An explicit algorithm for solving the acoustic tomography problem - PowerPoint PPT Presentation

An explicit algorithm for solving the acoustic tomography problem for a moving fluid Alexey Agaltsov agaltsov @ cmap.polytechnique.fr Moscow September 12, 2016 Alexey Agaltsov Algorithm for acoustic tomography of moving fluid

Acoustic tomography of moving fluid A moving fluid in a bounded domain D ⊂ R d , d ≥ 2, is characterized by sound speed c = c ( x ) , density ρ = ρ ( x ) , velocity v = v ( x ) and absorption α = ω ζ ( x ) α 0 ( x ) There are acoustic transducers on ∂ D . A transducer produces time-harmonic acoustic waves which are scattered by the fluid. Scattered acoustic waves are recorded by other transducers. Acoustic tomography problem. Given this data, recover fluid parameters. Main applications in ocean tomography ( determine the ocean temperature and heat transferring currents ) and in medi- cal diagnostics ( determine scalar inhomo- geneities and the blood flow ) image: (Burov et al. ’13) Alexey Agaltsov Algorithm for acoustic tomography of moving fluid

Acoustic tomography of moving fluid � ω v � · ∇ − ω 2 c 2 + i c 2 − 2 i ωα L ω = − ∆ − 2 i 2 ∇ ln ρ (AC) c Data from point sources: G ω | X × Y , ω ∈ Ω , where X , Y ⊂ ∂ D , Ω ⊂ R ≥ 0 , � x ∈ R d , L ω G ω ( x , y ) = − δ y ( x ) , G ω ( · , y ) radiates at ∞ Acoustic tomography problem Given G ω | X × Y for ω ∈ Ω and c 0 , find c , image: (Burov et al. ’13) v , ∇ ρ and α in D acoustic data from fluid point sources = = = = = = = = = = = = = = ⇒ tomography parameters problem Alexey Agaltsov Algorithm for acoustic tomography of moving fluid

Mathematical framework We consider the following operator with smooth coefficients: � ∂ � 2 d � L A , Q = − + iA j ( x ) + Q ( x ) , (OP) ∂ x j j = 1 where x = ( x 1 , . . . , x d ) ∈ D , A = ( A 1 , . . . , A d ) , A j ( x ) ∈ M n ( C ) , Q ( x ) ∈ M n ( C ) , D is an open bounded domain in R d with boundary ∂ D L A , Q acts on C n -valued functions in D Alexey Agaltsov Algorithm for acoustic tomography of moving fluid

Mathematical framework � ∂ � 2 � d L A , Q = − + iA j ( x ) + Q ( x ) , (OP) ∂ x j j = 1 Suppose that E ∈ C is not a DE for L A , Q in D : � L A , Q ψ = E ψ in D , ψ | ∂ D = f , is uniquely solvable for any sufficiently regular f on ∂ D . The Dirichlet-to-Neumann map Λ A , Q = Λ A , Q ( E ) : � d � � j = 1 ν j ( ∂ Λ A , Q f = ∂ x j + iA j ) ψ ∂ D , (DN) where ν = ( ν 1 , . . . , ν d ) is the unit exterior normal to ∂ D . Alexey Agaltsov Algorithm for acoustic tomography of moving fluid

Mathematical framework � ∂ � 2 d � L A , Q = − + iA j ( x ) + Q ( x ) , (OP) ∂ x j j = 1 � d � � ∂ � � Λ A , Q f = j = 1 ν j ∂ x j + iA j ψ ∂ D , (DN) Conjugation of L A , Q by a smooth GL n ( C ) -valued function g :  gL A , Q g − 1 = L A g , Q g ,   j = gA j g − 1 + i ∂ g A g ∂ x j g − 1 , j = 1 , . . . , d , (GT)   Q g = gQg − 1 . The following formula holds: Λ A g , Q g = g | ∂ D Λ A , Q ( g | ∂ D ) − 1 . Alexey Agaltsov Algorithm for acoustic tomography of moving fluid

Mathematical framework � d � ∂ � 2 + Q , L A , Q = − ∂ x j + iA j (OP) j = 1 � d � � ∂ � � Λ A , Q ( ψ | ∂ D ) = j = 1 ν j ∂ x j + iA j ψ ∂ D , L A , Q ψ = E ψ, (DN)  gL A , Q g − 1 = L A g , Q g ,   Λ A g , Q g = Λ A , Q , (GT)   g is smooth GL n ( C ) -valued, g | ∂ D = Id The inverse Dirichlet-to-Neumann problem Given Λ A , Q at fixed E , find L A , Q modulo (GT). inverse conjugacy class > D-to-N map of Schr¨ odinger Dirichlet-to-Neumann operators problem Alexey Agaltsov Algorithm for acoustic tomography of moving fluid

The IDN problem: scalar case A j , Q are scalar functions, d ∈ { 2 , 3 } , A = ( A 1 , . . . , A d ) � � 2 + Q , L A , Q = − ∇ + iA (OP) � � � � Λ A , Q f = ν · ( ∇ + iA ) ψ ∂ D , (DN)  e i ϕ L A , Q e − i ϕ = L A ϕ , Q ϕ ,   A ϕ = A + ∇ ϕ, (GT)   Q ϕ = Q F = curl A and Q are gauge invariant and are uniquely determined by Λ A , V ( E ) , see [10] ( d ≥ 3) and [9] ( d = 2) ( A − ( A · ν ) ν ) | ∂ D is uniquely determined by Λ A , V ( E ) , see [6] Alexey Agaltsov Algorithm for acoustic tomography of moving fluid

Acoustic scattering: reduction to the IDN problem Use the second Green formula (Nachman ’88): � � G ω ( x , y ) − G 0 G 0 ω ( x , z )(Λ ω − Λ 0 ω ( x , y ) = ω )( z , w ) G ω ( w , y ) dy dw ∂ D ∂ D where G 0 ω , Λ 0 ω correspond to v = 0, ∇ ρ = 0, c = c 0 , α = 0. acoustic data from fluid point sources = = = = = = = = = = = = = = = = ⇒ tomography parameters problem second Green formula ∨ D-to-N map(s) Alexey Agaltsov Algorithm for acoustic tomography of moving fluid

Gauge fixing Question. Suppose that we know how to solve the IDN problem. How to complete the following diagram? acoustic data from fluid point sources = = = = = = = = = = = = = = = = = = ⇒ tomography parameters problem ∧ second Green formula ? ∨ inverse conjugacy class(es) > D-to-N map(s) Dirichlet-to-Neumann of Schr¨ odinger operators problem Alexey Agaltsov Algorithm for acoustic tomography of moving fluid

Gauge fixing � ω v � · ∇ − ω 2 c 2 + i c 2 − 2 i ωα L ω = − ∆ − 2 i 2 ∇ ln ρ (AC) c functions F and q ω are invariants of the conjugacy class: F = curl v c 2 , q ω = f 1 − ω 2 f 2 + i ω f 3 − 2 i ω 1 + ζ α 0 , � v � 1 2 ∆ ρ − 1 f 2 = 1 2 , f 1 = ρ c 2 + v c 2 v c 2 , f 3 = ∇ · − v c 2 · ∇ ln ρ c 2 The fluid parameters can be recovered as follows: conjugacy classes Λ ω 1 ,. . . , Λ ω N > F , q ω 1 , . . . , q ω N > of L ω 1 , . . . , L ω N � � � ∨ v , c , ρ , ζ , α 0 < F , f 1 , f 2 , f 3 , ζ , α 0 Alexey Agaltsov Algorithm for acoustic tomography of moving fluid

Gauge fixing: summary � ω v � · ∇ − ω 2 c 2 + i c 2 − 2 i ω 1 + ζ α 0 L ω = − ∆ − 2 i 2 ∇ ln ρ (AC) c , � � Λ ω ( ψ | ∂ D ) = ∂ψ ∂ D , L ω ψ = 0 . ∂ν ρ ≡ ρ 0 , α 0 ≡ 0 = ⇒ Λ ω at fixed ω determines v , c α 0 ≡ 0 = ⇒ Λ ω at 2 ω ’s determines v , c , ρ ζ � = 0 = ⇒ Λ ω at 3 ω ’s determines v , c , ρ , ζ , α 0 Explicit examples of non-uniquenes when ζ ≡ 0 [Agaltsov, Bull. Sci. Math. ’15]: uniqueness [Agaltsov, Novikov, JIIP ’15] : uniqueness and invisible fluids [Agaltsov, EJMA ’16]: algorithms Alexey Agaltsov Algorithm for acoustic tomography of moving fluid

Solving the acoustic tomography problem So far we have the following scheme with vertical arrows being explicit algorithms: acoustic data from fluid point sources = = = = = = = = = = = = = = = = = = = = = ⇒ tomography parameters > problem uniqueness and non-uniqueness ∧ [Agaltsov, Bull. Sci. Math. ’15] second [Agaltsov, Novikov, JIIP ’15] gauge fixing Green [Agaltsov, EJMA ’16] formula ∨ inverse conjugacy class(es) > D-to-N map(s) Dirichlet-to-Neumann of Schr¨ odinger operators problem Question. How to solve constructively the inverse Dirichlet-to-Neumann problem? Alexey Agaltsov Algorithm for acoustic tomography of moving fluid

Solving the acoustic tomography problem A common project with Moscow University Acoustical Physics group (Burov et al.) acoustic data from fluid point sources = = = = = = = = = = = = = = = = = = = = = ⇒ tomography parameters > problem ∧ [Agaltsov, Bull. Sci. Math. ’15] uniqueness second [Agaltsov, Novikov, JIIP ’15] gauge fixing Green [Agaltsov, EJMA ’16] formula ∨ inverse conjugacy class(es) > D-to-N map(s) Dirichlet-to-Neumann of Schr¨ odinger operators problem Alessandrini identity, > Lippman–Schwinger e s r g e v n [Agaltsov, J. Inv. Ill-Posed Problems ’15] equation n i r i [Agaltsov, Novikov, e t m t a e J. Math. Phys. ’14] c l s b o r p > scattering amplitude(s) Alexey Agaltsov Algorithm for acoustic tomography of moving fluid

The inverse scattering problem � d � ∂ � 2 + Q , L A , Q = − ∂ x j + iA j (OP) j = 1 A j , Q are smooth M n ( C ) -valued in D Set A , Q equal to zero outside of D E = { κ ∈ R d | κ 2 = E } : Consider functions ψ + ( · , k ) , k ∈ S d − 1 √  L A , Q ψ + ( x , k ) = E ψ + ( x , k ) , x ∈ R d ,   ψ + ( x , k ) = e ikx Id n + ψ + sc ( x , k ) ,   ψ + sc radiates at ∞ The scattering amplitude f A , Q on M E = S d − 1 E × S d − 1 E : √ √ � universal � � � � � ψ + k , | k | sc ( x , k ) = · f A , Q | x | x 1 + o ( 1 ) , | x | → ∞ . spherical wave Alexey Agaltsov Algorithm for acoustic tomography of moving fluid

Direct scattering problem � d � ∂ � 2 + Q L A , Q = − ∂ x j + iA j (OP) j = 1 Direct scattering problem Given L A , Q , find f A , Q . ψ + ( · , k ) satisfies the Lippmann-Schwinger equation: � � � ψ + ( x , k ) = e ikx Id n + G + ( x − y , k ) ψ + ( y , k ) dy , (LS) L A , Q − L 0 , 0 D � e i ξ x d ξ 1 G + ( x , k ) = − ( 2 π ) − d ξ 2 − k 2 − i 0“ ≃ ” E − L 0 , 0 R d The scattering amplitude f A , Q can be found from: � f A , Q ( k , l ) = ( 2 π ) − d R d e − ilx ( L A , Q − L 0 , 0 ) ψ + ( x , k ) dx (SA) Alexey Agaltsov Algorithm for acoustic tomography of moving fluid