Scientific Computing I Module 10: Algorithms and Data Storage for - - PowerPoint PPT Presentation

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Scientific Computing I

Module 10: Algorithms and Data Storage for PDE Solvers

Miriam Mehl

Winter 2011/2012

Miriam Mehl: Scientific Computing I Module 10: Algorithms and Data Storage for PDE Solvers, Winter 2011/2012 1

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Part I: Algorithms for PDE Solvers

Model + Discretization + Grid: What’s Missing? Grid Traversal

Miriam Mehl: Scientific Computing I Module 10: Algorithms and Data Storage for PDE Solvers, Winter 2011/2012 2

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What’s Missing?

Scenario Model: ut = ∆u − f Spatial discreti- sation: 1 h2   1 1 −4 1 1   Time discretization: un+1 = un + ∂t 1

h2 [∆] un − f

• .

Grid: What else do we need?

Miriam Mehl: Scientific Computing I Module 10: Algorithms and Data Storage for PDE Solvers, Winter 2011/2012 3

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What’s Missing (2)?

• How to travers the grid?
• How to store and access data?
• How to assemble and store the system matrix?
• How to evaluate stencils?
• How to solve the (non-)linear system of equations?

Miriam Mehl: Scientific Computing I Module 10: Algorithms and Data Storage for PDE Solvers, Winter 2011/2012 4

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Grid Traversal – Two Examples

line-/row-wise space-filling curves (also applicable to un- structured grids)

Miriam Mehl: Scientific Computing I Module 10: Algorithms and Data Storage for PDE Solvers, Winter 2011/2012 5

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Space-Filling Curve Grid Traversal

• broad applicability (regular, adaptive, structures, unstructured

grids)

• efficient recursive construction
• locality properties
• quasi-optimal grid partitioning
• locality of data-access (cache!)

Miriam Mehl: Scientific Computing I Module 10: Algorithms and Data Storage for PDE Solvers, Winter 2011/2012 6

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Iterator Grid Traversal

• travers lists of elements, faces, edges, nodes, boundary

elements,...

• possible for all kinds of grids
• very flexible
• traverses a single grid level

→ extra work for restrictions/interpolations

• widely used in frameworks allowing for the usage of different grid

types (DUNE, Sundance,...)

Miriam Mehl: Scientific Computing I Module 10: Algorithms and Data Storage for PDE Solvers, Winter 2011/2012 7

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Recursive Grid Traversal

• follow the tree-structure of a grid (usually depth-first)
• applicable for tree-structured grids
• traverses all grid-levels

→ allows for inter-level operations (restriction, interpolation)

Miriam Mehl: Scientific Computing I Module 10: Algorithms and Data Storage for PDE Solvers, Winter 2011/2012 8

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Part II: Data Storage for PDE Solvers

Data Storage and Access Storage of the Computational Grid Storage of the Unknowns Storage of the System Matrix

Miriam Mehl: Scientific Computing I Module 10: Algorithms and Data Storage for PDE Solvers, Winter 2011/2012 9

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Data Storage and Access

What do we have to store?

• the computational grid
• values of unknowns and auxiliary variables (residuals,...)
• system matrices

Miriam Mehl: Scientific Computing I Module 10: Algorithms and Data Storage for PDE Solvers, Winter 2011/2012 10

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Storage of the Computational Grid

• regular grid:

→ nothing to store except Nx, Ny, and Nz

• tree-structured grid:

linearized storage: 1

• ref.root

1

• ref. cell l1

000000000

• unref. cells l2

1

• ref. cell l1

000000000

• unref. cells l2

0000000

• unref. cells l1

Miriam Mehl: Scientific Computing I Module 10: Algorithms and Data Storage for PDE Solvers, Winter 2011/2012 11

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Storage of the Computational Grid (2)

• unstructured grid:

linked lists of grid components:      elem1 elem2 . . . elemM      ց . . . ր . . .      face1 face2 . . . faceN      → ց . . . →      edge1 edge2 . . . edgeQ      ց → . . . . . .      node1 node2 . . . nodeP     

Miriam Mehl: Scientific Computing I Module 10: Algorithms and Data Storage for PDE Solvers, Winter 2011/2012 12

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Storage of the Unknowns

... and auxiliary values such as residuals, old time step values,...

• arrays/vectors

→ regular structured grids

• hashtable

→ unstructured / (dynamically) adaptive grids

• streams / stacks

→ optimal data locality (cache!) → possible in special cases

Miriam Mehl: Scientific Computing I Module 10: Algorithms and Data Storage for PDE Solvers, Winter 2011/2012 13

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Storage of the Unknowns (2)

Storage in streams and stacks in Peano:

• utput stream

input stream temporary stacks input stream

• utput stream

u, v, res, crit ref, crit p., res Miriam Mehl: Scientific Computing I Module 10: Algorithms and Data Storage for PDE Solvers, Winter 2011/2012 14

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Storage of the System Matrix

• dense matrix format
• sparse matrix format
• matrix-free

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