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A Memory-Efﬁcient Algorithm for Large-Scale Symmetric Tridiagonal Eigenvalue Problem on Multi-GPU Systems Hyunsu Cho and Peter A. Yoon Trinity College, Hartford, CT, USA

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Symmetric Eigenvalue Problem Many interesting applications require eigenvectors Ax = λx where A is symmetric

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Divide and Conquer Yields full spectrum of eigenvalues and eigenvectors Is numerically stable Gives rise to independent subproblems Often faster than O(n 3 ) due to deflation

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Divide and Conquer Apply orthogonal similarity transformation to reduce A to tridiagonal form Existing work on single-node, multi-GPU: MAGMA (UTK) Q T AQ = A’ where A’ is symmetric tridiagonal and Q is orthogonal

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Solve subproblems Merge solutions Repair Divide and Conquer A A1A1 A2A2

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Merging solutions Suppose where (subproblem #1) (subproblem #2)

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Merging solutions Then where and

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Merging solutions Then where and Rank-one modifier

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Rank-one update where

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Rank-one update where Need eigen-decomposition of inner system

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Decompose 1. Sort entries in D; permute z likewise 2. Filter some entries in D and z via deflation (next slide)

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Decompose 1. Sort entries in D; permute z likewise 2. Filter some entries in D and z via deflation (next slide) 3. Compute all roots of the secular equation [1] giving the m eigenvalues. 4. Compute corresponding eigenvectors stably [2] [1] Li 1994 [2] Gu & Eisenstat 1994,

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Decompose 1. Sort entries in D; permute z likewise 2. Filter some entries in D and z via deflation (next slide) 3. Compute all roots of the secular equation [1] giving the m eigenvalues. 4. Compute corresponding eigenvectors stably [2] 5. Multiply each eigenvector by Q Recall: [1] Li 1994 [2] Gu & Eisenstat 1994,

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Deflation Recall: Entries of D are eigenvalues of two subproblems If two entries are nearly identical, we throw one away Fewer columns when multiplying eigenvectors by Q Same thing for small entries in z Reduce work complexity to O(n 2.3 )

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GPU computing General-purpose computation on GPUs Bulk parallelism w/ many small threads Cost effective; widely available

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Mapping work to GPU 1. Sort entries in D; permute z likewise 2. Filter some entries in D and z via deflation 3. Compute all roots of the secular equation, giving the m eigenvalues. 4. Compute corresponding eigenvectors stably 5. Multiply each eigenvector by Q → Done in bulk via DGEMM Parallel but not as work-intense

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GPU memory High-bandwidth dedicated memory Separate from main memory Limited in size Main memory CPU GPU memory GPU PCI-E

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Memory requirement Eigenvectors are dense → O(n 2 ) storage Intermediate workspace: eigenvectors of inner system Matrix dimension Memory required 81921.5 GB 163845.8 GB 3276823.4 GB 3600028.2 GB 5000054.4 GB

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Our contribution Overcome limitation in GPU memory while retaining adequate performance

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Strategies 1. Use multiple GPUs GPU workspace

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Strategies 2. Keep most of workspace in main memory (out-of-core approach) GPU CPU

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Strategies 3. Shape work to fit GPU workspaces GPU work

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Block matrix multiplication Use a fine partition to fit submatrices into GPU memory X = Q Eigenvectors of Eigenvectors of A

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Hybrid computation Allocate subproblems to both GPUs and CPUs Model performance as a power function Profiler fits parameters using least-squares performance subproblem sizelog(subproblem size) log(performance)

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Hybrid computation Solve many subproblems in parallel

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Hybrid computation Solve each subproblem by parts

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Results Scales to 50k * 50k matrix With 4 GB of GPU memory Main memory: 64 GB GPU memory: 5 GB per GPU

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Results 9x CPU: dual Intel® Xeon® E5-2620 GPU: 4 Nvidia Tesla® K20c

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Conclusion Out-of-core approach overcomes memory limitation on the GPU Hybrid computation with profiling delivers reasonable performance

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Acknowledgment Trinity College, Student Research Program Nvidia Corporation, CUDA Teaching Center Program

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