Support-Graph Preconditioners for 2-Dimensional Trusses

We use support theory, in particular the fretsaw extensions of Shklarski and Toledo [ST06a], to design preconditioners for the sti ness matrices of 2-dimensional truss structures that are sti y connected. Provided that all the lengths of the trusses are within constant factors of each other, that the angles at the corners of the triangles are bounded away from 0 and , and that the elastic moduli and cross-sectional areas of all the truss elements are within constant factors of each other, our preconditioners allow us to solve linear equations in the sti ness matrices to accuracy in time O(n5=4(log n log logn)3=4 log(1= )). 1 Preconditioning When solving a linear system in an n n positive semide nite matrix A, the running time of an iterative solver can often be sped up by supporting A with another matrix B, called a preconditioner. An e ective preconditioner B has the properties that it is much easier to solve than A, and that A has a low condition number relative to B. We de ne here generalized eigenvalues and condition numbers: De nition 1.1. For positive semide nite A;B, the maximum eigenvalue, minimum eigenvalue, and condition number of A relative to B are de ned respectively as max(A;B) = max x :x?null(B) xTAx xTBx min(A;B) = min x :x?null(A) xTAx xTBx (A;B) = max(A;B)= min(A;B) where x ? null(S) means that x is orthogonal to the null space of S. Note that the standard condition number of A can be expressed as (A) = (A; I). The conjugate gradient method is an example of a linear solver that can be sped up using a preconditioner. The precise analysis of the running time can be found, for example, in [Axe85]: Partially supported by NSF grant CCR-0324914. Partially supported by NSF grant CCR-0324914. Theorem 1.2 ([Axe85]). For positive semide nite A;B, and vector b, let x satisfy Ax = b. Each iteration of the preconditioned conjugate gradient method multiplies one vector by A, solves one linear system in B, and performs a constant number of vector additions. For > 0, it requires at most O( p (A;B) log(1= )) such iterations to produce a ~ x that satis es k~ x xkA kxkA 1.1 Using a Larger Matrix In certain situations it may be easier to nd a good preconditioner for a matrix A if we treat A as being larger than it really is. That is, if we pad A with zeros to form a larger square matrix A0 = A 0 0 0 , it may be simpler to nd a good preconditioner B for A0. We then need to show how to use B to yield a preconditioner for the original matrix A. To this end, we de ne the Schur complement: De nition 1.3. For square matrices A and B = B11 B12 BT 12 B22 , where square submatrix B11 is the same size as A, and such that B22 is nonsingular, the Schur complement of B with respect to A is BS = B11 B12B 1 22 B T 12 While BS will not automatically be a good preconditioner for A simply because B is a good preconditioner for A0, we do know that the maximum eigenvalue will be the same: Lemma 1.4. For positive semide nite A;B, max(A; B) = max(A;BS). We also know that solving a linear system in BS is as easy as solving a linear system in B: Lemma 1.5. B x y = b 0 implies BSx = b For completeness, we give proofs for these lemmas in Appendix A. 1.2 Congestion-Dilation Suppose that we have matrices A and B that can be expressed as the sums of other matrices, i.e. A = P iAi and B = P j Bj, and that we know how to support each Ai by a subset of the Bj matrices. In this situation, we can use the following lemma to show how B supports A: Lemma 1.6 (Congestion-Dilation Lemma). Given the symmetric positive semide nite matrices A1; :::; An; B1; :::; Bm and A = P iAi and B = P j Bj and given sets i [1; :::;m] and real values si that satisfy max(Ai; X