A comparison of Algorithms for Sparse Matrix. Real-time Multibody Dynamic Simulation

Transcription

1 A comparison of Algorithms for Sparse Matrix Factoring and Variable Reordering aimed at Real-time Multibody Dynamic Simulation Jose-Luis Torres-Moreno, Jose-Luis Blanco, Javier López-Martínez, Antonio Giménez-Fernández 1-4 July, 2013 Zagreb (Croatia) ECCOMAS Multibody Dynamics 2013

2 Goal of this work Problem under study Multibody dynamic formulation: Best numerical solver algorithm? = = \ 2 It depends Ax b x A b

3 Talk outline 1. Problem introduction 2. Tested algorithms 3. Benchmark 4. Results 3

4 Problem introduction Modeling decisions: coordinates, formulation, etc. (,&, ) q&& t = f q t q t t ( ) ( ) ( ) (, &, ) && z t = f z t z t t ( ) ( ) ( ) Our choices for this work: Dependent coordinates vs. Independent coordinates Coordinate type for modeling: Dynamic formulation: Natural coordinates Index-1 Lagrange w. Baumgarte 4 1/4: Introduction

5 Problem introduction A x = b Problem structure? N dependent coordinates M constraints (N+M) x (N+M) (N+M M) x 1 = (N+M M) x 1 Square, symmetric, Dense columns psd and sparse 5 1/4: Introduction

6 Structure of matrix A A Most important features - M is constant - Static non-zero structure - SPARSITY 6 1/4: Introduction

7 Why sparsity matters? Naive matrix inversion: T M φq q&& Q = φq 0 λ c A q && λ 1 T M φ q Q = φq 0 c A 1 7 1/4: Introduction

8 Why sparsity matters? nnz=528 (93.76% are zeros) nnz=8266 (2.34% are zeros) M φq φ T q 0 8 1/4: Introduction M φq In general, the inverse of a sparse matrix is NOT sparse! ( 3 ( + ) ) Cost: O N M φ T q 0 1

9 Why sparsity matters? How to keep matrices sparse? Avoid direct matrix inversion. Alternatives? Sparse matrix factorization algorithms: 1) Method: LU, Cholesky. 2) Variable ordering 9 1/4: Introduction

10 Why variable reordering matters? nnz=528 (93.76% are zeros) ~ 94% zeros Natural ordering A = ~ 70% zeros nnz=2599 (69.29% are zeros) nnz=2456 (70.98% are zeros) A = L U = /4: Introduction

11 Why variable reordering matters? Reordering = permutation of variable indices nnz=528 (93.76% are zeros) nnz=528 (93.76% are zeros) A ( :, : ) A( p, p ) 11 1/4: Introduction

12 Why variable reordering matters? nnz=528 (93.76% are zeros) ~ 94% zeros Optimized ordering ~97% and ~93% zeros nnz=244 (97.12% are zeros) nnz=620 (92.67% are zeros) /4: Introduction

13 Talk outline 1. Problem introduction 2. Tested algorithms 3. Benchmark 4. Results 13

14 Tested algorithms Approaches: LU decomposition Cholesky decomposition 14 3/4: Algorithms

15 Tested algorithms Approaches: LU decomposition Cholesky decomposition Ax = b } A 1) L Ux { = L y= b, solve for y ( LU) x = b = y 2) Ux = y, solve for x nnz=244 (97.12% are zeros) nnz=620 (92.67% are zeros) /4: Algorithms

16 Tested algorithms Benchmarked LU implementations (6 variations): 1) Dense LU Eigen C++ library 2) Sparse LU Suitesparse s KLU a) AMD ordering b) COLAMD ordering 3) Sparse LU Suitesparse s UMFPACK a) Natural ordering b) AMD ordering c) METIS ordering Symbolic vs. numeric factorization! 16 3/4: Algorithms

17 Tested algorithms Approaches: LU decomposition Cholesky decomposition (only when M is definite positive) T M φq q&& Q = φq 0 λ c A Apply Cholesky to M. Use Schür complement to build two reduced systems of equations. [R. von Schwerin, Mathematical Methods for MBS in Descriptor Form, 1999] 17 3/4: Algorithms

18 Tested algorithms Approaches: LU decomposition Cholesky decomposition (only when M is definite positive) Benchmarked Cholesky implementations (3 variations): 1) Sparse Cholesky Suitesparse s CHOLMOD a) AMD ordering b) COLAMD ordering c) METIS ordering 18 3/4: Algorithms

19 Talk outline 1. Problem introduction 2. Tested algorithms 3. Benchmark 4. Results 19

20 Our benchmark model Test mechanism: Parameterized 2D lattice of N x N y articulated quadrangles. Example: Nx = 5 Ny = /4: Benchmark

21 Our benchmark model Irregular node locations to avoid singular configurations: 21 3/4: Benchmark

22 Benchmark C++ implementation MB model Solver-specific tasks to be profiled Assemble Solver Assembled model Preprocessing 1. Update Jacobians 2. Build RHS 3. Triplet -> CCS (*) 4. Numeric factoring 5. Solve Ax=b N.I.S. 22 3/4: Benchmark

23 The benchmark For each solver For each ordering For N=1 32 Build mechanism with N x = N, N y =N. Run thousands of time steps for each configuration. Measure average execution time of each calculation. 23 3/4: Benchmark

24 Talk outline 1. Problem introduction 2. Tested algorithms 3. Benchmark 4. Results 24

25 25 4/4: Results

26 Benchmark C++ implementation MB model Solver-specific tasks to be profiled Assemble Solver Assembled model Preprocessing 1. Update Jacobians 2. Build RHS 3. Triplet -> CCS (*) 4. Numeric factoring 5. Solve Ax=b N.I.S. 26 3/4: Benchmark

27 Results: Preprocessing (symbolic factorization) Time (ms) Dense LU UMFPACK Natural UMFPACK AMD UMFPACK METIS KLU AMD KLU COLAMD CHOLMOD AMD/AMD CHOLMOD COLAMD/COLAMD CHOLMOD METIS/METIS Number of redundant coordinates 27 4/4: Results

28 Benchmark C++ implementation MB model Solver-specific tasks to be profiled Assemble Solver Assembled model Preprocessing 1. Update Jacobians 2. Build RHS 3. Triplet -> CCS (*) 4. Numeric factoring 5. Solve Ax=b N.I.S. 28 3/4: Benchmark

29 Results: Distribution of time (KLU+COLAMD) Update Jacobians Build RHS Solve Ax=b Triplet->CCS Numeric factoring 4% 7% 4% 4% Triplet -> CCS can be eliminated with smarter coding! 81% 29 4/4: Results

30 Results: num. factoring + solve Total numeric factorization & solve time Dense LU UMFPACK Natural UMFPACK AMD UMFPACK METIS KLU AMD KLU COLAMD CHOLMOD AMD/AMD CHOLMOD COLAMD/COLAMD CHOLMOD METIS/METIS 200 Time (ms) UMFPACK with different orderings KLU with different orderings Number of redundant coordinates 30 4/4: Results

31 A comment on the Cholesky approach E matrix chol( E * E ) dense!! 31 4/4: Results

32 Results: num. factoring + solve (log scale) 10 2 Dense LU UMFPACK+AMD 10 1 Time (ms) KLU+COLAMD Dense LU UMFPACK Natural UMFPACK AMD UMFPACK METIS KLU AMD KLU COLAMD CHOLMOD AMD/AMD CHOLMOD COLAMD/COLAMD CHOLMOD METIS/METIS Number of redundant coordinates 4/4: Results

33 Results: num. factoring + solve (log scale) 10 2 Dense LU UMFPACK+AMD 10 1 ~ 50 coords Time (ms) 10 0 ~ 2000 coords KLU+COLAMD Dense LU UMFPACK Natural UMFPACK AMD UMFPACK METIS KLU AMD KLU COLAMD CHOLMOD AMD/AMD CHOLMOD COLAMD/COLAMD CHOLMOD METIS/METIS Number of redundant coordinates 4/4: Results

34 Results: num. factoring + solve (log scale) 10 2 Dense LU UMFPACK+AMD 10 1 Time (ms) 10 0 T=1ms KLU+COLAMD Dense LU UMFPACK Natural UMFPACK AMD UMFPACK METIS KLU AMD KLU COLAMD CHOLMOD AMD/AMD CHOLMOD COLAMD/COLAMD CHOLMOD METIS/METIS Number of redundant coordinates 4/4: Results

35 Conclusions MBS dynamics leads to very sparse systems. Exploiting sparsity is crucial for performance unless N<50 approx. 50<N<2000 Use KLU+COLAMD 2000<N Use UMFPACK+AMD In any case: reuse symbolic factorizations! Slides available online: Publications 35 4/4: Results

36 A comparison of Algorithms for Sparse Matrix Factoring and Variable Reordering aimed at Real-time Multibody Dynamic Simulation Thanks for your attention! Slides available online: Publications