STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Lower bounds for graph embeddings and combinatorial preconditioners
Proceedings of the sixteenth annual ACM symposium on Parallelism in algorithms and architectures
Finding effective support-tree preconditioners
Proceedings of the seventeenth annual ACM symposium on Parallelism in algorithms and architectures
Algebraic analysis of high-pass quantization
ACM Transactions on Graphics (TOG)
Proceedings of the twentieth annual symposium on Parallelism in algorithms and architectures
ISVC '09 Proceedings of the 5th International Symposium on Advances in Visual Computing: Part I
VECPAR'06 Proceedings of the 7th international conference on High performance computing for computational science
Hierarchical Diagonal Blocking and Precision Reduction Applied to Combinatorial Multigrid
Proceedings of the 2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis
Computer Vision and Image Understanding
Spectral Sparsification of Graphs
SIAM Journal on Computing
Parallel support graph preconditioners
HiPC'06 Proceedings of the 13th international conference on High Performance Computing
Power grid analysis with hierarchical support graphs
Proceedings of the International Conference on Computer-Aided Design
The laplacian paradigm: emerging algorithms for massive graphs
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
Algorithms, graph theory, and the solution of laplacian linear equations
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part II
Generalized subgraph preconditioners for large-scale bundle adjustment
Proceedings of the 15th international conference on Theoretical Foundations of Computer Vision: outdoor and large-scale real-world scene analysis
Proceedings of the International Conference on Computer-Aided Design
Efficient preconditioning of laplacian matrices for computer graphics
ACM Transactions on Graphics (TOG) - SIGGRAPH 2013 Conference Proceedings
A simple, combinatorial algorithm for solving SDD systems in nearly-linear time
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Proceedings of the International Conference on Computer-Aided Design
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We present support theory, a set of techniques for bounding extreme eigenvalues and condition numbers for matrix pencils. Our intended application of support theory is to enable proving condition number bounds for preconditioners for symmetric, positive definite systems. One key feature sets our approach apart from most other works: We use support numbers instead of generalized eigenvalues. Although closely related, we believe support numbers are more convenient to work with algebraically. This paper provides the theoretical foundation of support theory and describes a set of analytical tools and techniques. For example, we present a new theorem for bounding support numbers (generalized eigenvalues) where the matrices have a known factorization (not necessarily square or triangular). This result generalizes earlier results based on graph theory. We demonstrate the utility of this approach by a simple example: block Jacobi preconditioning on a model problem. Also, our analysis of a new class of preconditioners, maximum-weight basis preconditioners, in [E. G. Boman, D. Chen, B. Hendrickson, and S. Toledo, Numer. Linear Algebra Appl., to appear] is based on results contained in this paper.