Maria Ugryumova Direct Solution Techniques in Spectral Methods CASA Seminar, 13 December 2007.

Maria Ugryumova

Direct Solution Techniques in Spectral Methods

CASA Seminar, 13 December 2007

Outline

1. Introduction

2. Ad-hoc Direct Methods

4. Direct methods

5. Conclusions

2/30

3. The matrix diagonalization techniques

1. Introduction

3/30

,u u f , ( ), 0 and BCdR f f x

• Constant-coefficient Helmholtz equation

L u b

• Some generalizations

• Spectral descretization methods lead to the system

• Steady and unsteady problems;

Outline

1. Introduction



4. Direct methods

5. Conclusions

4/30


• Fourier

• Chebyshev

• Legendre

1. To performe appropriate transform

2. To solve the system

3. To performe an inverse transform on to get .

Solution process:

L u b

ku ju

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Approximations:

2.1 Fourier Approximations

2

2 in (0,2 ), is 2 periodic

d uu f u

dx

2 , ,..., 1, (1)2 2

k k k

N Nk u u f k

ku

Problem

- the Fourier coefficients;

Solution 1a - The Fourier Galerkin approximation

• The solution is 2/( ), ,..., 1.2 2

k k

N Nu f k k

- the trancated Fourier series; / 2 1

/ 2

( )N

ikxkN

k N

P u x u e

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Solution 1b - a Fourier collocation approximation

Given2

, 0,..., 1.j

jx j N

N

2

2| =0, 0,..., -1 (2)

jx x

d uu f j N

dx

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2 , / 2,..., / 2 1, (3)k k kk u u f k N N k ku fand - the discrete Fourier coefficients;


• Using the discrete Fourier transform (DFT is a mapping between )

Given2

, 0,..., 1.j

jx j N

N

2

2| =0, 0,..., -1 (2)

jx x

d uu f j N

dx

( ) and kju x u

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2 , / 2,..., / 2 1, (3)k k kk u u f k N N k ku fand - the discrete Fourier coefficients;


• Using the discrete Fourier transform (DFT is a mapping between )

Given2

, 0,..., 1.j

jx j N

N

2

2| =0, 0,..., -1 (2)

jx x

d uu f j N

dx

( ) and kju x u

, / 2,..., / 2 1;ku k N N • (3) is solved for

( ) , 0,..., 1.jikxkju x u e j N • Reversing the DFT

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Galerkin and collocation approximation to Helmholz problem are equally

straightfoward and demand operations. 2( log )dO N N

8/30

2.3 Chebyshev Tau Approximation

, 0,1,... 2,(2)

k k k-u u f k N 0 0

0, ( 1) 0N N

kk k

k k

u u

Solution 1 - Chebyshev Tau approximation:

2

2 in ( 1,1),

( 1) (1) 0

d uu f

dxu u

Problem

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2.3 Chebyshev Tau Approximation

, 0,1,... 2,(2)

k k k-u u f k N 0 0

0, ( 1) 0N N

kk k

k k

u u

Solution 1 - Chebyshev Tau approximation:

2

2 in ( 1,1),

( 1) (1) 0

d uu f

dxu u

2 2

2 even

1( ) , 0,1,... 2.

N

p k kp kkp k

p p k u u f k Nc

• Rewriting the second derivative

, where L is upper triangular.

• Solution process requires operations 2N

Lu b

Problem

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(1)

1 11 1 12 , 1,..., 3 (5)k k kk k kku c f u f u k N

( 1) ( ) ( )

1 112 , 0 (4)q q q

k k kkku c u u k

Solution 2 - To rearrange the equations

2.3.1 More efficient solution procedure

For q=2

For q=1 in combination with (5) will lead

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After simplification

2 2 2 2 , 2,..., (6)k k kk k k k k kk k ku u u f f f k N

0

2 0

4 2

4 6

2 4

2

01 1 1 ...

* * *

* * *.... .

* * *

* * *

* *

N N

N N

kN

u

gu

gu

gu

gu

uu

• To minimize the round-off errors;

• quasi-tridiagonal system;

• not diagonally dominant;

• Nonhomogeneous BC.

For even coefficients:

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1. Discrete Chebyshev transform;

2. To solve quasi-tridiagonal system;

3. Inverse Chebyshev transform on to get .

2.4 Mixed Collocation Tau Approximation

ku ju

12/30

Solution process:

0

1

( ) ( ), 2 even( )

( ) ( ), 3 oddk

kk

L x L x kx

L x L x k

2.5 Galerkin Approximation

2

,N

Nk k

k

u u

2

2 in ( 1,1), ( 1) (1) 0

d uu f u u

dx

Solution: Legendre Galerkin approx.

Problem

13/30

0

1

( ) ( ), 2 even( )

( ) ( ), 3 oddk

kk

L x L x kx

L x L x k

2.5 Galerkin Approximation

2

,N

Nk k

k

u u

2

2 in ( 1,1), ( 1) (1) 0

d uu f u u

dx

2

2, , , , 2,..., (5)

NN

h h h

d uu f h N

dx

Solution: Legendre Galerkin approx.

Ku Mu b

,k hhk

d dK

dx dx

, , , 2,...hk k hM k h N

After integration by parts

Problem

(full matrices)

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• The same system but

• An alternative set of basis functions produces tridiagonal system:

2

1( ) ( ) ( ) , where

40,

6k k k k kx s L x L x s k

k

• Then expension is2

0

,N

Nk

kku u

K M u u b 0 21( , ,..., )N

Tu u u u

10 2( , ,..., )NTb b b b

1, ,

0, ,hk

k hK

k h

1

2

( , , ), ,

( , , ), 2,

0, otherwise

h k

hk kh h k

G s s h k h

M M G s s k k h

• Two sets of tridiagonal equations; O(N) operations

14/30

• Transformation between spectral space and physical space: 2( )O N

-2 -2

, 0,1,

- , 2,..., .k k

k

k k k k

s u ku

s u s u k N

• The standard Legendre coefficients of the solution can be found viaNu

15/30

2.6 Numerical example for Ad Hoc Methods in 1-D

2

2 in ( 1,1), ( 1) (1) 0

d uu f u u

dx

• Galerkin method is more accurate than Tau methods

• Roundoff errors are more for Chebyshev methods, significantly for N>1024

Exact solution is ( ) sin(4 )u x x

( ) sin(4 )u x x

16/30

( 0) 5( 10 )

Outline

1. Introduction


4. Direct methods

5. Conclusions

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3.1 Schur Decomposition

0 on u

(6)Tx yDU UD U F

dim( ) = ( 1) ( 1)x yU N N

2 in ( 1,1) ,u u f

• Collocation approx and Legendre G-NI approxim. lead

Problem:

• Solving (6) by Schur decomposition [Bartels, Stewart, 1972]

' ' ' 'p QD U U D U F xD

TyD

lower-triangular

upper-triangular

Tp xD P D P

T TQ yD Q D Q

' TU P UQ' TF P FQ

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Computational cost:

Solution process:

• Reduction and to Schur form

• Construction of F’

• Solution of for U’

• Transformation from U’ to U.

' ' ' 'P QD U U D U F

TyD

3 320( ) 5 ( )x y x y x yN N N N N N

xD

3if = then comp.cost is 50x yN N N N

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3.2 Matrix Digitalization Approach

• Similar to Schur decomposition. The same solution steps.

• and are diagonalizedTyDxD

' ' ' 'TDx DyU U U F

1

1

,

,

P x Dx

TQ y Dy

D P D P

D Q D Q

1

1

' ,

' .

U P UQ

F P FQ

• Operation cost:3 324( ) 4 ( ) 3x y x y x y x yN N N N N N N N

3if = then comp.cost is 56x yN N N N

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3.3 Numerical example for Ad Hoc Methods in 2-D

2 in ( 1,1) ,

0 on

u u f

u

Problem:

( ) sin(4 )sin(4 ), 0u x x y

• Matrix diagonalization was used for the solution procedure

Haidvogel and Zang (1979), Shen (1994)

• Results are very similar to 1-D case

21/30

Outline

1. Introduction


4. Direct methods

5. Conclusions

22/30


4. Direct Methods

• Matrix structure produced by Galerkin and G-NI methods ;

• How the tensor-product nature of the methods can be used efficiently to build matrices;

• How the sparseness of the matrices in 2D and in 3D can be accounted in direct techniques

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4.1 Multidimensional Stiffness and Mass Matrices

, ,

in d d

dij i

i j i ji i i

u uu f R

x x x

+ homogen. BC on

. allfor ,1,1,

Vvdxfvdxuvvdxx

udx

x

v

x

u

j

d

jiij

ij

d

jiij

ˆu ( ), b ,k hu f dx

ˆ{ }k

Problem:

Integral formulation:

Galerkin solution: ˆ , , Nk k

k

u u Lu b

– stiffness matrix

Let be a finite tensor-product basis in .

The trial and test function will be chosen in { }N kV span

( )hkL K

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Decomposition of K into its 1st, 2nd, 0 – order components

(2) (1) (0)K K K K

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then(0) (0; )

1

ld

x

lK K

* *

1

if ( )d

l ll

x const


(2) (1) (0)K K K K

*(0)hk k hd xK

• for a general , the use of G-NI approach with Lagrange nodal basis*

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will lead to diagonal marix(0)hkK


(1(2) (0))K KK K *)

1

(1d

khr

r ihk xK d

x

• - tensor-product function,

*r ( ; )(1; )

1

llr

dxr

lK K

• - arbitrary, G-NI approach leads to sparse matrix *r

(a matrix-vector multiply requires operations)

1( )dO N

1( )dO N

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2D 3D

In 2D: matrix is in general full for arbitrary nonzero

*

, 1

(2)d

k hrs

r s shk

r

d xx

Kx


(1 )(2) ) (0K KK K

(2)hkK *{ }rs

( ; )(2; , )

1

llr ls

dxr s

lK K

• for arbitrary , G-NI approach (with Lagange nodal basis) leads

*rs

(a matrix-vector multiply requires operations)1( )dO N

In 3D: has sparse structure(2)hkK

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to sparse matrix

• - decomposition

4. Gaussian Elimination Techniques

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• LU - decomposition

TCC

3

2

- for decomposituin

- solution phase

d

d

O N

O N

2D: special cases of Ad-hoс methods have lower cost

• Frontal and multifrontal [Davis and Duff 1999]

• To get benifit from sparsyty, reodering of matrix to factorization have to be done [Gilbert, 1992, Saad, 1996]

Outline

1. Introduction


4. Direct methods

5. Conclusions

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5. Conclusions

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• Approximation techniques;

• Galerkin approximations give more accurate results than other methods;

• Techniques, which can eliminate the cost of solution on prepocessing stage;

• Sparcity matrices

Thank you for attention.

Thank you for attention

Maria Ugryumova Direct Solution Techniques in Spectral Methods CASA Seminar, 13 December 2007.

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