Inverse problems andtheory of reproducing
kernels – numericalexperiments(abstract)
Tsutomu Matsuura
Gunma University Center for Mathematics and Data Sciencematsuura@gunma-u.ac.jp

1

Formulations and solutions

Here, for the three inverse problems previously shown by Prof. Saito in his abstract, we present formurations and solutions for those problems using Tikhonov
regularization.
1) Inverse source problem in the Poisson equation([1])
We obtained very and surprisingly simple approximate solutions for the Poisson
equation, for any L2 (Rn ) function g,on Rn∆u = g(1.1)

in the class of the functions of the s order Sobolev Hilbert space H s on thewhole real space Rn (n ≥ 1, s ≥ 2, s > n/2).
We shall use the n order Sobolev Hilbert space H n comprising functions Fon Rn with the norm (Here, of course, r1 + r2 + · · · + rn = ν.))2∫ (nν∑∑
ν!
∂ ν F (x)2
∥F ∥HS =dx.
n Cν
r1 !r2 ! · · · rn ! Rn ∂xr11 ∂xr22 · · · ∂xrnn
ν=0
r1 ,r2 ,...,rn ≥0(1.2)
This Hilbert space admits the reproducing kernel∫
11ei(x−y)·ξ dξ
K(x, y) =(2π)n Rn (1 + |ξ|2 )n(1.3)

as we see easily by using Fourier’s transform.Note that the Sobolev Hilbert
space H s admitting the reproducing kernel (1.3) for n = s can be deﬁned for1any positive number s in term of Fourier integrals F̂ of F∫
1e−iξ·x F (x)dxF̂ (ξ) =(2π)n/2 Rn∫

as follows:∥F ∥2H s =|F̂ (ξ)|2 (1 + |ξ|2 )s dξ
Rnfor any s > n/2.
Under these conditions our formulations and results are stated as follows:
Theorem 1 Let n ≥ 1, s ≥ 2 and s > n/2. For any function g ∈ L2 (Rn ) and∗for any λ > 0, the best approximate function Fλ,s,gin the sense{}∗∗
2inf s λ∥F ∥2H s + ∥g − ∆F ∥L= λ∥Fλ,s,g∥2H s + ∥g − ∆Fλ,s,g∥2L2 (Rn )n2 (R )F ∈H(1.4)∗exists uniquely and Fλ,s,g
is represented by∫∗
Fλ,s,g (x) =g(ξ)Qλ,s (ξ − x)dξ(1.5)

RnforQλ,s (ξ − x) =1(2π)n∫−|p|2 e−ip·(ξ−x) dp.
2s4
Rn λ(|p| + 1) + |p|(1.6)

If, for F ∈ H s we consider the solution uF (x): ∆uF (x) = F (x) and we takeuF (ξ) as g, then we have the favourable result: as λ → 0∗Fλ,s,g→ F,(1.7)

uniformly.
2) The problem in the heat conduction([2]); that is, from some heat u(x, t)observation at a time t, look for the initial heat u(x, 0).
We gave simple approximate real inversion formulas for the Gaussian convolution (the Weierstrass transform){}∫1|ξ − x|2uF (x, t) = (Lt F )(x) =F(ξ)
exp
−dξ
(1.8)
4t(4πt)n/2 Rn
for the functions of the s order Sobolev Hilbert space H s on the whole real
space Rn (n ≥ 1, s > n/2). This integral transform which represents the solution
u(x, t) of the heat equationut (x, t) = uxx (x, t) onRn × {t > 0}(u(x, 0) = F (x)on Rn ).

(1.9)

In this problem we can set same norm and reproducing kernel as (1.2) and (1.3).
Under those situations our formulations and results are stated as follows:
2Theorem 2 For any function g ∈ L2 (Rn ) and for any λ > 0, the best approx∗
imate function Fλ,s,gin the sense{}
inf s λ∥F ∥2H s + ∥g − uF (·, t)∥2L2 (Rn )F ∈H∗∗= λ∥Fλ,s,g∥2H s + ∥g − uFλ,s,g(·, t)∥2L2 (Rn )(1.10)∗exists uniquely and Fλ,s,gis represented by∗
Fλ,s,g(x) =forQλ,s (ξ − x) =1(2π)n∫g(ξ)Qλ,s (ξ − x)dξ(1.11)

Rn∫e−ip·(ξ−x) dp

2s |p|2 t + e−|p|2 t
Rn λ(|p| + 1) e.

(1.12)

If, for F ∈ H s we consider the output uF (x, t) and we take uF (ξ, t) as g, thenwe have the favourable result: as λ → 0∗Fλ,s,g→ F,(1.13)

uniformly.
3) Real inversion formulas for the Laplace transform([3])
We obtained a very natural and numerical real inversion formula of theLaplace transform∫ ∞(LF )(p) = f (p) =e−pt F (t)dt, p > 0(1.14)
0for functions F of some natural function space. The inversion of the Laplacetransform is, in general, given by a complex form, however, we are interested
in and are requested to obtain its real inversion in many practical problems.
However, the real inversion will be very involved and one might think that its
real inversion will be essentially involved, because we must catch ”analyticity”
from the real or discrete data.
We shall introduce the simple reproducing kernel Hilbert space (RKHS) HK
comprised of absolutely continuous functions F on the positive real line R+
with ﬁnite norms{∫ ∞}1/2′21 t|F (t)| e dt( F (0) = 0 ).
(1.15)
t0This Hilbert space admits the reproducing kernel
K(t, t′ ) =∫ min(t,t′ )0Then we see that3ξe−ξ dξ.

(1.16)∫ ∞1∥F ∥2HK ;(1.17)
20that is, the linear operator on HK ,(LF )(p)p into L2 (R+ , dp) = L2 (R+ ) isbounded([4]). For the reproducing kernel Hilbert spaces HK satisfying (1.17),
we can ﬁnd some general spaces.Therefore, from the general theory in [4], weobtain|(LF )(p)p|2 dp ≤Theorem 3 ([4]). For any g ∈ L2 (R+ ) and for any α > 0, the best approxi∗
mation Fα,gin the sense{ ∫ ∞}′21 t
2infα|F (t)| e dt + ∥(LF )(p)p − g∥L2 (R+ )F ∈HK
t0∫ ∞
1∗′
∗
=α|Fα,g(t)|2 et dt + ∥(LFα,g)(p)p − g∥2L2 (R+ )(1.18)
t0exists uniquely and we obtain the representation∫ ∞∗Fα,g (t) =g(ξ) (LKα (·, t)) (ξ)ξdξ.
(1.19)
0Here, Kα (·, t) is determined by the functional equation
11K(t, t′ ) − ((LKα,t′ )(p)p, (LKt )(p)p)L2 (R+ )ααfor Kα,t′ = Kα (·, t′ ) and Kt = K(·, t).
Kα (t, t′ ) =(1.20)∗(t) by using (1.19). For thisWe shall look for the approximate inversion Fα,gpurpose, we take the Laplace transfrom of (1.20) in t and change the variables
t and t′ as in
1
1(LKα (·, t))(ξ) = (LK(·, t′ ))(ξ) − ((LKα,t′ )(p)p, (L(LKt )(p)p))(ξ))L2 (R+ ) .
αα(1.21)Note that{−te−t − e−t + 1for t ≤ t′′′−t′ e−t − e−t + 1
for t ≥ t′ .
[′′−t′−1 ]1(LKα (·, t))(ξ) = e−t p e−t++.
(1.22)
2p(p + 1) p(p + 1)
p(p + 1)2
∫ ∞′1.
(1.23)
e−qt (LK(·, t′ ))(p)dt′ =pq(p + q + 1)20
Therefore, by setting (LKα (·, t))(ξ)ξ = Hα (ξ, t), which is needed in (1.19), we
obtain the Fredholm integral equation of the second type()∫ ∞e−tξ e−t
111dp=−.
αHα (ξ, t) +Hα (p, t)t++2(p+ξ+1)
ξ+1ξ
+1(ξ+1)20(1.24)′K(t, t ) =4

2

Some examples of numerical experiments

1) Inverse source problem in the Poisson equation([1])

01-0.25
10-0.5
-0.75
-1
0

2.5

0.5

8
2

6

0
1.5

20

4

0.5

4

1
1

2

60.5

1.5

82

10 0

2.50

Figure 1: For g(x1 , x2 ) = χ[−1,1] (x1 ) × χ[−1,1] (x2 ) on R2 , the ﬁgures of∗∗
Fλ,2,g(x1 , x2 ) and ∆Fλ,2,g(x1 , x2 ) for λ = 10−2 .
This numerical result shows that the new method ((1.5),(1.6)) is working
eﬀectively and is useful.
2) The problem in the heat conduction([2])

181·101000.5-2·10

488-1·10

82060
20-4

44-2-226082-410 0

4Figure 2: For g(x1 , x2 ) = χ[−1,1] (x1 ) × χ[−1,1] (x2 ) on R2 , the ﬁgures of−22∗∗.
Fλ,s,g(x1 , x2 ) and uFλ,s,g(x1 ,x2 ;t) for t = 1, s = 2, λ = 10The results of this numerical experiment prove the usefulness and correctnessof our method.

53) Real inversion formulas for the Laplace transform([3])

Figure 3: For F (t) = χ(t, [1/2, 3/2]), the characteristic function and for α =10−1 , 10−4 , 10−8 , 10−12 , 10−16 .

Figure 4: For F (t) = χ(t, [1/2, 3/2]), the characteristic function and for α =10−1 , 10−4 , 10−8 , 10−12 , 10−16 .
The results of these numerical experiments show that our method is eﬀective
even when there are jumps in the target function, and in Figure 4 we use a highprecision numerical algorithm developed by our collaborator Professor Fujiwara.

6References[1] T. Matsuura, S. Saitoh and D. D. Trong, Numerical Solutions of the Poisson
Equation, Trans. Applicable Analysis, 83(10)(2004), 1037–1051.
[2] T. Matsuura，S. Saitoh and D.D. Trong, Approximate and analytical inversion formulas in heat conduction on multidimensional spaces, Journalof Inverse and Ill-posed Problems, 13(2005), 1–15.
[3] T. Matsuura, A. Al-Shuaibi and H. Fujiwara, Numerical Real Inversion
Formulas of the Laplace Transform by using a Fredholm integral equationof the second kind, Journal of Analysis and Applications, 5(2007), 123–136.
[4] S. Saitoh, Approximate real inversion formulas of the Laplace transform,
Far East J. Math. Sci.,11(2003), 53–64.

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