One-dimensional oscillatory integrals
The multi-dimensional case
A Numerical Approach To
The Steepest Descent Method
Stefan Vandewalle and Daan Huybrechs
K.U. Leuven, Division of Numerical and Applied Mathematics
Isaac Newton Institute, Cambridge, Feb 15, 2007
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Outline
1
One-dimensional oscillatory integrals
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
2
The multi-dimensional case
Motivating examples
Theory
Numerical results
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
Outline
1
One-dimensional oscillatory integrals
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
2
The multi-dimensional case
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
Introduction
The model problem
Z
I :=
b
f (x)e iωg (x) dx
a
with...
ω: frequency parameter; f : amplitude; g : oscillator
f , g : smooth real functions
classical quadrature deteriorates rapidly as ω increases
take fixed number of points per oscillation
amount of operations scales linearly with ω
new oscillatory quadrature methods
asymptotic, Filon, Levin, numerical steepest descent
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
2
What determines the value of the integral?
10
10
10
8
8
8
6
6
6
4
4
4
2
2
0
0
0
−2
−2
−2
−4
−4
−4
−6
2
−6
−6
−8
−8
−8
−10
−10
−10
−2
−1
0
1
2
Example: (10 − x2 )ei ω x
−2
−1
0
1
2
−2
−1
0
1
2
regions where the oscillations do not cancel:
⇒ boundary points
regions where the integrand is (locally) not oscillatory:
⇒ stationary points: solutions to g0 (x) = 0
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
Introduction
What is the size of the integral (asymptotically) ?
|I | = O(ω −1/(r +1) )
with r : the largest order of any stationary point ξ in [a, b]
g (j) (ξ) = 0, j = 1, . . . , r ; g (r +1) (ξ) 6= 0
Our goal: to find a decomposition of the integral
Rb
a
f (x)e iωg (x) dx = F (a) + F (b) +
P
i
F (ξi )
with ξi ∈ [a, b] the stationary points
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
The steepest descent approach
Basic idea: select a new integration path
assume f and g are analytic
Cauchy’s theorem: the value of I does not depend on the
complex path taken
C
a
Stefan Vandewalle and Daan Huybrechs
b
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
The steepest descent approach
The path of ”steepest descent”
(Cauchy(1827), Riemann (1863))
eiωg (x) = e iω(<g (x)+i=g (x)) = e −ω=g (x) e iω<g (x)
1
The function e iωg (x) does not oscillate if <g (x) is fixed
2
The function e iωg (x) decays exponentially fast if =g (x) > 0
⇒ new path at point a: ha (p) such that
g(ha (p)) = g(a) + p i , p ≥ 0
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
One-dimensional oscillatory integrals
The multi-dimensional case
The steepest descent approach
Example 1: Fourier oscillator g (x) = x (f (x) = 10 − x 2 )
g(ha (p)) = g(a) + pi ⇒ ha (p) = a + pi
30
20
C
10
0
−0.5
0
ha
hb
0.5
1
11
00
00
11
a
1
0
0
1
1.5
3
2
1
0
−1
−2
−3
b
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
The steepest descent approach
Decomposition: I =
Z
Rb
a
f (x)e iωg (x) = F (a) − F (b)
∞
f (ha (p))e iωg (ha (p)) ha0 (p)dp
0
Z ∞
iωg (a)
=e
f (ha (p))e −ωp ha0 (p)dp
0
Z ∞
iωg (a)
=e
f (a + pi)e −ωp idp
F (a) =
0
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
One-dimensional oscillatory integrals
The multi-dimensional case
A new integration path
Example 2: Quadratic oscillator g (x) = x 2 (f (x) = 10 − x 2 )
p
g(ha (p)) = g(a) + pi ⇒ ha (p) = ± a2 + pi
40
30
20
10
0
−10
−20
−30
−40
−50
1
0.8
0.6
1
0.4
0.8
0.2
0.6
0.4
0
0.2
−0.2
0
−0.4
−0.2
−0.4
−0.6
−0.6
−0.8
−0.8
−1
Stefan Vandewalle and Daan Huybrechs
−1
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
The steepest descent approach
Decomposition: I = F1 (a) − F1 (ξ) + F2 (ξ) − F2 (b)
Z
1
2
f (x)e iωx dx =
−1
Z ∞
Z ∞
0
0
f (h0,1 (p))e −ωp h0,1
(p)dp
f (h−1,1 (p))e −ωp h−1,1
(p)dp −
e iω
0
0
Z ∞
Z ∞
0
−ωp 0
iω
f (h1,2 (p))e −ωp h1,2
(p)dp
f (h0,2 (p))e
h0,2 (p)dp − e
+
0
0
Numerical singularity of the path: hξ0 (p) ∼ p −1/2 , p → 0
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
One-dimensional oscillatory integrals
The multi-dimensional case
The steepest descent approach
Example 3: Cubic oscillator g (x) = x 3
p
g(ha (p)) = g(a) + pi ⇒ ha (p) = 3 a3 + pi · e i2π/3k , k = 0, 1, 2
C
i
1
0
−1
1
0
11
00
11
00
0
1
I = F1 (a) − F1 (ξ) + F2 (ξ) − F2 (b)
Numerical singularity of the path: hξ0 (p) ∼ p −2/3 , p → 0
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
One-dimensional oscillatory integrals
The multi-dimensional case
The steepest descent approach
Example 3: Cubic oscillator g (x) = x 3 (and f (x) = 10 − x 2 )
30
20
10
0
−10
−20
−30
−40
1
0.8
0.6
1.5
0.4
1
0.2
0
0.5
−0.2
0
−0.4
−0.5
−0.6
−1
−0.8
−1
Stefan Vandewalle and Daan Huybrechs
−1.5
A Numerical Approach To The Steepest Descent Method
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
One-dimensional oscillatory integrals
The multi-dimensional case
The numerical steepest descent method
Implementation issue 1. How to evaluate Fj ?
Fj (x) = e
iωg (x)
∞
Z
f (hx (p))hx0 (p)e −ωp dp
0
e iωg (x)
=
ω
Z
∞
0
q
q
f (hx ( ))hx0 ( )e −q dq
ω
ω
if exponential decay: Gauss-Laguerre (w (q) = e −q )
if singularity: generalized Gauss-Laguerre (w (q) = q −α e −q )
n
or, generalized Gauss-Hermite (w (q) = e −q )
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
One-dimensional oscillatory integrals
The multi-dimensional case
The numerical steepest descent method
Fj (x) ≈ QF [f , g , hx ] :=
n
e iωg (x) X
wi f (hx (xi /ω))hx0 (xi /ω)
ω
i=1
Fj (ξ) ≈ QFr [f , g , hξ,j ] :=
e iωg (ξ)
ω 1/(r +1)
n
X
0
wi f (hξ,j (xir +1 /ω)) hξ,j
(xir +1 /ω) xir
i=1
absolute error: O(ω (−2n−1)/(r +1) )
relative error: O(ω −2n/(r +1) )
Gaussian convergence rate 2n as a function of 1/ω (r = 0)
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
The numerical steepest descent method
Example:
R1
1
iωx dx
0 1+x e
I ≈ QF [f , g , ha ] − QF [f , g , hb ]
ω\n
10
20
40
80
rate
1
1.0E − 3
1.2E − 4
1.7E − 5
2.0E − 6
3.1(3)
2
3.1E − 5
1.1E − 6
3.9E − 8
1.2E − 9
5.0(5)
3
1.9E − 6
2.3E − 8
2.1E − 10
1.7E − 12
6.9(7)
Stefan Vandewalle and Daan Huybrechs
4
1.7E − 7
7.5E − 10
2.0E − 12
4.2E − 15
8.9(9)
5
2.1E − 8
3.2E − 11
2.8E − 14
1.6E − 17
10.8(11)
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
The numerical steepest descent method
Example:
R1
1
iωx 3 dx
−1 x+2 e
I ≈ Q[f , g , ha ] − QF2 [f , g , hξ,1 ] + QF2 [f , g , hξ,2 ] − QF [f , g , hb ]
ω\n
40
80
160
320
rate
1
1.5E − 4
5.8E − 5
2.3E − 5
9.1E − 6
1.3 (3/3)
2
2.4E − 6
7.1E − 7
2.1E − 7
6.7E − 8
1.7 (5/3)
3
1.3E − 8
2.3E − 9
4.2E − 10
8.1E − 11
2.4 (7/3)
Stefan Vandewalle and Daan Huybrechs
4
6.9E − 11
6.4E − 12
6.1E − 13
5.8E − 14
3.4 (9/3)
5
7.6E − 13
4.7E − 14
3.1E − 15
2.1E − 16
3.9 (11/3)
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
The numerical steepest descent method
Implementation issue 2. How to determine the path ?
if inverse of g is available: ha (p) = g −1 (g (a) + pi)
otherwise...compute ha (xi /ω) and ha0 (xi /ω) numerically
Apply Newton-Raphson iteration to g (ha (p)) − g (a) − pi = 0
initial guess by truncated Taylor series of g
only very few iterations necessary
Derivative: g (ha (p)) − g (a) − pi = 0 ⇒ g 0 (ha (p))ha0 (p) − i = 0
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
The numerical steepest descent method
Example:
R1
1
iω(x 2 +x+1)1/3 dx
0 1+x e
I ≈ QF [f , g , ha ] − QF [f , g , hb ] with 2nd order Taylor path
ω\n
20
40
80
160
320
rate
1
1.4E − 2
2.5E − 3
3.8E − 4
5.2E − 5
6.7E − 6
3.0
2
2.7E − 3
2.6E − 4
1.8E − 5
1.1E − 6
6.8E − 8
4.0
3
7.4E − 4
4.6E − 5
1.7E − 6
4.0E − 8
7.7E − 10
5.7
Stefan Vandewalle and Daan Huybrechs
4
2.4E − 4
1.0E − 5
2.0E − 7
2.1E − 9
1.6E − 11
7.0
5
8.9E − 5
2.5E − 6
2.9E − 8
1.5E − 10
4.4E − 13
8.4
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
The numerical steepest descent method
Example:
R1
1
iω(x 2 +x+1)1/3 dx
0 1+x e
I ≈ QF [f , g , ha ] − QF [f , g , hb ] with 2nd order Taylor path, followed
by 1 to 4 Newton iteration steps
ω\n
20
40
80
160
320
640
rate
1
1.1E − 2
2.1E − 3
3.3E − 4
4.5E − 5
5.9E − 6
7.2E − 7
3.0
2
2.4E − 3
2.4E − 4
1.5E − 5
6.1E − 7
2.1E − 8
6.7E − 10
5.0
3
7.4E − 4
4.4E − 5
1.2E − 6
1.8E − 8
1.8E − 10
1.5E − 12
6.9
Stefan Vandewalle and Daan Huybrechs
4
2.5E − 4
1.0E − 5
1.5E − 7
8.7E − 10
2.7E − 12
6.3E − 15
8.8
5
7.5E − 5
2.4E − 6
2.3E − 8
6.2E − 11
6.2E − 14
4.3E − 17
10.5
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
The numerical steepest descent method
Relation to Levin type methods
Levin method: write I = F (b)e iωg (b) − F (a)e iωg (a)
with F (x) the non-oscillatory solution to
F 0 (x) + iωg 0 (x)F (x) = f (x)
It can be verified that the exact solution is given by
Z ∞
f (hx (p))hx0 (p)e −ωp dp
F (x) = −
0
Numerical steepest descent: evaluate the analytical
solution to the Levin ODE numerically !
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
A Filon type quadrature rule
The Generalized Filon’s method (Iserles and Nørsett)
Idea: approximate f globally on [a, b] by Hermite interpolation
P
Assume: f (x) ≈ N
i=1 ci φi (x)
Rb
P
iωg (x) dx
Then: I ≈ N
i=1 wi ci with wi = a φi (x)e
Iserles and Nørsett: convergence O(ω −s−1/(r +1) )
interpolate derivatives of order 0 . . . s − 1 at corner points
interpolate derivatives of order 0 . . . (s − 1)(r + 1) at
stationary points
computation of moments, e.g., by numerical steepest descent
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
A Filon type quadrature rule
Localized Filon’s method
Idea: apply local Hermite (or Taylor) approximation of f for each
integral contribution Fj
Fj [f ](x) := e iωg (x)
Define:
f (z) ≈
From:
We have:
with:
0
f (hx (p))hx0 (p)e −ωp dp
(i) (x) (z−x)
i!
i
Pdj
(i) (x)
Pdj
i=0 f
Fj [f ](x) ≈
R∞
i=0 wi,j f
i
wi,j := Fj [ (z−x)
i! ](x)
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
One-dimensional oscillatory integrals
The multi-dimensional case
A Filon type quadrature rule
A “classical” quadrature rule
I ≈
Pl
j=0
Pdj
i=0 wi,j f
(i) (x
j)
Convergence result:
Let dj = s − 1 at a non-stationary point, and let
dj = s(r + 1) − 1 at a stationary point then our rule has
an absolute error of O(ω −s−1/(r +1) ) and a relative error
of O(ω −s ), asymptotically for large ω.
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
One-dimensional oscillatory integrals
The multi-dimensional case
Two more examples
R1
1
iω(x−1/2)2 dx
0 1+x 2 e
Example 1:
0
10
Filon−type 3 weights
localized Filon−type
numerical steepest descent
Filon−type 5 weights
−2
10
Filon (3 evals): O(ω −3/2 )
Local Filon (3 evals): O(ω −3/2 )
−4
10
−6
NSD (4 evals): O(ω −5/2 )
−8
Filon (5 evals): O(ω −2 )
10
10
−10
10
1
10
2
10
3
ω
10
4
10
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
Two more examples
Example 2: The Hankel oscillator
Z
IH [f ] :=
b
f (x)Hν(1) (ωg1 (x))e iωg2 (x) dx,
a
For large arguments
r
2 i(z− 1 νπ−1/4π)
2
Hν(1) (z) ∼
e
, −π < argz < 2π, |z| → ∞.
πz
Approximate oscillator: g (x) = g1 (x) + g2 (x)
P P l
H f (j) (x ).
Quadrature rule: IH [f ] ≈ QH [f ] := Ll=0 dj=0
wl,j
l
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Introduction
The steepest descent approach
The numerical steepest descent method
A Filon type quadrature rule
Two more examples
Two more examples
R1
0
(1)
cos(x − 1)H0 (ωx)e iω(x
2 +x 3 −x)
dx.
Two quadrature points:
x = 0: a singularity and a stationary point of order 1
x = 1: a regular endpoint
ω \ (d0 , d1 )
100
200
400
800
rate
(0, 0)
1.2E − 3
5.1E − 4
2.2E − 4
9.3E − 5
1.23 (1.25)
(1, 0)
2.8E − 5
8.6E − 6
2.6E − 6
7.8.1E − 7
1.73 (1.75)
Stefan Vandewalle and Daan Huybrechs
(2, 0)
1.3E − 6
2.9E − 7
6.4E − 8
1.4E − 8
2.20 (2.25)
(3, 1)
2.6E − 8
4.1E − 9
6.2E − 10
9.7E − 11
2.68 (2.75)
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Outline
1
One-dimensional oscillatory integrals
2
The multi-dimensional case
Motivating examples
Theory
Numerical results
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Multivariate oscillatory integrals
Model form
Z
In :=
f (x)e iωg (x) dx
S
Contributing points?
corner points
critical points: ∇g = 0
resonance points: ∇g ⊥ ∂S
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
Motivating examples
Theory
Numerical results
One-dimensional oscillatory integrals
The multi-dimensional case
Multivariate oscillatory integrals
Main approach: repeated one-dimensional integration
deform onto path of steepest descent for inner variable
Z
b(x)
f (x, y )e iωg (x,y ) dy
I1 (x) :=
a(x)
= F (x, a(x)) − F (x, b(x))
the function F is evaluated only in points on the boundary
F (x, a(x)) = e iωg (x,a(x))
Z
∞
f (x, u(x, p))
0
∂u(x, p) −ωp
e
dp
∂p
oscillator of F (x, a(x)) is exactly known: g (x, a(x)) !
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Example 1
Rectangular domain in two dimensions
Z
b
Z
I :=
a
d
f (x, y )e iω(x+y ) dy dx
c
Step 1: deform onto path of steepest descent for y
Z
I :=
b
G (x, c)e iω(x+c) − G (x, d)e iω(x+d) dx
a
Smooth function G is given by
Z ∞
G (x, y ) =
f (x, y + ip)ie −ωp dp
0
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Example 1 (continued)
Rectangular domain in two dimensions
b
Z
G (x, c)e iω(x+c) = G̃ (a, c)e iω(a+c) − G̃ (b, c)e iω(b+c)
a
Total decomposition
I := F (a, c) − F (b, c) − F (a, d) + F (b, d)
Contributions are given by non-oscillatory double integrals
with exponential decay in both variables
Z ∞Z ∞
iω(x+y )
F (x, y ) = e
f (x + ip, y + iq)i 2 e −ω(p+q) dq dp
0
0
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Example 1 (continued)
Rectangular domain in two dimensions
b
Z
Z
I :=
a
d
f (x, y )e iω(x+y ) dy dx
c
(a,d)
(b,d)
(a,c)
(b,c)
I := F (a, c) − F (b, c) − F (a, d) + F (b, d)
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Example 2: smooth boundaries
1. A two-dimensional simplex
Z
b
Z
I :=
a
x
f (x, y )e iω(x+y ) dy dx
a
Step 1: deform onto path of steepest descent for y
Z
I :=
b
G (x, a)e iω(x+a) − G (x, x)e iω(x+x) dx
a
oscillators are different
Result: contributions of the three corner points
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Example 2: smooth boundaries (continued)
2. More general boundaries
Z
b
Z
d(x)
I :=
Z
a
b
=
f (x, y )e iω(x+y ) dy dx
c(x)
G (x, c(x))e iω(x+c(x)) − G (x, d(x))e iω(x+d(x)) dx
a
new oscillator x + c(x) may have stationary points!
resonance points: stationary point of oscillator g (x, c(x))
evaluated along the boundary
happens when ∇g ⊥ ∂S
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Example 2: smooth boundaries (continued)
Fourier integral on a circle
Z
1
√
Z
1−x 2
f (x, y )e iω(x+y ) dy dx
−1 −
√
√ √ !
√ !
2
2
2 2
=F −
−F
,−
,
2
2
2 2
I :=
√
1−x 2
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Example 3: critical points
A rectangular domain with a critical point
Z
b
Z
I :=
a
d
f (x, y )e iω(x
2 −xy −y 2 )
dy dx
c
g (x, y ) = x 2 − xy − y 2
∇g (0, 0) = 0
∂g
∂x (x, y )
∂g
∂y (x, y )
= 0 ⇐⇒ x = y /2
= 0 ⇐⇒ y = −x/2
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Example 3: critical points (continued)
A rectangular domain with a critical point
(a,d)
(d/2,d)
(b,d)
(a,−a/2)
(0,0)
(b,−b/2)
(a,c)
(c/2,c)
Stefan Vandewalle and Daan Huybrechs
(b,c)
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Example 3: critical points (continued)
A rectangular domain with a critical point
Step 1: deform onto path of steepest descent for y
Z
I :=
b
G1 (x, c)e iωg (x,c)
a
− G1 (x, −x/2)e iωg (x,−x/2)
+ G2 (x, −x/2)e iωg (x,−x/2)
− G2 (x, d)e iωg (x,d) dx
g11 (x) := g (x, c) = x 2 − cx − c 2
g12 (x) := g (x, −x/2) = 54 x 2
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Example 3: critical points (continued)
A rectangular domain with a critical point
Step 2: deform onto path of steepest descent for x
I = F111 (a, c) − F111 (c/2, c) + F112 (c/2, c) − F112 (b, c)
− F121 (a, −a/2) + F121 (0, 0) − F122 (0, 0) + F122 (b, −b/2)
+ F211 (a, −a/2) − F211 (0, 0) + F212 (0, 0) − F212 (b, −b/2)
− F221 (a, d) + F221 (d/2, d) − F222 (d/2, d) + F222 (b, d).
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
What can be proved
Theorem 4.2 (Huybrechs and Vandewalle, 2006)
Z
X
In :=
f (x)e iωg (x) dx =
sλ Fλ0 (xλ ) + O(e −ωd0 ),
S
size(λ)=2n
Conditions are:
analyticity of f , g in a ’complex neighbourhood’ of S
piecewise analytic parameterisation of S
stationary points may be degenerate
two additional conditions exclude special cases
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Additional conditions
1. Exclude curves of resonance points and critical points
If for some λ we have
∂gλ
(x, y ) ≡ 0,
∂y
then the integral in y is not oscillatory.
Solution: integration in y can be performed numerically on
the real line
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Additional conditions (continued)
Example curve of resonance points on circular boundary
R
2
2
Example: x 2 +y 2 <=1 f (x, y )e iω(x +y ) dV
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Additional conditions (continued)
2. Each lower-dimensional integral in x is either:
an integral along a curve of stationary points (of the same
order) in y
an integral along a curve that has no stationary point in y
In particular: this excludes critical points on the boundary
Reason: existence of complex stationary points in y that may (or
may not) be arbitrarily close to the real line
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Additional conditions (continued)
Example: integral on the part in the upper right halfplane
(a,d)
(d/2,d)
(b,d)
(a,−a/2)
(0,0)
(b,−b/2)
(a,c)
(c/2,c)
Stefan Vandewalle and Daan Huybrechs
(b,c)
A Numerical Approach To The Steepest Descent Method
Motivating examples
Theory
Numerical results
One-dimensional oscillatory integrals
The multi-dimensional case
Example 1: 3D balls and ellipsoids
Sphere with Fourier oscillator
No stationary points, two boundary points
−1
q
1−x 2
1
q
− 1−x 2
1
Z 1 Z
I3 =
q
Z
−
1−x 2 −x 2
1
2
q
1−x 2 −x 2
1
2
2
x +x x
iω(x1 +x2 +x3 )
e 1 2 3 (3x3 + cos(x2 ))e
dx3 dx2 dx1 .
Contributing points: ∇g ⊥ ∂S
√
√
√
√
√
√
(− 3/3, − 3/3, − 3/3) and ( 3/3, 3/3, 3/3)
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
Motivating examples
Theory
Numerical results
One-dimensional oscillatory integrals
The multi-dimensional case
Example 1: 3D balls and ellipsoids
Cubature rule
I3 ≈
2 XXX
X
i=1
j
k
l
wi,j,k,l
∂ j+k+l f
∂x1j ∂x2k ∂x3l
(xi )
Convergence
ω\d
100
200
400
800
1600
rate
0
2.6e − 5
3.2e − 6
3.9e − 7
5.0e − 8
6.3e − 9
3.0 (2.5)
Stefan Vandewalle and Daan Huybrechs
1
2.4e − 6
3.6e − 7
5.2e − 8
3.8e − 9
5.2e − 10
2.9 (3.0)
2
1.2e − 7
8.0e − 9
5.5e − 10
2.7e − 11
1.7e − 12
4.0 (3.5)
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Example 1: 3D balls and ellipsoids
Generalisation to an ellipsoid
Z
1 2
I3 :=
k (n(x)2 − 1)e iω a·x dx3 dx2 dx1 .
E 4π
length scales R1 , R2 , R3 along X , Y and Z axis
oscillator: a · x = a1 x1 + a2 x2 + a3 x3
two resonance points
application: scattering of light due to propagation in an object
with refractive index n(x) (Sigal Trattner)
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Example 1: an ellipsoid (continued)
Absolute errors
R1 = 1, R2 = 2, R3 = 1
tensor-product Gauss-Laguerre with m points per dimension
ω \ m (4m3 )
1
2
4
8
16
rate
1 (4)
2.2e − 2
6.1e − 3
1.1e − 3
6.7e − 6
1.6e − 6
2.0
2 (32)
4.9e − 3
7.5e − 5
6.0e − 7
3.8e − 8
2.4e − 9
4.0
Stefan Vandewalle and Daan Huybrechs
3 (108)
3.8e − 3
7.8e − 5
5.9e − 7
6.6e − 11
1.1e − 12
6.0
4 (256)
1.7e − 2
6.7e − 4
5.9e − 7
3.5e − 13
1.5e − 14
4.6
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Example 2: a degenerate stationary point
A rectangular domain with a degenerate stationary point
Z
1
Z
1
I2 :=
−1
−1
1
3
3
e iω(x +y ) dy dx,
3+x +y
(a,d)
(b,d)
(a,c)
(b,c)
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Example 2 (continued)
Localised Filon-type quadrature rule
nine quadrature points
use d function values and derivatives at each point
ω\d
50
100
200
400
800
rate
0
5.3e − 03
2.7e − 03
1.3e − 03
6.7e − 04
3.4e − 04
0.98 (1.0)
1
2.5e − 04
9.9e − 05
3.9e − 05
1.6e − 05
6.1e − 06
1.34 (1.33)
Stefan Vandewalle and Daan Huybrechs
2
2.9e − 05
9.0e − 06
2.8e − 06
8.9e − 07
2.9e − 07
1.63 (1.66)
A Numerical Approach To The Steepest Descent Method
One-dimensional oscillatory integrals
The multi-dimensional case
Motivating examples
Theory
Numerical results
Concluding remarks
1
One-dimensional integrals
two types of points: stationary points and endpoints
integration along the path of steepest descent
construction of Filon-type quadrature rules
2
Multi-dimensional integrals
three types of points: corners, critical points, resonance points
integration on a manifold of steepest descent
construction of Filon-type cubature rules
3
Problems not treated here
functions with complex poles or stationary points
oscillatory integral equations
Stefan Vandewalle and Daan Huybrechs
A Numerical Approach To The Steepest Descent Method