A New Non-Iterative Reconstruction Method for a Class of Inverse Problems

Transcription

1 New Trends in Parameter Identification for Mathematical Models A New Non-Iterative Reconstruction Method for a Class of Inverse Problems Antonio André Novotny Laboratório Nacional de Computação Científica, LNCC/MCTI Av. Getúlio Vargas 333, Petrópolis - RJ, Brasil IMPA, 31th October, 2017 A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

2 Outline 1 Motivation 2 Topological Derivative Concept 3 Why is making holes a good idea 4 Applications of the Topological Derivative 5 Second Order Topological Derivative 6 Problem Setting Topological Asymptotic Expansion Non-Iterative Reconstruction Algorithm Numerical Experiments 7 Conclusions Remarks A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

3 Motivation inf ψ(ω) Ω E ψ(ω): shape functional Ω: geometrical domain E: set of admissible domains A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

4 Motivation inf ψ(ω) Ω E ψ(ω): shape functional Ω: geometrical domain E: set of admissible domains A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

5 Topological Derivative Concept Sokolowski & Zochowski, 1999 A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

6 Topological Derivative Concept Sokolowski & Zochowski, 1999 ψ(ω ε ( x)) = ψ(ω) + f (ε)t ( x) + o(f (ε)), where Ω ε ( x) = Ω \ ω ε ( x) and f (ε) 0, when ε 0. A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

7 Topological Derivative Concept Sokolowski & Zochowski, 1999 ψ(ω ε ( x)) = ψ(ω) + f (ε)t ( x) + o(f (ε)), where Ω ε ( x) = Ω \ ω ε ( x) and f (ε) 0, when ε 0. T ( x) = lim ε 0 ψ(ω ε ( x)) ψ(ω) f (ε). In general, f (ε) = ω ε. It depends on the boundary condition on ω ε. A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

8 Topological Derivative Concept ψ(ω ε ( x)) = ψ(ω) + f (ε)t ( x) + o(f (ε)) The topological sensitivity analysis gives the topological asymptotic expansion of a shape functional with respect to a singular domain perturbation, like the insertion of holes, inclusions or cracks. A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

9 Why is making holes a good idea ψ(ω) = 1 2 B 1 (u z d ) 2, Find u, such that u = b in Ω = B 1 R 2, u = 0 on B 1. A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

10 Why is making holes a good idea ψ(ω) = 1 2 B 1 (u z d ) 2, Find u, such that u = b in Ω = B 1 R 2, u = 0 on B 1. z d = u Ω, where Ω = B 1 \ B ρ, with ρ = 1/4. A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

11 Why is making holes a good idea ψ(ω) = 1 2 B 1 (u z d ) 2, Find u, such that u = b in Ω = B 1 R 2, u = 0 on B 1. z d = u Ω, where Ω = B 1 \ B ρ, with ρ = 1/ Figure: solution u Figure: target z d A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

12 Why is making holes a good idea Figure: topological derivative T A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

13 Why is making holes a good idea / Figure: topological derivative T Figure: optimal domain Ω A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

14 Applications of the Topological Derivative T ( x) = lim ε 0 ψ(ω ε ( x)) ψ(ω) f (ε) The topological derivative T ( x) is now of common use for resolution of several problems, such as: Topology Design: Amstutz, Canelas, Leugering, Zochowski... Inverse Problems: Ammari, Capdeboscq, Hintermüller, Kang, Laurain, Prakash... Multi-Scale Material Design: Giusti, Souza Neto, Toader... Image Processing: Auroux, Belaid, Drogoul, Masmoudi... Fracture and Damage Modeling: Allaire, Jouve, Van Goethem, Xavier... Theory Development: Amstutz, Nazarov, Sokolowski... A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

15 Second Order Topological Derivative ψ(ω ε ( x)) = ψ(ω) + f (ε)t ( x) + f 2 (ε)t 2 ( x) + R(f 2 (ε)), where f (ε) 0 and f 2 (ε) 0 with ε 0, and f 2 (ε) lim ε 0 f (ε) = 0, lim R(f 2 (ε)) ε 0 f 2 (ε) = 0. A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

16 Second Order Topological Derivative ψ(ω ε ( x)) = ψ(ω) + f (ε)t ( x) + f 2 (ε)t 2 ( x) + R(f 2 (ε)), where f (ε) 0 and f 2 (ε) 0 with ε 0, and f 2 (ε) lim ε 0 f (ε) = 0, lim R(f 2 (ε)) ε 0 f 2 (ε) = 0. (first order) topological derivative T ( x) := lim ε 0 ψ(ω ε ( x)) ψ(ω) f (ε). A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

17 Second Order Topological Derivative ψ(ω ε ( x)) = ψ(ω) + f (ε)t ( x) + f 2 (ε)t 2 ( x) + R(f 2 (ε)), where f (ε) 0 and f 2 (ε) 0 with ε 0, and f 2 (ε) lim ε 0 f (ε) = 0, lim R(f 2 (ε)) ε 0 f 2 (ε) = 0. (first order) topological derivative T ( x) := lim ε 0 ψ(ω ε ( x)) ψ(ω) f (ε) second order topological derivative T 2 ( x) := lim ε 0 ψ(ω ε ( x)) ψ(ω) f (ε)t ( x) f 2 (ε).. A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

18 A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

19 ? A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

20 A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

21 A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

22 Problem Setting Find k ω, such that div(k ω z) = 0 } in Ω, z = U on Γ n z = Q M { 1 in Ω \ ω k ω = γ in ω A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

23 Difficulties The problem is over determined and highly ill-posed; Lack of uniqueness if the contrast γ and the region ω are unknown simultaneously. A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

24 Difficulties The problem is over determined and highly ill-posed; Lack of uniqueness if the contrast γ and the region ω are unknown simultaneously. We assume that the contrast γ is known A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

25 Minimize ω Ω J ω (u) = (u z) 2 Γ M Find u, such that div(k ω u) = 0 in Ω n u = Q on Γ M u = z Γ M Γ M { 1 in Ω \ ω k ω = γ in ω A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

26 Minimize ω Ω J ω (u) = (u z) 2 Γ M Find u, such that div(k ω u) = 0 in Ω n u = Q on Γ M u Γ M = z Γ M { 1 in Ω \ ω k ω = γ in ω ω A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

27 Topological Asymptotic Expansion { 1 in Ω \ Bε (ξ) k ε = γ in B ε (ξ) m B ε (ξ) = B(x i, ε i ) i=1 ε := {ε 1, ε 2,...ε m } and ξ := {x 1, x 2,..., x m } J ε (u ε ) = (u ε z) 2 Γ M A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

28 Theorem J ε (u ε ) = J 0 (u 0 ) α d(ξ) H(ξ)α α + o( α 2 ), where the vector α = (α 1,, α m ), with α i = B(x i, ε i ). div(k ε u ε ) = 0 in Ω n u ε = Q on Γ M Γ M u ε = Γ M z u 0 = 0 in Ω n u 0 = Q on Γ M Γ M u 0 = Γ M z Joint work with A.D. Ferreira, A. Laurain & M. Hintermüller A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

29 The vector d R m and the matrix H R m R m are defined as d i := 2 ρ(u 0 z)(g i + ũ i ), Γ M H ii := 4 (u 0 z)(ρh i + ρ g i + ũ i ) + 2 Γ M (ρg i + ũ i ) 2, Γ M with H ij := 2 (u 0 z)(ρθ j i + ρθj i + u j i + uj i ) Γ M + 2 (ρg i + ũ i )(ρg j + ũ j ), j i. Γ M ρ = 1 γ 1 + γ, A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

30 ũ i = 0 in Ω n ũ i = ρ n g i on Ω Γ M ũ i = ρ Γ M g i ũ i = 0 in Ω n ũ i = ρ n (h i + g i ) on Ω Γ M ũ i = ρ Γ M h i + g i u j i = 0 in Ω n u j i = ρ n θ j i on Ω Γ M u j i = ρ Γ M θ j i A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

31 1 g i (x) = x x i u 0(x 2 i ) (x x i ), h i (x) = x x i 4 2 u 0 (x i )(x x i ) 2, 1 g i (x) = x x i ũ i(x 2 i ) (x x i ), θ j i (x) = 1 x x j 2 A(x j) u 0 (x i ) (x x j ). A(x) = [ 1 I 2 (x x ] i) (x x i ). x x i 2 x x i 2 A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

32 Theorem J ε (u ε ) = J 0 (u 0 ) α d(ξ) H(ξ)α α + o( α 2 ), where the vector α = (α 1,, α m ), with α i = B(x i, ε i ). δj(α, ξ, m) := α d(ξ) H(ξ)α α A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

33 Theorem J ε (u ε ) = J 0 (u 0 ) α d(ξ) H(ξ)α α + o( α 2 ), where the vector α = (α 1,, α m ), with α i = B(x i, ε i ). δj(α, ξ, m) := α d(ξ) H(ξ)α α α(ξ) = H(ξ) 1 d(ξ) A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

34 Theorem J ε (u ε ) = J 0 (u 0 ) α d(ξ) H(ξ)α α + o( α 2 ), where the vector α = (α 1,, α m ), with α i = B(x i, ε i ). δj(α, ξ, m) := α d(ξ) H(ξ)α α α(ξ) = H(ξ) 1 d(ξ) ξ = argmin ξ X δj(α(ξ), ξ, m), with δj(α(ξ), ξ, m) = 1 d(ξ) α(ξ) 2 A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

35 Theorem J ε (u ε ) = J 0 (u 0 ) α d(ξ) H(ξ)α α + o( α 2 ), where the vector α = (α 1,, α m ), with α i = B(x i, ε i ). δj(α, ξ, m) := α d(ξ) H(ξ)α α α(ξ) = H(ξ) 1 d(ξ) ξ = argmin ξ X δj(α(ξ), ξ, m), with δj(α(ξ), ξ, m) = 1 d(ξ) α(ξ) 2 (ξ, α(ξ )) A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

36 Complexity order of the Resulting Algorithm ( ) n C(n, m) = m 3 n! = m m!(n m)! m3 A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

37 Complexity order of the Resulting Algorithm m log 10 (C(n, m)), for n = 100 in blue and n = 400 in red A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

38 Numerical Experiments (a) electrodes (b) measurements A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

39 A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

40 Numerical Results (a) WGN = 5% (b) M = 64 A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

41 Numerical Results (a) WGN = 10% (b) M = 64 A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

42 Numerical Results (a) WGN = 15% (b) M = 64 A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

43 Numerical Results (a) WGN = 20% (b) M = 64 A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

44 Numerical Results (a) target (b) result A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

45 Numerical Results (a) target (b) result A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

46 Conclusions Remarks Non-iterative reconstruction method, robust with respect to noisy data and free of any initial guess; Specifically designed for a class of inverse problems where the unknown is given by a set of anomalies; Gravimetry, inverse potential problem, seismic, obstacle reconstruction... The main drawback is the combinatorial nature of the algorithm, summarized as follows: 1 If m n and m small, the complexity is treatable; 2 If m n, m can assume high values and the complexity remains treatable, since the number of combinations becomes small; 3 If m < n (m n/2) and m high, the complexity blows up and the combinatorial search becomes unfeasible; Actually, we are thinking about different possibilities to explore these features of the algorithm. A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

47 References A.A. Novotny & J. Soko lowski. Topological Derivatives in Shape Optimization. Mechanics and Mathematics Iteraction Series. 432p. Springer, A.A. Novotny & J. Soko lowski. Análise de Sensibilidade Topológica: Teoria e Aplicações. Notas em Matemática Aplicada. SBMAC, A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35

48 Muito Obrigado! A.A. Novotny (LNCC) Non-Iterative Reconstruction Method IMPA, 31th October, / 35