# Frederic Weidling Thorsten Hohage

## Transcription

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