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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

1

Introduction

2

Probe method for acoustic equation Statement of probe method for acoustic equation Ingredients of proof for Proposition 2

3

BC type probe method for acoustic wave equation 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 ⊂ R3 be bounded domains with C2 boundaries ∂Ω, ∂D such that D ⊂ Ω and Ω \ D is connected. Also let LD := ∆ + ω2ρ be an acoustic operator with angular frequency ω > 0 and density ρ. For simplicity assume that ρ > 0 has the form ρ = 1 + kχD with a constant k > 0, (1) where χD 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) ∈ H1(Ω) to the boundary value problem: (BP)    LDu = 0 in Ω, u = f ∈ H1/2(∂Ω) on ∂Ω. (2)

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 uH1(Ω) ≤ CfH1/2(∂Ω) (f ∈ H1/2(∂Ω). (3) Based on this we take as our measurement the Dirichlet to Neumann map (DN map) ΛD : H1/2(∂Ω) → H−1/2(∂Ω) defined by ΛDf = ∂νu(f)

• ∂Ω,

(4) where ν is the outer unit normal to ∂Ω. Λ∅ = ΛD 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 ΛD. Theorem 1 There is a reconstruction method called the probe method to reconstruct D from ΛD. 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 Cλ := {c(µ) : 0 ≤ µ ≤ λ}. Let Φ(x, y) := (4π|x − y|)−1eiω|x−y| (x = y) be the fundamental solution of ∆ + ω2.

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

D Ω C c(0) c(λ − δ) c(1) c(λ)

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 ∆ + ω2. we have the so called Runge’s approximation functions {vλ

j,ℓ}∞ ℓ=1 ⊂ H1(Ω): solutions of L∅v = 0 in Ω,

vλ

j,ℓ(x) → vλ j := aj · ∇Φ(x, c(λ)) (ℓ → ∞) in H1 loc(Ω \ Cλ),

(6) where 0 = aj ∈ R3 (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): I(λ, C) := lim

ℓ→∞ 3

• j=1
• ∂Ω
• ΛD − Λ∅
• vλ

j,ℓ

• ∂Ωvλ

j,ℓ

• ∂Ω
• (7)

whenever the limit exists. Here the precise meaning of the above integral is the pairing between

• ΛD − Λ∅
• vλ

j,ℓ

• ∂Ω ∈ H−1/2(∂Ω) and

vλ

j,ℓ

• ∂Ω ∈ H1/2(∂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 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) lim

λ→λ0 |I(λ, C)| = ∞ if C ∩ D = ∅,

(8) where λ0 is defined as c(λ0) 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
• ΛD − Λ∅
• f f

= −ω2

Ω (ρ − 1)u(f) v(f)

= −kω2

D u(f) v(f),

(9) where v(f) = u(f) for the case D = ∅. Let uλ

j,ℓ = u(vλ j,ℓ

• ∂Ω), wλ

j,ℓ := uλ j,ℓ − vλ j,ℓ with Runge approximation

functions {vλ

j,ℓ}∞ ℓ=1 approximating vλ j in H1 loc(Ω \ Cλ).

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 Cλ ∩ D = ∅. Then, there exists the so called reflected solution wλ

j ∈ H1(Ω) such that wλ j,ℓ → wλ j (ℓ → ∞) in H1(Ω) and it satisfies

   LDwλ

j = −kχD

• aj · ∇
• Φ(·, c(λ)) in Ω

wλ

j = 0 in ∂Ω.

(10) Can complete the proof by the following observation:

• I(λ, C)
• = kω2

3

• j=1
• D

(wλ

j + vλ j ) vλ j

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

≥ 1 2kω2

3

• j=1

D

|vλ

j |2 −

• D

|wλ

j |2

, where at least one of the first term blows up while the second terms stay bounded as λ → λ0.

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

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

Set up of BC type probe method for acoustic wave equation

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

Set up I

Let Ω ⊂ R3 be a bounded domain with C2 boundary ∂Ω and D be a domain with C2 boundary ∂D such that Ω \ D is connected. Physically, we consider Ω as an acoustic medium with an inclusion D inside. Hereafter, we assume that all the functions are real valued. We assume for simplicity the density ρ of Ω is piecewise constant and it

• nly changes across ∂D. More precisely, we assume that ρ = 1 + kχD

with a constant k > −1, where χD denotes the characteristic function of

• D. For simplicity we consider the case k > 0, which means the speed of

the acoustic wave inside D is slower than that of outside. If an acoustical pressure f ∈ L2(∂Ω × (0, T)) is given at ∂Ω over the time interval (0, T) and the medium Ω is at rest in the past t < 0, then an acoustic wave will propagate inside the medium and its displacement u(x, t) = uf(x, t) ∈ L2((0, T) × Ω) is given as a weak unique solution to the initial boundary value problem

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

Set up II

           Lu = ρ ∂2u

∂t2 − ∆u = 0 in ΩT = Ω × (0, T) ∂u ∂ν = f ∈ L2(∂Ω × (0, T)) on (∂Ω)T = ∂Ω × (0, T)

u(·, 0) = ∂u

∂t (·, 0) = 0 in Ω,

(11) where ν denotes the outer unit normals to ∂Ω and u = uf is called the weak solution to (11) if it satisfies

• Q

u Lv dx dt −

• ΣT

fv ds dt = 0 (12) for any v ∈ H2(ΩT ) with ∂v

∂ν = 0 on (∂Ω)T and v

• t=T = ∂v

∂t

• t=T = 0.

It can be shown that uf ∈ L2((0, T); H1(Ω)) ∩ C0([0, T]; L2(Ω)) depends continuously on f ∈ L2((∂Ω)T ).

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

Set up III

For any fix proper open set Σ ⊂ ∂Ω, this regularity of solutions allows us to define the localized Neumann to Dirichlet map ΛT : F T → L2(ΣT ) by ΛT f = uf

• ΣT

, (13) where F T = {f ∈ L2((∂Ω)T ) supp f ⊂ Σ} and ΣT = Σ × (0, T). The further consequences of the regularity of solutions are that the

• perators defined by W T : F T ∋ f → uf(·, T) ∈ H = L2

ρ(Ω) is a

bounded operator and CT = (W T )∗W T : F T → F T is a bounded selfadjoint operator, respectively, where L2

ρ(Ω) denotes L2 space in Ω

with measure ρdx. The inverse problem which we are interested in is to reconstruct D from Λ2T if we do not know D and k.

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

Preliminaries

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

Preliminaries I

To study this inverse problem, we first prepare some preliminary facts we need. Let f, g ∈ F 2T and uf, ug ∈ L2((0, 2T); H1(Ω)) be the corresponding unique solutions to (11) with Neumann data f, g. Then, by the definition

• f Λ2T , the real inner product of uf(·, t) and ug(·, s) in H denoted by

q(t, s) = (uf(·, t), ug(·, s))H ∈ C([0, 2T] × [0, 2T]) is a weak solution of ∂2 ∂t2 − ∂2 ∂s2

• q(t, s) =
• Σ
• f(x, t)(Λ2T g)(x, s)−g(x, s)(Λ2T f)(x, t)
• ds(x)

(14) in (0, 2T) × (0, 2T), where the right hand side is a known bounded measurable function in (0, 2T) × (0, 2T) and by the initial conditions for uf and ug, q(t, s) = 0 at t = 0 and s = 0. Hence, by integrating this equation and setting t = s = T, we conclude that CT is determined by Λ2T .

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

Preliminaries II

The so called unique continuation property (UCP) is known for our

• perator L. That is the continuation of the zero set of weak solutions of

Lu = 0 in C0([0, T]; H1(Ω)) ∩ C1([0, T]; L2(Ω)) is known. (cf. Oksanen

• r Stefanov-Uhlmann)

Based on this, we define our domain of influence as follows. Definition 1 We define the domain of influence ˜ ET as the maximal subdomain of Ω in which all the weak solutions of Lu = 0 in C0([0, 2T]; H1(Ω)) ∩ C1([0, 2T]; L2(Ω)) will become zero at t = T if their Cauchy data on Σ2T are zero.

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

Preliminaries III

Remark Let ET = {x ∈ Ω : dist(x, Σ) < T}. If ET ∩ D = ∅, then ˜ ET = ET by the global Holmgren-John-Tataru uniqueness theorem which says that all the solutions with zero Cauchy data on Σ2T vanish in the double cone {(x, t) : dist(x, Σ) ≤ T − |t − T|} Further, by the finite speed of propagation and UCP for wave equation with jump in density, we have ˜ ET ⊂ ET .

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

Approximate controllability

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

Approximate controllability I

We have the following two very important theorems for the BC method Theorem 2 (approximate controllability) The closure cl (W T F T ) of W T F T in H is equal to L2( ˜ ET ) ⊂ H. That is cl (W T F T ) = L2( ˜ ET ). Proof It is enough to show that ψ ∈ L2( ˜ ET ) satisfies (uf(·, T), ψ)H = 0 for any f ∈ C∞

0 (ΣT ) implies ψ = 0. In order to do that, let

e ∈ C0([0, T]; H1(Ω)) ∩ C1([0, T]; L2(Ω)) be the weak solution to            Le = 0 in ΩT

∂e ∂ν = 0 on ΣT

e = 0,

∂e ∂t = ψ at t = T.

(15)

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

Approximate controllability II

Then, integrating by parts, we have =

• ΩT {uf(Le) − (Luf)e} dx dt

=

• Ω uf(·, T)ψρ dx +
• (∂Ω)T fe ds dt =
• ΣT fe ds dt,

for any f ∈ C∞

0 (ΣT ). Hence, the Cauchy data of e on ΣT are zero.

Now let E(x, t) = e(x, t) (t ≤ T), −e(x, 2T − t) (t > T). Then, due to e = 0 at t = T, E ∈ C0([0, 2T]; H1(Ω)) ∩ C1([0, 2T]; L2(Ω)), and it satisfies LE = 0 in Ω2T and E = ∂E ∂ν = 0 on Σ2T . By the definition of ˜ ET and its continuity with respect to T, we have ψ = ∂te(·, T) = ∂tE(·, T) = 0 on ˜ ET .

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

Approximate controllability III

Theorem 3 Let T < T∗ := inf{T : ET = Ω}. Then, the kernel Ker CT of CT just consists of 0. Proof Let f ∈ F T satisfy CT f = 0 which immediately implies uf(·, T) = 0 in Ω. Then consider an extension u0(·, t) of uf(·, t) to Ω × R defined by u0(·, t) =        (t < 0) uf(·, t) (0 ≤ t ≤ T) −uf(·, 2T − t) (T < t ≤ 2T) (2T < t). It is easy to see that u0 ∈ L2(R; H1(Ω)) satisfies    Lu0 = 0 in Ω × R

∂u0 ∂ν = f0

• n

∂Ω × R, (16)

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

Approximate controllability IV

where f0 is the extension of f by a similar extension as u0. Then, the Fourier transform ˆ u0(·, ω) of u0(·, t) with respect to t satisfies    (∆ + ρω2)ˆ u0 = 0 in Ω

∂ˆ u0 ∂ν = ˆ

f0

• n

∂Ω (17) for any ω ∈ R, where ˆ f0(·, ω) is the Fourier transform of f(·, t) with respect to t. Since by assumption T < T∗ and Theorem 2, ˆ u0(·, ω) is zero in the non-empty set Ω \ ET for any ω ∈ R. Hence, by a UCP for ∆ + ρω2, ˆ u0(·, ω) = 0 in Ω for any ω ∈ R. Therefore, ˆ f0(·, ω) = 0 for any ω ∈ R and hence f = 0.

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

Approximate controllability V

Fix any T ∈ (0, T∗). Let ΦT be a Hilbert space obtained by completing F T with respect to the new inner product f, gΦT = CT f, gF T (f, g ∈ F T ). Also, by Theorem 2, let WT be the continuous extension of W T to ΦT and CT = (WT )∗WT . Then, WT : F T → L2( ˜ ET ) becomes an onto isometry and CT : ΦT → ΦT becomes an isomorphism. Observe that for f ∈ F T and V ∈ H1(Ω) with ∂V

∂ν

• ∂Ω ∈ L2(∂Ω) and

∆V = 0 in Ω, we have the Blagovescenskii formula by the Green formula uf(·, T), V H =

• Ω V (x){

T

0 (T − t) ∂2uf ∂t2 (x, t) dt} dx

=

• ΣT
• (T − t)V (x) − (ΛT )∗

(T − t) ∂V

∂ν (x)

• f(x, t) ds(x) dt,

(18) with the adjoint Λ∗

T of ΛT .

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

Approximate controllability VI

Based on this observation, we define qT

V by

qT

V = (CT )−1Y with Y = (T − t)V − (ΛT )∗

(T − t)∂V ∂ν

• n ΣT . (19)

Then, qT

V satisfies

WT qT

V = V in ˜

ET . (20) In fact, for any f ∈ F T , we have Y, fF T = (CT )−1Y, fΦT = WT (CT )−1Y

• , WT fH.

On the other hand, from (18), Y, fF T = W T f, V H = V, WT fH. Then, (20) follows from these two equations and recalling that WT f (f ∈ F T ) is dense in C∞

0 ( ˜

ET ) with respect to the L2( ˜ ET ) norm.

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

BC method

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

BC method I

We will build our reconstruction scheme to reconstruct D from Λ2T based on these preliminaries. Let T < T∗, x0 ∈ Ω \ ˜ ET , aj ∈ R3 (j = 1, 2, 3) be linearly independent vectors and Ex0(x) be the fundamental solution of ∆ with singularity at x0. Further, let Gaj,x0 = aj · ∇Ex0. By the Runge approximation theorem, for each j (j = 1, 2, 3), there exists a sequence {hj,ℓ}∞

ℓ=1 ⊂ H2(Ω) such that

∆hj,ℓ = 0 in Ω, hj,ℓ → Gaj,x0 (ℓ → ∞) in L2( ˜ ET ).

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

BC method II

Then, by taking qj,ℓ = qT

V of (19) with V = hj,ℓ, we define our indicator

function σ(x0, T) by σ(x0, T) =

3

• j=1

(CT qj,ℓ0, qj,ℓ0)F T , (21) where ℓ0 = inf{ ℓ : hj,ℓ − Gaj,x0L2( ˜

ET ) ≤ 1 (j = 1, 2, 3)}. Then,

clearly σ(x0, T) is finite. To see the behavior of σ(x0, T) as x0 → ˜ ET ,

• bserve the estimate

σ(x0, T) = 3

j=1

• Ω
• uqj,ℓ0 (x, T)
• 2

ρ(x) dx ≥ 3

j=1

• ˜

ET

• uqj,ℓ0 (x, T)
• 2

ρ(x) dx ≥ 3

j=1

• ˜

ET

• Gaj,x0(x)
• 2

ρ(x) dx − K (22) with some constant K > 0 which is independent of x0 and T.

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

BC method III

We will use this indicator function to detect ∂D as follows. Let C = {c(α) : α ∈ [0, 1]} be a needle which connects c(0) ∈ Σ to c(1) ∈ ∂Ω \ Σ and γ(T) be the first touching point of C to ∂ ˜ ET when we traces C from c(1). By the estimate (22), γ(T) is given by γ(T) = c(αT ) with αT = inf{α′ ∈ (0, 1) : σ(c(α), T) < ∞ (α′ < α ≤ 1)}. (23) Further, by the previous Remark on the influence domain we have T = dist(γ(T), Σ) if ˜ ET ∩ D = ∅ T > dist(γ(T), Σ) if ˜ ET ∩ D = ∅. (24) Hence, by these (23), (24), we can in principle recover the point at which C touches ∂D at the first time by tracing C from its end point c(1) ∈ Σ.

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

BC method IV

By taking several needles, we can recover ∂D. Of course ∂ ˜ ET may becomes complicated as T becomes large and Σ has a complicated shape. These cases have to be avoided. Remark The idea given here works even in the case the density ρ in Ω \ D is unknown and C∞ smooth. For this case we can even reconstruct ρ there.

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

Thank you for your attention.