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Sparse Inverse Incidence Matrices for Schilders’ Factorization Applied to Resistor Network Modeling

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Sparse Inverse Incidence Matrices for Schilders’ Factorization Applied to Resistor Network Modeling NUMERICAL ALGEBRA, doi:10.3934/naco.2014.4.227 CONTROL AND OPTIMIZATION Volume 4, Number 3, September 2014 pp. 227{239 SPARSE INVERSE INCIDENCE MATRICES FOR SCHILDERS' FACTORIZATION APPLIED TO RESISTOR NETWORK MODELING Sangye Lungten y, Wil H. A. Schilders and Joseph M. L. Maubach Center for Analysis, Scientific Computing and Applications Department of Mathematics and Computer Science Eindhoven University of Technology 5600 MB, Eindhoven, The Netherlands (Communicated by Xiaoqing Jin) Abstract. Schilders' factorization can be used as a basis for precondition- ing indefinite linear systems which arise in many problems like least-squares, saddle-point and electronic circuit simulations. Here we consider its appli- cation to resistor network modeling. In that case the sparsity of the matrix blocks in Schilders' factorization depends on the sparsity of the inverse of a permuted incidence matrix. We introduce three different possible permuta- tions and determine which permutation leads to the sparsest inverse of the incidence matrix. Permutation techniques are based on types of sub-digraphs of the network of an incidence matrix. 1. Introduction and motivation. Indefinite linear systems of the form n m " # n A BT x f = (1) m B 0 y g | {z } A arise in electronic circuit simulations and many other applications where A is sym- metric and positive (semi) definite, and BT is of maximal column rank m (≤ n). Preconditioning techniques to solve (1) have become very important, especially for problems which arise from Stokes equation resulting in saddle point problems [6]. Schilders' factorization [7] can be used as a basis for such preconditioners. A pos- sibly permuted A~ B~T A~ = B~ 0 2010 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35. Key words and phrases. Schilders' factorization, lower trapezoidal, digraph, incidence matrix, nilpotent. y The first author's PhD research is supported by the Erasmus Mundus IDEAS project and the Department of Mathematics and Computer Science, Eindhoven University of Technology. 227 228 SANGYE LUNGTEN, WIL H. A. SCHILDERS AND JOSEPH M. L. MAUBACH is split into a block 3 × 3 structure and factorized: 2 ~ ~ ~T 3 2 ~T 3 2 3 2 ~ ~ 3 A11 A12 B1 B1 0 L1 D1 0 I B1 B2 0 ~ ~ ~T ~T T 4A21 A22 B2 5 = 4B2 L2 M 5 4 0 D2 05 4 0 L2 05 ; (2) ~ ~ T T B1 B2 0 0 0 I I 0 0 L1 M I ~−T where the blocks L1;L2 & M depend on B1 . The permutation has to be such that BT is permuted into a lower trapezoidal form B~T B~T = 1 = ; (3) ~T B2 ~T ~T where the top part B1 is a lower triangular matrix of size m × m and B2 is an (n − m) × m matrix. Generally, A is sparse and so is A~. But (2) can have blocks ~−T which are more dense because of B1 . In other words, sparsity of the blocks in ~−T ~−T (2) depends on the sparsity of B1 . In order to illustrate the involvement of B1 while computing the blocks in (2), we consider (4) and (5). The details for deriving these formulas can be found in [7]. ~T ~−T ~ ~−1 L1 = B1 lower B1 A11B1 ; (4) ~ ~T T ~−1 ~T M = A21 − B2 L1 B1 − B2 D1: (5) ~ ~T ~−T ~ ~−1 Although A11 and B1 in (4) are sparse, the product B1 A11B1 can become dense ~−T where B1 is dense. Eventually this will lead to even more dense blocks L1 and M. ~−T This entails to have a permutation such that B1 is sparse. Therefore, we develop an algorithm which permutes an incidence matrix B~T into a lower trapezoidal form ~−T (3) such that it results sparse B1 . In this paper we consider Schilders' factorization applied to resistor network modeling in which A is a diagonal matrix with entries of resistance values, and BT is an incidence matrix. The modeling can be done by applying Kirchhoff's current law and Ohm's law for resistors [5]. In order to explain the definition of an incidence matrix, and to understand some of its properties, a brief summary on graph theory and incidence matrices is given in Section2. In Section3, aspects of the sparsity ~−T of B1 are discussed. Then we introduce three different possible permutations and determine which permutation leads to the sparsest inverse of the incidence matrix. ~−T Section4 gives the permutation algorithm which leads to a sparse inverse, B1 . Numerical experiments performed on industrial resistor networks are presented in Section5. 2. A brief summary on graph theory and incidence matrices. A graph G consists of a finite set V = fη0; η1; : : : ; ηmg called the vertex set and a finite set E = fξ1; ξ2; : : : ; ξng called the edge set. An η 2 V is called a vertex (or node) of G and a ξ 2 E an edge of G. An edge ξ is represented by an unordered pair of nodes fηi; ηjg, which are said to be adjacent to each other, and called the end points of ξ. A directed graph (or digraph) G consists of a node set V and an edge set E, where each edge ξ is an ordered pair (ηi; ηj) known as an arc (or directed edge). We write ηi ! ηj to represent that the arc ξ connects the two nodes in the direction from ηi to ηj. Here, call ηi the initial node and ηj the terminal node of ξ. If the direction between the two nodes is unknown, we shall write it as ηi ηj to avoid ambiguity. SPARSE INVERSE INCIDENCE MATRICES 229 A path in a graph G is an alternating sequence of distinct nodes and edges. For example the path between η0 and ηr is given by η0; ξ1; η1; ξ2; : : : ; ξr; ηr. If there is a path between every pair of nodes, then G is said to be connected. A digraph is called weakly connected if its underlying graph is connected. A digraph is strongly connected if for every pair distinct nodes ηi and ηj, there is a directed path starting from ηi to ηj.A Hamiltonian path in a digraph is a path in a single direction that visits each node exactly once. A tournament is a digraph in which every pair of nodes is connected by a single directed edge [1]. A digraph is called loopless if it contains none of (ηi; ηi). Theorem 2.1. [3] Every tournament on a finite number of m nodes contains a Hamiltonian path. Proof. See page 149, Section 5.3 in [3]. We exploit this theorem later for the generation of test examples. Definition 2.2. Consider a loopless digraph G of m + 1 nodes and n arcs. The T node-arc incidence matrix of G is an n × (m + 1) matrix B^ = [bij], where 8 < 1; if ηj is the initial node of arc ξi bij = −1; if ηj is the terminal node of arc ξi : 0; if arc ξi does not contain node ηj : To obtain a reduced incidence matrix BT of size n×m from B^T , we call one node the reference or ground node, and delete its related column. This causes all rows of BT , related to the arcs connected to the reference node, to only have one nonzero entry. This is necessary for the construction of a permutation to a lower trapezoidal matrix. Moreover, systems of the form (1) resulting from resistor network modelings are made consistent by grounding a node and removing the corresponding column [5]. Theorem 2.3 gives an important property of an incidence matrix which relates to its underlying digraph. Theorem 2.3. [2] Suppose G is a digraph with m+1 nodes. Then the column rank of the incidence matrix BT is m if and only if G is weakly connected. Proof. See page 12, Section 2.1 in [2]. 3. Sparsity of inverse incidence matrix. Before exploring different types of permutations, we first give an overview of interdependence between the sparsity of inverse and nilpotency degree of a lower triangular matrix which contains maximally two nonzero entries in each row. For this we consider a unit lower bidiagonal matrix −1 H and compute its inverse using the expansion of (I + NH ) , where I is the identity matrix and NH is the strictly lower triangular part of H (the nilpotent part of H). The inverse of unit lower bidiagonal matrices can also be computed by using Gaussian elimination [8]. However, the former gives an insight on how the interdependence between the sparsity of inverse and nilpotency degree can be conceived. A unit lower bidigonal matrix H of size n × n is of the form 2 1 3 6β1 1 7 H = 6 7 ; where β 6= 0; 1 ≤ i ≤ n − 1: 6 .. .. 7 i 4 . 5 βn−1 1 230 SANGYE LUNGTEN, WIL H. A. SCHILDERS AND JOSEPH M. L. MAUBACH By defining the nilpotent part 2 0 3 6β1 0 7 N = 6 7 ; H 6 .. .. 7 4 . 5 βn−1 0 it is trivial to observe that −1 2 n−1 n−1 H = I − NH + NH − · · · + (−1) NH 2 1 3 (1) 6 α 1 7 6 1 7 6 . (1) 7 6 . α 1 7 6 2 7 6 . 7 6 α(r) . 7 = 6 1 7 ; 6 . 7 6 . α(r) .. .. 7 6 2 7 6 . 7 6α(n−2) . .. 7 4 1 5 (n−1) (n−2) (r) (1) α1 α2 : : : αn−r : : : αn−1 1 where each rth subdiagonal entry, 1 ≤ r ≤ n − 1, is defined by r+j−1 (r) r Y αj = (−1) βi 6= 0 ; 1 ≤ j ≤ n − r: (6) i=j −1 That is every subdiagonal entry of H is a product of some βi's and not equal to −1 zero.
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