is integration in 2d or 3d really different from - tu/e · 18 september 2002 seminar scg 1 is...
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18 september 2002 Seminar SCG 1
Is integration in 2D or 3D
reallydifferent from
integration in 1D?Pieter Heres
18 september 2002 Seminar SCG 2
Overview! Numerical integration in 1D! Numerical integration in xD! Literature! Available software
18 september 2002 Seminar SCG 3
Numerical integration1. Partition complex region into
fundamental ones
2. Use numerical integration on fundamental region
3. Adaptive: let partition depend on function
18 september 2002 Seminar SCG 6
Errors1. In basic region, due to
approximation error
2. Error due to non-exact covering of the region with basic regions
18 september 2002 Seminar SCG 7
Num. Integration of Basic Regions! Standard
! Advanced
Degrees of freedom: and
∑∫ =i
ii xfwf )(!
∑∑∫ =k p
kp
pk xfwf )()(,!
kw kx
18 september 2002 Seminar SCG 8
Error! A rule is called exact for f(x) if the
error, given by is zero.
! A rule is exact for degree n if it is exact for polynomials of degree up to n and not for n+1
∫ ∑− )( kk xfwf
18 september 2002 Seminar SCG 9
The best integration rule! Minimal amount of points, such
that the rule is exact for specific degree p
18 september 2002 Seminar SCG 10
Rules for integration1. Choose xk determine wk: Newton-
Cotes (demand degree p exact)2. Interpolate the function
then:
3. Determine xk and wk: Gauss-Legendre (demand degree p exact)
∑=
=n
kii xgfxf
1
)()()( α!
∑ ∫∫∑∫==
==n
iii
n
iii xgfxgfxf
11
)()()()()( αα!
18 september 2002 Seminar SCG 11
Rules for integration! Newton-Cotes via interpolation
" Error also via interpolation error
! Gauss-Legendre via orthogonal polynomials" Error also via orthogonal
polynomials
18 september 2002 Seminar SCG 13
Orthogonal polynomialsIntegration over interval [a,b]! The optimal points are the zeros of
the orthogonal polynomial Pn(x) on [a,b].
for all polynomials Qn-1(x) of degree §n-1.
Proof in [1]
∫ =−
b
ann dxxQxP 0)()( 1
18 september 2002 Seminar SCG 14
Proof[1] A.H. Stroud “ Numerical Quadrature and
Solution of Ordinary Differential Equations”
and
0)(
)()(0)()( 11
=⇒
==∫ ∑ −−
kn
b
aknknknn
xP
xQxPwdxxQxP
"" #"" $%"#"$%0
1
exact
1
1212
)()()(
exact)()(
∫∫∫ ∑
−−
−−
+
=
dxxRxPdxxS
xQAdxxQ
nnn
knkn
18 september 2002 Seminar SCG 15
Orthogonal polynomials! It can be proven that:
" Pn(x) is unique (normalized)
" That zeros are real and distinct and lie in the open interval (a,b)
" Zeros distributed symmetrically?! Pn(x) can be found efficiently via
recursion relation.
1)()()()()()(
011
21
=−=−−= −−
xPxxPxPxPxxP nnnnn
βγβ
18 september 2002 Seminar SCG 16
Error Newton-Cotes! Error made with Newton-Cotes can
be determined with the interpolation error:
so:!
)()()()()()(
1 nfxxxxxpxf
n
nfξ−−=− …
∫∫∫ −−=−!
)()()()()()(
1 nfxxxxxpxf
n
nfξ…
18 september 2002 Seminar SCG 17
Error Gauss-Legendre! Also: weights are positive! The error made for arbitrary f(x) for
a simple region:
)())(()()!2(
1
)()()(
)2(2)2(
1
θθ nb
an
n
n
iii
cfdxxPfn
xfwfIfR
=
=−=
∫
∑=
18 september 2002 Seminar SCG 18
Integration in 2D! For basic regions some formulae
exist or can be determined from 1D method
! Fundamental geometries
! For non-trivial regions:" Use Monte Carlo" Partition into basic regions
18 september 2002 Seminar SCG 19
Lagrange for basic region in xD! Let two Lagrangian polynomials be
given:
! Then the 2-dimensional interpolating function:
∑∑==
==m
jjjm
n
iiin yxfyyfLyxfxxfL
11
),()(),(,),()(),( µλ
∑∑= =
=≡n
i
m
jjijimnnm yxfyxyxfLLyxf
1 1
),()()()),,((),,( µλL
18 september 2002 Seminar SCG 21
Newton-Cotes for basic region in xD
! For simple domains only! Domain specific, example simplex
! Newton-Cotes cubatures can be found via cardinal functions [3]
18 september 2002 Seminar SCG 22
Newton-Cotes for basic region in xD
! Cardinal functions:
! Interpolating
! Which leads to a first-degree rule
,),(,),(),1(),( 3,12,11,1 yyxxyxyxyx ==−+−= λλλ
)1,0()0,1()0,0()1(),,(1 fyfxfyxyxfL ++−−=
)]1,0()0,1()0,0([61),,(
1
0
1
013 fffdydxyxfLfC
x
++== ∫ ∫−
18 september 2002 Seminar SCG 23
Given rules for 2D! Stroud [2] gives some rules for a
set of basic regions" Degree" Number of points
! With * are “particularly useful”
18 september 2002 Seminar SCG 24
Given rules for 2D! *-Example:
C2:5-1 degree 5, with 7 points:with weight
with weight
with weight
±±
31,
53
±
1514,0
( )0,0
V365
V72
V635
18 september 2002 Seminar SCG 25
The optimal choice?! Problem remains: is the choice of
your points optimal?
18 september 2002 Seminar SCG 26
Gauss for xD! Zeros of xD-orthogonal polynomials! Example 2D:! Square . Find cubature rule
with degree 2.! Orthogonal polynomials can be found:
! But, how many points to choose?
31),(),(
31),( 2)2,0()1,1(2)0,2( −==−= yyxpxyyxpxyxp
1, ≤yx
18 september 2002 Seminar SCG 27
Open problemGiven a fundamental geometryThen find the least amount of points
(and weights) such that
is exact for degree d.
Next session more about this problem
∑∫ =i
iie
xfwf )(ˆ
!
18 september 2002 Seminar SCG 28
Monte Carlo approach! First order method:
with N randomly chosen numbers
∑∫=Ω
=N
iixf
Nf
1
)(1
18 september 2002 Seminar SCG 29
LiteratureLibrary (CUL):[1] A.H. Stroud “ Numerical
Quadrature and Solution of Ordinary Differential Equations”,1974
[2] A.H. Stroud, “Appr. Calculation of Multiple Integrals”, 1971
[3] H. Engels, “Numerical Quadrate and Cubature”, 1980
Articles:
18 september 2002 Seminar SCG 30
@ARTICLEAllgGeor4,
AUTHOR="E. Allgower, K. Georg and R. Widmann",
TITLE="Volume integrals for boundary element methods",
JOURNAL="Journal of Computational and Applied Mathematics",
PAGES="17--29",
VOLUME="38",
YEAR="1991"
@ARTICLECooRab,
AUTHOR="R. Cools and P. Rabinowitz",
TITLE="Monomial cubature rules since ``Stroud'': a compilation",
JOURNAL="Journal of Computational and Applied Mathematics",
PAGES="309--326",
VOLUME="48",
YEAR="1993"
@ARTICLEDuve,
AUTHOR="D.A.~Dunavant",
TITLE="High degree efficient symmetric gauss quadrature rules for thetriangle",
JOURNAL="International Journal for Numerical Methods in Engineering",
PAGES="1129--1148",
VOLUME="21",
YEAR="1985"
@ARTICLEGeorWidm,
AUTHOR="K. Georg and R. Widmann",
TITLE="Adaptive quadratures over volumes",
JOURNAL="Computing",
PAGES="121--136",
VOLUME="47",
YEAR="1991"
18 september 2002 Seminar SCG 31
@ARTICLEGrund78,
AUTHOR="Axel Grundmann and H.M. M\"oller",
TITLE="Invariant integration formulas for the n-simplex by combinatorial methods",
JOURNAL="SIAM J. Numer. Anal.",
VOLUME="15",
NUMBER="2",
PAGES="282-290",
YEAR="1978"
@ARTICLEKaha91,
AUTHOR="D.K. Kahaner",
TITLE="A Survey of Existing multidimensional quadrature
Routines",
JOURNAL="Contemporary Mathematics",
VOLUME="155",
YEAR="1991"
@BOOKReich,
AUTHOR="S. Reich",
BOOKTITLE="Backward Error Analysis for Numerical Integrators",
YEAR="1996",
PUBLISHER="Preprint SC of the Konrad Zuse-Zentrum f\umlaut ur Informationstechnik Berlin,Berlin, October Germany”
@BOOKZumb1,
AUTHOR="G. W. Zumbusch",
BOOKTITLE="Adaptive h-p approximation procedures, graded meshes and anisotropic refinementfor Numerical Quadrature",
YEAR="1995",
PUBLISHER="Preprint SC of the Konrad Zuse-Zentrum f\umlaut ur Informationstechnik Berlin,Berlin, October Germany“
18 september 2002 Seminar SCG 32
Available software! NAG-Lib! QUADPACK! Net-lib! Mathematica
" Packages:# NumericalMath`GaussianQuadra-ture`
# NumericalMath`NewtonCotes`
# More…
" Normally Gauss-Konrod based! Matlab
18 september 2002 Seminar SCG 33
The answer is….
Yes!Integrating in 2D or 3D is really
different from integrating in 1D!
18 september 2002 Seminar SCG 36
Map to fundamental geometry! The domain Ω is divided into elements:
[ ] )(0
)(0
)(2
)(0
)(1 ˆ
ˆˆˆ lllll
l vyx
vvvvyx
F +
−−=