HLIBpro
2.9

In some applications geometrical data is not available or was lost in some previous computation step. Fortunately, there is another source for describing the relation between indices: the system matrix. In case of a sparse matrix \( A \in R^{I \times I} \) over the index set \( I \), a (matrix) graph can be constructed with nodes corresponding to indices and edges in case of a nonzero matrix entry, e.g. \( G = (V,E) \) with \( V = I \) and
\[ (i,j) \in E \Leftrightarrow a_{ij} \ne 0 \vee a_{ji} \ne 0 \]
Please note, that the symmetrised matrix graph is used, leading to an undirected graph.
By partitioning the graph \( G \), the index set \( I \) is divided into subset and hence, a cluster tree is constructed by recursivly applying the procedure. This method provides a purely algebraic way for constructing Hmatrices.
Before starting with clustering the index set, the sparse matrix has to be available:
The partitioning of graphs is a standard problem in graph theory. The aim is to split the graph into two or more sub graphs, such that the number of edges connecting the parts is minimized. Albeit this problem is NPhard, good approximation algorithms have been developed and are available in the form of open source libraries, e.g. METIS or Scotch.
While being able to use these external software libraries, 𝖧𝖫𝖨𝖡𝗉𝗋𝗈 also contains two graph partitioning algorithms:
In both cases, the FiducciaMattheyses algorithm is applied to optimise the results.
THe actual cluster tree construction algorithms in 𝖧𝖫𝖨𝖡𝗉𝗋𝗈 rely on these graph partitioning strategies, which are implemented in the following classes:
The standard cluster tree construction algorithm is implemented in TAlgCTBuilder and uses a graph partitioning strategy to construct a binary cluster tree.
The second argument for the cluster tree constructor defines the size of leaves in the cluster tree, e.g. no leaf with at most \( n_{\min} \) indices will be further partitioned. By default, \( n_{\min} \) is of size 20 to 40.
In many applications, e.g. if a LU factorisation is planned, a more efficient Harithmetic can be obtained by using nested dissection instead of standard bisection for clustering. TNDAlgCTBuilder uses a given cluster tree constructor and extends it by computing an interface (or vertex separator) between the sub graphs, which decouples to indices. The prodedure is applied recursivly on the sub graphs, whereas the nodes in the interface are the divided using the predefined clustering algorithm, e.g. usually bisection.
Another variation from the bisection algorithm uses a predefined partitioning to divide the indices of the root of the cluster tree. This is typically used, if the underlying problem already has a block structure, which shall also be present in the Hmatrices. The user supplied partition is defined by an array of the same size as the number of indices and the value at position i
defines the block index i
should belong to.
Again, this algorithm extends a given cluster tree constructor, thereby providing a chaining, e.g. using nested dissection for all other levels beside the first: