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CSC 2016, Albuquerque, USA, October 11, 2016 Enabling Implicit Time Integration for Compressible Flows by Partial Coloring: A Case Study of a Semi-matrix-free Preconditioning Technique H. Martin Bcker, M. Ali Rostami Mathematics and Computer

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Enabling Implicit Time Integration for Compressible Flows by Partial Coloring: A Case Study of a Semi-matrix-free Preconditioning Technique

  • H. Martin Bücker, M. Ali Rostami

Mathematics and Computer Science Friedrich Schiller University Jena CSC 2016, Albuquerque, USA, October 11, 2016 Michael Lülfesmann Düsseldorf, Germany

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Outline

  • Problem in Scientific Computing
  • Combinatorial Model
  • Heuristic Coloring Algorithm
  • Experimental Results

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QUADFLOW

Josef Ballmann Mechanics

  • Finite Volume
  • Implicit Time Integration
  • Unstructured Grids
  • Adaptivity via Multiscale Analysis

(Wolfgang Dahmen, Sigfried Müller, Mathematics, RWTH)

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Large Sparse Linear System

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= = +

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Automatic Differentiation (AD)

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Given , , code to evaluate and seed matrix , generate code to evaluate matrix-matrix product Relative runtime overhead:

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Preconditioning (PC)

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Let Rather than Solve

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Missing Connections: AD and PC

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  • Access to individual elements
  • Access to chunks of rows/columns

Preconditioning: Automatic Differentiation:

  • Access to complete row/column
  • Access to groups of complete rows/columns
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Sparsification

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Main Idea

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sparsification preconditioning

J J M  ) (J  ) ( ~ J M  

preconditioning

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Full vs Partial

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Problem :

Scientific Computing Problem

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Let be a sparse Jacobian matrix with known nonzero pattern and let denote its sparsification using blocks on the diagonal of . Find binary seed matrix with minimal number of columns such that all nonzeros

  • f also appear in .
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Outline

  • Problem in Scientific Computing
  • Combinatorial Model
  • Heuristic Coloring Algorithm
  • Experimental Results

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Full Coloring

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Full Coloring

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Partial Coloring

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Partial Coloring

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Definition: ρ-Orthogonality

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structurally -orthogonal to There is no row position in which and are nonzeros and at least one

  • f them belongs to .

: ≝

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Definition: ρ-Column Intersection Graph

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Jacobian matrices and , where associated with a pair of represents not structurally -orthogonal. ∈ iff and are

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Combinatorial Problem

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Problem : Find a coloring of the -column intersection graph with a minimal number of colors. Equivalent to problem .

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Outline

  • Problem in Scientific Computing
  • Combinatorial Model
  • Heuristic Coloring Algorithm
  • Experimental Results

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Greedy Partial Coloring Heuristic

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Outline

  • Problem in Scientific Computing
  • Combinatorial Model
  • Heuristic Coloring Algorithm
  • Experimental Results

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Convergence Behavior with GMRES, ILU(0)

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Number of Nonzeros

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Iterations and Colors

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Zoom into Previous Figure

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Execution Times

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Concluding Remarks

  • Formulation of a combinatorial problem

arising from preconditioning using automatic differentiation

  • Graph model encoding this situation as

a partial coloring problem

  • Design of heuristic partial coloring

algorithm

  • Application to case study from CFD

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Major References

  • QUADFLOW: Bramkamp, Lamby, and Müller. An

adaptive multiscale finite volume solver for unsteady and steady state ow computations. Journal of Computational Physics, 197(2):460-490, 2004.

  • Sparsity: Gebremedhin, Manne, and Pothen. What

color is your Jacobian? Graph coloring for computing derivatives. SIAM Review, 47(4):629- 705, 2005

  • Sparsification: Cullum and Tuma. Matrix-free

preconditioning using partial matrix estimation. BIT Numerical Mathematics, 46(4):711-729, 2006.

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