A non-iterative sampling approach for EIT using noise subspace projection Cédric Bellis 1 Andrei Constantinescu 2 Armin Lechleiter 3 Thomas Coquet 4 Thomas Jaravel 4 1 Dept Applied Physics & Applied Mathematics · Columbia University · USA 2 Solid Mechanics Laboratory (UMR CNRS 7649) · École Polytechnique · France 3 Center for Industrial Mathematics · University of Bremen · Germany 4 Dept of Mechanics · École Polytechnique · France PICOF’12 · Wednesday, April 4 th
Context • Let Ω ⊂ R d , d = 2 , 3 bounded connected background domain, ∂ Ω Lipschitz • Finite union I = � K j = 1 Ω j of disjoint inclusions Ω j ⊂ Ω and Ω \I connected. � 1 if ξ ∈ Ω \ I γ ( ξ ) = γ j ( ξ ) if ξ ∈ Ω j , j = 1 , . . . , K . with γ j ∈ L ∞ (Ω , R ) and 0 < c ≤ γ j < 1 (could be changed into 1 < γ j ≤ C) • Mean-free current density f ∈ L 2 ( ∂ Ω) generates potential u solution of ∇ · ( γ ∇ u ) = 0 in Ω ( γ ∇ u ) · n = f on ∂ Ω , → Determine γ from knowledge of NtD operator Λ γ : f �→ u | ∂ Ω • Geometric Prior: Inclusions embedded in known background medium C. Bellis & al. Noise Subspace Projection for EIT PICOF’12 · 4/4/12 2 / 18
Context • Let Ω ⊂ R d , d = 2 , 3 bounded connected background domain, ∂ Ω Lipschitz • Finite union I = � K j = 1 Ω j of disjoint inclusions Ω j ⊂ Ω and Ω \I connected. � 1 if ξ ∈ Ω \ I γ ( ξ ) = γ j ( ξ ) if ξ ∈ Ω j , j = 1 , . . . , K . with γ j ∈ L ∞ (Ω , R ) and 0 < c ≤ γ j < 1 (could be changed into 1 < γ j ≤ C) • Mean-free current density f ∈ L 2 ( ∂ Ω) generates potential u solution of ∇ · ( γ ∇ u ) = 0 in Ω ( γ ∇ u ) · n = f on ∂ Ω , → Determine γ from knowledge of NtD operator Λ γ : f �→ u | ∂ Ω • Geometric Prior: Inclusions embedded in known background medium C. Bellis & al. Noise Subspace Projection for EIT PICOF’12 · 4/4/12 2 / 18
Motivations • The map γ �→ Λ γ is non-linear • Severe ill-posedness: Given γ, γ ′ ∈ H 2 + s (Ω) with s > d / 2 (dimension d = 2 , 3 ) ◦ If γ, γ ′ piecewise-constant and Ω j , Ω ′ j known Lipschitz domains � � γ − γ ′ � � � � Λ − 1 − Λ − 1 � � L ∞ (Ω) ≤ β γ γ ′ H 1 / 2 ( ∂ Ω) → H − 1 / 2 ( ∂ Ω) G . Alessandrini , S . Vessella , Adv . Appl . Math ., 2005 . ◦ Standard logarithmic stability for generic configurations � � γ − γ ′ � � � � � � − α � log � Λ − 1 − Λ − 1 � � L ∞ (Ω) ≤ β γ γ ′ H 1 / 2 ( ∂ Ω) → H − 1 / 2 ( ∂ Ω) G . Alessandrini , Appl . Anal ., 1988 . → Qualitative sampling methods: computationally-effective approaches Factorization method, point source method, topological sensitivity, MUSIC, . . . C. Bellis & al. Noise Subspace Projection for EIT PICOF’12 · 4/4/12 3 / 18
Motivations • The map γ �→ Λ γ is non-linear • Severe ill-posedness: Given γ, γ ′ ∈ H 2 + s (Ω) with s > d / 2 (dimension d = 2 , 3 ) ◦ If γ, γ ′ piecewise-constant and Ω j , Ω ′ j known Lipschitz domains � � γ − γ ′ � � � � Λ − 1 − Λ − 1 � � L ∞ (Ω) ≤ β γ γ ′ H 1 / 2 ( ∂ Ω) → H − 1 / 2 ( ∂ Ω) G . Alessandrini , S . Vessella , Adv . Appl . Math ., 2005 . ◦ Standard logarithmic stability for generic configurations � � γ − γ ′ � � � � � � − α � log � Λ − 1 − Λ − 1 � � L ∞ (Ω) ≤ β γ γ ′ H 1 / 2 ( ∂ Ω) → H − 1 / 2 ( ∂ Ω) G . Alessandrini , Appl . Anal ., 1988 . → Qualitative sampling methods: computationally-effective approaches Factorization method, point source method, topological sensitivity, MUSIC, . . . C. Bellis & al. Noise Subspace Projection for EIT PICOF’12 · 4/4/12 3 / 18
Outline 1 Sampling approaches and noise subspace projection 2 Finite dimensional approximation of NtD operators 3 Numerical implementations and examples C. Bellis & al. Noise Subspace Projection for EIT PICOF’12 · 4/4/12 4 / 18
Identification framework � � � • Let L 2 ϕ ∈ L 2 ( ∂ Ω) : ⋄ ( ∂ Ω) := ∂ Ω ϕ d S = 0 → NtD map Λ : L 2 ⋄ ( ∂ Ω) → L 2 ⋄ ( ∂ Ω) s.t. Λ f = u | ∂ Ω � � ∀ ϕ ∈ H 1 with u ∈ H 1 ⋄ (Ω) solution of: γ ∇ u · ∇ ϕ d V = f ϕ d S , ⋄ (Ω) Ω ∂ Ω • Reference homogeneous counterpart, i.e. γ = 1 in Ω : NtD map Λ 1 f = u 1 | ∂ Ω • NtD operators Λ and Λ 1 are compact on L 2 ⋄ ( ∂ Ω) → Measurement operator Π = Λ − Λ 1 • Extract the informations synthesized in Π → Probe range R (Π) using a fundamental solution g z g z has singular behavior at chosen sampling point z ∈ Ω • Π is self-adjoint and compact ⇒ ∃{ λ j , ψ j } with λ j > 0 and ψ j ∈ L 2 ⋄ ( ∂ Ω) s.t. ∞ � Π f = λ j ( f , ψ j ) L 2 ( ∂ Ω) ψ j j = 1 C. Bellis & al. Noise Subspace Projection for EIT PICOF’12 · 4/4/12 5 / 18
Identification framework � � � • Let L 2 ϕ ∈ L 2 ( ∂ Ω) : ⋄ ( ∂ Ω) := ∂ Ω ϕ d S = 0 → NtD map Λ : L 2 ⋄ ( ∂ Ω) → L 2 ⋄ ( ∂ Ω) s.t. Λ f = u | ∂ Ω � � ∀ ϕ ∈ H 1 with u ∈ H 1 ⋄ (Ω) solution of: γ ∇ u · ∇ ϕ d V = f ϕ d S , ⋄ (Ω) Ω ∂ Ω • Reference homogeneous counterpart, i.e. γ = 1 in Ω : NtD map Λ 1 f = u 1 | ∂ Ω • NtD operators Λ and Λ 1 are compact on L 2 ⋄ ( ∂ Ω) → Measurement operator Π = Λ − Λ 1 • Extract the informations synthesized in Π → Probe range R (Π) using a fundamental solution g z g z has singular behavior at chosen sampling point z ∈ Ω • Π is self-adjoint and compact ⇒ ∃{ λ j , ψ j } with λ j > 0 and ψ j ∈ L 2 ⋄ ( ∂ Ω) s.t. ∞ � Π f = λ j ( f , ψ j ) L 2 ( ∂ Ω) ψ j j = 1 C. Bellis & al. Noise Subspace Projection for EIT PICOF’12 · 4/4/12 5 / 18
Existing sampling approach 1/2 • Low-dimensional parameterization: K inclusions of small volume j = z j + ε ˆ → Ω j ≡ Ω ε Ω j • Measurement operator Π ≡ Π ε = Λ ε − Λ 1 converges to finite-rank operator ˆ Π � � � Π ε − ε d ˆ ⋄ ( ∂ Ω) = O ( ε d + 1 � 2 ) . Π L 2 ⋄ ( ∂ Ω) → L 2 with range: R (ˆ Π) = span { e k · ∇ z N ( · , z j ) , k = 1 , . . . , d ; j = 1 , . . . , K } using mean-free Green’s function N ( · , z ) of − ∆ , i.e. − ∆ ξ N ( ξ , z ) = δ ( ξ − z ) in Ω ξ N ( ξ , z ) · n ( ξ ) = −| ∂ Ω | − 1 on ∂ Ω , ∇ • Theorem: For any d ∈ S d − 1 and z ∈ Ω , let g z , d = d · ∇ z N ( · , z ) | ∂ Ω then z ∈ { z 1 , . . . , z K } if and only if g z , d ∈ R (ˆ Π) → MUSIC algorithm M . Bruhl , M . Hanke , M . Vogelius , Numer . Math ., 2003 C. Bellis & al. Noise Subspace Projection for EIT PICOF’12 · 4/4/12 6 / 18
Existing sampling approach 1/2 • Low-dimensional parameterization: K inclusions of small volume j = z j + ε ˆ → Ω j ≡ Ω ε Ω j • Measurement operator Π ≡ Π ε = Λ ε − Λ 1 converges to finite-rank operator ˆ Π � � � Π ε − ε d ˆ ⋄ ( ∂ Ω) = O ( ε d + 1 � 2 ) . Π L 2 ⋄ ( ∂ Ω) → L 2 with range: R (ˆ Π) = span { e k · ∇ z N ( · , z j ) , k = 1 , . . . , d ; j = 1 , . . . , K } using mean-free Green’s function N ( · , z ) of − ∆ , i.e. − ∆ ξ N ( ξ , z ) = δ ( ξ − z ) in Ω ξ N ( ξ , z ) · n ( ξ ) = −| ∂ Ω | − 1 on ∂ Ω , ∇ • Theorem: For any d ∈ S d − 1 and z ∈ Ω , let g z , d = d · ∇ z N ( · , z ) | ∂ Ω then z ∈ { z 1 , . . . , z K } if and only if g z , d ∈ R (ˆ Π) → MUSIC algorithm M . Bruhl , M . Hanke , M . Vogelius , Numer . Math ., 2003 C. Bellis & al. Noise Subspace Projection for EIT PICOF’12 · 4/4/12 6 / 18
Existing sampling approach 2/2 • Extended inclusions: Separate influence of unknown inclusion from that of known background Using that Π f = (Λ − Λ 1 ) f = ( u − u 1 ) | ∂ Ω with � � ∀ ϕ ∈ H 1 γ ∇ ( u − u 1 ) · ∇ ϕ d V = ( 1 − γ ) ∇ u 1 · ∇ ϕ d V ⋄ (Ω) , Ω I → Factorization of the type: Π = A ∗ T A • Characterization of R ( A ∗ ) = R (Π 1 / 2 ) Functions harmonic in Ω \I with homog. Neumann B.C. on ∂ Ω • Theorem: For any d ∈ S d − 1 and z ∈ Ω , let g z , d = d · ∇ z N ( · , z ) | ∂ Ω then 1 / 2 ) z ∈ I if and only if g z , d ∈ R (Π • Picard criterion: ∞ | ( g z , d , ψ j ) L 2 ( ∂ Ω) | 2 � z ∈ I if and only if the series converges λ j j = 1 Bruhl , Hanke , Kirsch , . . . C. Bellis & al. Noise Subspace Projection for EIT PICOF’12 · 4/4/12 7 / 18
Existing sampling approach 2/2 • Extended inclusions: Separate influence of unknown inclusion from that of known background Using that Π f = (Λ − Λ 1 ) f = ( u − u 1 ) | ∂ Ω with � � ∀ ϕ ∈ H 1 γ ∇ ( u − u 1 ) · ∇ ϕ d V = ( 1 − γ ) ∇ u 1 · ∇ ϕ d V ⋄ (Ω) , Ω I → Factorization of the type: Π = A ∗ T A • Characterization of R ( A ∗ ) = R (Π 1 / 2 ) Functions harmonic in Ω \I with homog. Neumann B.C. on ∂ Ω • Theorem: For any d ∈ S d − 1 and z ∈ Ω , let g z , d = d · ∇ z N ( · , z ) | ∂ Ω then 1 / 2 ) z ∈ I if and only if g z , d ∈ R (Π • Picard criterion: ∞ | ( g z , d , ψ j ) L 2 ( ∂ Ω) | 2 � z ∈ I if and only if the series converges λ j j = 1 Bruhl , Hanke , Kirsch , . . . C. Bellis & al. Noise Subspace Projection for EIT PICOF’12 · 4/4/12 7 / 18
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