On a new notion of the solution to an ill-posed problem

0.11 MB·English

by  A. G. Ramm

On a new notion of the solution to an ill-posed
problem
∗†
0
1 A.G. Ramm
0
2 Mathematics Department, Kansas State University,

n Manhattan, KS 66506-2602, USA
a
J [e-mail]

3

]
A
N
Abstract
.
h
A new understanding of the notion of the stable solution to ill-posed problems
t
a
is proposed. The new notion is more realistic than the old one and better fits
m
the practical computational needs. A method for constructing stable solutions in
[
the new sense is proposed and justified. The basic point is: in the traditional

1
definition of the stable solution to an ill-posed problem Au = f, where A is a linear
v
6 ornonlinearoperatorinaHilbertspaceH, itisassumedthatthenoisydata fδ,δ
{ }
6 are given, f f δ, and a stable solution u := R f is defined by the relation
δ δ δ δ
3 || − || ≤
lim R f y = 0, where y solves the equation Au = f, i.e., Ay = f. In this
0 δ→0|| δ δ − ||
. definition y and f are unknown. Any f B(fδ,δ) can be the exact data, where
1 ∈
B(f ,δ) := f : f f δ .
0 δ δ
{ || − || ≤ }
0 The new notion of the stable solution excludes the unknown y and f from the
1
definition of the solution.
:
v
i
X
1 Introduction
r
a
Let
Au = f, (1.1)
where A : H H is a linear closed operator, densely defined in a Hilbert space H.
→
Problem (1.1) is called ill-posed if A is not a homeomorphism of H onto H, that is, either
equation (1.1) does not have a solution, or the solution is non-unique, or the solution
does not depend on f continuously. Let us assume that (1.1) has a solution, possibly
non-unique. Let N(A) be the null space of A, and y be the unique normal solution to
∗key words: ill-posed problems, regularizer, stable solution of ill-posed problems
†AMS2010 subject classification: 47A52, 65F22, 65J20
(1.1), i.e., y N(A). Given noisy data f , f f δ, one wants to construct a stable
δ δ
⊥ k − k ≤
approximation u := R f of the solution y, u y 0 as δ 0.
δ δ δ δ
k − k → →
Traditionally(see, e.g., [2])onecallsafamilyofoperatorsR aregularizerforproblem
h
(1.1) (with not necessarily linear operator A) if
a) R A(u) u as h 0 for any u D(A),
h
→ → ∈
b) R f is defined for any f H and there exists h = h(δ) 0 as δ 0 such that
h δ δ
∈ → →
R f y 0 as δ 0. ( )
h(δ) δ
k − k → → ∗
Inthisdefinitiony isfixedand( )mustholdforanyf B(f,δ) := f : f f δ .
δ δ δ
∗ ∈ { k − k ≤ }
In practice one does not know the solution y and the exact data f. The only available
information is a family f and some a priori information about f or about the solution
δ
y. This a priori information often consists of the knowledge that y , where is a
∈ K K
compactum in H. Thus
y S := v : A(v) f δ, v .
δ δ
∈ { k − k ≤ ∈ K}
WeassumethattheoperatorAisknownexactly, andwealwaysassumethatf B(f,δ),
δ
∈
where f = A(y).
Definition: We call a family of operators R(δ) a regularizer if
sup R(δ)f v η(δ) 0 as δ 0. (1.2)
δ
k − k ≤ → →
v S
∈ δ
There is a crucial difference between our new Definition (1.2) and the standard definition
( ):
∗
In ( ) u is fixed, while in (1.2) v is an arbitrary element of S and the supremum of
δ
∗
the norm in (1.2) over all such v must tend to zero as δ 0.
→
The new definition is more realistic and better fits computational needs because not
only the solution y to (1.1) satisfies the inequality Ay f δ, but any v S satisfies
δ δ
k − k ≤ ∈
this inequality Av f δ, v . The data f may correspond to any f = Av, where
δ δ
k − k ≤ ∈ K
v S , and not only to f = Ay, where y is a solution of equation (1.1). Therefore it is
δ
∈
more natural to use definition (1.2) than ( ).
∗
Our goal is to illustrate the practical difference between these two definitions, and to
constructregularizerinthesense(1.2)forproblem(1.1)withanarbitrary, notnecessarily
bounded, linear operator A, which is closed and densely defined in H. This is done in
Section 2. In Section 1 this is done for a class of equations (1.1) with nonlinear operators
A : X Y, where X and Y are Banach spaces. In this case we assume that
→
A1) A : X Y is a closed, nonlinear, injective map, f (A), (A) it is the range of
→ ∈ R R
A,
and
A2) φ : D(φ) [0, ), φ(u) > 0 if u = 0, D(φ) D(A), the sets = := v : φ(v)
c
→ ∞ 6 ⊆ K K { ≤
c are compact inX for everyc = const > 0,and ifv v,thenφ(v) liminf φ(v ).
n n n
} → ≤ →∞
The last inequality holds if φ is lower semicontinuous. In Hilbert spaces and in
reflexive Banach spaces norms are lower semicontinuous.
2
Let us give some examples of equations for which assumptions A1) and A2) are
satisfied.
Example 1. A is a linear injective compact operator, f (A), φ(v) is a norm on
∈ R
X X, where X is densely imbedded in X, the embedding i : X X is compact,
1 1 1
⊂ →
and φ(v) is lower semicontinuous.
Example 2. A is a nonlinear injective continuous operator f (A), A 1 is not
−
∈ R
continuous, φ is as in Example 1.
Example 3. A is linear, injective, densely defined, closed operator, f (A), A 1 is
−
∈ R
unbounded, φ is as in Example 1, X D(A).
1
⊆
Let us demonstrate by Example A that a regularizer in the sense ( ) may be not a
∗
regularizer in the sense (1.2).
In Example B a theoretical construction of a regularizer in the sense (1.2) is given for
some equations (1.1) with nonlinear operators.
In Section 2 a novel theoretical construction of a regularizer in the sense (1.2) is given
for a very wide class of equations (1.1) with linear operators A.
Example A: Stable numerical differentiation.
In this Example the results from [3] - [11] are used. This Example is borrowed from
[10].
Consider stable numerical differentiation of noisy data. The problem is:
x
Au := u(s)ds = f(x), f(0) = 0, 0 x 1. (1.3)
Z ≤ ≤
0
The data are: f and a constant M , which defines a compact , where f f δ, the
δ a δ
K k − k ≤
norm is L (0,1) norm, and consists of the L functions which satisfy the inequality
∞ ∞
K
u M , a 0. The norm
a a
k k ≤ ≥
u(x) u(y)
u := sup | − | + sup u(x) if 0 a 1,
k ka x y a | | ≤ ≤
x,y [0,1] 0 x 1
x∈=y | − | ≤ ≤
6
u(x) u(y)
′ ′
u := sup ( u(x) + u(x) )+ sup | − |, 1 < a 2.
k ka | | | ′ | x y a 1 ≤
0 x 1 x,y [0,1] −
≤ ≤ x∈=y | − |
6
If a > 1, then we define
fδ(x+h(δ))−fδ(x−h(δ)), h(δ) x 1 h(δ),
2h(δ) ≤ ≤ −
R(δ)fδ := fδ(x+hh((δδ)))−fδ(x), 0 ≤ x < h(δ), (1.4)
fδ(x)−fδ(x−h(δ)), 1 h(δ) < x 1,
h(δ) − ≤


where
1
h(δ) = caδa, (1.5)
and c is a constant given explicitly (cf [4]).
a
3
We prove that (1.4) is a regularizer for (1.3) in the sense (1.2), and := v : v
a
K { k k ≤
M , a > 1 . In this example we do not use lower semicontinuity of the norm φ(v) and
a
}
do not define φ.
Let S := v : Av f δ, v M . To prove that (1.4)-(1.5) is a regularizer
δ,a δ a a
{ k − k ≤ k k ≤ }
in the sense (1.2) we use the estimate
δ
sup R(δ)f v sup R(δ)(f Av) + R(δ)Av v +M ha 1(δ)
δ δ a −
k − k ≤ {k − k k − k} ≤ h(δ) ≤
v S v S
∈ δ,a ∈ δ,a
caδ1−a1 := η(δ) 0 as δ 0.
≤ → →
(1.6)
Thus we have proved that (1.4)-(1.5) is a regularizer in the sense (1.2).
If a = 1, and M < , then one can prove the following result:
1
∞
Claim: There is no regularizer for problem (1.3) in the sense (1.2) even if the regu-
larizer is sought in the set of all operators, including nonlinear ones.
More precisely, it is proved in [5], p.345, (see also [8], pp 197-235, where the stable
numerical differentiation problem is discussed in detail) that
inf sup R(δ)f v c > 0,
δ
R(δ) v∈Sδ,1 k − k ≥
where c > 0 is a constant independent of δ and the infimum is taken over all operators
R(δ) acting from L (0,1) into L (0,1), including nonlinear ones.
∞ ∞
On the other hand, if a = 1 and M < , then a regularizer in the sense ( ) does
1
∞ ∗
exist, but the rate of convergence in (*) may be as slow as one wishes, if u(x) is chosen
suitably (see [4], [8]).
Example B: Construction of a regularizer in the sense (1.2) for some nonlinear
equations.
Assuming A1) and A2), let us construct a regularizer for (1.1) in the sense (1.2). We
use the ideas from [10] and [11].
Define F (v) := Av f +δφ(v) and consider the minimization problem of finding
δ δ
k − k
the infimum m(δ) of the functional F (v) on a set S :
δ δ
m(δ) := inf F (v), S := v : Av f δ, φ(v) c . (1.7)
δ δ δ
v S { k − k ≤ ≤ }
∈ δ
Here
= := v : φ(v) c .
c
K K { ≤ }
The constant c > 0 can be chosen arbitrary large and fixed at the beginning of the
argument, and then one can choose a smaller constant c , specified below. Since F (u) =
1 δ
δ +δφ(u) := c δ, c := 1+φ(u), where u solves (1.1), one concludes that
1 1
m(δ) c δ. (1.8)
1
≤
Let v be a minimizing sequence and F (v ) 2m(δ). Then φ(v ) 2c . By assumption
j δ j j 1
≤ ≤
A2), as j , one has:
→ ∞
v v , φ(v ) 2c . (1.9)
j δ δ 1
→ ≤
4
Take δ = δ 0 and denote v := w . Then (1.9) and Assumption A2) imply the
m → δm m
existence of a subsequence, denoted again w , such that:
m
w w, A(w ) A(w), A(w) g = 0. (1.10)
m m
→ → k − k
Thus A(w) = g and, since A is injective by Assumption A1), it follows that w = u, where
u is the unique solution to (1.1).
Define now R(δ)f by the formula R(δ)f := v , where v is defined in (1.9).
δ δ δ δ
Theorem 1.1. R(δ) is a regularizer for problem (1.1) in the sense (1.2).
Proof. Assume the contrary:
sup R(δ)f v = sup v v γ > 0, (1.11)
δ δ
k − k k − k ≥
v S v S
∈ δ ∈ δ
where γ > 0 is a constant independent of δ. Since φ(v ) 2c by (1.9), and φ(v) c, one
δ 1
≤ ≤
can choose convergent in X sequences w := v w˜, δ 0, and v v˜, such that
m δm → m → m →
w v γ, w˜ v˜ γ, and A(w˜) = g, A(v˜) = g. By the injectivity of A it follows
k m− mk ≥ 2 k − k ≥ 2
that w˜ = v˜ = u. This contradicts the inequality w˜ v˜ γ > 0. This contradiction
k − k ≥ 2
proves the theorem.
The conclusions A(w˜) = g and A(v˜) = g, that we have used above, follow from the
inequalities A(v ) f δ and A(v) f δ after passing to the limit δ 0, using
δ δ δ
k − k ≤ k − k ≤ → 2
assumption A2).
2 Construction of a regularizer in the sense (1.2) for
linear equations
If A is a linear closed densely defined in H operator, then T = A A is a densely defined
∗
selfadjoint operator. Let T := T + aI, where a = const > 0. The operator T 1A is
a a− ∗
densely defined and closable. Its closure is a bounded operator, defined on all of H, and
T 1A 1 . See [12]-[15] for details and other results. Let E be the resolution of the
|| a− ∗|| ≤ 2√a s
identity of the selfadjoint operator T, dρ := d(Esy,y), and K := {u : 0∞s−2pdρ ≤ kp2},
where p (0,1) and k > 0 are constants. R
p
∈
Our basic result is:
Theorem 2.1. The operator R = T 1 A is a regularizer for problem (1.1) in the
δ a−(δ) ∗
2
sense (1.2) if limδ→0 a(δδ)1/2 = 0 and limδ→0a(δ) = 0. Moreover, if a(δ) = bpδ2p+1, then
2p
sup R(δ)fδ y Cpδ2p+1, (2.1)
k − k ≤
y , Ay f δ
∈K|| − δ||≤
where
1
Cp = 2 b +cpkpbpp, cp = pp(1−p)1−p, bp := (4pcpkp)−2p2+1.
p
p
5
The above choice of a(δ) is optimal in the sense that the right-hand side of (2.2) (see
below) is minimal for this choice of a(δ).
Proof.
Let
ǫ := sup T 1A f y := sup T 1A f y .
|| a− ∗ δ − || || a− ∗ δ − ||
y , Ay f δ
∈K|| − δ||≤
Then, with Ay = f, one has
ǫ sup T 1A (f f) +sup T 1A Ay y := J +J ,
≤ || a− ∗ δ − || || a− ∗ − || 1 2
where
δ
J ,
1
≤ 2√a
and
a2
J2 sup a2 T 1y 2 sup ∞ d(E y,y).
2 ≤ { || a− || } ≤ Z (s+a)2 s
0
Thus,
asp
J2 max 2k2 = c2k2a2p,
2 ≤ s 0 a+s p p p
(cid:0) ≥ (cid:1)
because max asp is attained at s = pa and is equal to c ap, where
s 0 a+s 1 p p
≥ −
c := pp(1 p)1 p, k2 := sup ∞s 2pd(E y,y).
p − − p Z − s
y 0
∈K
Consequently,
J c k ap,
2 p p
≤
and
δ
ǫ +c k ap. (2.2)
p p
≤ 2√a
Minimizing the right-hand side of (2.2) with respect to a > 0, one obtains inequality
(2.1).
The minimizer of the right-hand side of (2.2) is
2 2
a = a(δ) = bpδ2p+1, bp := (4pcpkp)−2p+1,
2p
and the minimum of the right-hand side of (2.2) is Cpδ2p+1, where
1
C := +c k bp. (2.3)
p p p p
2 b
p
p
2
Theorem 2.1 is proved.
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References
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