Introduction to Simulation - Lecture 21...Introduction to Simulation - Lecture 21 Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy Boundary Value Problems - Solving 3-D

Introduction to Simulation - Lecture 21

Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy

Boundary Value Problems - Solving 3-D Finite-Difference problems

Jacob White

Outline

• Reminder about FEM and F-D– 1-D Example

• Finite Difference Matrices in 1, 2 and 3D– Gaussian Elimination Costs

• Krylov Method– Communication Lower bound

– Preconditioners based on improving communication

1-D ExampleNormalized 1-D Equation

Normalized Poisson Equation

2

2

( ) ( ) ( )sT x u xh f x

x x xκ∂ ∂ ∂

= − ⇒ − =∂ ∂ ∂

Heat Flow

( ) ( )xxu x f x− =

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FD Matrix properties Finite Differences

1-D Poisson

0x 1x 2x nx 1nx +

12

ˆ ˆ ˆ2( )j j jj

u u uf x

x+ − +

− =∆

xxu f− =

1 1

2

ˆ ( )2 1 0 01 2 1

1 0 01

0 0 1 2ˆ ( )n n

u f x

x

u f x

− − − = ∆ − −

M ML

M MO M

M MO O O

M MM O O O

M ML

A

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Residual EquationUsing Basis Functions

2

2

u fx∂

− =∂

Partial Differential Equation form

(0) 0 (1) 0u u= =

Basis Function Representation

Plug Basis Function Representation into the Equation

( ) ( ) ( )2

21

ni

ii

d xR x f x

dxϕ

ω=

= +∑

( ) ( ) ( ){1 Basis Functions

n

h i ii

u x u x xω ϕ=

≈ =∑

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Basis WeightsUsing Basis functions

Galerkin Scheme

Generates n equations in n unknowns

( ) ( ) ( )21

210

0n

il i

i

d xx f x dx

dxϕ

ϕ ω=

+ =

∑∫

Force the residual to be “orthogonal” to the basis functions

{ }1,...,l n∈

( ) ( )1

0

0l x R x dtϕ =∫

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Basis WeightsUsing Basis Functions Galerkin with integration by

parts

Only first derivatives of basis functions

( ) ( )( ) ( )1

1 1

0 0

0

n

i iil

i

xdd xdx x f x dx

dx dx

ωϕϕϕ= − =

∑∫ ∫

{ }1,...,l n∈

Structural Analysis of Automobiles

• Equations– Force-displacement relationships for

mechanical elements (plates, beams, shells) and sum of forces = 0.

– Partial Differential Equations of Continuum Mechanics

Drag Force Analysis of Aircraft

• Equations– Navier-Stokes Partial Differential Equations.

Engine Thermal Analysis

• Equations– The Poisson Partial Differential Equation.

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FD Matrix properties Discretized Poisson

2-D Discretized Problem

1x 2x mx

1mx + 2mx

m

m

1 12 2

2 2( )j j j j m j j m

j

u ux

u u u u

x

u uf x

x y

yy

+ − + −− + − ++ =

∆ ∆1442443 1442443

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FD Matrix properties Matrix Nonzeros, 5x5 example

2-D Discretized Problem

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FD Matrix properties

3-D Discretization

mm

jx

j mx −2j m

x−

1jx +

1jx −

j mx +

2j mx

+

2 21 12 2 2

ˆ ˆ ˆ2ˆ ˆ ˆ ˆ ˆ ˆ2 2( )

( ) ( ) ( )jj j j j m j j m j m j m

j

xx yy zz

u u uu u u u u uf x

x y zu u u

+ − + − + −− +− + − +

+ + =∆ ∆ ∆

1442443 1442443 144424443

m

Discretized Poisson

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FD Matrix properties Matrix nonzeros, m = 4 example

3-D Discretization

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Summary

2 2 2

1 1 11 2, 2 4, 3 6,D D D− → − → − →∆ ∆ ∆

Matrix is symmetric positive definite

Assuming uniform discretization, diagonal is

| | | |ii ijj i

A A≠

≥

∑

Numerical Properties

Matrix is Irreducibly Diagonally Dominant

Each row is strictly diagonally dominant, or path connected to a strictly diagonally dominant row

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Summary

2 2 3 31 , 2 , 3D m m D m m D m m− → × − → × − → ×

Nonzeros per row 1 3, 2 5, 3 7D D D− − −

2

1 0 | | 1

2 0 | |

3 0 | |

ij

ij

ij

D A i j

D A i j m

D A i j m

− = − >

− = − >

− = − >

Structural Properties

Matrices in 3-D are LARGE

100x100x100 grid in 3-D = 1 million x 1 million matrixMatrices are very sparse

Matrices are banded

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11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

A A A AA A A AA A A AA A A A

44A43A43A 44A44A

34A33A32A

Basics of GEPicture

Triangularizing

22A 23A 24A

42A33A 34A0

0

000 0

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Pivot

GE BasicsAlgorithm

Triangularizing

Multiplier

Form n-1 reciprocals (pivots)

Form 21

1

( )2

n

i

nn i−

=

− =∑ multipliers

For i = 1 to n-1 { “For each Row”For j = i+1 to n { “For each Row below pivot”

For k = i+1 to n { “For each element beyond Pivot”

}}

}

jijk jk ik

ii

AA A A

A⇐ −

Perform1

2 3

1

2( )3

n

in i n

−

=

− ≈∑Multiply-adds

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Complexity of GE

3 3

3 6

3

6

12

19 8

1 D ( ) ( ) 100 pt grid 2 D ( ) ( ) 100 10

(10 ) ops(0 grid

3 D ( ) ( ) 100 100 110 ) ops

(10 )00 g sr opid

O n O mO n O mO n

OO

O m O

− = ←

− = ← ×

− = ← × ×

For 2- D and 3-D problems Need a Faster Solver !

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For i = 1 to n-1 {For j = i+1 to n {

For k = i+1 to n {

}}

}

Banded GEAlgorithm

Triangularizing

NONZEROS

0

0b

jijk jk ik

ii

AA A A

A⇐ −

b i+b-1 {i+b-1 {

Perform( )

12 2

1(min( 1, ))

n

ib n i O b n

−

=

− − ≈∑Multiply-adds

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Complexity of Banded GE

2

2 4

2 7

8

14

1 D ( ) ( ) 100 pt grid 2 D ( ) ( ) 100 100 grid 3 D

(100) ops(10 ) ops

(10 ) ops( ) ( ) 100 100 100 grid

O b n OO

O mO b n

OO m

O b n O m

− = ←

− = ← ×

− = ← × ×

For 3 - D problems Need a Faster Solver !

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The World According to Krylov

Determine the Krylov Subspace 0r Pb PAx= −

Select Solution from the Krylov Subspace( ){ }1 0 0 0 0, , ,..., kk k kx x y y span r PAr PA r+ = + ∈kGCR picks a residual minimiz ng yi .

Start with , Form Ax b PAx Pb= =Preconditioning

( ){ }0 0 0Krylov Subspace , ,..., kspan r PAr PA r≡

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Krylov MethodsDiagonal Preconditioners

Preconditioning

Let A ndD A= +

( ) ( )1 1 1Apply GCR to ndD A x I D A x D b− − −= + =

• The Inverse of a diagonal is cheap to compute• Usually improves convergence

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Krylov MethodsConvergence Analysis

Optimality of GCR poly

GCR Optimality Property

( )k+1 polynomial such that 0 =1℘%

ThereforeAny polynomial which satisfies the

constraints can be used to get an upper bound on

1

0

kr

r

+

1 0k+1 k+1( ) r where is any orderk thr kPA+ ≤ ℘ ℘% %

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( )Keep as small as possible:k iλ℘Easier if eigenvalues are clustered!

Residual Poly Picture for Heat Conducting Bar MatrixNo loss to air (n=10)

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10

0

M


Krylov Vectors for diagonal Preconditioned A

1

1 0.5 0 00.5 1 00 0.50 0 0.5 1

D A−

− − − −

O

O O

1444442444443

1-D DiscretizedPDE0b r= 1 0 0 0 0 0 0

1 -.5 0 0 0 0 0

1.25 -1 -.5 0 0 0 0

1D A−

1D A−

10.5

0

−

M=

(1 digit)exactx 0.9 -0.7 0.6 -0.5 0.4 -0.3 0.1

10.5

0

−

M

1.251

0.50

− −

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0b r= 1 0 0 0 0 0 0

1 -.5 0 0 0 0 0

1.25 -1 -.5 0 0 0 0

1D A−

1D A−

(1 digit)exactx 0.9 -0.7 0.6 -0.5 0.4 -0.3 0.1

Krylov Vectors for Diagonal Preconditioned A

Communication Lower Bound for m gridpoints( )1 0 is nonzero in entry after itersthk

m mD A r k− =

Communication Lower Bound

( )1 0

1Need at least iterations for + 0

kk j

jmj m

x x rm α+

=

= ≠

∑

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Krylov Vectors for Diagonal Preconditioned A

Two Dimensional Case

For an mxm Grid

m

m

0

10

If

0

r

=

M

( )Takes iters2m O m≈ =

( )12or 0f k

mx + ≠

m2

1

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Convergence for GCR

Eigenanalysis

1Recall Eigenvalues of 1 cos1

kD Amπ− = − +

10

0

M

1

1 0.5 0 00.5 1 00 0.50 0 0.5 1

D A−

− − − −

O

O O

1444442444443

10.5

0

−

M=

10.5

0

−

M

1.251

0.50

− −

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Convergence for GCR

Eigenanalysis

# Iters for GCR to achieve convergence

log2

1log1

( )m

k O mγ

κκ

→ ∞= ≅

−

+

1 max

min

1For , 1 cos 1 cos

1 1mD Am m

λ π πλ

κ−−

≡ = − − + +

0

121

kkr convergencetolerancer

κκ

γ −≤ ≤ +

GCR achieves Communication lower bound O(m)!

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Work for Banded Gaussian Elimination, Relaxation and GCR

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

2

4 3 3

7 6 4

Dimension Banded GE Sparse GE GCR

1

2

3

O m O m O m

O m O m O m

O m O m O m

GCR faster than banded GE in 2 and 3 dimensions Could be faster, 3-D matrix only m3 nonzeros.Must defeat the communication lower bound!

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How to get Faster Converging GCR

Preconditioner must accelerate communication

Preconditioning is the Only Hope

Multiplying by PA must move values by more than one grid point.

GCR already achieves Communication Lowerbound for a diagonally preconditioned A

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1-D Discretized PDE0u (old)

2u(new)1u

(new)1ˆnu −

(new)ˆnu1ˆnu +

Gauss Seidel(new)1u

(new)2u (old)

3u

Each Iteration of Gauss-Seidel Relaxationmoves data across the grid

Preconditioning Approaches

Gauss-Seidel Preconditioning

Physical View

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1-D DiscretizedPDE0b r= 1 0 0 0 0 0 0

x x x x x x x

x x x x x x x

( ) 1D L A−+

( ) 1D L A−+

(1 digit)exactx 0.9 -0.7 0.6 -0.5 0.4 -0.3 0.1

0b r= 0 0 0 0 0 0 1

0 0 0 0 0 x x

0 0 0 0 x x x

( ) 1D L A−+

( ) 1D L A−+

X=nonzero

Gauss-Seidel communicates quickly in only one direction



Krylov Vectors

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0u (old)2u

(new)1u

(new)1ˆnu −

(new)ˆnu1ˆnu +

(new)1u

(new)2u (old)

3u

(new)2ˆnu −

(newer)1ˆnu − ( )ˆ new

nu

0u (newer)2u

(newer)1u



Symmetric Gauss-Seidel

This symmetric Gauss-Seidel PreconditionerCommunicates both directions

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Symmetric Gauss-Seidel

( ) ( ) 1/ 2Forward Sweep half step : k kD L x Ux b++ + =

Derivation of the SGS Iteration Equations

( ) ( ) 1/ 2Backward Sweep half step : k kD U Lx x b+++ =

( ) ( ) 11 1 k kD L UUx xL D− −+ +⇒ +=( ) ( ) ( )1 1 1D U D U Lb D L b−− −+ − ++ +

( ) ( )1 11 k k kx x D U D D L Ax− −+ = − + +⇒

( ) ( )1 1D U D D L b− −+ + +

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Block Diagonal Preconditioners

Line SchemesGrid

Matrix

m2

1

Tridiagonal Matrices factor quickly

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Line Schemes

m2

1

The Preconditioner is now two Tridiagonal solves, with variable reordering in between.

Grid

Do lines first in x, then in y.

Solution

ProblemLines preconditioners

communicate rapidly in only one direction

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Domain Decomposition

Fewer blocks means faster convergence, but

more costly iterates

The trade-off

ApproachBreak the domain into

small blocks each with the same # of grid points

m2

1

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Domain Decompositionm points

mpointsm

l

1 2

2l

l

( )2

2B factoring grids, , sparse GElock cost: m l l O m ll

×

Communication bound gives iterations.GCR iters: mOl

( )3Suggests insensitivity t Algorithm is : .o l O m

Do you have to refactor every GCR iteration?

Block index

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Siedelerized Block Diagonal Preconditioners

Line SchemesGrid

Matrix

m2

1

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GridMatrix

m2

1

Bigger systems to solve, but can have faster convergence on tougher problems (not just Poisson).


Overlapping Domain Preconditioners

Line based Schemes

i

Reminder about Gaussian Elimination Computational Steps

Fill-in for Sparse MatricesGreatly increases factorization cost

Fill-in in a 2-D gridIncomplete Factorization Idea


Incomplete Factorization Schemes

Outline

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Sparse MatricesFill-In

Example

X X XX X 0X 0 X

X X XX X 0X 0 X

XX

X X

X= Non zero

Matrix Non zero structure Matrix after one GE step

X X

Fill-ins

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Sparse MatricesFill-In

Second Example

X X X XX X 0 00 X X 0

X 0 00

Fill-ins Propagate

XX

X

X

X

X X

X

X X

Fill-ins from Step 1 result in Fill-ins in step 2

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Sparse Matrices Fill-In

Pattern of a Filled-in Matrix

Very Sparse

Very Sparse

Dense

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Sparse Matrices Fill-InUnfactored Random Matrix

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Sparse Matrices Fill-InFactored Random Matrix

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Factoring 2-D Finite-Difference matrices

Generated Fill-in Makes Factorization Expensive

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FD Matrix properties Matrix nonzeros, m = 4 example

3-D Discretization

iPreconditioning Approaches

Incomplete Factorization Schemes

Key idea

THROW AWAY FILL-INS!Throw away all fill-ins

Throw away fill-ins produced by other fill-insThrow away only fill-ins with small values

Throw away fill-ins produced by fill-ins of other fill-ins, etc.

Summary

• 3-D BVP Examples– Aerodynamics, Continuum Mechanics, Heat-Flow

• Finite Difference Matrices in 1, 2 and 3D– Gaussian Elimination Costs

• Krylov Method– Communication Lower bound

– Preconditioners based on improving communication

Documents

Introduction to Simulation - Lecture 21...Introduction to Simulation - Lecture 21 Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy Boundary Value Problems - Solving 3-D