Real Inversion Formulas of the Laplace Transform on Weighted - - PDF document

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Real Inversion Formulas

• f the Laplace Transform
• n Weighted Function Spaces

Hiroshi Fujiwara, Saitoh Saburoh and Yoshihiro Sawano August, 2008

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1 Introduction

The aim of this talk is to consider a numerical inver- sion formula of the Laplace transform Lf(p) = ∫ ∞ e−p tf(t) dt.

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2 Our new approach

We define Lf(p) := p ∫ ∞ f(t)e−p t dt. This operator is rather easy to deal with. Indeed, assuming some integrability conditions and smooth- ness, we obtain Lf(p) = ∫ ∞ f ′(t)e−p t dt.

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2.1 Model case Here we shall present our model case. First we define ∥f : HK∥ := {∫ ∞ |f ′(t)|2et t dt }1

2

when f : R → R is absolutely continuous. Below we postulate the function on normalization conditions. Namely, we assume f(0) = 0.

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With this in mind, we define HK to be the set of all absolutely continuous funtions f : R → R for which f(0) = 0 and ∥f : HK∥ < ∞.

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We formulate our main theorem. Theorem 1 The mapping L : f ∈ HK → Lf ∈ L2([0, ∞)) is an injective and compact linear operator.

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The injectivity of L is clear because L itself is injec-

• tive. As for the compactness, we argue by approxi-
• mation. We define

LRf(p) := p ∫ ∞

R

e−p tf(t) dt.

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Note that Lf(p) − LRf(p) = p ∫ R e−p tf(t) dt and that f ∈ HK → p ∫ R e−p tf(t) dt is compact.

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Therefore, to establish that L itself is compact, it suffices to show that LR shrinks to 0 in the operator topology as R → ∞.

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We proceed as follows : An integration by parts yields LRf(p) = ∫ ∞

R

e−p tf ′(t) dt + e−R pf(R). Hence it follows that ∥LRf∥2 ≤ ∫ ∞

R

∥e−p t∥L2([0,∞)p)|f ′(t)| dt + ∥e−R p∥L2([0,∞)p) · |f(R)| = ∫ ∞

R

|f ′(t)| dt t + |f(R)| R by the triangle inequality.

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Equality f(R) = ∫ R f ′(t) dt yields ∥LRf∥2 ≤ ∫ ∞ |f ′(t)| dt max(t, R) ≤ √∫ ∞ |f ′(t)|2et dt t ∫ ∞ t e−t dt max(t, R)2 ≤ ∥f : HK∥ √∫ ∞ t e−t dt max(t, R)2.

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Therefore, it follows that ∥LR∥HK→L2([0,∞)) ≤ √∫ ∞ t e−tdt max(t, R)2. As a consequence we obtain LR shrinks to 0 in the norm topology of L2.

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Now that L is established to be a compact operator, we can take {vn}n∈N and {un}n∈N such that Lvn = λn un, L∗un = λn vn, where λn is a singular value of L and we assume that λ1 ≥ λ2 ≥ · · · ≥ 0.

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Under this notation, we establish the following : Theorem 2 Suppose that F ∈ L2 is a function that can be written as F = Lf. Then we have L−1F (t) =

∞

∑

n=1

1 λn (∫ ∞ F (p)p un(p) dp ) vn(t).

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We prove our main theorem by calculating the right- hand side.

∞

∑

n=1

1 λn (∫ ∞ F (p)p un(p) dp ) vn(t) =

∞

∑

n=1

1 λn (∫ ∞ Lf(p)p un(p) dp ) vn(t) =

∞

∑

n=1

1 λn (∫ ∞ Lf(p) un(p) dp ) vn(t)

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=

∞

∑

n=1

1 λn ⟨Lf, un⟩L2([0,∞))vn(t) =

∞

∑

n=1

1 λn ⟨f, L∗un⟩HKvn(t) =

∞

∑

n=1

⟨f, vn⟩HKvn(t) = f(t) = L−1F (t). This is the desired result.

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We shall consider the original question. Lf = F, F ∈ L2([0, ∞)). In general, as we have seen before, this problem does not admit a solution unless we assume that F ∈ L(HK).

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The Tikhonov regularization is a method to over- come this difficulty. Instead of considering the above quesion directly, we consider min

f∈HK

( α∥f : HK∥2 + ∥Lf − F : L2([0, ∞))∥2) , where α is the smoothing parameter.

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It is not so hard to solve the minimizing problem min

f∈HK

( α∥f : HK∥2 + ∥Lf − F : L2([0, ∞))∥2) because we can complete the square.

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After completing the square, we see that fα = (α + L∗L)−1L∗[p · F ] is the desired element.

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If we expand fα = (α + L∗L)−1L∗F, then we obtain fα =

∞

∑

j=1

1 α + λn2⟨L∗[p · F ], vn⟩HKvn =

∞

∑

j=1

λn α + λn2⟨p · F, un⟩L2([0,∞))vn =

∞

∑

j=1

λn α + λn2 (∫ p · F (p)un(p) dp ) vn.

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To realize our scheme numerically, we use ∫ ∞ f(x) dx ≃

N

∑

i=0

f(xi)wi, where {xi}N

i=1 and {wi}N i=1 is a collection of points

and a correction of weights respectively.

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Finally, to conclude our first part of the talk, we present two examples. Example 1 f(t) :=          t 0 ≤ t < 1 3/2 − t/2 1 ≤ t < 3

• therwise

is the first example of our result.

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This is our numerical result.

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The second example is f(t) = δ(t − 1). Observe that f does not belong to HK. However, Lf(p) = p e−p does belong to L2. So we are in the position of applying our result of the Tikhonov regularization.

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This is our numerical result.

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ラプラス変換の実逆変換への再生核空間の応用

Hiroshi Fujiwara, Naotaka Kajino and Yoshihiro Sawano August, 2008

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Based on the previous talk, we consider the above result in the generalized framework.

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Below we shall consider weighted function space L2([0, ∞), u) whose norm is given by ∥f : L2([0, ∞), u)∥ := (∫ ∞ |f(t)|2u(t) dt )1

2

.

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We define H0

K(w)

:= { f ∈ C1([0, ∞)) : f ′ ∈ L2(w−1), f(0) = 0 } and HK(w) := the completion of H0

K(w).

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We want to consider the following problem. What is the condition for L to be compact ? L : HK(w) → L2(u). Here L is given by Lf(p) := p ∫ ∞ f(t)e−p t dt.

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In this problem, it is tacitly included that we need to give a sufficient condition for Lf, f ∈ HK(w) to make sense.

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A closer look at the proof, which we gave in our earlier paper, makes us notice that the quantity M is a key quantity, where M := ∫ ∞ ∫ ∞ e−t pu(p)w(t) dp dt < ∞.

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If M < ∞, then we proved that L : HK(w) → L2(u) is a Hilbert-Schmidt class operator and that the Hilbert- Schmit norm is M. Under the assumption M < ∞, we see that L is well-defined on HK(w).

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To show that L : HK(w) → L2(u) belongs to the Hilbert-Schmidt class, it suffices to show that LL∗ : L2(u) → L2(u) is a trace class operator.

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First, we obtain L∗g(t) = ⟨L∗g, Kw(·, t)⟩HK(w) = ⟨g, L[Kw(·, t)]⟩L2(u) = ∫ ∞ g(p)L[Kw(·, t)](p) u(p) dp = ∫ ∞ g(p) (∫ t exp(−s p)w(s) dp ) u(p) dt.

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Hence it follows that LL∗g(p) = L[(L∗g)′](p) = ∫ ∞ ∫ ∞ g(q) exp(−t (p + q))w(t)u(p) dq dt.

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Let ϕj, j = 1, 2, · · · be eigenvectors of LL∗. Then the integral kernel can be written as

∞

∑

j=1

λjϕj(p)ϕj(q) = ∫ ∞ w(t) exp(−t (p + q)) dt.

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As a result we obtain

∞

∑

j=1

λj =

∞

∑

j=1

λj ∫ ∞ ϕj(p)2u(p) dp = ∫ ∞ ∫ ∞ w(t)u(p) exp(−2p t) dp dt.