## Flow Field Clustering via Algebraic Multigrid

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Citations: | 20 - 3 self |

### BibTeX

@MISC{Griebel_flowfield,

author = {M. Griebel and T. Preusser and M. Rumpf and M. A. Schweitzer and A. Telea},

title = {Flow Field Clustering via Algebraic Multigrid},

year = {}

}

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### Abstract

We present a novel multiscale approach for flow visualization. We define a local alignment tensor that encodes a measure for alignment to the direction of a given flow field. This tensor induces an anisotropic differential operator on the flow domain, which is discretized with a standard finite element technique. The entries of the corresponding stiffness matrix represent the anisotropically weighted couplings of adjacent nodes of the domain mesh. We use an algebraic multigrid algorithm to generate a hierarchy of fine to coarse descriptions for the above coupling data. This hierarchy comprises a set of coarse grid nodes, a multiscale of basis functions and their corresponding supports. We use these supports to obtain a multilevel decomposition of the flow structure. Standard streamline icons are used to visualize this decomposition at any user-selected level of detail. The method provides a single framework for vector field decomposition independent on the domain dimension or mesh type. Applications are shown in 2D, for flow fields on curved surfaces, and for 3D volumetric flow fields.

### Citations

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(Show Context)
Citation Context ...select a local level of detail. Finally, we draw conclusions in Section 9. 2 AN OPERATOR ENCODING THE FLOW ALIGNMENT Any clustering algorithm relies on a given coupling of the objects to be clustered =-=[15]-=-. Our goal is to cluster flow fields, so we first define a local coupling on the flow domain. We desire a strong coupling in the direction of the flow and a weak coupling orthogonal to the flow. Let u... |

362 |
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Citation Context ... steps. For every domain D l,i on a given level l, we construct a closed triangle mesh that bounds D l,i . Next, we relax (smooth) these meshes using e.g. a Laplacian filter or a windowed sync filter =-=[19]-=-. As a result, the meshes become slightly smaller, which allows us to better separate them visually. Next, we implement an interactive navigation scheme in which domains D l,i can be made half or comp... |

152 | Image-Guided Streamline Placement
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(Show Context)
Citation Context ...ds, alext@win.tue.nl IEEE Visualization 2004 October 10-15, Austin, Texas, USA 0-7803-8788-0/04/$20.00 ©2004 IEEE user-prescribed spacing. Different heuristics [16] and energy minimization stragegies =-=[26]-=- are used to compute the icons’ positions. Several multiscale clustering approaches for vector data have been proposed. These methods build a hierarchical clustering tree and visualize the clusters wi... |

127 |
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(Show Context)
Citation Context ...lgebraic multigrid algorithm (AMG) and the heuristics which led to its development. We refer to [25] for a detailed introduction. Algebraic multigrid methods were first introduced in the early 1980’s =-=[2, 3, 4, 5, 18]-=- for the solution of discrete linear systems AU = F of equations coming from the discretization of a linear differential equation A u = f on a domain Ω with suitable boundary conditions. Here U is sup... |

117 | Creating evenly-spaced streamlines of arbitrary density
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Citation Context ...sity, Den Dolech 2, Eindhoven, Netherlands, alext@win.tue.nl IEEE Visualization 2004 October 10-15, Austin, Texas, USA 0-7803-8788-0/04/$20.00 ©2004 IEEE user-prescribed spacing. Different heuristics =-=[16]-=- and energy minimization stragegies [26] are used to compute the icons’ positions. Several multiscale clustering approaches for vector data have been proposed. These methods build a hierarchical clust... |

109 | Image Based Flow Visualization
- Wijk
- 2002
(Show Context)
Citation Context ...ructed by the AMG can be used to design different visualization techniques as well (cf. Fig. 12 c). One idea is to use them as color and/or transparency information in texture advection based methods =-=[27, 20]-=- to produce multiscale textured animations of flow fields. 41Figure 11: Climate dataset decomposition, five coarsest levels (left to right). Domains (top row) and flow texture overlaid with curved ar... |

78 | Algebraic multigrid based on element interpolation (AMGe
- Brezina, Cleary, et al.
(Show Context)
Citation Context ...el l −1 appropriately to define the coarse basis {Ψ l,i } on level l. This merging is actually encoded in the columns of the prolongation matrix P l . Common to nowadays algebraic smoothness criteria =-=[18, 6, 7]-=- is the general observation that a simple relaxation scheme – most often Gauss–Seidel smoothing – efficiently damps components in the direction of eigenvectors associated with large eigenvalues. Conse... |

69 | DESBRUN M.: Discrete multiscale vector field decomposition
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(Show Context)
Citation Context ...od avoids some of the critical point detection problems in standard topological methods. A similar method that also incorporates multiple scales via component filtering has been recently presented by =-=[23]-=-. Finally, we mention the class of physics-based, continuous clustering approaches. These methods simulate a physical process, such as anisotropic diffusion [9, 11], applied to an initial fine-grained... |

63 |
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(Show Context)
Citation Context ...lgebraic multigrid algorithm (AMG) and the heuristics which led to its development. We refer to [25] for a detailed introduction. Algebraic multigrid methods were first introduced in the early 1980’s =-=[2, 3, 4, 5, 18]-=- for the solution of discrete linear systems AU = F of equations coming from the discretization of a linear differential equation A u = f on a domain Ω with suitable boundary conditions. Here U is sup... |

55 | Image Based Flow Visualization for Curved Surfaces
- Wijk
- 2003
(Show Context)
Citation Context ...n Sec. 4. Figure 10: MHD flow decomposition As an illustration, we give the multiscale decomposition of the average wind stress field on the surface of the Earth in Fig. 11 (the dataset is taken from =-=[28]-=-). The flow texture in the bottom row was produced with the IBFV method described in [28]. 7 LOCAL LEVEL OF DETAIL Our multiscale method presented so far displays a flow field at a user-selected globa... |

52 |
Simplified Representation of Vector Fields
- Telea, Wijk
- 1999
(Show Context)
Citation Context ...n by the cluster is bisected by a plane, using principal component analysis. Although this guarantees convex clusters, an accurate representation may require a large cluster count. Telea and Van Wijk =-=[21]-=- place each data point in a cluster. Next, similar clusters are merged bottom-up, using a metric of the difference in position and orientation of the vectors that represent the clusters. The cluster s... |

42 |
Anisotropic Diffusion in Vector Field Visualization on Euclidean Domains and Surfaces
- DIEWALD, PREUSSER, et al.
- 2000
(Show Context)
Citation Context ...ltering has been recently presented by [23]. Finally, we mention the class of physics-based, continuous clustering approaches. These methods simulate a physical process, such as anisotropic diffusion =-=[9, 11]-=-, applied to an initial fine-grained, noise-like signal. The initial clusters, represented implicitly by the fine-grained noise, are coarsened and aligned to the flow, which determines the anisotropy ... |

41 | Construction of Vector Field Hierarchies
- Heckel, Weber, et al.
- 1999
(Show Context)
Citation Context ...ions. Several multiscale clustering approaches for vector data have been proposed. These methods build a hierarchical clustering tree and visualize the clusters with curved arrow icons. Heckel et al. =-=[14]-=- place all data points in a single cluster, which is recursively split in a top-down manner. At each step, the cluster with the strongest discrepancy between streamlines generated by the original fiel... |

40 |
Feature Flow Fields
- Theisel, Seidel
- 2003
(Show Context)
Citation Context ...matching approach using Clifford algebra convolution operations is proposed. Vortex features are found by matching a particular vortex mask against the flow field. Similar approaches are presented by =-=[22]-=- and [1]. Here, the multiscale is given in a scalespace fashion by the convolution operator’s size. Overall, such methods are quite sensitive to the feature definition (’what is a vortex?’) and the de... |

33 |
3D IBFV: hardwareaccelerated 3D flow visualization
- Telea, Wijk
- 2003
(Show Context)
Citation Context ...ructed by the AMG can be used to design different visualization techniques as well (cf. Fig. 12 c). One idea is to use them as color and/or transparency information in texture advection based methods =-=[27, 20]-=- to produce multiscale textured animations of flow fields. 41Figure 11: Climate dataset decomposition, five coarsest levels (left to right). Domains (top row) and flow texture overlaid with curved ar... |

30 | Spectral AMGe (ρAMGe
- CHARTIER, FALGOUT, et al.
(Show Context)
Citation Context ...el l −1 appropriately to define the coarse basis {Ψ l,i } on level l. This merging is actually encoded in the columns of the prolongation matrix P l . Common to nowadays algebraic smoothness criteria =-=[18, 6, 7]-=- is the general observation that a simple relaxation scheme – most often Gauss–Seidel smoothing – efficiently damps components in the direction of eigenvectors associated with large eigenvalues. Conse... |

30 |
Clifford Convolution and Pattern Matching on Vector Fields
- Ebling, Scheuermann
- 2003
(Show Context)
Citation Context ... detects features such as saddles, sources, sinks, and vortices. A multiscale simplification is then obtained by successively removing pairs of critical points based on some relevance metric [24]. In =-=[10]-=-, a pattern matching approach using Clifford algebra convolution operations is proposed. Vortex features are found by matching a particular vortex mask against the flow field. Similar approaches are p... |

25 | H.: Additive multilevel preconditioners based on bilinear interpolation, matrixdependent geometric coarsening and algebraic multigrid coarsening for second order elliptic PDEs
- Grauschopf, Griebel, et al.
- 1997
(Show Context)
Citation Context ...on tensor a of Sec. 4, AMG joins flow-aligned basis functions automatically in the construction of coarser basis functions. For further details on the specific AMG implementation we used, we refer to =-=[12, 13]-=-. 4 A MATRIX ENCODING THE FLOW STRUCTURE Based on our local tensor a(v), which encodes the flow-aligned coupling, we have defined the differential operator A which globally represents the flow structu... |

23 | Variational approach to vector field decomposition, Scientific Visualization
- Polthier, Preuß
- 2000
(Show Context)
Citation Context ... operator’s size. Overall, such methods are quite sensitive to the feature definition (’what is a vortex?’) and the detection process used, especially for 3D fields. A different approach, proposed by =-=[17]-=-, decomposes the field into a divergence free, a rotation free, and a harmonic component, which are separately simplified by detecting critical points corresponding to the components’ extrema. This me... |

17 | A Phase Field Model for Continuous Clustering on Vector Fields
- Garcke, Preu¨ser, et al.
(Show Context)
Citation Context ...ltering has been recently presented by [23]. Finally, we mention the class of physics-based, continuous clustering approaches. These methods simulate a physical process, such as anisotropic diffusion =-=[9, 11]-=-, applied to an initial fine-grained, noise-like signal. The initial clusters, represented implicitly by the fine-grained noise, are coarsened and aligned to the flow, which determines the anisotropy ... |

14 | An algebraic multigrid method for linear elasticity
- Griebel, Oeltz, et al.
- 2003
(Show Context)
Citation Context ...on tensor a of Sec. 4, AMG joins flow-aligned basis functions automatically in the construction of coarser basis functions. For further details on the specific AMG implementation we used, we refer to =-=[12, 13]-=-. 4 A MATRIX ENCODING THE FLOW STRUCTURE Based on our local tensor a(v), which encodes the flow-aligned coupling, we have defined the differential operator A which globally represents the flow structu... |

13 | A.: Feature sensitive multiscale editing on surfaces
- Clarenz, Griebel, et al.
- 2004
(Show Context)
Citation Context ...domains either by direct color coding or by using curved arrow icons, as in [21, 11]. Our method shares its conceptual origin with an AMG-based multiscale, feature-sensitive editing tool for surfaces =-=[8]-=-. In summary, our method solves the question ”show a vector field with n flow icons, for a given n” not only very effectively but also very efficiently: Given AMG’s high computational performance, we ... |

11 |
Vortex tracking in scale space
- Bauer, Peikert
- 2002
(Show Context)
Citation Context ...approach using Clifford algebra convolution operations is proposed. Vortex features are found by matching a particular vortex mask against the flow field. Similar approaches are presented by [22] and =-=[1]-=-. Here, the multiscale is given in a scalespace fashion by the convolution operator’s size. Overall, such methods are quite sensitive to the feature definition (’what is a vortex?’) and the detection ... |

11 |
Algebraic multigrid for automatic multigrid solutions with application to geodetic computations
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(Show Context)
Citation Context ...lgebraic multigrid algorithm (AMG) and the heuristics which led to its development. We refer to [25] for a detailed introduction. Algebraic multigrid methods were first introduced in the early 1980’s =-=[2, 3, 4, 5, 18]-=- for the solution of discrete linear systems AU = F of equations coming from the discretization of a linear differential equation A u = f on a domain Ω with suitable boundary conditions. Here U is sup... |

9 |
Multigrid, Appendix A: An Introduction to Algebraic Multigrid by K. Stüben
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Citation Context ...s of algebraic multigrid. 3 A BRIEF REVIEW OF AMG Here we give an overview of the basic aspects of the algebraic multigrid algorithm (AMG) and the heuristics which led to its development. We refer to =-=[25]-=- for a detailed introduction. Algebraic multigrid methods were first introduced in the early 1980’s [2, 3, 4, 5, 18] for the solution of discrete linear systems AU = F of equations coming from the dis... |

5 |
Algebraic Multigrid for Sparse Matrix Equations
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Citation Context |

2 |
Continuous simplification of planar vector fields
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(Show Context)
Citation Context ...xplicitly detects features such as saddles, sources, sinks, and vortices. A multiscale simplification is then obtained by successively removing pairs of critical points based on some relevance metric =-=[24]-=-. In [10], a pattern matching approach using Clifford algebra convolution operations is proposed. Vortex features are found by matching a particular vortex mask against the flow field. Similar approac... |