary observables of lattice QCD. These have a finite extent in the timedimension and are able to appear and disappear, unlike real particles.In order to reveal the structure of (anti-)instanton observables, the effect ofquantum fluctuations must be minimised through a noise reduction techniqueknown as cooling . After applying cooling what should remain are long range6hysical interactions that characterise the existence of an (anti-)instanton; how-ever, overly aggressive use of cooling can result in the destruction of the coreobservables.
3. Implementation
In this work we use the JCN implementation supplied as part of the MultifieldExtension of Topological Analysis (META) project [14]. This provides a numberof filters for multivariate data that can be applied to the visualisation pipelineas part of a VTK [15] workflow. Filters are included for creating an initial JCNdecomposition of the input fields that are able to be presented in graph form.As part of the process of creating the JCN individual joint contour slabs, in theform of polygon meshes, are created as the union of multiple smaller fragments.Input fields are placed on to a common set of sampling points in three di-mensions and each cubic cell is subdivided into 6 tetrahedra using a Freudenthalsubdivison. In order to handle periodicity an additional cell is constructed tolink the minimum and maximum samples on each axis.
This case study was performed using a modified version of the interactive toolused by Geng et al. for the analysis of hurricane data [8]. Modifications of thesoftware were largely made to facilitate collection of quantitative measurementsfor analysis. In addition, the transfer functions used to colour the glyphs havebeen modified to enhance feedback for lattice QCD fields that are often centredon zero [16, 17, 18]. Figures 3 and 4 give a visual overview of the user interface.
4. Visual analysis of lattice objects in four dimensions.
In the following section we give details of how an (anti-)instanton can be locatedand tracked using the JCN. The entire workflow is given, beginning with thesteps a domain scientist would use to locate a potentially interesting observablein a typical study. We also discuss important considerations such as the slabsize parameter which defines the resolution the scalar field is captured at.7 igure 3: The user interface used to explore temporal lattice QCD data as captured by theJCN. The JCN captures the topology as a graph structure (1) — displayed here in a springlayout. Each JCN glyph (2) represents a slab (or quantised contour) using scale to providefeedback on the overall size of the slab. Glyphs are coloured by isovalue (3), where the tophalf of each glyph represents the first input field and the bottom the second input. Also visibleis the Reeb skeleton (4) which captures the JCN in a simplified form; red glyphs representmajor changes in topological connectivity and blue glyphs represent relatively stable regionsof topological structure. Slabs are only rendered (5) to reflect any selections made by theuser; in this view nothing has been selected.Figure 4: The user interface allows the user to select nodes in the Reeb skeleton (1) or JCN(2) using rubber band selection. Green glyphs in the Reeb skeleton correspond directly to theblue glyphs in the JCN view. Selected vertices are displayed as quantised contours (3) usinga colour transfer function (4) determined by the bivariate input fields. ools S MAX Q MAX Q MIN
27 (11, 4, 13, 5) (11, 4, 13, 5) (6, 2, 7, 2)28 (9, 8, 10, 5) (9, 8, 10, 5) (6, 2, 7, 2)29 (9, 8, 10, 5) (9, 8, 10, 5) (6, 2, 7, 2)30 (9, 8, 10, 5) (9, 8, 10, 5) (6, 2, 7, 2)31 (9, 8, 10, 5) (9, 8, 10, 5) (6, 2, 7, 2)
Cools S MAX Q MAX Q MIN
32 (9, 8, 10, 5) (9, 8, 10, 5) (6, 2, 7, 2)33 (9, 8, 10, 5) (9, 8, 10, 5) (6, 2, 7, 2)34 (9, 8, 10, 5) (9, 8, 10, 5) (6, 2, 7, 2)35 (9, 8, 10, 5) (9, 8, 10, 5) (6, 2, 7, 2)36 (9, 8, 10, 5) (9, 8, 10, 5) (6, 2, 7, 2)
Table 1: The location on the lattice of global minima and maxima in the conp0050 configu-ration. Changes between cooling iterations are highlighted in bold . Data used in this case study originates from a lattice with 16 spatial sites and8 time sites, otherwise denoted as a 16 × t = 8 and t = 1. We also use the short temporal axis to push the limits of the JCN byqueuing multiple time slices on the multi-field to see if we are able to observe asignature for the entire 4D hyper-volume (Sec. 6).The configuration used in this study conp0050 originates from an ensemblewith a chemical potential µ = 0 .
7. The lattice is pre-cooled for 30 iterationsto a stable state, validated by inspection of the total topological charge densityand peak Wilson action graphs. At this point in the cooling process the totaltopological charge Q T OT remains flat for many cooling iterations (Fig. 5) andthe peak Wilson action S P EAK follows a smooth trajectory (Fig. 6).The stability of the lattice is also reflected by monitoring the location ofminima and maxima in the topological charge density Q and Wilson action S fields (Table 1). In this interval of the cooling process the predicted locations offield minima and maxima are extremely stable, indicating that the same objectis persisting throughout. The location of maxima in the Wilson action S MAX and topological charge density Q MAX coincide predicting the presence of an(anti-)instanton. 9
10 20 30 40 50 cools . . . . . . . . . T o t a l t opo l og i c a l c ha r geden s i t y Q T O T A L Total topological charge density - [config.b190k1680mu0700j02s16t08] conp0050
Figure 5: Total topological charge Q TOT cools . . . . . . . . . P ea k W il s ona c t i on S P E A K Peak Wilson action - [config.b190k1680mu0700j02s16t08] conp0050
Figure 6: Peak Wilson action S PEAK
In order to use the 4D topological charge density with the JCN algorithm [14]the dimensionality is reduced to 3D by slicing along the the t axis. Each JCNin this case study is constructed from two neighbouring time-slices ( t n , t n +1 ),additionally as the time axis is also periodic in lattice QCD the JCN ( t max ( n ) , t min ( n ) ) is a valid input configuration. Hence, for the 16 × t , t ).10 .2.1. Slab size parameter Throughout this study the JCN slab size parameter is fixed dividing the topolog-ical charge density, with approximate range [ − . , . = 512 intervalsto give a slab size of 0 . Topo-logicalchargeden-sity t = 4 t = 5(maxima) t = 6Figure 7: View of the main instanton observable as univariate slabs. Figure ?? presents the eight separate JCNs created by evaluating each pair( t n , t n +1 ) of temporal fields. A fixed colour transfer function is used, basedupon on the peak magnitude in four-dimensions, in order to present a truerepresentation of potential (anti-)instantons in the data. This technique waschosen as lattice QCD observables must be considered as global extrema in 4D,rather than as localised between two time steps.11nitially it is possible to locate an anti-instanton in the JCN for t = (1 , t = (2 ,
3) using the same approach; however,the bottom half of the glyph turns green to indicate a neutral isovalue. Thestructure of the anti-instanton continues to persist in the data beyond this timeslice despite the change in isovalue. Recovery of the anti-instanton slab structurein later time slices requires some exploration of JCN branches using knowledgeof the objects location. The anti-instanton becomes a prominent feature in thebivariate topology at t = (8 ,
1) where the coloured glyphs indicate the presenceof a global minima.
Topo-logicalchargedensity
Figure 8: An anti-instanton can be located in the t = (1 ,
2) JCN as the branch with blueglyphs. This object continues to exist for several time steps.
Also present in the JCN overview (Fig. ?? ) is a global maxima which firstbecomes prominent in the multivariate glyphs at t = (4 , t = 5. The instanton quickly begins to fade into the surrounding latticestructure at t = (5 ,
6) where the multivariate glyphs indicate a return to moreneutral isovalues (Fig. 9).
Topo-logicalchargedensity
Figure 9: An instanton can be located in the t = (5 ,
6) JCN as the branch with red & yellowglyphs. The bottom half of the glyphs indicate that the topological charge density quicklydrops off from a global peak at t = 5. Several other localised features exist in the data at various points in thetemporal JCNs. A second potential anti-instanton appears in the JCN at t =(7 , t = (8 , .4. Visually tracking an instanton across the temporal axis In the following section the bivariate topology, as captured by the JCN and Reebskeleton, is examined in greater detail. We begin by looking at the identifiedglobal maxima in four dimensions, predicted by the cooling code as being at(9 , , ,
5) and continue across the periodic boundary back to the origin.
Time steps t = 4 and t = 5 . The instanton is located by examining the JCNvertices using the coloured glyphs relating to isovalue. The JCN, when drawn indomain layout, shows the approximate location of barycentre of the slabs makingup the instanton. Displaying the slab geometry makes it possible to validatethat the location of the object agrees with the predicted location (9 , , , Time steps t = 5 and t = 6 . The Reeb skeleton highlights two significantfeatures for t = (5 , ime steps t = 6 and t = 7 . The instanton is located using the JCN as signifi-cant feature of the topology. At t = 6 the instanton appears as a maxima in thetopological charge density field; however, at t = 7 the isovalue of the instantonis drastically reduced to neutral. The Reeb skeleton also identifies a potentialanti-instanton, existing as a local minima at t = 7. Also present is a secondarymaxima that replaces the instanton structure as the local maxima at time slice t = 7. Time steps t = 7 and t = 8 . The instanton structure continues to remain visiblein the t = (7 ,
8) JCN. The object becomes a less significant feature of the JCNwith green glyphs revealing that the isovalue has reduced to near zero. How-ever, enough topological structure remains to separate the instanton from thesurrounding region of percolation. Due to reduction of isovalue range represent-ing the object the Reeb skeleton discards the instanton, instead determining itto be topological noise.
Time steps t = 8 and t = 1 . At standard resolution the JCN, with inputs setto t = 8 and t = 1, failed to reveal any structure relating to the instantonobservable. However, when the slab size is decreased the instanton can beisolated from the surrounding topology charge density. A halving of the slabsize to 0 . = 1024 intervals, is sufficient to allow the objectto be located. Examination of the geometric structure of the instanton, throughthe joint contour slabs, reveals that the shape found in earlier time slices beginsto merge into the surrounding lattice field. However, it is possible to confirmthat the selected object relates to the instanton by observing the JCN in domainlayout (Figure 10). Time steps t = 1 and t = 2 . The JCN for time-steps t = (1 ,
2) reveals thatthe instanton can still be differentiated from the surrounding topological chargedensity by again halving the slab size to 0 . opo-logicalchargedensity Figure 10: The JCN in domain layout for t = (8 ,
1) displayed alongside the joint contourslabs. The selected glyphs (11) represent the instanton and correlate with the slab structure. skeleton. This coincides with the global minima Q MIN estimated by the coolingcode to be present on the lattice at (6 , , , Time steps t = 2 and t = 3 . The main instanton observable reappears in thebivariate topology at standard resolution with inputs t = (2 , Time steps t = 3 and t = 4 . The Reeb skeleton detects the instanton as the mostprominent feature. The slab structure captured by the JCN is well defined andresembles that of other time slices. The isovalues associated with the instantonobject at t = (3 ,
4) shows a large jump in isovalue between the two time-slices,captured by the colour change from red to green in the JCN.
5. A quantitative approach to instanton tracking
We have shown how the instanton can be tracked by the JCN visually; however,for domain scientists a more quantitative approach is required. In the following16ection we look at what statistical measurements are available using bivariatetopological structures. It should be noted that each of these process requiremanual locating of the target object.
A basic measure of persistence can be computed from the JCN by countingthe number of slabs making up a topological region. Each vertex in the JCNrepresents a joint contour slab, or region of the quantised Reeb space, associatedwith a pair of isovalues. Throughout the case study it was found that the numberof slabs linked to an object tended to vary with two properties of the object;the volume of the slabs, and the isovalue range of the object. Sparse objectswith a wide range of isovalues frequently appear as sheet-like structures in theJCN, and densely packed objects appear with a branch-like structure.We found that it was possible to get a rough estimate of properties of latticeobjects in the quantised Reeb space by calculating the percentage of vertices inthe JCN contributing to the object, as detected through manual selection.
Input fields Instanton Entire JCN Percentagesub-graph structure t = (1 ,
2) 0 572 0 t = (2 ,
3) 5 371 1 . t = (3 ,
4) 39 339 11 . t = (4 ,
5) 199 584 34 .
08 Input fields Instanton Entire JCN Percentagesub-graph structure t = (5 ,
6) 212 472 44 . t = (6 ,
7) 44 237 18 . t = (7 ,
8) 6 360 1 . t = (8 ,
1) 0 806 0
Table 2: Number of JCN vertices contributing to instanton structureAll values are taken using a slab size of 0 . Table 2 presents the number of vertices present in each JCN across thetemporal axis at the standard slab size. When plotted as a histogram, as inFigure 11, a peak in the number of vertices in the manually identified instantonsub-graph coincides with Q MAX .The JCNs show an increase in the number of total vertices on the lead upto Q MAX and Q MIN , followed by a drop in graph complexity after each event.The peaks were expected to coincide with the emergence of the anti-instanton17 = (1, 2) t = (2, 3) t = (3, 4) t = (4, 5) t = (5, 6) t = (6, 7) t = (7, 8) t = (8, 1)input fields nu m be r o f node s Number of nodes in joint contour net that relate to instantonInstanton sub-graphEntire jcn
Figure 11: Number of slabs in the JCN representing the instanton structure. and instanton; however, both peaks seem to proceed the time-steps containingthe (anti-)instanton structures rather than directly matching them.Viewing the number of vertices in the instanton sub-graph as a percentageof the entire JCN (Fig. 12) suggests that the structure dominates the quantisedReeb space at its peak value, where t = 5, agreeing with the predicted globalmaxima Q MAX . In the t = (1 ,
2) and t = (8 ,
1) JCNs the percentage of verticesis zero — this is where the instanton structure was not detected at all at thestandard resolution.
The Reeb skeleton summarises the multivariate topology in a more compactgraph structure. Adjacent slabs are merged, provided no critical events coincidewith the slab. This allows entire path connected regions to be summarise as asingle unit, represented by a vertex in the Reeb skeleton. Besides simplification18 = (1, 2) t = (2, 3) t = (3, 4) t = (4, 5) t = (5, 6) t = (6, 7) t = (7, 8) t = (8, 1)input fields pe r c en t ageo f node s Percentage of nodes in joint contour net that relate to instanton
Figure 12: Percentage of slabs in the JCN representing the instanton structure. of the quantised Reeb space, the Reeb skeleton enables the generalisation ofpersistence measures used in univariate topology to multivariate topology. Eachvertex in the Reeb skeleton has persistence measures attached representing theconnected components (the joint contour slabs), allowing us to further analysethe instanton structure.This section examines how different persistence measures, used during sim-plification, affect the ability of the Reeb skeleton to detect the instanton struc-ture. The simplification measures are defined as follows:
Reeb skeleton.
Collapses adjacent slabs into a single vertex, meaning entire sheetor branch-like structures in the JCN can be summarised by a single branch ofthe Reeb skeleton. Topological events, such as splits and merges, are capturedas vertices with degree 3 or higher. No simplification is performed, frequentlyleading to the creation of multiple leaf vertices along branches of the Reeb19keleton.
Simplified Reeb skeleton.
This is the full Reeb skeleton except a basic pruningof non-critical vertices is performed. First, all degree one singular vertices (leafnodes) are removed from the Reeb skeleton and replaced with regular vertices.Next any degree two singular vertices are replaced with regular nodes. Finally,any regular leaf vertices are merged with their neighbours until a singular vertexis encountered. This has the effect of collapsing large branches of regular verticesto a single vertex representing the entire set of connected components.
Persistence simplified Reeb skeleton.
The Reeb skeleton is first pruned of non-critical vertices, as described above. Following this each of the remaining com-ponents in the graph are assigned a level of persistence based upon the quantityof JCN nodes that make up the sub-graph represented by the vertex. Next reg-ular vertices of the JCN that fall below a specified threshold are removed fromthe graph. Finally, any remaining non-critical vertices are removed by repeatingthe basic pruning technique.
Volume simplified Reeb skeleton.
This method of simplification is similar to“persistence simplification” except instead of counting the number of JCN nodesin a sub-graph of connected components, the volumes are approximated bycounting the number of fragments passed through. Each fragment representsa tetrahedra cell in the quantised Reeb space, counting the number of these ineach slab gives a rough estimate of volume.Table 3 shows the effect that various forms of simplification have on the Reebskeleton, also visualised in Figure 15. This confirms that under simplificationthe Reeb skeleton often reduces regions of the bivariate topology representingthe instanton to a single vertex.Simplification using basic leaf pruning and volume measures at first appearto give similar results. However, when viewing the effect by cross referencing theJCN (Table 4, Fig. 16) variations appear, as smaller slabs are filtered out undersimplification. The simplification preserves the observed forking behaviour in20 = (1 , t = (2 , t = (3 , t = (4 ,
5) t = (5 , t = (6 , t = (7 , t = (8 , Table 3: Number of vertices in Reeb skeleton after simplification t = (1 , t = (2 , t = (3 , t = (4 , t = (5 , t = (6 , t = (7 , t = (8 , Table 4: Number of node in JCN relating to instanton after simplification the instanton structure at t = (5 , igure 13: Reeb skeleton for t = (5 , Topo-logicalchargedensity
Slab structure Outer core Inner coreFigure 14: The forking behaviour visualised using the slab geometry. The full instantonstructure (1) splits into outer (3) and inner (4) parts. based upon the number of connected components. This removed the intendedinstanton observable from the simplified topology several time-steps earlier thanother simplification techniques. We believe this was likely caused by an overlystrict threshold level and highlights the need for careful selection of simplificationparameters. 22 = (1, 2) t = (2, 3) t = (3, 4) t = (4, 5) t = (5, 6) t = (6, 7) t = (7, 8) t = (8, 1)input fields nu m be r o f node s Number of nodes in Reeb skeleton relating to instantonNo simplificationBasic simplificationPersistence SimplificationVolume simplification
Figure 15: Number of nodes in the
Reeb skeleton representing theinstanton under simplification. t = (1, 2) t = (2, 3) t = (3, 4) t = (4, 5) t = (5, 6) t = (6, 7) t = (7, 8) t = (8, 1)input fields nu m be r o f j c nnode s Number of nodes in joint contour net relating to instantonas identified by Reeb skeleton No simplificationBasic simplificationPersistence simplificationVolume simplificationManual selection
Figure 16: Number of nodes in the
JCN representing the instantonunder simplification.
6. Feature detection using the entire lattice data
The final use of the JCN is an experimental attempt to detect the instantonusing all eight time-slices as input. Whilst the Reeb space is only well defined23 anual selection. Unsimplified Reeb skeletonselection. Volume simplified Reeb skele-ton selection.Figure 17: Comparison of simplification techniques on the t = (6 ,
7) joint contour net. for situations where m > n , the JCN algorithm can be used on data wherethis condition is not met. In this situation we will be dealing with data definedon an m = 3 dimensional mesh with n = 8 function values. The output ofthe algorithm is a subdivision of the input field with regard to all 8 functions;embedded amongst this we expect to find structure relating to the instanton.The JCN for t = (1 , , , , , , ,
8) is shown in Figure 18, featuring 36450vertices and 104373 edges. The structure clearly splits the input space intodistinct topological objects, meaning it is possible to select a single branch ofthe JCN to isolate the instanton structure (1). Viewing the instanton as jointcontour slabs (Fig. 19) shows an object very similar to that output by thebivariate JCNs.Stepping through the data, colouring the graph by isovalue for each time-step, allows the variation in isovalues of each branch to be observed. Whenreaching peak topological charge density Q MAX at time-step t = 5 all othervertices in the graph have very low relative isovalues, indicated by blue glyphs.This is largely a side-effect of the colour transfer function being relative to thedata set in view — for Q MAX this offsets the values further from zero thanother time-steps. The same effect is less obvious for Q MIN at t = 2, but canstill be observed. 24 opo-logicalchargedensity Figure 18: Top: The JCN for t = (1 , , , , , , , t = 5. It is also possible to observeelevated levels in the topological charge density in neighbouring time slices. opo-logicalchargedensity Figure 19: The slab structure of the instanton recovered from the JCN of t = (1 , , , , , , ,
7. Conclusion
We have shown in this paper how recent advances in multivariate topologicalanalysis can be applied to lattice QCD data sets. We have demonstrated howmultivariate topology can be used for comparing data with a temporal elementin order to observe critical topological events as the time step is varied. Thiscase study also considered some basic multivariate persistence measures avail-able in the data, such as the number of slabs. These measures are suited toevaluation as lattice simulation parameters are varied as their values can becollected autonomously.In summary this paper makes the following contributions:26
The JCN was used to track and observe the structure of an instanton in4D space-time. • Quantitative measures were taken directly from the JCN and Reeb skele-ton to evaluate the importance of lattice observables in the context of thetopology of the lattice. • We demonstrated how multivariate simplification metrics could potentiallybe utilised to locate important observables in lattice QCD. • It was also demonstrated how the JCN can be used to reduce the structureof a 4D object to a 3D approximation.The study carried out in this paper forms part of a much larger body of workstudying the use of topological visualisation techniques in lattice QCD [19].The quantitative approaches demonstrated here can be used to perform anal-ysis on ensembles of hundreds of configurations. This is a typical use case forlattice QCD scientists, where averages across a large sample of configurationsare needed to evaluate the effect of changing simulation parameters. We give amore detailed review of how a domain expert may perform such a study in [20].
Acknowledgements
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