Uncertainty principles for far field patterns and applications to inverse source problems

Transcription

1 for far field patterns and applications to inverse source problems (joint work with J. Sylvester) Paris, September 17

2 Outline Source problems and far field patterns AregularizedPicardcriterion Corollaries of the uncertainty principles Numerical examples

3 Source problems and far field patterns

4 Far fields of compactly supported sources 3 k > : wave number (= π/wave length) 1 k F : source term ( L ( R )) U : time-harmonic radiated wave Direct source problem: U k U = k F in R and SRC Rescaling: Rewriting u(x) = U(kx), f (x) =F (kx) we can w.l.o.g. set k = 1 (i.e., distances are measured in wavelengths)

5 Far fields of compactly supported sources 3 k = 1 : wave number (= π/wave length) 1 f : source term ( L ( R )) u : time-harmonic radiated wave Direct source problem: u u = f in R and SRC Far field expansion: u(x) = C eir r α( x) +O(r 3/ ), r, x = r x, where α(θ) = R e iθ y f (y) dy = f (θ), θ S 1

6 Facts about far fields The far field radiated by a source f is its restricted Fourier transform: α = f S 1 Translations and Fourier transforms: f ( + c)(θ) = e ic θ f (θ), θ S 1, c R, i.e., if f radiates α(θ), thenf ( + c) radiates e ic θ α(θ) Far field translation operator: T c : L (S 1 ) L (S 1 ), (T c α)(θ) := e ic θ α(θ) Note that T c = T c

7 A regularized Picard criterion

8 SVD of the restricted Fourier transform Consider restriction of F to sources supported in B R (): F BR () : L (B R ()) L (S 1 ), F BR ()f := f S 1 Singular value decomposition: (F BR() f )(θ) = n πsn(r) f (x), in J n( x )e inϕx s n(r) where sn (R) = B R () J n (x) dx e inθ π Asymptotically: 5 i.e., { sνr lim (R) 1 ν ν 1 R = R ν 1 { sn (R) 5 R n n R n R n R =

9 SVD of the restricted Fourier transform Consider restriction of F to sources supported in B R (): F BR () : L (B R ()) L (S 1 ), F BR ()f := f S 1 Singular value decomposition: (F BR() f )(θ) = n πsn(r) f (x), in J n( x )e inϕx s n(r) where sn (R) = B R () J n (x) dx e inθ π Asymptotically: 5 i.e., { sνr lim (R) 1 ν ν 1 R = R ν 1 { sn (R) 5 R n n R n R n R =

10 The Picard criterion Fourier expansion of the far field: α(θ) = einθ n αn, θ S 1 π Radiated power of the far field: α L (S 1 ) = n αn Picard criterion: α R(F BR() ) 1 α n π n sn (R) < Minimal power source: fα (x) = 1 α n π n s n(r) in J n( x )e inϕx, x B R () Input power required to radiate the far field: fα L (B R ()) = 1 α n π n sn (R)

11 AregularizedPicardcriterion Picard criterion: α R(F BR() ) 1 α n π n sn (R) < Input power required to radiate the far field: f α L (B R ()) Radiated power of the far field: α L (S 1 ) Regularizing assumptions: Not every source/farfield combination is equally relevant! physical sources have limited power P > areceiverhasapowerthresholdp > Define N(R, P, p) := sup πs n (R) p P n The space of non-evanescent far fields is given by: { V NE := α L (S 1 ) α(θ) = } N n= N αneinθ For a wide range of p and P: N R

12 Questions

13 Far field splitting and data completion Far field splitting: Suppose γ = γ γ m, γ j is radiated from B rj (c j ) i.e., γ = Tc 1 α Tc m α m, α j is radiated from B rj () Can we stably recover the non-evanescent part of γ 1,...,γ m? Data completion: Suppose we cannot measure γ on a subset Ω S 1,wemeasure γ = γ + β, β = γ Ω Can we stably recover the non-evanescent part of γ on Ω?

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15 Far field translation Translation of the far field: The far field translation operator T c : L (S 1 ) L (S 1 ), (T c α)(θ) := e ic θ α(θ) acts on the Fourier coefficients {α n} of α as a convolution operator T c : l l, (T c {α n}) m = n α ( m n i n J ) n( c )e inϕc We have estimates T c L p,l p = 1 and Tc l 1,l 1 c 1/3

16 for far field translation T c L p,l p = 1 and Tc l 1,l 1 c 1/3 Theorem: Let α, β L (S 1 ) and let c R.Then T cα, β α l β l c 1/3 α β Proof: T cα, β T cα l β l 1 1 c 1/3 α l 1 β l 1 1 c 1/3 α l α β l β

17 for far field translation Assuming that the supports of the individual source components are well-separated, we can improve the first estimate: T c l 1 [ N,N],l [ M,M] 1 c 1/ if c > (M + N + 1) Theorem: and let Then Suppose that α l ( M, M), β l ( N, N) with M, N 1 c R such that c > (M + N + 1) T cα, β (N + 1)(M + 1) c 1/ α β

18 Uncertainty principle for data completion T c L p,l p = 1 and Tc l 1,l 1 c 1/3 Theorem: Let α, β L (S 1 ) and let c R.Then T cα, β α l β L α β π Proof: T cα, β T c α L β L 1 α L β L 1 1 π α l 1 β L 1 1 π α l α β L β

19 l corollaries of the uncertainty principle

20 Stability of far field splitting by least squares Theorem: Suppose that γ,γ 1 L (S 1 ), c 1, c R and N 1, N N such that c 1 c > (N 1 + N + 1) and (N 1 + 1)(N + 1) c 1 c < 1 and let Then, for i = 1, γ LS = T c 1 α 1 + T c α, α i l ( N i, N i ) γ 1 LS = T c 1 α T c α 1, α1 i l ( N i, N i ) ( α 1 i α i 1 (N ) 1 + 1)(N + 1) 1 γ 1 γ c 1 c

21 Stability of data completion by least squares Theorem: Suppose that γ,γ 1 L (S 1 ), c R, N N and Ω S 1 such that (N + 1) Ω π < 1 and let Then and γ LS = β + T c α, α l ( N, N) and β L (Ω) γ 1 LS = β 1 + T c α1, α 1 l ( N, N) and β 1 L (Ω) α 1 α ( 1 β 1 β ( 1 ) (N + 1) Ω 1 γ 1 γ π ) (N + 1) Ω 1 γ 1 γ π

22 l 1 corollaries of the uncertainty principle

23 Stability of far field splitting by basis pursuit Theorem: Suppose that γ,α 1,α L (S 1 ) and c 1, c R such that 4 α i l c 1 c 1/3 < 1 for i = 1, and γ T c 1 α 1 T c α δ for some δ If δ and γ L (S 1 ) with and δ δ + γ γ (α 1,α )=argmin α 1 l 1 + α l 1 then, for i = 1, s.t. γ T c 1 α 1 T c α δ, α 1,α L (S 1 ), α i α i ( 1 4 α i l c 1 c 1/3 ) 14δ

24 Stability of data completion by basis pursuit Theorem: that Suppose that γ,α L (S 1 ), Ω S 1, β L (Ω) and c R such π α l Ω < 1 and γ Tc α β δ for some δ If δ and γ L (S 1 ) with δ δ + γ γ and α = argmin α l 1 s.t. γ β Tc α δ, α L (S 1 ), β L (Ω) then ( ) α α 1 α l Ω 14δ π and ( ) β β 1 α l Ω 14δ π

25 Stability of data completion by basis pursuit Corollary: that and Suppose that γ,α L (S 1 ), Ω S 1, β L (Ω) and c R such 4 π 1 τ α l < 1 and If δ and γ L (S 1 ) with and 4 π τ Ω < 1 for some τ> γ T c α β δ for some δ δ δ + γ γ (α, β) =argmin 1 τ α l 1 + τ β L 1 (S 1 ) s.t. γ T c α β δ, α,β L (S 1 ) then and ( ) α α δ π τ α l β β ( 1 4 π τ Ω ) 14δ

26 Numerical examples

27 The setup 3 Geometry and a priori information 6 Exact farfield π/ π 3π/ π Wavenumber: k = 1

28 Data completion using the l approach Observed far field data: γ = γ S 1 \Ω, Galerkin condition: γ = β + 3 i=1 T c i α i, β = γ Ω β + T c 1 α T c 3 α 3,φ = γ, φ for all φ L (Ω) T c 1 l ( N 1, N 1 ) T c 3 l ( N 3, N 3 ) 6 Observed farfield 6 Reconstructed missing data 6 Absolute error π/ π 3π/ π π/ π 3π/ π π/ π 3π/ π Condition number:

29 The setup 3 Geometry and a priori information 6 Exact farfield π/ π 3π/ π Wavenumber: k = 1

30 Data completion using the l 1 approach Observed far field data: γ = γ S 1 \Ω, γ = β + 3 i=1 T c i α i, β = γ Ω Tikhonov functional: 3 3 Ψ µ(α 1,α,α 3 ) = γ (I P Ω )( Tc i α i ) l + µ α i l 1 i=1 i=1 with [α 1,α,α 3 ] l l l 6 Observed farfield 6 Reconstructed missing data 6 Absolute error π/ π 3π/ π π/ π 3π/ π π/ π 3π/ π Apply (fast) iterated soft shrinkage ((F)ISTA) to minimize Ψ µ;hereµ = 1 3

31 Thanks for listening!