CS 855 - Symbolic Parallel Computation
and Algebraic Geometry
University of Western Ontario
Computer Science Department
Date: September 7, 2009
Since the discovery of Groebner bases, the algorithmic advances in Commutative Algebra have made possible to tackle many classical problems in Algebraic Geometry that were previously out of reach. However, algorithmic progress is still desirable, for instance when solving symbolically a large system of algebraic (non-linear) equations is needed.
For such a system, in particular if its solution set consists of geometric components of different nature (surfaces, curves, points) it is necessary to combine Groebner bases with decomposition techniques, such as triangular decompositions.
The efficiency of this approach depends on its ability to detect and exploit geometrical information during the solving process. Its implementation, which naturally involves symbolic parallel computations, is another challenging topic.
The motivation of this course is the parallelization of triangular decompositions for solving systems of algebraic equations. The topics covered are the computer science techniques needed for achieving this goal.
The first third of this course will be devoted to the complexity analysis of parallel algorithms. In the second third, we will discuss parallel algorithms for scientific computations, with a view toward symbolic computations. Then, we will present a triangular decomposition algorithm and its applications.
|Week Jan. 9-15||Introduction to Parallel Symbolic Computations|
|Week Jan. 16-22||No lecture.|
|Week Jan. 23-29||Performance and scalability of parallel algorithms, PRAM models|
|Week Jan. 30-5||Parallel Complexity Theory|
|Week Feb. 6-12||Parallel Sorting I|
|Week Feb. 13-19||Parallel Sorting II|
|Week Feb. 20-26||Parallel Dense Matrix Operations|
|Week Mar. 6-12||Solving systems of non-linear equations in parallel|
|Week Mar. 13-19||Solving systems of linear equations in parallel|
|Week Mar. 20-26||Triangular decomposition algorithms|
|Week Mar. 27-2||No lecture.|
|Week Apr. 3-9||Parallel algorithms for GCD computations|
|Week Apr. 10-16||Processor Efficient Parallel algorithms for Linear Algebra|
|Week Apr. 17-23||Project Presentations|