Mikyoung LIM(KAIST) Optimization algorithm for the reconstruction of a conductivity in

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1 Optimization algorithm for the reconstruction of a conductivity inclusion Mikyoung LIM (KAIST)

2 Inclusion Ω in R d For a given entire harmonic function H, consider { (χ(r d \ Ω) + kχ(ω)) u = in R d, u(x) H(x) = O( x d ) as x. Multipolar expansions: u(x) = H(x) + α β ( ) β α H()M αβ (k, Ω) β Γ(x), x. α!β! {M αβ } : Generalized Polarization Tensors (GPT). (Γ(x): the fundamental solution for the Laplacian.)

3 Small Inclusion Let D = δb + z be a single inclusion or multiple inclusions contained in Ω and < k < + be a constant. Consider (χ(ω \ D) + kχ(d)) u =, in Ω, For d = 2, 3, u(x) = U(x) + α u ν = g, on Ω. ( ) β δ α + β α U(z)M αβ (k, B) β N(x, z) + O(δ 2d ). α!β! β A complete asymptotic expansion is possible, but terms higher than 2d involves not only GPTs but also the boundary-inclusion interaction. (U(x): the back ground solution, N(x): the Neumann function)

4 Equivalent ellipse There is a canonical correspondence between the class of ellipses (ellipsoids) and the class of PTs: If E is an ellipse x2 + y 2, then a 2 b 2 a + b M(k, E) = (k ) E a + kb a + b. b + ka

5 Equivalent ellipse (ellipsoid)= ellipse with the same PT:

6 GPT and Imaging by Ammari-Kang-L-Zribi Aim: Make use of α + β K of the shape of the inclusion. a α b β m αβ for a fixed K 2 to image finer details If K = 2, it is imaging by PT (equivalent ellipse). Optimization Problem: Let Ω be the target domain. Minimize over D J[D] := w α + β 2 a αb β M αβ (k, D) 2 a αb β M αβ (k, Ω). α,β α,β α + β K w α + β are binary weights: w α + β = (on) or (off). A good choice for the initial guess: the equivalent ellipse.

7 An asymptotic expansion of GPT due to small boundary change: ϵ-perturbation of D: D ϵ := { x = x + ϵh(x)ν(x) x D}. Asymptotic formula: Suppose that H = α a αx α and F = β b βx β are harmonic polynomials. Then a α b β M αβ (k, D ϵ ) a α b β M αβ (k, D) α,β α,β [ v u = ϵ(k ) h + ] u v dσ D ν ν k T T +O(ϵ 2 ),

8 where and u =, in D (R 2 \D), u + u =, on D, u k u =, ν + ν on D, (u H)(x) = O( x ) as x, v =, in D (R 2 \D), kv + v =, v v =, ν + ν on D, on D, (v F )(x) = O( x ) as x.

9 Shape derivative: Let [ ϕ HF v u D (x) = (k ) + ] u v, ν ν k T T and Then δ HF D = α,β a α b β M αβ (k, D) α,β d S J[D], h L 2 ( D) = α + β K w α + β δ HF D a α b β M αβ (k, B). ϕ HF D, h L 2 ( D). Determination of location: Let i l := e l and j l := 2e l, l =,..., d. Then ( m i l j l m il i l ) is the center of mass of B if B is a ball.

10 Gradient Descent Method To get a minimum of F : R m R, one starts with an initial guess x and modify it as x n+ = x n γ n F (x n ), where γ n is a positive real number. Note that F (x) = m d j= F (x + te dt j) e j. t= To approximate the inclusion, modify D () (initial guess) as D (n+) = D (n) γ n ( j d S J[D (n) ], ψ j ψ j ) ν, where ν is the outward normal direction to D (n) and d S J is the shape derivative.

11 Figure: K = 6, 6 iterations.

12 Figure: Reconstruction of clusters of inclusions. The upper images: the equivalent ellipses, and the lower ones: results after 6 iterations.

13 Figure: After 6 iterations

14 Related Works Conductivity Equation, NtD map Ω D ε D ϵ = { x = x + ϵh(x)ν x } f (u ϵ u) dσ = ϵ( k) Ω D [ v u h T T + v + u ] + dσ + O(ϵ 2 ), k ν ν v ν = f, where v satisfies { (( + (k )χ(d) v) =, in Ω, on Ω.

15 proof Note that Ω ( + (k )χ D ϵ ) u ϵ v = Ω u ϵ ν v ( + (k )χ Ω D) u ϵ v = v uϵ Ω ν ( ) v u u v = Ω ν ν We have ( v (u ϵ u) v ) Ω ν ν (u ϵ u) ) = (k ) (χ Dϵ χ D u ϵ v Ω = (k ) uϵ i v e (k ) uϵ e v i D ϵ\d D\D ϵ (k )ϵ hm u e : v e, D where M = τ τ + k ν ν.

16 Similarly, we can derive Modal Measurements: the Conductivity case g(u ϵ u) + (ω 2 ωϵ) 2 w g u Ω Ω ϵ(k ) h(x)m u e : wg e dσ x, where M = τ τ + ν ν k Modal Measurements: the Elastic case g C ( u ϵ u)ν + (ω 2 ωϵ) 2 w g u Ω ϵ D D h(x)m[ u e ](x) : w e g (x)dσ(x) where M is defined from the lame constants. Ω

17 Basis functions: Optimal vectors by Ammari-Beretta-Francini-Kang-L Remind g C ( u ϵ u )ν + (ω 2 ωϵ) 2 Ω = ϵ D Ω w g u h(x)m[ u e ](x) : w e g (x)dσ(x) + O(ϵ +β ). { Let V := g L 2 ( Ω) : g (C u } Ω D )ν = and define Λ : V L 2 ( D) by If where Λ(g) := M[ u e ] : w e g h(x) = L α l v gl, l= on D. v gl := M[ u e ] : w e g l on D, l =,..., L, and g l are the significant singular vectors of Λ, then we have good reconstructed images.

18 Incomplete Measurements Suppose that C ( u ϵ u )ν is measured only in an open part Γ of the boundary Ω. For g L 2 ( Ω) such that g = on Γ 2 and g (C Γ D u)ν =, we can prove that the following asymptotic formula holds as ϵ : g C ( u ϵ u )ν + (ω 2 ω2 ϵ ) w g u Γ Ω = ϵ h(x)m[ u e ](x) : w e g (x)dσ(x) + O(ϵ+β ) D for some β >... Figure: Incomplete measurements reconstruction: we use the data only measured on the part of Ω, that is {e iθ : θ [, π]}.

19 Thank you!