Frederic Weidling Thorsten Hohage

Verification of a variational source condition
for inverse medium scattering problems
Frederic Weidling
3. Near field inverse scattering
Assumption (Solution regularity) Let 32 < m < s, s , 2m + 3/2. Suppose that the true
contrast f † satisfies f † ∈ D ∩ H0s(B(π)) with k f †kH s ≤ C s for some C s ≥ 0.
• Choose R > π.
• For each y ∈ ∂B(R):
– Use a point source as incident wave
1. Introduction
iκ|x−y|
1
e
uiy(x) =
.
4π |x − y|
Regularization theory deals with the solution of ill-posed operator equations
F( f ) = yδ
– For all x ∈ ∂B(R) measure the total field
w f (x, y).
where F : dom(F) ⊂ X → Y. A prominent example to solve these equation given perturbed data yδ with kF( f †) − yδkY ≤ δ is Tikhonov regularization
#
"
1
1
2
δ
δ
fα ∈ argmin F( f ) − y Y + k f k2X .
2
f ∈dom(F) α
For many interesting problems the convergence rates are unknown, while stability estimates
exists. We try to answer the question:
Can stability estimates be sharpened to
variational source conditions?
• Define the operator
Fn : D → L2(∂B(R)2),
f 7→ w f .
Theorem (VSC for near field) Let R > π and the assumption be fulfilled.
holds true for the operator Fn with dom(Fn) := D ∩ H0m(B(π)) with
)
(
−2µ
s−m
−1
,
β=
ψn(t) := A ln(3 + t )
,
µ := min 1,
m + 3/2
Then (VSC)
1
,
2
where the constant A > 0 depends on m, s, C s, κ and R.
We consider acoustic medium scattering given by
∆u + κ u = f u
!
s
1
∂u
s
− iκu = O 2
∂r
r
2
4. Far field inverse scattering
in R ,
3
• For each direction d ∈ ∂B(1):
as r = |x| → ∞,
– Use a incident plane wave propagating
in direction d
where κ > 0, f is the contrast of the medium and
n
o
f ∈ D := f ∈ L∞(R3) : =( f ) ≤ 0, <( f ) ≤ 1, supp( f ) ⊂ B(π) .
uid,∞(x) = eiκd·x.
– For all directions x̂ ∈ ∂B(1) measure the
far field u∞( x̂, d) defined by
!
iκ|x|
e
∞ x
s
u
, d + O(|x|−3)
ud,∞(x) =
|x|
|x|
The problem is to reconstruct f from the knowledge of the incident wave and measurements
of the scattered field.
Employing methods used to proof the stability estimates (e.g. [4, 6]), for example geometrical optical solutions, we show that for this problem the answer is yes.
2. Why variational source conditions?
• Define the operator
In our setting condition of the form: There exists β ∈ (0, 1] and an index function ψ s.t.:
∀ f ∈ dom(F) :
1
β 1
2
2
2
2
†
†
†
f − f X ≤ k f kX − f X + ψ F( f ) − F( f )Y
2
2
2
(VSC)
Ff : D → L2(∂B(1)2),
f 7→ u∞.
Theorem (VSC for far field) Let the assumption be satisfied and 0 < θ < 1. Then the Ff
with dom(Ff ) := D ∩ H0m(B(π)) fulfills (VSC) with
−1 −2µθ
ψf (t) := B ln(3 + t )
,
Advantages over spectral source conditions:
1
β= ,
2
where µ is given as before and the constant B > 0 depends on m, s, C s, κ, θ and R.
• Yields for Tikhonov regularization the convergence rate
2
β δ
†
fα − f X ≤ 4ψ(δ2)
2
5. Outlook
if the index function ψ is concave and the regularization parameter chosen as −1/(2α) ∈
∂(−ψ(4δ2)) (see [3, 7]).
• Extend the results to other scattering problems, for example electromagnetic inverse
medium scattering.
• Specify the dependenc on the variational source condition of the parameter κ to obtain
Hölder-logarithmic results (see [6]).
• If (VSC) holds for all f † ∈ K ⊂ X they imply the stability estimate
β
k f1 − f2k2X ≤ ψ kF( f1) − F( f2)k2Y ,
2
∀ f1, f2 ∈ K.
• Use Banach space penalty functional.
• In Hilbert spaces with linear operators necessary and sufficient for certain rates of convergence for a big class of regularization methods (see [2]).
6. Reference
• No further differentiability assumptions on the forward operator no restrictive assumptions connecting the operators F and F 0 like the tangential cone condition (usually not
verifiable).
[1] M. Burger, J. Flemming, and B. Hofmann. Inverse Problems, 29 (2013), p. 025013.
[2] J. Flemming, B. Hofmann, and P. Mathé. Inverse Problems, 27 (2011), p. 025006.
[3] M.Grasmair. Inverse Problems, 26 (2010), p. 115014.
[4] P. Hähner and T. Hohage. SIAM J. Math. Anal., 33 (2001), pp. 670–685.
[5] B. Hofmann, B. Kaltenbacher, C. Pöschl, and O. Scherzer. Inverse Problems, 23
(2007), p. 987–1010.
[6] M. I. Isaev and R. G. Novikov. Inverse Problems, 30 (2014), p. 095006.
[7] F. Werner and T. Hohage. Inverse Problems, 28 (2012), p. 104004.
• Usable in in a Banach space setting.
• Suitable for more general noise models and data fidelity terms/penalty terms.
Hence popular in regularization theory but few results on the verification of such conditions. [1, 5]
joint work with:
Frederic Weidling
Any questions?
Just ask me:
Institute for Numerical and Applied
Mathematics
University of Goettingen
Email: [email protected]
Thorsten Hohage
accepted for puplication in
Inverse Problems
available on arxiv
1503.05763