Combinatorial problems in solving linear systems

42 pages, available as LIP research report RR-2009-15Numerical 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

Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure