Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works Increasing stability in the inverse source problem with attenuation and many frequencies Shuai Lu (Fudan University) Joint work with Gang Bao, Jin Cheng, Victor Isakov and William Rundell IAS Workshop on Inverse Problems, Imaging and Partial Differential Equations May 20–24, 2019 – The Hong Kong University of Science and Technology Shuai Lu (Fudan University) Increasing stability in the inverse source problem 1 / 45
Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works Outline Introduction 1 Related work Non-radiating source Helmholtz equation Main result Increasing stability with attenuation 2 Analytic functions Exact observability bounds Proof of the main theorem Numerical algorithm for source identification 3 Regularization method Numerical examples Ongoing works 4 Shuai Lu (Fudan University) Increasing stability in the inverse source problem 2 / 45
Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works Outline Introduction 1 Related work Non-radiating source Helmholtz equation Main result Increasing stability with attenuation 2 Analytic functions Exact observability bounds Proof of the main theorem Numerical algorithm for source identification 3 Regularization method Numerical examples Ongoing works 4 Shuai Lu (Fudan University) Increasing stability in the inverse source problem 3 / 45
Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works Related work Multifrequency (highfrequency) regimes Inverse medium problems Chen & Rokhlin (97); Chen (97a, 97b); Bao, Chen & Ma(00); Bao & Liu (03), Bao & Li (07); Bao & Triki (10); Nagayasu, Uhlmann & Wang (13); Bao & Triki (19)... Inverse obstacle/interface problems Coifman, Goldberg, Hrycak & Rokhlin (99); Bao, Hou & Li (07); Bao & Lin (10, 11); Sini & Th` anh (12); Borges & Greengard (15); Rondi & Sini (15)... Inverse source problems Bao, Lin & Triki (10, 11); Bao, Lu, Rundell & Xu; Cheng, Isakov & Lu (16); Li & Yuan (17); & Isakov & Lu (18)... Bao, Gang; Li, Peijun; Lin, Junshan; Triki, Faouzi Inverse scattering problems with multi-frequencies . Inverse Problems 31 (2015), no. 9, 093001, 21 pp. Shuai Lu (Fudan University) Increasing stability in the inverse source problem 4 / 45
Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works Related work Multifrequency (highfrequency) regimes Continuation problems Subbarayappa & Isakov (07, 10); Isakov & Kindermann (11); Isakov (14)... Schr¨ odinger problems Isakov (11); Isakov, Nagayasu, Uhlmann & Wang (14); Isakov & Wang (14); Isakov, Lai & Wang (16); Isakov, Lu & Xu (19)... Multifrequency electric impedance tomography Ammari & Triki (17); Cheng, Choulli & Lu (19)... Shuai Lu (Fudan University) Increasing stability in the inverse source problem 5 / 45
Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works Types of measurements Frequency sweep: Inverse medium problems; inverse obstacle/interface problems; inverse source problems; multifrequency electric impedance tomography Dirichlet-to-Neumann map at a fixed high wavenumber Inverse medium problems; Schr¨ odinger problems Cauchy data at a fixed high wavenumber Continuation problems Shuai Lu (Fudan University) Increasing stability in the inverse source problem 6 / 45
Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works Types of measurements Frequency sweep: Inverse medium problems; inverse obstacle/interface problems; inverse source problems; multifrequency electric impedance tomography Dirichlet-to-Neumann map at a fixed high wavenumber Inverse medium problems; Schr¨ odinger problems Cauchy data at a fixed high wavenumber Continuation problems Shuai Lu (Fudan University) Increasing stability in the inverse source problem 7 / 45
Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works Forward source problems Helmholtz equation with inner source terms Radiated field u ( x ; k ) satisfies Helmholtz equation and Sommerfeld radiation condition ∆ u ( x ; k )+ k 2 u ( x ; k ) = − f ( x ) , x ∈ R d � ∂ u ( x , k ) � d − 1 r x → ∞ r lim 2 − i ku ( x , k ) = 0 x ∂ r x where k � 0 is the wavenumber. We assume f ( x ) ∈ L 2 ( R d ) and compactly supported in a bounded open domain Ω . Outgoing radiated field u ( x ; k ) is generated by the source function f ( x ) and measured at ∂ Ω . Shuai Lu (Fudan University) Increasing stability in the inverse source problem 8 / 45
Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works Forward map ∂Ω Definition: Forward operator L k If we assume f † ( x ) ∈ L 2 ( Ω ) satisfying supp f † ⊂⊂ Ω , ( L k f † )( x ) = u ( x ; k ) | ∂ Ω = g ( x ; k ) � Ω f † ( y ) Φ ( x , y ; k ) d y , = x ∈ ∂ Ω 1 where Φ ( x , y ; k ) is the fundamental solution. L k : L 2 ( Ω ) → H 2 ( ∂ Ω ) . Shuai Lu (Fudan University) Increasing stability in the inverse source problem 9 / 45
Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works Non-radiating source Non-radiating source at a wavenumber k The outgoing radiated field u ( x ; k ) ∈ H 1 ( Ω ) corresponding to the non-zero source f ( x ) ∈ L 2 ( Ω ) satisfies u ( x ; k ) | ∂ Ω = 0 . We call f ( x ) the non-radiating source for the wavenumber k . For instance � � f ( x ) ∈ L 2 ( Ω ) | f ( x ) = − ∆ w ( x ) − k 2 w ( x ) , w ( x ) ∈ C ∞ ∅ � = 0 ( Ω ) . Bleistein and Cohen [J. Math. Phy. 1977]; Kim and Wolf [Optics Comm. 1986]; Marengo and Ziolkowski [PRL 1999]. Shuai Lu (Fudan University) Increasing stability in the inverse source problem 10 / 45
Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works Spherical Fourier transform Spherical Fourier basis Let Ω = B ρ in R 3 with ρ > 0 . Choose ψ nm ℓ ( x ) = ( N n ℓ ) − 1 / 2 j n ( ζ n ℓ r x ) Y m n ( ˆ x ) with ζ n ℓ = z nl ρ , ℓ = 1 , 2 ,... , z n ℓ is the ℓ -th positive zero of the spherical � d Bessel function j n ( r ) , N n ℓ = ρ 3 � 2 , r x = | x | and ˆ x = x dr j n ( ζ n ℓ ρ ) | x | . 2 Spherical Fourier transform For all f ∈ L 2 ( Ω ) , � a nm ℓ = Ω f ( y ) ψ nm ℓ ( y ) dy . Shuai Lu (Fudan University) Increasing stability in the inverse source problem 11 / 45
Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works Spherical Fourier transform Notice the fact that ∞ ∞ n ∑ ∑ ∑ f ( x ) = a nm ℓ ψ nm ℓ ( x ) , m = − n n = 0 ℓ = 1 e i k | x − y | ∞ n h ( 1 ) ∑ ∑ n ( kr x ) Y m x ) j n ( kr y ) Y m Φ ( x , y ; k ) = 4 π | x − y | = i k n ( ˆ n ( ˆ y ) . m = − n n = 0 by choosing k = ζ n ℓ , we obtain ∞ n � � � i kh ( 1 ) ∑ ∑ n ( kr x ) Y m Ω ψ n ′ m ′ ℓ ′ ( y ) j n ( kr y ) Y m ( L k ψ n ′ m ′ ℓ ′ )( x ) = n ( ˆ x ) n ( ˆ y ) d y m = − n n = 0 ∞ n � 1 2 δ n = n ′ , m = m ′ δ k = ζ n ′ ℓ ′ i kh ( 1 ) n ( kr x ) Y m N S ∑ ∑ � = n ( ˆ x ) n ′ ℓ ′ m = − n n = 0 � 1 = i kh ( 1 ) n ′ ( kr x ) Y m ′ 2 δ k = ζ n ′ ℓ ′ , N S � x ∈ ∂ Ω . n ′ ( ˆ x ) n ′ ℓ ′ Shuai Lu (Fudan University) Increasing stability in the inverse source problem 12 / 45
Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works Singular value decomposition (SVD) SVD of the forward operator L k Assume Ω = B ρ in R 3 , the singular value system of the forward 1 operator L k : L 2 ( Ω ) → H 2 ( ∂ Ω ) is { σ nm ℓ , ψ nm ℓ , φ nm ℓ } when k = ζ n ℓ . Here ψ nm ℓ ( x ) = ( N n ℓ ) − 1 / 2 j n ( ζ n ℓ r x ) Y m n ( ˆ x ) with x ∈ Ω . Furthermore, for all ℓ = 1 , 2 , ··· , φ nm ℓ ( x ) = Y m n ( ˆ x ) with x = ρ ˆ x , and for all m = 0 , ± 1 , ··· , ± n , n = 0 , 1 , ··· , the singular values σ nm ℓ are � � � i kh ( 1 ) 1 2 δ k = ζ n ℓ � 0 . σ nm ℓ ( k ) = n ( k ρ ) � ( N n ℓ ) � � σ nml 0 5 10 15 20 Shuai Lu (Fudan University) Increasing stability in the inverse source problem 13 / 45
Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works Inverse source problems Ideal inverse source problems Identify the source function f ( x ) from the measured data (Dirichlet, Neumann, Cauchy) on the boundary ∂ Ω for wavenumbers k ∈ ( 0 , + ∞ ) . Real situation Identify the source function f ( x ) from the measured data (Dirichlet, Neumann, Cauchy) on the boundary ∂ Ω for wavenumbers k ∈ ( 0 , K ) , K >> 0 . Hoenders and Ferwerda [PRL 2001]; Devaney et. al [SIAP 2007]; Bao, Lin and Triki [JDE 2010]. Shuai Lu (Fudan University) Increasing stability in the inverse source problem 14 / 45
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