Parallel Discontinuous Galerkin Method
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1 Parallel Discontinuous Galerkin Method Yin Ki, NG The Chinese University of Hong Kong Aug 5, 2015 Mentors: Dr. Ohannes Karakashian, Dr. Kwai Wong
2 Overview Project Goal Implement parallelization on Discontinuous Galerkin Method (DG-FEM) scaled on existing supercomputers
3 Overview Mathematics behind 1D Parallelization Future Work Acknowledgement
4 Discontinuous Galerkin Method (DG-FEM) Discontinuous Galerkin Method (DG-FEM) A class of Finite Element Method (FEM) Finding approximate solutions to system of differential equations
5 Example: Solving Heat Equation (1D) 1D Poisson s Equation on domain I = [a, b] u = f u: test function (to be solved) u a = u b = 0
6 Example: Solving Heat Equation (1D) u: test function (to be solved) Multiply by an arbitrary trial function v (satisfying v a = v b = 0 ) u v fv = 0 Integration (Strong-form) u v fv = 0
7 Example: 1D Finite Element Method u: test function (to be solved) v : trial function (our choice) Suppose v is continuous over I, integration by parts u v + u v I fv = 0 Obtain Weak-form of Finite Element Method (FEM) u v fv = 0
8 Example: 1D FEM Basis Functions u: test function (to be solved) v : trial function (our choice) (FEM) Choose continuous basis functions θ over I to approximate u Example: Hat function (Linear) u
9 Example: 1D FEM Basis Functions u: test function (to be solved) v : trial function (our choice) (FEM) Choose v to be the the basis functions θ Approximate u i = j u j v j : linear combination of v j u v fv = 0 v 0 v 0 v 1 v 0 v 2 v 0 u 0 fv 0 I I I I v 0 v 1 v 1 v 1 v 2 v 1 u 1 fv 1 = 0 I I I I v 0 v 2 v 1 v 2 v 2 v 2 u 2 fv 2 I I I I
10 Example: 1D DG-FEM u: test function (to be solved) v : trial function (our choice) Suppose v is discontinuous on x j v (I 1) + x j + : start of the left interval I j v (I 0) + v (I 2) x j : start of the right interval I j 1 + x 0 x 1 x 2 x 3 Integration by parts on each interval x j+1 ) u v = j Ij u v = j ( Ij u v ) j u v xj
11 Example: 1D DG-FEM u: test function (to be solved) v : trial function (our choice) u v xj x j+1 = u x j + v x j + u x j+1 v(x ) j+1 Jumps on internal nodes j j = u x 0 + v x 0 + u x 1 v x 1 u v (I 1) + u x 1 + v x 1 + u x 2 v x 2 u v (I 0) + + u v (I 2) + + u x 2 + v x 2 + u x 3 v x 3 x 0 x 1 x 2 x 3 = u x 0 + v x ( u x j + v x j + u x j v(x j ) ) u x 3 v x 3 interiorj u v (I 1) from u v (I 0) + + u v (I 2) interval-oriented + to x 0 x 1 x 2 x 3 node-oriented
12 Example: 1D DG-FEM u: test function (to be solved) v : trial function (our choice) u v (I 0) u v (I 1) + u v (I 2) + x j + : start of the left interval I j + x j : start of the right interval I j 1 x 0 x 1 x 2 x 3 Non-zero jumps at interior nodes Transfer intervals information between intervals Discontinuous Galerkin Methods
13 Example: 1D DG-FEM u: test function (to be solved) v : trial function (our choice) Recall our Jumps and rewrite { } : Average of jump : Difference of jump 0 Define Bilinear Function Construct Stiffness Matrix from a(u, v)
14 Example: 1D DG-FEM Basis Functions u: test function (to be solved) v : trial function (our choice) Choose basis function φ to approximate u Example: Linear Lagrange polynomial uh = k u k (I h ) φk (I h ) span(φ) u 0 u 1 u 2 u k (I h ) : to be solved I 0 I 1 I 2 x x 0 1 x x 2 3
15 Example: 1D DG-FEM Basis Functions (I u = u h ) (I h k φk h ) (I, u h ) to be solved v: trial function (our choice) k k u span(φ) v = φ Also choose v to be φ Stiffness Matrix from a u, v => from a φ ku, φ kv (I 0 ) φ1 (I 0 ) (I 1 ) φ 1 (I 1 ) (I 2 ) φ1 (I 2 ) (I 0 ) u 0 (I 0) φ 1 (I 0 ) u 1 (I 0) (I 1 ) φ 1 (I 1 ) u 0 (I 1) u 1 (I 1) = (I 2 ) φ 1 (I 2 ) u 0 (I 2) u 1 (I 2)
16 Example: 1D DG-FEM (I 0 ) φ1 (I 0 ) (I 1 ) φ1 (I 1 ) (I 2 ) φ1 (I 2 ) (I 0 ) φ 1 (I 0 ) Info within Interval Info across Interval Zeros (Non adjacent I 0 I 0 and I 1 intervals) (I 1 ) Info across Interval Info within Interval Info across Interval φ 1 (I 1 ) I 1 and I 0 I 1 I 1 and I 2 (I 2 ) Zeros (Non adjacent intervals) Info across Interval I 2 and I 1 Info within Interval I 2 φ 1 (I 2 ) Each block contains basis functions interaction w.r.t intervals Diagonal blocks: within an interval Sub-diagonal blocks: adjacent intervals Others: ZEROS (non-adjacent intervals)
17 Example: 1D DG-FEM u h = k u k (I h ) φk (I h ) v: trial function (our choice) u span(φ) v = φ 0 x j + : start of the left interval I j x j : start of the right interval I j
18 Example: 1D DG-FEM Example (I 0 ) φ 1 (I 0 ) Zeros (Non adjacent intervals) j = 1 (on x 1 ) (I 1 ) φ 1 (I 1 ) Info across Interval I 1 and I 2 (I 2 ) Zeros (Non adjacent Info across Interval Info within Interval φ 1 (I 2 ) intervals) I 2 and I 1 I
19 Example: 1D DG-FEM Correctness u: test function (to be solved) v : trial function (our choice) { } : Average of jump Recall Original Equation : Difference of jump u v = j Ij u v = j ( I j u v ) j u v xj x j+1 ) Bilinear Function: terms added
20 Example: 1D DG-FEM Correctness u: test function (to be solved) v : trial function (our choice) Symmetric term Added for the matrix to be symmetric Penalty term Added for the matrix to be positive-definite
21 Example: 1D DG-FEM Correctness u: test function (to be solved) v : trial function (our choice) Show stiffness matrix postive-definite Show V h : DG Finite Element Space u 1,h : well-defined norm
22 Example: 1D DG-FEM Correctness u: test function (to be solved) v : trial function (our choice) Show V h : DG Finite Element Space u 1,h : well-defined norm 1. Define - can be shown well-defined.
23 Example: 1D DG-FEM Correctness u: test function (to be solved) v : trial function (our choice) Show 2. Important Inequalities used Trace Inequality Trace Inequality (1D) H: Length of Interval a H b
24 Example: 1D DG-FEM Correctness u: test function (to be solved) v : trial function (our choice) Show 2. Important Inequalities used Inverse Inequality Trace + Inverse: can show that
25 Example: 1D DG-FEM Correctness u: test function (to be solved) v : trial function (our choice) Show After proof of
26 Example: 1D DG-FEM Correctness u: test function (to be solved) v : trial function (our choice) Choice of γ As long as it is large enough Dependent on degree of basis functions
27 Example: 1D DG-FEM Tri-diagonal Matrix Symmetric Positive Definite Sparse
28 Overview Mathematics behind 1D Parallelization Future Work Acknowledgement
29 1D Parallelization: Why Higher accuracy leads to larger matrix (and cost)
30 1D Parallelization(1): Matrix Construction Observe: each interval/node contributes to a local matrix (I 0 ) φ 1 (I 0 ) (I 1 ) φ 1 (I 1 ) (I 2 ) φ 1 (I 2 ) (I 2 ) φ 1 (I 2 )
31 1D Parallelization(1): Matrix Construction Observe: each interval/node contributes to a local matrix (I 0 ) φ 1 (I 0 ) (I 1 ) φ 1 (I 1 ) (I 2 ) φ 1 (I 2 ) (I 2 ) φ 1 (I 2 )
32 1D Parallelization(1): Matrix Construction Observe: each interval/node contributes to a local matrix (I 0 ) φ 1 (I 0 ) (I 1 ) φ 1 (I 1 ) (I 2 ) φ 1 (I 2 ) (I 2 ) φ 1 (I 2 )
33 1D Parallelization(1): Matrix Construction Algorithm 1. Initialization: for each processor, receive information of P cells 2. Local Matrix Construction for each processor Cell matrices for each of P cells, calculate the cell matrix Processor matrices calculate the processor matrix 3. Global Matrix Construction for root processor receive information from other processors calculate the global matrix (assembly) Assembly done by MPI
34 1D Parallelization(1): Matrix Construction MPI Commands: Assembly Processors: MPI_Send(pointerToProcessorMatrix, size, MPI_DOUBLE, root, senderid, MPI_COMM_WORLD); Root Processors: MPI_Recv(pointerToProcessorMatrix, size, MPI_DOUBLE, senderid, tag, MPI_COMM_WORLD, &status);
35 1D Parallelization(1): Matrix Construction After Global Matrix Construction Solve linear system C programming: dgesv on LAPACK LU factorization: dense matrix solver LAPACK dgesv MPI dense matrix solver Assembly... Cell Matrices Global Matrix Solution... Processor Matrices
36 1D Parallelization(1): Matrix Construction Overall Matrix Construction Time decreases with number of processors No. of Cells: 5,000,000 Degree of freedom: 2 No. of Procs Time (s) s s s Time (s) s s s Can we do better? No. of Processors
37 1D Parallelization(2): Linear System Solver Parallelize the linear system solving process AztecOO on Trilinos Each processor has access to global rows AztecOO on Trilinos... Cell Matrices Solution Global Matrix... Processor Matrices... Access to Global Rows
38 1D Parallelization(2): Linear System Solver Trilinos Commands 1. Define rows owned by each processor Epetra_Map map(globalnumberofrows, LocalNumberOfRows, GlobalIndicesOfRows, index, comm); 2. Send rows to global matrix position InsertGlobalValues(GlobalIndexOfRows, NumOfEntries, ValueOfEntries, GlobalColumnIndices); 3. Create solver x = new Epetra_Vector(map); problem = new Epetra_LinearProblem(GlobalMatrix, x, RHSvector); AztecOO solver(*problem); solver.iterate(maxnoofiterations, Tolerance);
39 1D Parallelization(2): Linear System Solver 4. Set solver parameters Solvers Direct (for dense matrix) or Iterative (for sparse matrix) LU, GMRES, CG Pre-conditioners Jacobi, sparse LU factorization, ML Library (Algebraic Multigrid Preconditioning) Large sparse linear system! More Linear Algebra
40 1D Parallelization(2): Linear System Solver AztecOO Commands solver.setaztecoption(az_solver, SolverType); solver.setaztecoption(az_precond, PreCondiType); ML preconditioner Commands ML_Epetra::MultiLevelPreconditioner* MLPrec = new ML_Epetra:: MultiLevelPreconditioner (GlobalMatrixPointer, Teuchos::ParameterList MLList, true); solver.setprecoperator(mlprec)
41 1D Parallelization: Overview
42 1D Parallelization(2): More More Linear Algebra: Check performance of different parameters
43 1D Parallelization(2): More Best performance found in ML No. of Cells: 5,000,000 Degree of freedom: 2 Domain-decomposition method Aggregation: Uncoupled Smoother: Aztec Coarse type: UMFPACK Solver Time/ No. of Iterations No. of Procs CG ML CG ilu s/ s/ s/ s/ s/ s/ s/ s/ s/ s/ s/ s/ s/ s/249 Can we do better?
44 Overview Mathematics behind 1D Parallelization Future Work Acknowledgement
45 Future Work: 2D DG-FEM Implement parallelization on 2D DG-FEM Partition of Domain: Triangles Jump condition over Edges
46 Future Work: 2D DG-FEM Implement parallelization on 2D DG-FEM Weak form of FEM: Bilinear Function of DG-FEM:
47 Future Work: 2D DG-FEM Correctness given by showing where norm defined as
48 Future Work and on 3D DG-FEM Adaptive Meshing local refinement Application e.g. Chemical Transport Equation
49 Overview Mathematics behind 1D Parallelization Future Work Acknowledgement
50 Acknowledgement This project was sponsored by Oak Ridge National Laboratory, Joint Institute for Computational Sciences, University of Tennessee, Knoxville and the Chinese University of Hong Kong. Also sincere gratitude to my mentors Dr. Kwai Wong and Dr. Ohannes Karakashian
51 Questions
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