Inverse problems in lithospheric flexure and viscoelasticity

Transcription

1 Inverse problems in lithospheric flexure and viscoelasticity Axel Osses in coll with M. de Buhan (CNRS, France), E. Contreras (Geophysics Department, Chile), B. Palacios (DIM, Chile) DIM - Departamento de Ingeniería Matemática CMM - Centro de Modelamiento Matemático Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago. [matematika mugaz bestalde] - [bcam] Bilbao, July 4 th 2011

2 Outline 1 Introduction The Maxwell viscoelastic model Relationship to other viscoelastic models 2 Inverse problems 3 Motivation 4 Stability Result (interior measurement) 5 Stability Result (boundary measurement) 6 Numerical resolution 7 Examples 2/57

3 Introduction The Maxwell viscoelastic model σ stress, ɛ strain springs: σ 0 = E 0 ɛ, σ e = Eɛ e, dashpot: σ v = ηɛ v (E, E 0 = Young s modulus, η = viscosity) in parallel: σ e = σ v, ɛ = ɛ e + ɛ v ηɛ v = Eɛ e = E(ɛ ɛ v ) ɛ v = t 0 1 t s η τ e τ ɛ(s) ds, τ = E (relaxation time) 3/57

4 Introduction The Maxwell viscoelastic model σ = σ 0 + σ e = E 0 ɛ + Eɛ e = E 0 ε + E(ɛ ɛ v ) = (E 0 + E)ɛ Eɛ v t 1 = E 0 ɛ E }{{} 0 elasticity τ e t s τ ɛ(s) ds } {{ } viscoelasticity PDE: 2 t u σ = f 4/57

5 Introduction The Maxwell viscoelastic model Let x Ω bounded domain in R 3, t 0, we consider the 3D-viscoelastic model (density ρ = 1), stress tensor: ɛ(u) = u + u T (1) Pu := 2 t u (µ 0 ɛ(u) + λ 0 ( u)i ) }{{} t elasticity + ( µ(t s)ɛ(u(s)) + λ(t s)( u)(s)i )ds = 0, } 0 {{ } viscoelasticity (one branch) u(0) = ū 0, t u(0) = ū 1 in Ω, u = 0, on Ω (0, + ). µ(x, t) = µ(x)h(t), λ(x, t) = λ(x)h(t) (e.g. h(t) = τ 1 e t/τ ). u(x, t) : displacement vector, (λ 0 (x), µ 0 (x), λ(x), µ(x)) : coefficients. 5/57

6 Introduction Relationship to other viscoelastic models Changing u by u s in the viscoelastic part... Pu := t 2 u (µ 0 ɛ(u) + λ 0 ( u)i ) }{{} (1 ) t elasticity ( µ(t s)ɛ(u s (s)) + λ(t s)( u s )(s)i )ds = 0, } 0 {{ } viscoelasticity (one branch) u(0) = ū 0, t u(0) = ū 1 in Ω, u = 0, on Ω (0, + ). µ(x, t) = µ(x)h(t), λ(x, t) = λ(x)h(t) (e.g. h(t) = e t/τ ). µ 0 (x) = µ 0 (x) + µ(x), λ 0 (x) = λ 0 (x) + λ(x), (λ 0 (x), µ 0 (x), λ(x), µ(x)) : coefficients. 6/57

7 Introduction Relationship to other viscoelastic models generalized Maxwell model... (1 ) µ(x, t) = Pu := t 2 u (µ 0 ɛ(u) + λ 0 ( u)i ) }{{} elasticity N t + 0 j=1 µ j (t s)ɛ(u(s))ds = 0, } {{ } viscoelasticity (N branchs) u(0) = ū 0, t u(0) = ū 1 in Ω, u = 0, N j=1 on Ω (0, + ). µ j (x)h j (t), (e.g. h j (t) = τ 1 j e t/τ j ). (λ 0 (x), µ 0 (x), {µ j (x)} N j=1 ) : coefficients. 7/57

8 Introduction Relationship to other viscoelastic models fractional Maxwell models (α (0, 1))... (1 ) µ α (x, t) = Pu := 2 t u (µ 0 ɛ(u) + λ 0 ( u)i ) }{{} 0 elasticity t + µ(ξ, t s)dξ ɛ(u(s))ds 0 0 }{{} viscoelasticity ( branchs) u(0) = ū 0, t u(0) = ū 1 in Ω, u = 0, = 0, on Ω (0, + ). µ α (ξ, x)h(ξ, t) dξ, (e.g. h(ξ, t) = ξ e ξ t, ξ = 1/τ). (λ 0 (x), µ 0 (x), µ α (ξ, x)) : coefficients. 8/57

9 Introduction Relationship to other viscoelastic models viscoelastic Kirchhoff plates... (D, u) = D yy u xx 2D xy u xy + D xx u yy (2) + Qu := 2 ttu ε 2 2 tt u + D 0 2 u(s) + ( D 0, u) }{{} t elasticity D(t s) 2 u + ( D(t s), u(s))ds = 0, 0 } {{ } viscoelasticity +i.c and b.c., on Ω (0, + ). D(x, t) = D(x)h(t), (e.g. h j (t) = τ 1 e t/τ ). 9/57

10 Outline 1 Introduction 2 Inverse problems Recover viscoelastic parameter from local displacement Recover viscoelastic parameter for plates (open problem) 3 Motivation 4 Stability Result (interior measurement) 5 Stability Result (boundary measurement) 6 Numerical resolution 7 Examples 10/57

11 Inverse problems Recover viscoelastic parameter from local displacement Recover p := µ(x) or p := λ(x) from measurements of the solution : u(x, t) in ω (0, T ) }{{} (single time dependent interior measurements) where ω is a (small) subset of Ω and T > 0 or u(x, t) on Γ (0, T ) }{{} (single time dependent boundary measurements) where Γ is a (small) part of Ω and T > 0. T Ω ω internal measure Γ boundary measure 0 11/57

12 Inverse problems Recover viscoelastic parameter for plates (open problem) Recover (uniqueness, stability) p := D 0 (x) (or p := D(x) for t ) in the stationary flexure plate model from the Cauchy data: ( u, u ) u, u, Ω n n Recover (uniqueness, stability) p := D(x) in the stationary flexure plate model from the Cauchy data: from internal (ω (0, T )) or boundary (Γ (0, T )) local measurements. 12/57

13 Outline 1 Introduction 2 Inverse problems 3 Motivation Motivation: elastography Lithospheric flexure 4 Stability Result (interior measurement) 5 Stability Result (boundary measurement) 6 Numerical resolution 7 Examples 13/57

14 Motivation Motivation: elastography McLaughlin and Yoon, Inverse Problems 2004 [1] elastography: diagram for identification of stiffness profile M. Fink, presentation in Valparaiso, 2010 quantitative shear wave elastography, Airexplorer /57

15 Motivation Lithospheric flexure Watts and Zhong, Geophys. J. Int flexure of a two-layer viscoelastic plate model, relaxation time=1myr Contreras and Osses, Geophys. J. Int [2] Nazca plate variable thickness flexure (1D) 15/57

16 Motivation Lithospheric flexure Contreras, Manriquez, Osses, 2011, work in progress Bathymetry 16/57

17 Motivation Lithospheric flexure Contreras, Manriquez, Osses, 2011, work in progress Nazca plate variable thickness flexure (2D) 17/57

18 Outline 1 Introduction 2 Inverse problems 3 Motivation 4 Stability Result (interior measurement) Setting Theorem Assumptions 5 Stability Result (boundary measurement) 6 Numerical resolution 7 Examples 18/57

19 Stability Result (interior measurement) Setting p in Ω? u in ω /57

20 Stability Result (interior measurement) Theorem Theorem (2) (Stability) [3] Let u (resp. ū) be the solution of (1) associated to the coefficient p (resp. p). Under Hypothesis 1, 2 and 3, there exists κ (0, 1) such that: p p H 2 (Ω) C u ū κ H 2 (ω (0,T )), where C depends on the W 2, (Ω)-norm of p and p and on the W 8, (Ω (0, T ))-norm of u and ū. Corollary (Uniqueness from interior measurement) u = ū in ω (0, T ) = p = p in Ω 20/57

21 Stability Result (interior measurement) Assumptions Hypothesis 1 on the coefficients and trajectories ( λ 0, µ 0 ) (W 2, (Ω)) 2 and ( λ, µ) (W 2, (Ω (0, T ))) 2 are such that u W 8, (Ω (0, T )), µ 0, λ 0 + 2µ 0, µ 0 h(0)µ, λ 0 + 2µ 0 h(0)(λ + 2µ) satisfy Condition 1, p is known in a neighborhood V of Ω, h(0) 0, ū 1 = h (0) h(0) ū0, (ū 1 = τ 1 ū 0 ). Condition 1 The scalar function q satisfies Condition 1 if : it exists K > 0 such that x Ω, q(x) K, it exists x 0 R 3 \ Ω such that x Ω : 1 2 q(x) q(x) (x x 0) 0. 21/57

22 Stability Result (interior measurement) Assumptions Hypothesis 2 on the observation part T > T 0 = d β }{{} large enough d = sup x x 0, β > 0 small. x Ω and ω } {{ V }, arbitrarily narrow, non trapping Ω Ω V Hypothesis 3 on the initial data We suppose that ū 0 is such that there exists M > 0 such that : ɛ(ū 0 (x))(x x 0 ) M, x Ω. 22/57

23 Stability Result (interior measurement) Proof: Carleman estimate for the viscoleastic system (1) u(0) t u(0) 0 1) Carleman weight: ϕ(x, t) = x x 0 2 βt 2 level sets cones. weighted energy source terms + internal measurements /57

24 Stability Result (interior measurement) Proof: Carleman estimate for the viscoleastic system (1) u(0) t u(0) 0 Write (1) as a system of 7 scalar equations for u, u and u (see McLaughlin-Yoon 2004 [1], Imanuvilov-Yamamoto 2005 [4]), 2 t u i (x, t) p i (x) u i (x, t)+ t h(t s) p i (x) u i (x, s)ds = L i (u, u, u) ε ε Ω χ = χ = 0 Change of variables to drop the integral term (similar to Cavaterra et al [5]): ( t ) ũ i (x, t) = (1+αt) p i (x)u i (x, t) h(t s) p i (x)u i (x, s)ds, α large. 0 = 2 t ũ i (x, t) q i (x) ũ i (x, t) = L i (u, u, u) 24/57

25 Stability Result (interior measurement) Proof: Carleman estimate for the viscoleastic system (1) u(0) t u(0) 0 Apply a modified pointwise Carleman inequality for a scalar hyperbolic equation (Klibanov and Timonov, 2004 [6]) in a cone. weigth = e σϕ(x,t), ϕ(x, t) = x x 0 2 βt 2, x 0 R 3 \Ω, β, σ > 0 t Q ϕ(x, t) = χ = 1 0 ε ε δ x 2δ χ = 0 Return to the initial variable and bound/absorb the integral terms: ( t ) 2 u(x, s) ds e 2σϕ(x,t) dx dt C u(x, t) 2 e 2σϕ(x,t) dx dt σ Q 0 Q 25/57

26 Stability Result (interior measurement) Proof: Bukhgeim and Klibanov method 2) We apply the method of Bukhgeim and Klibanov 1981 [7] : ŵ = t 2 (u ū), w = t 2 ū, t Pŵ = h(t s) (2(p p) ɛ( w(s))) ds 0 h (t) (2(p p) ɛ(ū 0 )), in Ω (0, + ), ŵ(0) = 0, in Ω, t ŵ(0) = h(0) (2(p p) ɛ(ū 0 )), in Ω, ŵ = 0, on Ω (0, + ). + Apply Carleman inequality : weighted initial energy interior measurement + sources = the stability result : p p H2 (Ω) C u ū κ H 2 (ω (0,T )) 26/57

27 Outline 1 Introduction 2 Inverse problems 3 Motivation 4 Stability Result (interior measurement) 5 Stability Result (boundary measurement) Setting Theorem Assumptions Proof: unique continuation Main scheme of the proof - elastic case Main scheme of the proof - viscoelastic case 6 Numerical resolution 27/57

28 Stability Result (boundary measurement) Setting p in Ω? u on Γ 28/57

29 Stability Result (boundary measurement) Theorem Theorem (3) (Logarithmic stability) [8] Let u (resp. ū) be the solution of (1) associated to the coefficient p (resp. p). Under Hypothesis 1, 2 and 3, there exists κ (0, 1) such that: [ ( C p p H 2 (Ω) C log α 2 δα x (u ū) 2 L 2 (Γ (0,T )) )] κ where C depends on the W 2, (Ω)-norm of p and p and on the W 8, (Ω (0, T ))-norm of u and ū. Corollary (Uniqueness from boundary measurement) α x u = α x ū on Γ (0, T ), α = 1, 2 = p = p in Ω. 29/57

30 Stability Result (boundary measurement) Assumptions Hypothesis 1 on the coefficients the same. Hypothesis 2 on the observation part T > T 0 }{{} large enough and } Γ {{ Ω }. arbitrarily small Hypothesis 3 on the initial data the same. 30/57

31 Stability Result (boundary measurement) Proof: unique continuation Theorem (1) (Unique continuation) [8] Let u be a regular solution of (1) starting from rest and with right hand side vanishing in a neighborhood V of Ω. Then [ ( )] 1 C u H2 (V (0,T /3)) C log α 2 δα x u 2 L 2 (Γ (0,3T )) where C depends on the W 2, (Ω)-norm of coefficients and on the W 4, (Ω (0, T ))-norm of u. Corollary (Unique continuation) α x u = 0 on Γ (0, 3T ), α = 1, 2 = u = 0 in V (0, T /3). 31/57

32 Stability Result (boundary measurement) Proof: unique continuation This is done by using a variant of Fourier-Bros-Iagolnitzer transform [9] for changing equations from hyperbolic to elliptic character as in Bellassoued 2008 [4]. applying a Carleman inequality for the resulting elliptic sytem. using interpolation results as in Robbiano 1995 [10]. 32/57

33 Stability Result (boundary measurement) Main scheme of the proof - elastic case 33/57

34 Stability Result (boundary measurement) Main scheme of the proof - viscoelastic case 34/57

35 Stability Result (boundary measurement) Main scheme of the proof - viscoelastic case 35/57

36 Stability Result (boundary measurement) Main scheme of the proof - viscoelastic case 36/57

37 Stability Result (boundary measurement) Main scheme of the proof - viscoelastic case 37/57

38 Stability Result (boundary measurement) Main scheme of the proof - viscoelastic case 38/57

39 Stability Result (boundary measurement) Main scheme of the proof - viscoelastic case 39/57

40 Outline 1 Introduction 2 Inverse problems 3 Motivation 4 Stability Result (interior measurement) 5 Stability Result (boundary measurement) 6 Numerical resolution 7 Examples 40/57

41 Numerical resolution Direct problem: finite elements We discretize the system (1): in space by P 1 Lagrange Finite Elements, in time by a θ-scheme with θ = 0.5 (implicit centered scheme), by using the Trapezium Formula for the integral term. We consider : µ 0 (x) = λ 0 (x) = 3, λ(x) = µ(x) = 1, h(t) = e t/τ with τ = /57

42 Numerical resolution Inverse problem: variational approach We are looking for the minimizer of the non-quadratic functional : J(p) = 1 T ( u(p) uobs 2 + (u(p) u obs ) 2) dxdt 2 0 ω with u(p) = M(p), M being the nonlinear operator associated to system (1). We solve the minimization problem by a BFGS algorithm, so we need: J(p, δp) = lim (J(p + εδp) J(p)) ε T = M p δp ((u u obs ) (u u obs )) ε ω with M p the linearized operator of M around p, i.e. M p δp = δu satisfies : t Pδu = 2δp h(t s)ɛ(u(s))ds 0 (2) δu(0) = 0, in Ω, t δu(0) = 0, in Ω, δu = 0, on Ω (0, + ). 42/57

43 Numerical resolution Inverse problem: adjoint problem and sensitivity We introduce the adjoint problem of (2) : P δu := t 2 δu (2µ 0 ɛ(δu ) + λ 0 ( δu )I ) (2) T ( + 2p h(s t)ɛ(δu (s)) + λ(s)( ) δu )(s)i ds t { P δu (u(p) uobs ) (u(p) u = obs ) in ω (0, T ) 0 in (Ω \ ω) (0, T ) δu (T ) = 0, in Ω, t δu (T ) = 0, in Ω, δu = 0, on Ω (0, + ). So we can compute : T T J(p, δp) = δu P δu = Pδu δu 0 Ω 0 Ω ( T ) t = δp 2h(t s)ɛ(u(s)) : ɛ(δu (t))ds dt dx Ω /57

44 Outline 1 Introduction 2 Inverse problems 3 Motivation 4 Stability Result (interior measurement) 5 Stability Result (boundary measurement) 6 Numerical resolution 7 Examples Example 1: Reference data A first result in recovering the coefficient Example II: More realistic domain 44/57

45 Examples Example 1: Reference data Resolution with FreeFem 2D, Visualization with Medit, µ(x) = λ(x) = 3, λ(x, t) = h(t) = e t/τ with τ = 0.3, { 1, in the healthy tissu, p(x) = 3, in the tumor, T = 4, N = T δt = 20, K = 25, δ = 0. tumor measurement region mesh 45/57

46 Examples A first result in recovering the coefficient p ΠK p p 46/57

47 Examples Changing the tumor size Error in the parameter in L2!norm 50 big tumor medium tumor small tumor % of error number of eigenvectors in the basis = K optimal depends on the tumor size 47/57

48 Examples Changing the noise level δ k uobs u kl (ω (0,T )) δ u 2% Error in the parameter in L2!norm % error in the parameter 5% % % error in the data 48/57

49 Examples Changing the time of observation T 2 Error in the parameter with respect to T = Nδt with N fixed Error in the parameter in L2!norm % error in the parameter observation time = T 0 49/57

50 Examples Changing the measurement region Error in the parameter with respect to ω Error in the parameter in L2!norm % error in the parameter observation zone 50/57

51 Examples Example II: More realistic domain 51/57

52 Examples Initial condition and trajectory 52/57

53 Examples Regularization: finite number K of eigenfunctions 53/57

54 Examples Recovering with fixed eigenfunctions 54/57

55 Examples Eigen and mesh adaptation 55/57

56 Examples Recovering with adaptative eigenfunctions 56/57

57 Examples Comparison 57/57

58 References: J.-R. Yoon J. R. McLaughlin. Unique identifiability of elastic parameters from time-dependent interior displacement measurement. Inverse Problems, 20:25 45, A. Osses J. Contreras-Reyes. Lithospheric flexure modeling seaward of the chile trench: Implications for oceanic plate weakening in the trench outer rise region. Geophys. J. Int., 182: , M. de Buhan and A. Osses. A stability result in parameter estimation of the 3d viscoelasticity system. C. R. Acad. Sci. Paris, Sér. 347: , M. Bellassoued, O. Imanuvilov, and M. Yamamoto. Inverse problem of determining the density and the Lamé coefficients by boundary data. SIAM J. Math. Anal., 40, C. Cavaterra, A. Lorenzi, and M. Yamamoto. A stability result via Carleman estimates for an inverse source problem related to a hyperbolic integro-differential equation. Computational and Applied Mathematics, 25: , M.V. Klibanov and A. Timonov. Carleman estimates for coefficient inverse problems and numerical applications. VSP, Utrecht, /57

59 A.L. Bukhgeim and M.V. Klibanov. Global uniqueness of class of multidimensional inverse problems. Soviet Math. Dokl., 24, M. de Buhan and A. Osses. Logarithmic stability in determination of a 3d viscoelastic coefficient and numerical examples. Inverse Problems, 26:38pp., D. Iagolnitzer. Microlocal essential support of a distribution and local decompositions - an introduction. Hyperfunctions and theoretical physics, volume 449, pp Lecture Notes in Mathematics, Springer-Verlag, L. Robbiano. Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptot. Anal., 10, /57