Probe type method for acoustic wave equations with discontinuous - PowerPoint PPT Presentation

Probe type method for acoustic wave equations with discontinuous coefficients Probe type method for acoustic wave equations with discontinuous coefficients Gen Nakamura nakamuragenn@gmail.com Hokkaido University, Japan Joint work with David Dos Santos Ferreira, Institute of ´ Elie Cartan, France Analysis and Numerics of Acoustic and Electromagnetic Problems, RICAM, October 17-22, 2016

Probe type method for acoustic wave equations with discontinuous coefficients Outline of my talk Introduction 1 Probe method for acoustic equation 2 Statement of probe method for acoustic equation Ingredients of proof for Proposition 2 BC type probe method for acoustic wave equation 3 Set up Preliminaries Approxiamte controllability BC type probe method

Probe type method for acoustic wave equations with discontinuous coefficients Introduction Introduction

Probe type method for acoustic wave equations with discontinuous coefficients Introduction Introduction I By adapting the idea of M. Belishev given in the paper below, we will show how the boundary control method abbreviated by BC method can be used to identify an unknown inclusion with unknown back ground inside a medium. This idea is quite similar to the probing method for the Helmholtz equation and heat equation. M. Belishev, Equations of the Gelfand-Levitan type in a multidimensional inverse problem for the wave equations Zap. Nauchn. Semin. LOMI, 1987 (Engl. transl.,J. Sov. Math. 55, 1991) Later we will give a brief review of probe method for acoustic equation which was introduced by Ikehata (1998) and it is exactly the same to singular sources method introduced by Potthast (1999).

Probe type method for acoustic wave equations with discontinuous coefficients Introduction Introduction II We list some other related results for some reconstruction methods for acoustic wave equations. Burkard-Potthast (2009), probe method for obstacle scattering Ikehata (2010), enclosure method for obstacle scattering Chen-Haddar-Lechleiter-Monk (2010), sampling method for obstacle scattering Kirchipnikova-Kurylev (2011), inverse spectral problem for Riemannian polyhedron Oksanen (2013), BC method for identifying an obstacle with unknown background (volume drop method)

Probe type method for acoustic wave equations with discontinuous coefficients Introduction IOP Expanding Physics Inverse Modeling An introduction to the theory and methods of inverse problems and data assimilation Gen Nakamura Roland Potthast

Probe type method for acoustic wave equations with discontinuous coefficients Probe method for acoustic equation Probe method for acoustic equation

Probe type method for acoustic wave equations with discontinuous coefficients Probe method for acoustic equation Statement of probe method for acoustic equation Statement of probe method for acoustic equation

Probe type method for acoustic wave equations with discontinuous coefficients Probe method for acoustic equation Statement of probe method for acoustic equation Statement of probe method for acoustic equation I Let D ⊂ Ω ⊂ R 3 be bounded domains with C 2 boundaries ∂ Ω , ∂D such that D ⊂ Ω and Ω \ D is connected. Also let L D := ∆ + ω 2 ρ be an acoustic operator with angular frequency ω > 0 and density ρ . For simplicity assume that ρ > 0 has the form ρ = 1 + kχ D with a constant k > 0 , (1) where χ D is the characteristic function of D . Forward problem for measurements : The the well-posedness (i.e. has existence, uniqueness, continuous dependence on the data) of the solution u = u ( f ) ∈ H 1 (Ω) to the boundary value problem:  L D u = 0 in Ω ,  ( BP ) (2) u = f ∈ H 1 / 2 ( ∂ Ω) on ∂ Ω . 

Probe type method for acoustic wave equations with discontinuous coefficients Probe method for acoustic equation Statement of probe method for acoustic equation Statement of probe method for acoustic equation II Assume that (BP) only has a trivial solution when f = 0 . Then (BP) is well-posed and hence has an estimate: ∃ a constant C > 0 such that � u � H 1 (Ω) ≤ C � f � H 1 / 2 ( ∂ Ω) ( f ∈ H 1 / 2 ( ∂ Ω) . (3) Based on this we take as our measurement the Dirichlet to Neumann map (DN map) Λ D : H 1 / 2 ( ∂ Ω) → H − 1 / 2 ( ∂ Ω) defined by � Λ D f = ∂ ν u ( f ) (4) ∂ Ω , � where ν is the outer unit normal to ∂ Ω . Λ ∅ = Λ D when D = ∅ , (5) where we have assumed (BP) with D = ∅ only has a trivial solution when f = 0 .

Probe type method for acoustic wave equations with discontinuous coefficients Probe method for acoustic equation Statement of probe method for acoustic equation Statement of probe method for acoustic equation III Inverse problem : Reconstruct D from Λ D . Theorem 1 There is a reconstruction method called the probe method to reconstruct D from Λ D . Let C = { c ( λ ) : 0 ≤ λ ≤ 1 } be a needle in Ω , i.e. non-self intersecting continuous curve in Ω joining two distinct points c (0) , c (1) ∈ ∂ Ω . For 0 < λ < 1 , let C λ := { c ( µ ) : 0 ≤ µ ≤ λ } . Let Φ( x, y ) := (4 π | x − y | ) − 1 e iω | x − y | ( x � = y ) be the fundamental solution of ∆ + ω 2 .

Probe type method for acoustic wave equations with discontinuous coefficients Probe method for acoustic equation Statement of probe method for acoustic equation c (0) C c ( λ − δ ) Ω c ( λ ) D c (1) Figure 1: Domains Ω , D , and a curve C

Probe type method for acoustic wave equations with discontinuous coefficients Probe method for acoustic equation Statement of probe method for acoustic equation Statement of probe method for acoustic equation IV By Runge’s approximation theorem which is a consequence of the weak unique continuation theorem for ∆ + ω 2 . we have the so called Runge’s approximation functions { v λ j,ℓ } ∞ ℓ =1 ⊂ H 1 (Ω) : solutions of L ∅ v = 0 in Ω , v λ j,ℓ ( x ) → v λ j := a j · ∇ Φ( x, c ( λ )) ( ℓ → ∞ ) in H 1 loc (Ω \ C λ ) , (6) where 0 � = a j ∈ R 3 ( j = 1 , 2 , 3) are fixed linearly independent vectors. Now for each needle C and λ ∈ [0 , 1] , define the indicator function I ( λ, C ) : 3 � � � � v λ � ∂ Ω v λ � � � I ( λ, C ) := lim Λ D − Λ ∅ (7) � � � � j,ℓ j,ℓ ∂ Ω � � ℓ →∞ ∂ Ω j =1 whenever the limit exists. Here the precise meaning of the above integral v λ � ∂ Ω ∈ H − 1 / 2 ( ∂ Ω) and � � is the pairing between Λ D − Λ ∅ � j,ℓ v λ � ∂ Ω ∈ H 1 / 2 ( ∂ Ω) . � j,ℓ

Probe type method for acoustic wave equations with discontinuous coefficients Probe method for acoustic equation Statement of probe method for acoustic equation Statement of probe method for acoustic equation V The definition of I ( λ, C ) is ”well-defined”, i.e. it does not depend on the choice of Runge’s approximation functions. Theorem 1 follows from the following proposition. Proposition 2 (i) | I ( λ, C ) | < ∞ ( λ ∈ [0 , 1]) if C ∩ D = ∅ . (ii) λ → λ 0 | I ( λ, C ) | = ∞ if C ∩ D � = ∅ , lim (8) where λ 0 is defined as c ( λ 0 ) gives the first touching point of C to ∂D . Remark There are other ”equivalent” reconstruction methods called the no-response test, range test (cf. book or papers by N-Potthast-Sini (2006), Honda-N-Potthast-Sini (2008)).

Probe type method for acoustic wave equations with discontinuous coefficients Probe method for acoustic equation Ingredients of proof for Proposition 2 Ingredients of proof for Proposition 2

Probe type method for acoustic wave equations with discontinuous coefficients Probe method for acoustic equation Ingredients of proof for Proposition 2 Ingredients of proof for Proposition 2 I There are two ingredients. They are Alessandrini’s argument and the reflected solution. Alessandrini’s identity = − ω 2 � � � � Λ D − Λ ∅ f f Ω ( ρ − 1) u ( f ) v ( f ) ∂ Ω (9) = − kω 2 � D u ( f ) v ( f ) , where v ( f ) = u ( f ) for the case D = ∅ . Let u λ j,ℓ = u ( v λ � ∂ Ω ) , w λ j,ℓ := u λ j,ℓ − v λ j,ℓ with Runge approximation j,ℓ � functions { v λ j,ℓ } ∞ ℓ =1 approximating v λ j in H 1 loc (Ω \ C λ ) . Then we have the following proposition.

Probe type method for acoustic wave equations with discontinuous coefficients Probe method for acoustic equation Ingredients of proof for Proposition 2 Ingredients of proof for Proposition 2 II Proposition 3 Assume C λ ∩ D = ∅ . Then, there exists the so called reflected solution w λ j ∈ H 1 (Ω) such that w λ j,ℓ → w λ j ( ℓ → ∞ ) in H 1 (Ω) and it satisfies  L D w λ � � j = − kχ D a j · ∇ Φ( · , c ( λ )) in Ω  (10) w λ j = 0 in ∂ Ω .  Can complete the proof by the following observation: 3 � � = kω 2 � ( w λ j + v λ � � � j ) v λ � � I ( λ, C ) � j � D j =1

Probe type method for acoustic wave equations with discontinuous coefficients Probe method for acoustic equation Ingredients of proof for Proposition 2 Ingredients of proof for Proposition 2 III 3 ≥ 1 � � � j | 2 − � 2 kω 2 | v λ | w λ j | 2 � , D D j =1 where at least one of the first term blows up while the second terms stay bounded as λ → λ 0 .

Probe type method for acoustic wave equations with discontinuous coefficients BC type probe method for acoustic wave equation BC type probe method for acoustic wave equation