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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
2. Ad-hoc Direct Methods
3. The matrix diagonalization techniques
4. Direct methods
5. Conclusions
4/30
2. Ad-hoc Direct Methods
• 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
5/30
Approximations:
2. Ad-hoc Direct Methods
• 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
5/30
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
6/30
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
7/30
2 , / 2,..., / 2 1, (3)k k kk u u f k N N k ku fand - the discrete Fourier coefficients;
Solution 1b - a Fourier collocation approximation
• 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
7/30
2 , / 2,..., / 2 1, (3)k k kk u u f k N N k ku fand - the discrete Fourier coefficients;
Solution 1b - a Fourier collocation approximation
• 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
7/30
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
9/30
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
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
9/30
(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
10/30
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:
11/30
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)
13/30
• 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
2. Ad-hoc Direct Methods
4. Direct methods
5. Conclusions
17/30
3. The matrix diagonalization techniques
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
18/30
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
19/30
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
20/30
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
2. Ad-hoc Direct Methods
4. Direct methods
5. Conclusions
22/30
3. The matrix diagonalization techniques
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
23/30
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
24/30
Decomposition of K into its 1st, 2nd, 0 – order components
(2) (1) (0)K K K K
25/30
then(0) (0; )
1
ld
x
lK K
* *
1
if ( )d
l ll
x const
Decomposition of K into its 1st, 2nd, 0 – order components
(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*
26/30
will lead to diagonal marix(0)hkK
Decomposition of K into its 1st, 2nd, 0 – order components
(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
27/30
2D 3D
In 2D: matrix is in general full for arbitrary nonzero
*
, 1
(2)d
k hrs
r s shk
r
d xx
Kx
Decomposition of K into its 1st, 2nd, 0 – order components
(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
28/30
to sparse matrix
• - decomposition
4. Gaussian Elimination Techniques
29/30
• 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
2. Ad-hoc Direct Methods
4. Direct methods
5. Conclusions
30/30
3. The matrix diagonalization techniques
5. Conclusions
31/31
• 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