Combinatorial problems in solving linear systems

Iain Duff (1, 2) , Bora Uçar (3, 4)

1 CERFACS - Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique
2 Numerical Analysis Group - STFC Rutherford Appleton Laboratory
3 GRAAL - Algorithms and Scheduling for Distributed Heterogeneous Platforms
4 LIP - Laboratoire de l'Informatique du Parallélisme

Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects. As the core of many of today's numerical linear algebra computations consists of the solution of sparse linear system by direct or iterative methods, we survey some combinatorial problems, ideas, and algorithms relating to these computations. On the direct methods side, we discuss issues such as matrix ordering; bipartite matching and matrix scaling for better pivoting; task assignment and scheduling for parallel multifrontal solvers. On the iterative method side, we discuss preconditioning techniques including incomplete factorization preconditioners, support graph preconditioners, and algebraic multigrid. In a separate part, we discuss the block triangular form of sparse matrices.

Mots clés: Combinatorial scientific computing; Graph theory; Combinatorial optimization; Sparse matrices; Linear system solution

Domaines: Combinatoire [math.CO]; Calcul parallèle, distribué et partagé [cs.DC]

Iain Duff, Bora Uçar. Combinatorial problems in solving linear systems. 2011. ⟨ensl-00411638v3⟩