d) This approach is generalizable to Maxwell's equa- tions [2], and constraints like 4 ... The full Maxwell's equations. ... [9] J. Sylvester and G. Uhlmann. A global ...
Using multiple frequencies to satisfy local constraints in PDE and applications to hybrid problems Giovanni S. Alberti OxPDE, Mathematical Institute, University of Oxford, UK.
Examples of hybrid imaging inverse problems (quantitative step)
Quantitative thermo-acoustic tomography
Microwave imaging by ultrasound deformation
∆uiω + (ω 2 + iωσ)uiω = 0 uiω = ϕi on ∂Ω.
in Ω,
• Internal measurements: σ(x)|uiω (x)|2 . • Unknown conductivity: σ(x) ≥ 0. • Explicit reconstruction [5] in x if
2
εuiω
+ω =0 on ∂Ω.
in Ω,
• Unknowns: a(x), ε(x) ≥ 0.
• Unknowns: ε(x), σ(x) ≥ 0.
in Ω, curlEωi = iωHωi curlHωi = −i(ωε + iσ)Eωi in Ω, i Eω × ν = ϕi × ν on ∂Ω. • Internal measurements: Hωi (x). • Unknowns: ε(x), σ(x) ≥ 0.
• Stability and convergence of optimal control algorithm (in two dimensions) [6] if 2 3 2. | det ∇uω ∇uω (x)| ≥ C.
1. |u1ω (x)| ≥ C, 2. |∇u2ω × ∇u3ω (x)| ≥ C.
ω
Magnetic resonance electrical impedance tomography
in Ω,
• Internal measurements: uiω (x).
• Explicit reconstruction [4, 1] in x if
• Stability [7] if 1 holds true.
div((ωε + iσ)∇uiω ) = 0 uiω = ϕi on ∂Ω.
• Internal measurements: ε(x)|uiω (x)|2 , a(x)|∇uiω (x)|2 .
1. |u1ω (x)| ≥ C > 0, h 1 i d+1 uω ··· uω 3. | det ∇u1 ··· ∇ud+1 (x)| ≥ C. ω
div(a∇uiω ) uiω = ϕi
Micro-electrical impedance tomography
• Stability [10] if 1 holds true.
• Explicit reconstruction [2] if 1 2 3 4. | det Eω Eω Eω (x)| ≥ C.
How can we find suitable boundary conditions ϕi such that the corresponding solutions to these PDE satisfy the required constraints 1, 2, 3 and 4?
The model problem d
• Ω ⊆ R , d = 2, 3: smooth bounded domain. • The Helmholtz equation: ∆uiω + (ω 2 ε + iωσ)uiω = 0 uiω = ϕi on ∂Ω.
Multiple frequencies
Additional results
Main idea (d = 1, Ω = (−π, π), σ = 0, constraint 1): the zero set {x ∈ Ω : u1ω (x) = 0} moves when ω varies...
a) Can we extend Theorem 1 to the case a 6= 1? div(a∇uiω ) + (ω 2 ε + iωσ)uiω = 0 in Ω, uiω = ϕi on ∂Ω.
ϕ(−π) = 1, ϕ(π) = 1
in Ω,
(1)
• ε, σ ∈ L∞ (Ω):
1
u10
−π
π
– 0 < Λ−1 ≤ ε ≤ Λ almost everywhere; – either Λ−1 ≤ σ ≤ Λ or σ = 0.
Proposition 3. Let Ω ⊆ R2 be strictly convex and a ∈ C 0,α (Ω; R2×2 ) be an elliptic tensor. There exist C > 0 and n ∈ N depending on Ω, Λ, α, kakC 0,α and A s.t. is C-complete.
u1ω
−1
K (n) × {1, x1 , x2 }
Based on the absence of critical points in 2D [3]. • ω ∈ A = [Kmin , Kmax ] ⊆ R+ : admissible frequencies. • ϕi ∈ C
1,α
(Ω): boundary conditions. 1
• Given a finite K ⊆ A and ϕ1 , . . . , ϕd+1 , we call K × {ϕ1 , . . . , ϕd+1 } a set of measurements. Definition. Take C > 0. A set of measurements K ×{ϕi }i=1,...,d+1 is C-complete if there exists an open cover Ω = ∪ω∈K Ωω such that for any ω ∈ K and x ∈ Ωω 1.
≥ C, ∇ud+1 ω
−π
(x)| ≥ C, ud+1 ω d+1 (x)| ≥ C. ∇uω
tx1
• By [8], kψt kC 1 ≤
u1ω
c) Is it possible to find an optimal value for the number of needed frequencies n in Theorem 1?
Main result
Proposition 5. If ε and σ are real analytic then n o (ωp ) ∈ Ad+1 : {ωp }p × {1, x1 , . . . , xd+1 } is complete is open and dense in Ad+1 .
K (n) : uniform partition of A with #K (n) = n. Theorem 1. There exist C > 0 and n ∈ N depending on Ω, Λ and A such that K (n) × {1, x1 , . . . , xd+1 }
(cos(tx2 ) + i sin(tx2 )) (1 + ψt ).
is C-complete.
c t
The construction is independent of ε and σ.
for some c > 0, whence
u(t) (x) ≈ etx1 (cos(tx2 ) + i sin(tx2 )) ,
Proposition 4. Suppose a, ε ∈ C 2 (R3 ) and σ = 0. For a generic C 2 bounded domain Ω and a generic ϕ ∈ C 2 (Ω) there exists a finite K ⊆ A such that X ϕ ∇uω (x) ≥ c > 0, in Ω. ω∈K
−1
• CGO solutions [9] are particular solutions to (1): u (x) = e
π
Complex geometric optics (t)
u10
It seems that all depends on the constraint 1 in ω = 0: the unknowns ε and σ have disappeared from (1)!
2 2. | det ∇uω · · · 1 uω · · · 3. | det ∇u1ω · · ·
b) What can we do if a 6≈ 1 in 3D?
ϕ(−π) = −1, ϕ(π) = 1
• uiω ∈ C 1 (Ω) (elliptic regularity).
|u1ω (x)|
...only if the boundary condition is suitably chosen.
t 1.
• Set ϕ1 ≈ u(t) |∂Ω , ϕ2 ≈ 0.
• Is it possible to lower the assumptions on the real analyticity of the coefficients in Proposition 5?
– very oscillatory functions: experimentally very challenging.
References [1] G. S. Alberti. On multiple frequency power density measurements. Inverse Problems, 29(11):115007, 2013. [2] G. S. Alberti. On multiple frequency power density measurements II. The full Maxwell’s equations. arXiv preprint:1311.7603, 2013. [3] G. Alessandrini. Critical points of solutions of elliptic equations in two variables. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14(2):229–256 (1988), 1987. [4] H. Ammari, Y. Capdeboscq, F. de Gournay, A. Rozanova-Pierrat,
and F. Triki. Microwave imaging by elastic deformation. SIAM J. Appl. Math., 71(6):2112–2130, 2011.
acoustics and related problems. Inverse Problems, 27(5):055007, 15, 2011.
[5] H. Ammari, J. Garnier, W. Jing, and L. H. Nguyen. Quantitative thermo-acoustic imaging: An exact reconstruction formula. J. Differential Equations, 254(3):1375–1395, 2013.
[8] G. Bal and G. Uhlmann. Inverse diffusion theory of photoacoustics. Inverse Problems, 26(8):085010, 20, 2010. [9] J. Sylvester and G. Uhlmann. A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. (2), 125(1):153– 169, 1987.
[6] H. Ammari, L. Giovangigli, L. H. Nguyen, and J. Seo. Admittivity imaging from multi-frequency micro-electrical impedance tomography. arXiv preprint:1403.5708, 2014. [10] F. Triki. Uniqueness and stability for the inverse medium problem [7] G. Bal, K. Ren, G. Uhlmann, and T. Zhou. Quantitative thermowith internal data. Inverse Problems, 26(9):095014, 11, 2010.
