Parallel Dense Matrix Algorithms II
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Building on matrix multiplication, this week tackles the parallel solution of Ax=b, concentrating on techniques like Gaussian Elimination and its reliance on broadcast and fan-in operations, as well as the parallel implementation of LU factorization. The focus will be on partitioning triangular matrices, managing dependencies, and minimizing the communication overhead inherent in these highly coupled computations.
Learning Outcomes
Learning Outcomes
The principles governing parallel Gaussian Elimination and LU factorization are understood, and the communication patterns arising from pivoting and data broadcasting are analyzed.
The effects of decomposition strategies and asynchronous communication on the scalability of direct solvers are evaluated.
The implementation and convergence behavior of iterative methods, including Jacobi and Gauss–Seidel, are compared in terms of parallel efficiency and numerical stability.
Topic Outline
Topic Outline
Handout: Parallel Dense Matrix Algorithms II*
Solving Together, One Pivot at a Time
Algorithms for Solving Systems of Linear Equations
Parallel GE and LU: Pivoting and Broadcast
Optimizing Elimination: The ScaLAPACK Approach
Parallel Iterative Methods
Jacobi and Gauss-Seidel: Dependency and Convergence
Synchronous vs. Asynchronous Iteration
From Linear Systems to Iterative Thinking
Note: Links marked with an asterisk (*) lead to materials accessible only to members of the University community. Please log in with your official University account to view them.
Readings
Readings
General Parallel Computing and Cost Models
Grama, A., Gupta, A., Karypis, G., & Kumar, V. (2003). Introduction to parallel computing. Addison-Wesley.
Dense Linear Algebra Libraries (ScaLAPACK Foundation)
Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., ... & Sorensen, D. (1999). LAPACK users' guide (3rd ed.). Society for Industrial and Applied Mathematics.
Parallel Iterative Methods and Sparse Solvers
Saad, Y. (2003). Iterative methods for sparse linear systems (2nd ed.). Society for Industrial and Applied Mathematics.
Access Note: Published research articles and books are linked to their respective sources. Some materials are freely accessible within the University network or when logged in with official University credentials. Others will be provided to enrolled students through the class learning management system (LMS).
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