SlideShare a Scribd company logo
1 / 23
Hyperfunction Method for Numerical Integration
and Fredholm Integral Equations of the Second Kind
Hidenori Ogata
The University of Electro-Communications, Japan
13 July, 2017
Aim of this study
2 / 23
Hyperfunction theory (M. Sato, 1958)✓ ✏
• A theory of generalized functions based on complex function theory.
• A “hyperfunction” is expressed in terms of complex analytic functions.
hyperfunctions
= functions with singularities
pole
discontinuity
delta impluse, ...
←−
complex analytic function
easy to treat
numerically
✒ ✑
In this talk, we propose hyperfunction methods for
• numerical integration
• Fredholm integral equations of the second kind.
Contents
3 / 23
1. Hyperfunction thoery
2. Hyperfunction method for numerical integration
3. Hyperfunction method for Fredholm integral equations
4. Summary
Contents
4 / 23
1. Hyperfunction thoery
2. Hyperfunction method for numerical integration
3. Hyperfunction method for Fredholm integral equations
4. Summary
1. Hyperfunction theory
5 / 23
Hyperfunction theory (M. Sato, 1958)✓ ✏
• hyperfunction on an interval I
. . . the difference between the values of a complex analytic funtion F(z) on I
f(x) = [F(z)] ≡ F(x + i0) − F(x − i0).
F(z) : defining function of the hyperfunction f(x)
analytic in D  I, where D is a complex neighborhood of I
✒ ✑
D
I
F(z)
=Re z
m z
1. Hyperfunctions: examples
6 / 23
Dirac’s delta function
δ(x) = −
1
2πi
1
x + i0
−
1
x − i0
.
1. Hyperfunctions: examples
6 / 23
Dirac’s delta function
δ(x) = −
1
2πi
1
x + i0
−
1
x − i0
.
O
D
a b
C
+ǫ
−ǫ
Suppose that φ(z) is analytic in D. By Cauchy’s integral formula,
φ(0) =
b
a
φ(x)δ(x)dx = −
1
2πi
b
a
φ(x)
1
x + i0
−
1
x − i0
dx.
1. Hyperfunctions: examples
6 / 23
Dirac’s delta function
δ(x) = −
1
2πi
1
x + i0
−
1
x − i0
.
O
D
a b
C
+ǫ
−ǫ
Suppose that φ(z) is analytic in D. By Cauchy’s integral formula,
φ(0) =
b
a
φ(x)δ(x)dx = −
1
2πi
b
a
φ(x)
1
x + i0
−
1
x − i0
dx.
1. Hyperfunction: examples
7 / 23
Heaviside step function
H(x) =
1 ( x > 0 )
0 ( x < 0 )
= F(x + i0) − F(x − i0), F(z) = −
1
2πi
log(−z).
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1Re z
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Im z
-1
-0.5
0
0.5
1
Re F(z)
The real part of F(z) = −
1
2πi
log(−z).
1. Hyperfunction theory: integral
8 / 23
integral of a hyperfunction✓ ✏
f(x) = F(x + i0) − F(x − i0) : hyperfunction on an interval I
I
f(x)dx ≡ −
C
F(z)dz,
C : closed path encircling I in the positive sense and included in D
(F(z) is analytic in D  I)
✒ ✑
D
C
I
1. Hyperfunction theory: integral
8 / 23
integral of a hyperfunction✓ ✏
f(x) = F(x + i0) − F(x − i0) : hyperfunction on an interval I
I
f(x)dx ≡ −
C
F(z)dz,
C : closed path encircling I in the positive sense and included in D
(F(z) is analytic in D  I)
✒ ✑
D
C
I
I
f(x)dx =
I
[F(x + i0) − F(x − i0)] dx.
Contents
9 / 23
1. Hyperfunction thoery
2. Hyperfunction method for numerical integration
3. Hyperfunction method for Fredholm integral equations
4. Summary
2. Hyperfunction method for numerical integration
10 / 23
We consider an integral of the form
I
f(x)w(x)dx,
f(x) : analytic in D (I ⊂ D ⊂ C, )
w(x) : weight function.
D
I
2. Hyperfunction method for numerical integration
10 / 23
We consider an integral of the form
I
f(x)w(x)dx,
f(x) : analytic in D (I ⊂ D ⊂ C, )
w(x) : weight function.
D
I
We can regard the integrand as a hyperfunction.
✓ ✏
f(x)w(x)χI(x) = −
1
2πi
{f(x + i0)Ψ(x + i0) − f(x − i0)Ψ(x − i0)}
with χI(x) =
1 (x ∈ I)
0 (x ∈ I)
, Ψ(z) =
I
w(x)
z − x
dx.
✒ ✑
2. Hyperfunction method for numerical integration
10 / 23
We consider an integral of the form
I
f(x)w(x)dx,
f(x) : analytic in D (I ⊂ D ⊂ C, )
w(x) : weight function.
D
C : z = ϕ(u)
I
We can regard the integrand as a hyperfunction.
✓ ✏
I
f(x)w(x)dx =
1
2πi C
f(z)Ψ(z)dz
=
1
2πi
τperiod
0
f(ϕ(τ))Ψ(ϕ(τ))ϕ′
(τ)dτ,
C : z = ϕ(τ) ( 0 ≦ τ ≦ τperiod ) periodic function (of period τperiod)
✒ ✑
Approximating the complex integral by the trapezoidal rule, we have ...
2. Hyperfunction method for numerical integration
11 / 23
Hyperfunction method✓ ✏
I
f(x)w(x)dx ≃
h
2πi
N−1
k=0
f(ϕ(kh))Ψ(ϕ(kh))ϕ′
(kh),
with Ψ(z) =
b
a
w(x)
z − x
dx and h =
τperiod
N
.
✒ ✑
D
C : z = ϕ(τ), 0 ≦ τ ≦ τperiod
I
2. Hyperfunction method for numerical integration
11 / 23
Hyperfunction method✓ ✏
I
f(x)w(x)dx ≃
h
2πi
N−1
k=0
f(ϕ(kh))Ψ(ϕ(kh))ϕ′
(kh),
with Ψ(z) =
b
a
w(x)
z − x
dx and h =
τperiod
N
.
✒ ✑
Ψ(z) for some typical weight functions w(x)
I w(x) Ψ(z)
(a, b) 1 log
z − a
z − b
∗
(0, 1) xα−1
(1 − x)β−1
B(α, β)z−1
F(α, 1; α + β; z−1
)∗∗
( α, β > 0 )
∗ log z is the branch s.t. −π ≦ arg z < π.
∗∗ F(α, 1; α + β; z−1
) can be easily evaluated using a continued fraction.
2. Hyperfunction method for numerical integration
11 / 23
Hyperfunction method✓ ✏
I
f(x)w(x)dx ≃
h
2πi
N−1
k=0
f(ϕ(kh))Ψ(ϕ(kh))ϕ′
(kh),
with Ψ(z) =
b
a
w(x)
z − x
dx and h =
τperiod
N
.
✒ ✑
If f(z) is real-valued on R, we can reduce the number of sampling points N by half
using the reflection principle.
2. Numerical integration: theoretical error estimate
12 / 23
theoretical error estimate✓ ✏
If f(ϕ(w)) and ϕ(w) are analytic in | Im w| < d0,
|error| ≦
τperiod
π
max
Im w=±d
|f(ϕ(w))Ψ(ϕ(w))ϕ′
(w)|
×
exp(−(4πd/τperiod)N)
1 − exp(−(4πd/uperiod)N)
( 0 < ∀d < d0 ).
. . . geometric convergence.
✒ ✑
2. Numerical integration: example
13 / 23
✓ ✏
1
0
ex
xα−1
(1 − x)β−1
dx = B(α, β)F(α; α + β; 1) ( α, β > 0 ).
✒ ✑
We computed this integral by
• hyperfunction method (with N reduction),
• DE formula (efficient for integrals with end-point singularities)
• Gauss-Jacobi formula
• C++ program, double precision
• complex integral path for the hyperfunction method (an ellipse)
z = ϕ(τ) =
1
2
+
1
4
ρ +
1
ρ
cos τ +
i
4
ρ −
1
ρ
sin τ ( ρ = 10 )
= 0.5 + 2.575 cos τ + i2.425 sin τ.
2. Numerical integration: example
14 / 23
-16
-14
-12
-10
-8
-6
-4
-2
0
0 10 20 30 40 50 60
log10(error)
N
hyperfunction
hyperfunction
Gauss-Jacobi
Gauss-Jacobi
DE
DE
-16
-14
-12
-10
-8
-6
-4
-2
0
0 20 40 60 80 100 120
log10(error)
N
hyperfunction
hyperfunction
Gauss-Jacobi
DE
DE
α = β = 0.5 α = β = 10−4
(very strong singularities)
The errors of the hyperfunction method, Gauss-Jacobi formula and the DE formula
hyperfunction Gauss-Jacobi DE
α = β = 0.5 O(0.025N
) O((8.2 × 10−4
)N
) O(0.36N
)
α = β = 10−4
O(0.029N
) — O(0.70N
)
2. Numerical integration: example
14 / 23
-16
-14
-12
-10
-8
-6
-4
-2
0
0 10 20 30 40 50 60
log10(error)
N
hyperfunction
hyperfunction
Gauss-Jacobi
Gauss-Jacobi
DE
DE
-16
-14
-12
-10
-8
-6
-4
-2
0
0 20 40 60 80 100 120
log10(error)
N
hyperfunction
hyperfunction
Gauss-Jacobi
DE
DE
α = β = 0.5 α = β = 10−4
(very strong singularities)
The hyperfunction method converges geometricaly,
and its performance is not affected by the end-point singularities.
Contents
15 / 23
1. Hyperfunction thoery
2. Hyperfunction method for numerical integration
3. Hyperfunction method for Fredholm integral equations
4. Summary
3. Hyperfunction method for integral equations
16 / 23
Fredholm integral equation for unknown u(x)✓ ✏
λu(x) −
b
a
K(x, ξ)u(ξ)w(ξ)dξ = g(x),
w(ξ) : weight function, K(x, ξ), g(x), λ(= 0) : given.
✒ ✑
We apply the hyperfunction method to this integral equation.
3. Hyperfunction method for integral equations
17 / 23
λu(x) −
b
a
K(x, ξ)u(ξ)w(ξ)dξ = g(x).
(Assumption)
• g(z) : analytic in D except for
a finite number of poles at a1, . . . , aK
• K(z, ζ) : analytic function in D w.r.t. z and ζ D
a b
ak
3. Hyperfunction method for integral equations
17 / 23
λu(x) −
b
a
K(x, ξ)u(ξ)w(ξ)dξ = g(x).
(Assumption)
• g(z) : analytic in D except for
a finite number of poles at a1, . . . , aK
• K(z, ζ) : analytic function in D w.r.t. z and ζ D
a b
ak
ua(z) ≡ u(z) − λ−1
g(z) is analytic in D.
ua(x) satisfies the integral equation
✓ ✏
λua(x) −
b
a
K(x, ξ)ua(ξ)w(ξ)dξ =
1
λ
b
a
K(x, ξ)g(ξ)w(ξ)dξ.
✒ ✑
1. We discretize the integral equation for ua(x) by the hyperfunction method.
2. We solve the discretized equation by the collocation method.
3. Integral equations: Collocation equation
18 / 23
h
2πi
N
k=1
λ
ϕ(kh) − zi
− K(zi, ϕ(kh))Ψ(ϕ(kh)) ϕ′
(ϕ(kh))ua(ϕ(kh))
=
1
2πiλ C
K(zi, ζ)g(ζ)Ψ(ζ)dζ−
1
λ
N
k=1
Res(K(zi, ·)Ψg, ak) (i = 1, . . . , N),
where
C : z = ϕ(τ) ( 0 ≦ τ ≦ τperiod ) closed path encircling [a, b],
periodic function (period τperiod)
z1, . . . , zN : the collocation points inside C, h = τperiod/N.
The collocation equation
... a system of linear equations for ua(ϕ(kh))
( k = 1, . . . , N ). a b
ak
C : z = ϕ(τ)
D
zi
3. Integral equations: Collocation equation
18 / 23
h
2πi
N
k=1
λ
ϕ(kh) − zi
− K(zi, ϕ(kh))Ψ(ϕ(kh)) ϕ′
(ϕ(kh))ua(ϕ(kh))
=
1
2πiλ C
K(zi, ζ)g(ζ)Ψ(ζ)dζ−
1
λ
N
k=1
Res(K(zi, ·)Ψg, ak) (i = 1, . . . , N),
where
C : z = ϕ(τ) ( 0 ≦ τ ≦ τperiod ) closed path encircling [a, b],
periodic function (period τperiod)
z1, . . . , zN : the collocation points inside C, h = τperiod/N.
The approximate solution u(z) is given by
u(z) =
1
2πi C
ua(ζ)
ζ − z
dζ + g(z)
≃
h
2πi
N
j=1
ua(ϕ(kh))
ϕ(kh) − z
ϕ′
(kh) + g(z).
a b
ak
C : z = ϕ(τ)
D
zi
3. Integral equations: example
19 / 23
✓ ✏
u(x) +
1
0
(x − ξ)u(ξ)ξα−1
(1 − ξ)β−1
dξ = g(x),
g(x) =
1
1 + x2
+ B(α, β) Re{F(α, 1; α + β; i)}x
− B(α + 1, β) Re{F(α + 1, 1; α + β + 1; i)} ( α = β = 0.5, 10−4
).
✒ ✑
We solved the integral equation by the hyperfunction method, DE-Nystr¨om method and
Gauss-Jacobi-Nystr¨om method.
• complex integral path
C : z = ϕ(τ) =
1
2
+
1
4
ρ +
1
ρ
cos τ +
i
4
ρ −
1
ρ
sin τ ( ρ = 200 )
• collocation points zi = ϕcol
2π(i − 1)
N
( i = 1, . . . , N )
ϕc(τ) =
1
2
+
1
4
ρc +
1
ρc
cos τ +
i
4
ρc −
1
ρc
sin τ ( 1 < ρc < ρ ).
3. Integral equations: example (α = β = 0.5)
20 / 23
-60
-50
-40
-30
-20
-10
0
0 20 40 60 80 100
log10(error)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
0
20
40
60
80
100
0 20 40 60 80 100
log10(cond)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
error ǫN condition number κN of
the collocation equation
(rhoc = ρc)
ρc/ρ 0.006 0.01 0.02 0.03 0.04
ǫN O(0.0058N
) O(0.010N
) O(0.020N
) O(0.030N
) O(0.040N
)
κN O(160N
) O(97N
) O(48N
) O(32N
) O(23N
)
3. Integral equations: example (α = β = 0.5)
20 / 23
-60
-50
-40
-30
-20
-10
0
0 20 40 60 80 100
log10(error)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
0
20
40
60
80
100
0 20 40 60 80 100
log10(cond)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
error ǫN condition number κN of
the collocation equation
(rhoc = ρc)
ρc/ρ 0.006 0.01 0.02 0.03 0.04
ǫN O(0.0058N
) O(0.010N
) O(0.020N
) O(0.030N
) O(0.040N
)
κN O(160N
) O(97N
) O(48N
) O(32N
) O(23N
)
• error ǫN = O[(ρc/ρ)N
], cond. number κN = O[(ρ/ρc)N
].
• converges faster than the DE-Nystr¨om method.
• The linear system of the collocation equation is very ill-conditioned.
3. Integral equations: example (α = β = 10−4
)
21 / 23
-60
-50
-40
-30
-20
-10
0
0 20 40 60 80 100
log10(error)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
0
20
40
60
80
100
0 20 40 60 80 100
log10(cond)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
error ǫN condition number κN of
the collocation equation
(rhoc = ρc)
ρcol/ρ 0.006 0.01 0.02 0.03 0.04
ǫN O(0.0058N
) O(0.010N
) O(0.020N
) O(0.030N
) O(0.040N
)
κN O(160N
) O(97N
) O(48N
) O(32N
) O(24N
)
3. Integral equations: example (α = β = 10−4
)
21 / 23
-60
-50
-40
-30
-20
-10
0
0 20 40 60 80 100
log10(error)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
0
20
40
60
80
100
0 20 40 60 80 100
log10(cond)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
error ǫN condition number κN of
the collocation equation
(rhoc = ρc)
ρcol/ρ 0.006 0.01 0.02 0.03 0.04
ǫN O(0.0058N
) O(0.010N
) O(0.020N
) O(0.030N
) O(0.040N
)
κN O(160N
) O(97N
) O(48N
) O(32N
) O(24N
)
• error ǫN = O[(ρcol/ρ)N
], cond. number κN = O[(ρ/ρcol)N
].
• The DE-Nystr¨om method does not work if the end-point singularities are
very strong.
Contents
22 / 23
1. Hyperfunction thoery
2. Hyperfunction method for numerical integration
3. Hyperfunction method for Fredholm integral equations
4. Summary
4. Summary
23 / 23
• We applied hyperfunction theory to numerical integration and Fredholm integral
equations of the second kind.
◦ Hyperfunction theory: a generalized function theory where a “hyperfunction” is
expressed in terms of complex analytic functions.
◦ A hyperfunction integral is given by a complex loop integral, which is evaluated
numerically in the hyperfunction method.
• Hyperfunction method
◦ (Theoretical error estimate) geometric convergence
◦ (Numerical examples) efficiency for problems with strong end-point singularities
◦ Integral equation: The linear system of the collocation equation is
very ill-conditioned.
• Problems for future study
◦ Volterra integral equations.
◦ theoretical error estimate.
4. Summary
23 / 23
• We applied hyperfunction theory to numerical integration and Fredholm integral
equations of the second kind.
◦ Hyperfunction theory: a generalized function theory where a “hyperfunction” is
expressed in terms of complex analytic functions.
◦ A hyperfunction integral is given by a complex loop integral, which is evaluated
numerically in the hyperfunction method.
• Hyperfunction method
◦ (Theoretical error estimate) geometric convergence
◦ (Numerical examples) efficiency for problems with strong end-point singularities
◦ Integral equation: The linear system of the collocation equation is
very ill-conditioned.
• Problems for future study
◦ Volterra integral equations.
◦ theoretical error estimate.
Thank you!

More Related Content

What's hot

QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
The Statistical and Applied Mathematical Sciences Institute
 
Probability Formula sheet
Probability Formula sheetProbability Formula sheet
Probability Formula sheet
Haris Hassan
 
Multilinear Twisted Paraproducts
Multilinear Twisted ParaproductsMultilinear Twisted Paraproducts
Multilinear Twisted Paraproducts
VjekoslavKovac1
 
Proximal Splitting and Optimal Transport
Proximal Splitting and Optimal TransportProximal Splitting and Optimal Transport
Proximal Splitting and Optimal Transport
Gabriel Peyré
 
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
The Statistical and Applied Mathematical Sciences Institute
 
Probability cheatsheet
Probability cheatsheetProbability cheatsheet
Probability cheatsheet
Suvrat Mishra
 
Levitan Centenary Conference Talk, June 27 2014
Levitan Centenary Conference Talk, June 27 2014Levitan Centenary Conference Talk, June 27 2014
Levitan Centenary Conference Talk, June 27 2014
Nikita V. Artamonov
 
slides CIRM copulas, extremes and actuarial science
slides CIRM copulas, extremes and actuarial scienceslides CIRM copulas, extremes and actuarial science
slides CIRM copulas, extremes and actuarial science
Arthur Charpentier
 
On Twisted Paraproducts and some other Multilinear Singular Integrals
On Twisted Paraproducts and some other Multilinear Singular IntegralsOn Twisted Paraproducts and some other Multilinear Singular Integrals
On Twisted Paraproducts and some other Multilinear Singular Integrals
VjekoslavKovac1
 
ABC based on Wasserstein distances
ABC based on Wasserstein distancesABC based on Wasserstein distances
ABC based on Wasserstein distances
Christian Robert
 
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
The Statistical and Applied Mathematical Sciences Institute
 
Classification with mixtures of curved Mahalanobis metrics
Classification with mixtures of curved Mahalanobis metricsClassification with mixtures of curved Mahalanobis metrics
Classification with mixtures of curved Mahalanobis metrics
Frank Nielsen
 
the ABC of ABC
the ABC of ABCthe ABC of ABC
the ABC of ABC
Christian Robert
 
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operators
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsA T(1)-type theorem for entangled multilinear Calderon-Zygmund operators
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operators
VjekoslavKovac1
 
Prml
PrmlPrml
Prml
syou6162
 
Multiple estimators for Monte Carlo approximations
Multiple estimators for Monte Carlo approximationsMultiple estimators for Monte Carlo approximations
Multiple estimators for Monte Carlo approximations
Christian Robert
 
The dual geometry of Shannon information
The dual geometry of Shannon informationThe dual geometry of Shannon information
The dual geometry of Shannon information
Frank Nielsen
 
Testing for mixtures by seeking components
Testing for mixtures by seeking componentsTesting for mixtures by seeking components
Testing for mixtures by seeking components
Christian Robert
 
Murphy: Machine learning A probabilistic perspective: Ch.9
Murphy: Machine learning A probabilistic perspective: Ch.9Murphy: Machine learning A probabilistic perspective: Ch.9
Murphy: Machine learning A probabilistic perspective: Ch.9
Daisuke Yoneoka
 
Poster for Bayesian Statistics in the Big Data Era conference
Poster for Bayesian Statistics in the Big Data Era conferencePoster for Bayesian Statistics in the Big Data Era conference
Poster for Bayesian Statistics in the Big Data Era conference
Christian Robert
 

What's hot (20)

QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
 
Probability Formula sheet
Probability Formula sheetProbability Formula sheet
Probability Formula sheet
 
Multilinear Twisted Paraproducts
Multilinear Twisted ParaproductsMultilinear Twisted Paraproducts
Multilinear Twisted Paraproducts
 
Proximal Splitting and Optimal Transport
Proximal Splitting and Optimal TransportProximal Splitting and Optimal Transport
Proximal Splitting and Optimal Transport
 
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
 
Probability cheatsheet
Probability cheatsheetProbability cheatsheet
Probability cheatsheet
 
Levitan Centenary Conference Talk, June 27 2014
Levitan Centenary Conference Talk, June 27 2014Levitan Centenary Conference Talk, June 27 2014
Levitan Centenary Conference Talk, June 27 2014
 
slides CIRM copulas, extremes and actuarial science
slides CIRM copulas, extremes and actuarial scienceslides CIRM copulas, extremes and actuarial science
slides CIRM copulas, extremes and actuarial science
 
On Twisted Paraproducts and some other Multilinear Singular Integrals
On Twisted Paraproducts and some other Multilinear Singular IntegralsOn Twisted Paraproducts and some other Multilinear Singular Integrals
On Twisted Paraproducts and some other Multilinear Singular Integrals
 
ABC based on Wasserstein distances
ABC based on Wasserstein distancesABC based on Wasserstein distances
ABC based on Wasserstein distances
 
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
 
Classification with mixtures of curved Mahalanobis metrics
Classification with mixtures of curved Mahalanobis metricsClassification with mixtures of curved Mahalanobis metrics
Classification with mixtures of curved Mahalanobis metrics
 
the ABC of ABC
the ABC of ABCthe ABC of ABC
the ABC of ABC
 
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operators
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsA T(1)-type theorem for entangled multilinear Calderon-Zygmund operators
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operators
 
Prml
PrmlPrml
Prml
 
Multiple estimators for Monte Carlo approximations
Multiple estimators for Monte Carlo approximationsMultiple estimators for Monte Carlo approximations
Multiple estimators for Monte Carlo approximations
 
The dual geometry of Shannon information
The dual geometry of Shannon informationThe dual geometry of Shannon information
The dual geometry of Shannon information
 
Testing for mixtures by seeking components
Testing for mixtures by seeking componentsTesting for mixtures by seeking components
Testing for mixtures by seeking components
 
Murphy: Machine learning A probabilistic perspective: Ch.9
Murphy: Machine learning A probabilistic perspective: Ch.9Murphy: Machine learning A probabilistic perspective: Ch.9
Murphy: Machine learning A probabilistic perspective: Ch.9
 
Poster for Bayesian Statistics in the Big Data Era conference
Poster for Bayesian Statistics in the Big Data Era conferencePoster for Bayesian Statistics in the Big Data Era conference
Poster for Bayesian Statistics in the Big Data Era conference
 

Similar to Hyperfunction method for numerical integration and Fredholm integral equations of the second kind

Maximum likelihood estimation of regularisation parameters in inverse problem...
Maximum likelihood estimation of regularisation parameters in inverse problem...Maximum likelihood estimation of regularisation parameters in inverse problem...
Maximum likelihood estimation of regularisation parameters in inverse problem...
Valentin De Bortoli
 
Tensor Train data format for uncertainty quantification
Tensor Train data format for uncertainty quantificationTensor Train data format for uncertainty quantification
Tensor Train data format for uncertainty quantification
Alexander Litvinenko
 
Differential Calculus
Differential Calculus Differential Calculus
Differential Calculus
OlooPundit
 
IVR - Chapter 1 - Introduction
IVR - Chapter 1 - IntroductionIVR - Chapter 1 - Introduction
IVR - Chapter 1 - Introduction
Charles Deledalle
 
The integral
The integralThe integral
Litvinenko_RWTH_UQ_Seminar_talk.pdf
Litvinenko_RWTH_UQ_Seminar_talk.pdfLitvinenko_RWTH_UQ_Seminar_talk.pdf
Litvinenko_RWTH_UQ_Seminar_talk.pdf
Alexander Litvinenko
 
ma112011id535
ma112011id535ma112011id535
ma112011id535
matsushimalab
 
Ece3075 a 8
Ece3075 a 8Ece3075 a 8
Ece3075 a 8
Aiman Malik
 
Maths AIP.pdf
Maths AIP.pdfMaths AIP.pdf
Maths AIP.pdf
YuvrajSingh835126
 
Functions
FunctionsFunctions
03 convexfunctions
03 convexfunctions03 convexfunctions
03 convexfunctions
Sufyan Sahoo
 
MUMS Opening Workshop - Panel Discussion: Facts About Some Statisitcal Models...
MUMS Opening Workshop - Panel Discussion: Facts About Some Statisitcal Models...MUMS Opening Workshop - Panel Discussion: Facts About Some Statisitcal Models...
MUMS Opening Workshop - Panel Discussion: Facts About Some Statisitcal Models...
The Statistical and Applied Mathematical Sciences Institute
 
Backpropagation in Convolutional Neural Network
Backpropagation in Convolutional Neural NetworkBackpropagation in Convolutional Neural Network
Backpropagation in Convolutional Neural Network
Hiroshi Kuwajima
 
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMS
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMSSOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMS
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMS
Tahia ZERIZER
 
Roots equations
Roots equationsRoots equations
Roots equations
oscar
 
Roots equations
Roots equationsRoots equations
Roots equations
oscar
 
Differentiation
DifferentiationDifferentiation
Differentiation
puspitaaya
 
exponen dan logaritma
exponen dan logaritmaexponen dan logaritma
exponen dan logaritma
Hanifa Zulfitri
 
A Szemeredi-type theorem for subsets of the unit cube
A Szemeredi-type theorem for subsets of the unit cubeA Szemeredi-type theorem for subsets of the unit cube
A Szemeredi-type theorem for subsets of the unit cube
VjekoslavKovac1
 
lesson10-thechainrule034slides-091006133832-phpapp01.pptx
lesson10-thechainrule034slides-091006133832-phpapp01.pptxlesson10-thechainrule034slides-091006133832-phpapp01.pptx
lesson10-thechainrule034slides-091006133832-phpapp01.pptx
JohnReyManzano2
 

Similar to Hyperfunction method for numerical integration and Fredholm integral equations of the second kind (20)

Maximum likelihood estimation of regularisation parameters in inverse problem...
Maximum likelihood estimation of regularisation parameters in inverse problem...Maximum likelihood estimation of regularisation parameters in inverse problem...
Maximum likelihood estimation of regularisation parameters in inverse problem...
 
Tensor Train data format for uncertainty quantification
Tensor Train data format for uncertainty quantificationTensor Train data format for uncertainty quantification
Tensor Train data format for uncertainty quantification
 
Differential Calculus
Differential Calculus Differential Calculus
Differential Calculus
 
IVR - Chapter 1 - Introduction
IVR - Chapter 1 - IntroductionIVR - Chapter 1 - Introduction
IVR - Chapter 1 - Introduction
 
The integral
The integralThe integral
The integral
 
Litvinenko_RWTH_UQ_Seminar_talk.pdf
Litvinenko_RWTH_UQ_Seminar_talk.pdfLitvinenko_RWTH_UQ_Seminar_talk.pdf
Litvinenko_RWTH_UQ_Seminar_talk.pdf
 
ma112011id535
ma112011id535ma112011id535
ma112011id535
 
Ece3075 a 8
Ece3075 a 8Ece3075 a 8
Ece3075 a 8
 
Maths AIP.pdf
Maths AIP.pdfMaths AIP.pdf
Maths AIP.pdf
 
Functions
FunctionsFunctions
Functions
 
03 convexfunctions
03 convexfunctions03 convexfunctions
03 convexfunctions
 
MUMS Opening Workshop - Panel Discussion: Facts About Some Statisitcal Models...
MUMS Opening Workshop - Panel Discussion: Facts About Some Statisitcal Models...MUMS Opening Workshop - Panel Discussion: Facts About Some Statisitcal Models...
MUMS Opening Workshop - Panel Discussion: Facts About Some Statisitcal Models...
 
Backpropagation in Convolutional Neural Network
Backpropagation in Convolutional Neural NetworkBackpropagation in Convolutional Neural Network
Backpropagation in Convolutional Neural Network
 
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMS
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMSSOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMS
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMS
 
Roots equations
Roots equationsRoots equations
Roots equations
 
Roots equations
Roots equationsRoots equations
Roots equations
 
Differentiation
DifferentiationDifferentiation
Differentiation
 
exponen dan logaritma
exponen dan logaritmaexponen dan logaritma
exponen dan logaritma
 
A Szemeredi-type theorem for subsets of the unit cube
A Szemeredi-type theorem for subsets of the unit cubeA Szemeredi-type theorem for subsets of the unit cube
A Szemeredi-type theorem for subsets of the unit cube
 
lesson10-thechainrule034slides-091006133832-phpapp01.pptx
lesson10-thechainrule034slides-091006133832-phpapp01.pptxlesson10-thechainrule034slides-091006133832-phpapp01.pptx
lesson10-thechainrule034slides-091006133832-phpapp01.pptx
 

Recently uploaded

Operational amplifiers and oscillators notes
Operational amplifiers and oscillators notesOperational amplifiers and oscillators notes
Operational amplifiers and oscillators notes
ShachiPGowda
 
5G Radio Network Througput Problem Analysis HCIA.pdf
5G Radio Network Througput Problem Analysis HCIA.pdf5G Radio Network Througput Problem Analysis HCIA.pdf
5G Radio Network Througput Problem Analysis HCIA.pdf
AlvianRamadhani5
 
一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理
一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理
一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理
upoux
 
Pressure Relief valve used in flow line to release the over pressure at our d...
Pressure Relief valve used in flow line to release the over pressure at our d...Pressure Relief valve used in flow line to release the over pressure at our d...
Pressure Relief valve used in flow line to release the over pressure at our d...
cannyengineerings
 
Digital Twins Computer Networking Paper Presentation.pptx
Digital Twins Computer Networking Paper Presentation.pptxDigital Twins Computer Networking Paper Presentation.pptx
Digital Twins Computer Networking Paper Presentation.pptx
aryanpankaj78
 
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODEL
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELDEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODEL
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODEL
ijaia
 
Introduction to Computer Networks & OSI MODEL.ppt
Introduction to Computer Networks & OSI MODEL.pptIntroduction to Computer Networks & OSI MODEL.ppt
Introduction to Computer Networks & OSI MODEL.ppt
Dwarkadas J Sanghvi College of Engineering
 
Introduction to verilog basic modeling .ppt
Introduction to verilog basic modeling   .pptIntroduction to verilog basic modeling   .ppt
Introduction to verilog basic modeling .ppt
AmitKumar730022
 
Mechanical Engineering on AAI Summer Training Report-003.pdf
Mechanical Engineering on AAI Summer Training Report-003.pdfMechanical Engineering on AAI Summer Training Report-003.pdf
Mechanical Engineering on AAI Summer Training Report-003.pdf
21UME003TUSHARDEB
 
smart pill dispenser is designed to improve medication adherence and safety f...
smart pill dispenser is designed to improve medication adherence and safety f...smart pill dispenser is designed to improve medication adherence and safety f...
smart pill dispenser is designed to improve medication adherence and safety f...
um7474492
 
Applications of artificial Intelligence in Mechanical Engineering.pdf
Applications of artificial Intelligence in Mechanical Engineering.pdfApplications of artificial Intelligence in Mechanical Engineering.pdf
Applications of artificial Intelligence in Mechanical Engineering.pdf
Atif Razi
 
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICSUNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
vmspraneeth
 
Bayesian Decision Theory details ML.pptx
Bayesian Decision Theory details ML.pptxBayesian Decision Theory details ML.pptx
Bayesian Decision Theory details ML.pptx
amrita chaturvedi
 
1FIDIC-CONSTRUCTION-CONTRACT-2ND-ED-2017-RED-BOOK.pdf
1FIDIC-CONSTRUCTION-CONTRACT-2ND-ED-2017-RED-BOOK.pdf1FIDIC-CONSTRUCTION-CONTRACT-2ND-ED-2017-RED-BOOK.pdf
1FIDIC-CONSTRUCTION-CONTRACT-2ND-ED-2017-RED-BOOK.pdf
MadhavJungKarki
 
AI + Data Community Tour - Build the Next Generation of Apps with the Einstei...
AI + Data Community Tour - Build the Next Generation of Apps with the Einstei...AI + Data Community Tour - Build the Next Generation of Apps with the Einstei...
AI + Data Community Tour - Build the Next Generation of Apps with the Einstei...
Paris Salesforce Developer Group
 
Transformers design and coooling methods
Transformers design and coooling methodsTransformers design and coooling methods
Transformers design and coooling methods
Roger Rozario
 
An Introduction to the Compiler Designss
An Introduction to the Compiler DesignssAn Introduction to the Compiler Designss
An Introduction to the Compiler Designss
ElakkiaU
 
Null Bangalore | Pentesters Approach to AWS IAM
Null Bangalore | Pentesters Approach to AWS IAMNull Bangalore | Pentesters Approach to AWS IAM
Null Bangalore | Pentesters Approach to AWS IAM
Divyanshu
 
Call For Paper -3rd International Conference on Artificial Intelligence Advan...
Call For Paper -3rd International Conference on Artificial Intelligence Advan...Call For Paper -3rd International Conference on Artificial Intelligence Advan...
Call For Paper -3rd International Conference on Artificial Intelligence Advan...
ijseajournal
 
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024
Sinan KOZAK
 

Recently uploaded (20)

Operational amplifiers and oscillators notes
Operational amplifiers and oscillators notesOperational amplifiers and oscillators notes
Operational amplifiers and oscillators notes
 
5G Radio Network Througput Problem Analysis HCIA.pdf
5G Radio Network Througput Problem Analysis HCIA.pdf5G Radio Network Througput Problem Analysis HCIA.pdf
5G Radio Network Througput Problem Analysis HCIA.pdf
 
一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理
一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理
一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理
 
Pressure Relief valve used in flow line to release the over pressure at our d...
Pressure Relief valve used in flow line to release the over pressure at our d...Pressure Relief valve used in flow line to release the over pressure at our d...
Pressure Relief valve used in flow line to release the over pressure at our d...
 
Digital Twins Computer Networking Paper Presentation.pptx
Digital Twins Computer Networking Paper Presentation.pptxDigital Twins Computer Networking Paper Presentation.pptx
Digital Twins Computer Networking Paper Presentation.pptx
 
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODEL
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELDEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODEL
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODEL
 
Introduction to Computer Networks & OSI MODEL.ppt
Introduction to Computer Networks & OSI MODEL.pptIntroduction to Computer Networks & OSI MODEL.ppt
Introduction to Computer Networks & OSI MODEL.ppt
 
Introduction to verilog basic modeling .ppt
Introduction to verilog basic modeling   .pptIntroduction to verilog basic modeling   .ppt
Introduction to verilog basic modeling .ppt
 
Mechanical Engineering on AAI Summer Training Report-003.pdf
Mechanical Engineering on AAI Summer Training Report-003.pdfMechanical Engineering on AAI Summer Training Report-003.pdf
Mechanical Engineering on AAI Summer Training Report-003.pdf
 
smart pill dispenser is designed to improve medication adherence and safety f...
smart pill dispenser is designed to improve medication adherence and safety f...smart pill dispenser is designed to improve medication adherence and safety f...
smart pill dispenser is designed to improve medication adherence and safety f...
 
Applications of artificial Intelligence in Mechanical Engineering.pdf
Applications of artificial Intelligence in Mechanical Engineering.pdfApplications of artificial Intelligence in Mechanical Engineering.pdf
Applications of artificial Intelligence in Mechanical Engineering.pdf
 
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICSUNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
 
Bayesian Decision Theory details ML.pptx
Bayesian Decision Theory details ML.pptxBayesian Decision Theory details ML.pptx
Bayesian Decision Theory details ML.pptx
 
1FIDIC-CONSTRUCTION-CONTRACT-2ND-ED-2017-RED-BOOK.pdf
1FIDIC-CONSTRUCTION-CONTRACT-2ND-ED-2017-RED-BOOK.pdf1FIDIC-CONSTRUCTION-CONTRACT-2ND-ED-2017-RED-BOOK.pdf
1FIDIC-CONSTRUCTION-CONTRACT-2ND-ED-2017-RED-BOOK.pdf
 
AI + Data Community Tour - Build the Next Generation of Apps with the Einstei...
AI + Data Community Tour - Build the Next Generation of Apps with the Einstei...AI + Data Community Tour - Build the Next Generation of Apps with the Einstei...
AI + Data Community Tour - Build the Next Generation of Apps with the Einstei...
 
Transformers design and coooling methods
Transformers design and coooling methodsTransformers design and coooling methods
Transformers design and coooling methods
 
An Introduction to the Compiler Designss
An Introduction to the Compiler DesignssAn Introduction to the Compiler Designss
An Introduction to the Compiler Designss
 
Null Bangalore | Pentesters Approach to AWS IAM
Null Bangalore | Pentesters Approach to AWS IAMNull Bangalore | Pentesters Approach to AWS IAM
Null Bangalore | Pentesters Approach to AWS IAM
 
Call For Paper -3rd International Conference on Artificial Intelligence Advan...
Call For Paper -3rd International Conference on Artificial Intelligence Advan...Call For Paper -3rd International Conference on Artificial Intelligence Advan...
Call For Paper -3rd International Conference on Artificial Intelligence Advan...
 
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024
 

Hyperfunction method for numerical integration and Fredholm integral equations of the second kind

  • 1. 1 / 23 Hyperfunction Method for Numerical Integration and Fredholm Integral Equations of the Second Kind Hidenori Ogata The University of Electro-Communications, Japan 13 July, 2017
  • 2. Aim of this study 2 / 23 Hyperfunction theory (M. Sato, 1958)✓ ✏ • A theory of generalized functions based on complex function theory. • A “hyperfunction” is expressed in terms of complex analytic functions. hyperfunctions = functions with singularities pole discontinuity delta impluse, ... ←− complex analytic function easy to treat numerically ✒ ✑ In this talk, we propose hyperfunction methods for • numerical integration • Fredholm integral equations of the second kind.
  • 3. Contents 3 / 23 1. Hyperfunction thoery 2. Hyperfunction method for numerical integration 3. Hyperfunction method for Fredholm integral equations 4. Summary
  • 4. Contents 4 / 23 1. Hyperfunction thoery 2. Hyperfunction method for numerical integration 3. Hyperfunction method for Fredholm integral equations 4. Summary
  • 5. 1. Hyperfunction theory 5 / 23 Hyperfunction theory (M. Sato, 1958)✓ ✏ • hyperfunction on an interval I . . . the difference between the values of a complex analytic funtion F(z) on I f(x) = [F(z)] ≡ F(x + i0) − F(x − i0). F(z) : defining function of the hyperfunction f(x) analytic in D I, where D is a complex neighborhood of I ✒ ✑ D I F(z) =Re z m z
  • 6. 1. Hyperfunctions: examples 6 / 23 Dirac’s delta function δ(x) = − 1 2πi 1 x + i0 − 1 x − i0 .
  • 7. 1. Hyperfunctions: examples 6 / 23 Dirac’s delta function δ(x) = − 1 2πi 1 x + i0 − 1 x − i0 . O D a b C +ǫ −ǫ Suppose that φ(z) is analytic in D. By Cauchy’s integral formula, φ(0) = b a φ(x)δ(x)dx = − 1 2πi b a φ(x) 1 x + i0 − 1 x − i0 dx.
  • 8. 1. Hyperfunctions: examples 6 / 23 Dirac’s delta function δ(x) = − 1 2πi 1 x + i0 − 1 x − i0 . O D a b C +ǫ −ǫ Suppose that φ(z) is analytic in D. By Cauchy’s integral formula, φ(0) = b a φ(x)δ(x)dx = − 1 2πi b a φ(x) 1 x + i0 − 1 x − i0 dx.
  • 9. 1. Hyperfunction: examples 7 / 23 Heaviside step function H(x) = 1 ( x > 0 ) 0 ( x < 0 ) = F(x + i0) − F(x − i0), F(z) = − 1 2πi log(−z). -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Re z -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Im z -1 -0.5 0 0.5 1 Re F(z) The real part of F(z) = − 1 2πi log(−z).
  • 10. 1. Hyperfunction theory: integral 8 / 23 integral of a hyperfunction✓ ✏ f(x) = F(x + i0) − F(x − i0) : hyperfunction on an interval I I f(x)dx ≡ − C F(z)dz, C : closed path encircling I in the positive sense and included in D (F(z) is analytic in D I) ✒ ✑ D C I
  • 11. 1. Hyperfunction theory: integral 8 / 23 integral of a hyperfunction✓ ✏ f(x) = F(x + i0) − F(x − i0) : hyperfunction on an interval I I f(x)dx ≡ − C F(z)dz, C : closed path encircling I in the positive sense and included in D (F(z) is analytic in D I) ✒ ✑ D C I I f(x)dx = I [F(x + i0) − F(x − i0)] dx.
  • 12. Contents 9 / 23 1. Hyperfunction thoery 2. Hyperfunction method for numerical integration 3. Hyperfunction method for Fredholm integral equations 4. Summary
  • 13. 2. Hyperfunction method for numerical integration 10 / 23 We consider an integral of the form I f(x)w(x)dx, f(x) : analytic in D (I ⊂ D ⊂ C, ) w(x) : weight function. D I
  • 14. 2. Hyperfunction method for numerical integration 10 / 23 We consider an integral of the form I f(x)w(x)dx, f(x) : analytic in D (I ⊂ D ⊂ C, ) w(x) : weight function. D I We can regard the integrand as a hyperfunction. ✓ ✏ f(x)w(x)χI(x) = − 1 2πi {f(x + i0)Ψ(x + i0) − f(x − i0)Ψ(x − i0)} with χI(x) = 1 (x ∈ I) 0 (x ∈ I) , Ψ(z) = I w(x) z − x dx. ✒ ✑
  • 15. 2. Hyperfunction method for numerical integration 10 / 23 We consider an integral of the form I f(x)w(x)dx, f(x) : analytic in D (I ⊂ D ⊂ C, ) w(x) : weight function. D C : z = ϕ(u) I We can regard the integrand as a hyperfunction. ✓ ✏ I f(x)w(x)dx = 1 2πi C f(z)Ψ(z)dz = 1 2πi τperiod 0 f(ϕ(τ))Ψ(ϕ(τ))ϕ′ (τ)dτ, C : z = ϕ(τ) ( 0 ≦ τ ≦ τperiod ) periodic function (of period τperiod) ✒ ✑ Approximating the complex integral by the trapezoidal rule, we have ...
  • 16. 2. Hyperfunction method for numerical integration 11 / 23 Hyperfunction method✓ ✏ I f(x)w(x)dx ≃ h 2πi N−1 k=0 f(ϕ(kh))Ψ(ϕ(kh))ϕ′ (kh), with Ψ(z) = b a w(x) z − x dx and h = τperiod N . ✒ ✑ D C : z = ϕ(τ), 0 ≦ τ ≦ τperiod I
  • 17. 2. Hyperfunction method for numerical integration 11 / 23 Hyperfunction method✓ ✏ I f(x)w(x)dx ≃ h 2πi N−1 k=0 f(ϕ(kh))Ψ(ϕ(kh))ϕ′ (kh), with Ψ(z) = b a w(x) z − x dx and h = τperiod N . ✒ ✑ Ψ(z) for some typical weight functions w(x) I w(x) Ψ(z) (a, b) 1 log z − a z − b ∗ (0, 1) xα−1 (1 − x)β−1 B(α, β)z−1 F(α, 1; α + β; z−1 )∗∗ ( α, β > 0 ) ∗ log z is the branch s.t. −π ≦ arg z < π. ∗∗ F(α, 1; α + β; z−1 ) can be easily evaluated using a continued fraction.
  • 18. 2. Hyperfunction method for numerical integration 11 / 23 Hyperfunction method✓ ✏ I f(x)w(x)dx ≃ h 2πi N−1 k=0 f(ϕ(kh))Ψ(ϕ(kh))ϕ′ (kh), with Ψ(z) = b a w(x) z − x dx and h = τperiod N . ✒ ✑ If f(z) is real-valued on R, we can reduce the number of sampling points N by half using the reflection principle.
  • 19. 2. Numerical integration: theoretical error estimate 12 / 23 theoretical error estimate✓ ✏ If f(ϕ(w)) and ϕ(w) are analytic in | Im w| < d0, |error| ≦ τperiod π max Im w=±d |f(ϕ(w))Ψ(ϕ(w))ϕ′ (w)| × exp(−(4πd/τperiod)N) 1 − exp(−(4πd/uperiod)N) ( 0 < ∀d < d0 ). . . . geometric convergence. ✒ ✑
  • 20. 2. Numerical integration: example 13 / 23 ✓ ✏ 1 0 ex xα−1 (1 − x)β−1 dx = B(α, β)F(α; α + β; 1) ( α, β > 0 ). ✒ ✑ We computed this integral by • hyperfunction method (with N reduction), • DE formula (efficient for integrals with end-point singularities) • Gauss-Jacobi formula • C++ program, double precision • complex integral path for the hyperfunction method (an ellipse) z = ϕ(τ) = 1 2 + 1 4 ρ + 1 ρ cos τ + i 4 ρ − 1 ρ sin τ ( ρ = 10 ) = 0.5 + 2.575 cos τ + i2.425 sin τ.
  • 21. 2. Numerical integration: example 14 / 23 -16 -14 -12 -10 -8 -6 -4 -2 0 0 10 20 30 40 50 60 log10(error) N hyperfunction hyperfunction Gauss-Jacobi Gauss-Jacobi DE DE -16 -14 -12 -10 -8 -6 -4 -2 0 0 20 40 60 80 100 120 log10(error) N hyperfunction hyperfunction Gauss-Jacobi DE DE α = β = 0.5 α = β = 10−4 (very strong singularities) The errors of the hyperfunction method, Gauss-Jacobi formula and the DE formula hyperfunction Gauss-Jacobi DE α = β = 0.5 O(0.025N ) O((8.2 × 10−4 )N ) O(0.36N ) α = β = 10−4 O(0.029N ) — O(0.70N )
  • 22. 2. Numerical integration: example 14 / 23 -16 -14 -12 -10 -8 -6 -4 -2 0 0 10 20 30 40 50 60 log10(error) N hyperfunction hyperfunction Gauss-Jacobi Gauss-Jacobi DE DE -16 -14 -12 -10 -8 -6 -4 -2 0 0 20 40 60 80 100 120 log10(error) N hyperfunction hyperfunction Gauss-Jacobi DE DE α = β = 0.5 α = β = 10−4 (very strong singularities) The hyperfunction method converges geometricaly, and its performance is not affected by the end-point singularities.
  • 23. Contents 15 / 23 1. Hyperfunction thoery 2. Hyperfunction method for numerical integration 3. Hyperfunction method for Fredholm integral equations 4. Summary
  • 24. 3. Hyperfunction method for integral equations 16 / 23 Fredholm integral equation for unknown u(x)✓ ✏ λu(x) − b a K(x, ξ)u(ξ)w(ξ)dξ = g(x), w(ξ) : weight function, K(x, ξ), g(x), λ(= 0) : given. ✒ ✑ We apply the hyperfunction method to this integral equation.
  • 25. 3. Hyperfunction method for integral equations 17 / 23 λu(x) − b a K(x, ξ)u(ξ)w(ξ)dξ = g(x). (Assumption) • g(z) : analytic in D except for a finite number of poles at a1, . . . , aK • K(z, ζ) : analytic function in D w.r.t. z and ζ D a b ak
  • 26. 3. Hyperfunction method for integral equations 17 / 23 λu(x) − b a K(x, ξ)u(ξ)w(ξ)dξ = g(x). (Assumption) • g(z) : analytic in D except for a finite number of poles at a1, . . . , aK • K(z, ζ) : analytic function in D w.r.t. z and ζ D a b ak ua(z) ≡ u(z) − λ−1 g(z) is analytic in D. ua(x) satisfies the integral equation ✓ ✏ λua(x) − b a K(x, ξ)ua(ξ)w(ξ)dξ = 1 λ b a K(x, ξ)g(ξ)w(ξ)dξ. ✒ ✑ 1. We discretize the integral equation for ua(x) by the hyperfunction method. 2. We solve the discretized equation by the collocation method.
  • 27. 3. Integral equations: Collocation equation 18 / 23 h 2πi N k=1 λ ϕ(kh) − zi − K(zi, ϕ(kh))Ψ(ϕ(kh)) ϕ′ (ϕ(kh))ua(ϕ(kh)) = 1 2πiλ C K(zi, ζ)g(ζ)Ψ(ζ)dζ− 1 λ N k=1 Res(K(zi, ·)Ψg, ak) (i = 1, . . . , N), where C : z = ϕ(τ) ( 0 ≦ τ ≦ τperiod ) closed path encircling [a, b], periodic function (period τperiod) z1, . . . , zN : the collocation points inside C, h = τperiod/N. The collocation equation ... a system of linear equations for ua(ϕ(kh)) ( k = 1, . . . , N ). a b ak C : z = ϕ(τ) D zi
  • 28. 3. Integral equations: Collocation equation 18 / 23 h 2πi N k=1 λ ϕ(kh) − zi − K(zi, ϕ(kh))Ψ(ϕ(kh)) ϕ′ (ϕ(kh))ua(ϕ(kh)) = 1 2πiλ C K(zi, ζ)g(ζ)Ψ(ζ)dζ− 1 λ N k=1 Res(K(zi, ·)Ψg, ak) (i = 1, . . . , N), where C : z = ϕ(τ) ( 0 ≦ τ ≦ τperiod ) closed path encircling [a, b], periodic function (period τperiod) z1, . . . , zN : the collocation points inside C, h = τperiod/N. The approximate solution u(z) is given by u(z) = 1 2πi C ua(ζ) ζ − z dζ + g(z) ≃ h 2πi N j=1 ua(ϕ(kh)) ϕ(kh) − z ϕ′ (kh) + g(z). a b ak C : z = ϕ(τ) D zi
  • 29. 3. Integral equations: example 19 / 23 ✓ ✏ u(x) + 1 0 (x − ξ)u(ξ)ξα−1 (1 − ξ)β−1 dξ = g(x), g(x) = 1 1 + x2 + B(α, β) Re{F(α, 1; α + β; i)}x − B(α + 1, β) Re{F(α + 1, 1; α + β + 1; i)} ( α = β = 0.5, 10−4 ). ✒ ✑ We solved the integral equation by the hyperfunction method, DE-Nystr¨om method and Gauss-Jacobi-Nystr¨om method. • complex integral path C : z = ϕ(τ) = 1 2 + 1 4 ρ + 1 ρ cos τ + i 4 ρ − 1 ρ sin τ ( ρ = 200 ) • collocation points zi = ϕcol 2π(i − 1) N ( i = 1, . . . , N ) ϕc(τ) = 1 2 + 1 4 ρc + 1 ρc cos τ + i 4 ρc − 1 ρc sin τ ( 1 < ρc < ρ ).
  • 30. 3. Integral equations: example (α = β = 0.5) 20 / 23 -60 -50 -40 -30 -20 -10 0 0 20 40 60 80 100 log10(error) N rhoc=1.2 rhoc=2.0 rhoc=4.0 rhoc=6.0 rhoc=8.0 DE Gauss-Jacobi 0 20 40 60 80 100 0 20 40 60 80 100 log10(cond) N rhoc=1.2 rhoc=2.0 rhoc=4.0 rhoc=6.0 rhoc=8.0 DE Gauss-Jacobi error ǫN condition number κN of the collocation equation (rhoc = ρc) ρc/ρ 0.006 0.01 0.02 0.03 0.04 ǫN O(0.0058N ) O(0.010N ) O(0.020N ) O(0.030N ) O(0.040N ) κN O(160N ) O(97N ) O(48N ) O(32N ) O(23N )
  • 31. 3. Integral equations: example (α = β = 0.5) 20 / 23 -60 -50 -40 -30 -20 -10 0 0 20 40 60 80 100 log10(error) N rhoc=1.2 rhoc=2.0 rhoc=4.0 rhoc=6.0 rhoc=8.0 DE Gauss-Jacobi 0 20 40 60 80 100 0 20 40 60 80 100 log10(cond) N rhoc=1.2 rhoc=2.0 rhoc=4.0 rhoc=6.0 rhoc=8.0 DE Gauss-Jacobi error ǫN condition number κN of the collocation equation (rhoc = ρc) ρc/ρ 0.006 0.01 0.02 0.03 0.04 ǫN O(0.0058N ) O(0.010N ) O(0.020N ) O(0.030N ) O(0.040N ) κN O(160N ) O(97N ) O(48N ) O(32N ) O(23N ) • error ǫN = O[(ρc/ρ)N ], cond. number κN = O[(ρ/ρc)N ]. • converges faster than the DE-Nystr¨om method. • The linear system of the collocation equation is very ill-conditioned.
  • 32. 3. Integral equations: example (α = β = 10−4 ) 21 / 23 -60 -50 -40 -30 -20 -10 0 0 20 40 60 80 100 log10(error) N rhoc=1.2 rhoc=2.0 rhoc=4.0 rhoc=6.0 rhoc=8.0 DE Gauss-Jacobi 0 20 40 60 80 100 0 20 40 60 80 100 log10(cond) N rhoc=1.2 rhoc=2.0 rhoc=4.0 rhoc=6.0 rhoc=8.0 DE Gauss-Jacobi error ǫN condition number κN of the collocation equation (rhoc = ρc) ρcol/ρ 0.006 0.01 0.02 0.03 0.04 ǫN O(0.0058N ) O(0.010N ) O(0.020N ) O(0.030N ) O(0.040N ) κN O(160N ) O(97N ) O(48N ) O(32N ) O(24N )
  • 33. 3. Integral equations: example (α = β = 10−4 ) 21 / 23 -60 -50 -40 -30 -20 -10 0 0 20 40 60 80 100 log10(error) N rhoc=1.2 rhoc=2.0 rhoc=4.0 rhoc=6.0 rhoc=8.0 DE Gauss-Jacobi 0 20 40 60 80 100 0 20 40 60 80 100 log10(cond) N rhoc=1.2 rhoc=2.0 rhoc=4.0 rhoc=6.0 rhoc=8.0 DE Gauss-Jacobi error ǫN condition number κN of the collocation equation (rhoc = ρc) ρcol/ρ 0.006 0.01 0.02 0.03 0.04 ǫN O(0.0058N ) O(0.010N ) O(0.020N ) O(0.030N ) O(0.040N ) κN O(160N ) O(97N ) O(48N ) O(32N ) O(24N ) • error ǫN = O[(ρcol/ρ)N ], cond. number κN = O[(ρ/ρcol)N ]. • The DE-Nystr¨om method does not work if the end-point singularities are very strong.
  • 34. Contents 22 / 23 1. Hyperfunction thoery 2. Hyperfunction method for numerical integration 3. Hyperfunction method for Fredholm integral equations 4. Summary
  • 35. 4. Summary 23 / 23 • We applied hyperfunction theory to numerical integration and Fredholm integral equations of the second kind. ◦ Hyperfunction theory: a generalized function theory where a “hyperfunction” is expressed in terms of complex analytic functions. ◦ A hyperfunction integral is given by a complex loop integral, which is evaluated numerically in the hyperfunction method. • Hyperfunction method ◦ (Theoretical error estimate) geometric convergence ◦ (Numerical examples) efficiency for problems with strong end-point singularities ◦ Integral equation: The linear system of the collocation equation is very ill-conditioned. • Problems for future study ◦ Volterra integral equations. ◦ theoretical error estimate.
  • 36. 4. Summary 23 / 23 • We applied hyperfunction theory to numerical integration and Fredholm integral equations of the second kind. ◦ Hyperfunction theory: a generalized function theory where a “hyperfunction” is expressed in terms of complex analytic functions. ◦ A hyperfunction integral is given by a complex loop integral, which is evaluated numerically in the hyperfunction method. • Hyperfunction method ◦ (Theoretical error estimate) geometric convergence ◦ (Numerical examples) efficiency for problems with strong end-point singularities ◦ Integral equation: The linear system of the collocation equation is very ill-conditioned. • Problems for future study ◦ Volterra integral equations. ◦ theoretical error estimate. Thank you!