Uncertainty principles for far field patterns and applications to - PowerPoint PPT Presentation

Uncertainty principles for far field patterns and applications to inverse source problems Roland Griesmaier roland.griesmaier@uni-wuerzburg.de (joint work with J. Sylvester) Paris, September 2017 Uncertainty principles Roland Griesmaier

Outline Source problems and far field patterns A regularized Picard criterion Uncertainty principles Corollaries of the uncertainty principles Numerical examples Uncertainty principles Roland Griesmaier

Source problems and far field patterns Uncertainty principles Roland Griesmaier

Far fields of compactly supported sources k > 0 : wave number ( = 2 π / wave length) 30 20 k 2 F ( ∈ L 2 0 ( R 2 ) ) : source term 10 U : time-harmonic radiated wave 0 − 10 − 20 − 30 − 30 − 20 − 10 0 10 20 30 Direct source problem: in R 2 and SRC − ∆ U − k 2 U = k 2 F 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) Uncertainty principles Roland Griesmaier

Far fields of compactly supported sources k = 1 : wave number ( = 2 π / wave length) 30 20 ( ∈ L 2 0 ( R 2 ) ) f : source term 10 u : time-harmonic radiated wave 0 − 10 − 20 − 30 − 30 − 20 − 10 0 10 20 30 Direct source problem: in R 2 and SRC − ∆ u − u = f Far field expansion: u ( x ) = C e i r x ) + O ( r − 3 / 2 ) , √ r α ( � r → ∞ , x = r � x , where � R 2 e − i θ · y f ( y ) d y = � θ ∈ S 1 α ( θ ) = f ( θ ) , Uncertainty principles Roland Griesmaier

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: θ ∈ S 1 , c ∈ R 2 , f ( · + c )( θ ) = e i c · θ � � f ( θ ) , i.e., if f radiates α ( θ ) , then f ( · + c ) radiates e i c · θ α ( θ ) Far field translation operator: T c : L 2 ( S 1 ) → L 2 ( S 1 ) , ( T c α )( θ ) := e i c · θ α ( θ ) Note that T ∗ c = T − c Uncertainty principles Roland Griesmaier

A regularized Picard criterion Uncertainty principles Roland Griesmaier

SVD of the restricted Fourier transform Consider restriction of F to sources supported in B R ( 0 ) : � F B R ( 0 ) f := � � F B R ( 0 ) : L 2 ( B R ( 0 )) → L 2 ( S 1 ) , f S 1 Singular value decomposition: � � √ ( F B R ( 0 ) f )( θ ) = � f ( x ) , i n J n ( | x | ) e i n ϕ x e i n θ 2 π s n ( R ) √ n s n ( R ) 2 π where n ( R ) = � s 2 B R ( 0 ) J 2 n ( x ) d x 25 Asymptotically: 20 �� 1 − ν 2 ν ≤ 1 s 2 ν R ( R ) lim R →∞ = 15 2 R 0 ν ≥ 1 10 i.e., � 5 � R 2 − n 2 2 n � R s 2 n ( R ) ∼ 0 − 10 0 10 0 n � R n R = 10 Uncertainty principles Roland Griesmaier

SVD of the restricted Fourier transform Consider restriction of F to sources supported in B R ( 0 ) : � F B R ( 0 ) f := � � F B R ( 0 ) : L 2 ( B R ( 0 )) → L 2 ( S 1 ) , f S 1 Singular value decomposition: � � √ ( F B R ( 0 ) f )( θ ) = � f ( x ) , i n J n ( | x | ) e i n ϕ x e i n θ 2 π s n ( R ) √ n s n ( R ) 2 π where n ( R ) = � s 2 B R ( 0 ) J 2 n ( x ) d x 250 Asymptotically: 200 �� 1 − ν 2 ν ≤ 1 s 2 ν R ( R ) = 150 lim R →∞ 2 R ν ≥ 1 0 100 i.e., � 50 � R 2 − n 2 2 n � R s 2 n ( R ) ∼ 0 − 100 0 100 0 n � R n R = 100 Uncertainty principles Roland Griesmaier

The Picard criterion Fourier expansion of the far field: α ( θ ) = � n α n e i n θ θ ∈ S 1 2 π , √ Radiated power of the far field: L 2 ( S 1 ) = � ∥ α ∥ 2 n | α n | 2 Picard criterion: � | α n | 2 1 α ∈ R ( F B R ( 0 ) ) ⇐ ⇒ n ( R ) < ∞ n s 2 2 π Minimal power source: � s n ( R ) 2 i n J n ( | x | ) e i n ϕ x , f ∗ 1 α n α ( x ) = x ∈ B R ( 0 ) √ n 2 π Input power required to radiate the far field: � | α n | 2 ∥ f ∗ 1 α ∥ 2 L 2 ( B R ( 0 )) = 2 π n s 2 n ( R ) Uncertainty principles Roland Griesmaier

A regularized Picard criterion Picard criterion: � | α n | 2 1 α ∈ R ( F B R ( 0 ) ) ⇐ ⇒ n ( R ) < ∞ n s 2 2 π ∥ f ∗ α ∥ 2 Input power required to radiate the far field: L 2 ( B R ( 0 )) ∥ α ∥ 2 Radiated power of the far field: L 2 ( S 1 ) Regularizing assumptions: Not every source/farfield combination is equally relevant! physical sources have limited power P > 0 a receiver has a power threshold p > 0 Define N ( R , P , p ) := sup n n ( R ) ≥ p 2 π s 2 P The space of non-evanescent far fields is given by: � � n = − N α n e in θ � � α ( θ ) = � N � α ∈ L 2 ( S 1 ) V NE := For a wide range of p and P : N � R Uncertainty principles Roland Griesmaier

Questions Uncertainty principles Roland Griesmaier

Far field splitting and data completion 30 30 30 30 20 20 20 20 10 10 10 10 0 0 0 0 − 10 − 10 − 10 − 10 − 20 − 20 − 20 − 20 − 30 − 30 − 30 − 30 − 30 − 20 − 10 0 10 20 30 − 30 − 20 − 10 0 10 20 30 − 30 − 20 − 10 0 10 20 30 − 30 − 20 − 10 0 10 20 30 Suppose Far field splitting: γ = γ 1 + · · · + γ m , γ j is radiated from B r j ( c j ) i.e., γ = T ∗ c 1 α 1 + · · · + T ∗ c m α m , α j is radiated from B r j ( 0 ) Can we stably recover the non-evanescent part of γ 1 , . . . , γ m ? Suppose we cannot measure γ on a subset Ω ⊂ S 1 , we measure Data completion: � γ = γ + β , β = − γ | Ω Can we stably recover the non-evanescent part of γ on Ω ? Uncertainty principles Roland Griesmaier

Uncertainty principles Uncertainty principles Roland Griesmaier

Far field translation Translation of the far field: The far field translation operator ( T c α )( θ ) := e i c · θ α ( θ ) T c : L 2 ( S 1 ) → L 2 ( S 1 ) , acts on the Fourier coe ffi cients { α n } of α as a convolution operator ( T c { α n } ) m = � � i n J n ( | c | ) e i n ϕ c � T c : ℓ 2 → ℓ 2 , n α m − n We have estimates 1 ∥ T c ∥ L p , L p = 1 and ∥ T c ∥ ℓ 1 , ℓ ∞ ≤ | c | 1 / 3 Uncertainty principles Roland Griesmaier

Uncertainty principles for far field translation 1 ∥ T c ∥ L p , L p = 1 and ∥ T c ∥ ℓ 1 , ℓ ∞ ≤ | c | 1 / 3 Let α , β ∈ L 2 ( S 1 ) and let c ∈ R 2 . Then Theorem: � ∥ α ∥ ℓ 0 ∥ β ∥ ℓ 0 | ⟨ T c α , β ⟩ | ≤ ∥ α ∥ 2 ∥ β ∥ 2 | c | 1 / 3 Proof: | ⟨ T c α , β ⟩ | ≤ ∥ T c α ∥ ℓ ∞ ∥ β ∥ ℓ 1 1 ≤ | c | 1 / 3 ∥ α ∥ ℓ 1 ∥ β ∥ ℓ 1 � � 1 ≤ ∥ α ∥ ℓ 0 ∥ α ∥ 2 ∥ β ∥ ℓ 0 ∥ β ∥ 2 | c | 1 / 3 � Uncertainty principles Roland Griesmaier

Uncertainty principles for far field translation Assuming that the supports of the individual source components are well-separated , we can improve the first estimate: 1 ∥ T c ∥ ℓ 1 [ − N , N ] , ℓ ∞ [ − M , M ] ≤ if | c | > 2 ( M + N + 1 ) | c | 1 / 2 Theorem: Suppose that α ∈ ℓ 2 ( − M , M ) , β ∈ ℓ 2 ( − N , N ) with M , N ≥ 1 and let c ∈ R 2 such that | c | > 2 ( M + N + 1 ) Then � ( 2 N + 1 )( 2 M + 1 ) | ⟨ T c α , β ⟩ | ≤ ∥ α ∥ 2 ∥ β ∥ 2 | c | 1 / 2 Uncertainty principles Roland Griesmaier

Uncertainty principle for data completion 1 ∥ T c ∥ L p , L p = 1 and ∥ T c ∥ ℓ 1 , ℓ ∞ ≤ | c | 1 / 3 Let α , β ∈ L 2 ( S 1 ) and let c ∈ R 2 . Then Theorem: � ∥ α ∥ ℓ 0 ∥ β ∥ L 0 | ⟨ T c α , β ⟩ | ≤ ∥ α ∥ 2 ∥ β ∥ 2 2 π Proof: | ⟨ T c α , β ⟩ | ≤ ∥ T c α ∥ L ∞ ∥ β ∥ L 1 ≤ ∥ α ∥ L ∞ ∥ β ∥ L 1 1 ≤ ∥ α ∥ ℓ 1 ∥ β ∥ L 1 √ 2 π � � 1 ≤ √ ∥ α ∥ ℓ 0 ∥ α ∥ 2 ∥ β ∥ L 0 ∥ β ∥ 2 2 π � Uncertainty principles Roland Griesmaier

ℓ 2 corollaries of the uncertainty principle Uncertainty principles Roland Griesmaier

Stability of far field splitting by least squares Suppose that γ 0 , γ 1 ∈ L 2 ( S 1 ) , c 1 , c 2 ∈ R 2 and N 1 , N 2 ∈ N such that Theorem: | c 1 − c 2 | > 2 ( N 1 + N 2 + 1 ) and ( 2 N 1 + 1 )( 2 N 2 + 1 ) < 1 | c 1 − c 2 | and let γ 0 LS = T ∗ c 1 α 0 1 + T ∗ c 2 α 0 α 0 i ∈ ℓ 2 ( − N i , N i ) 2 , γ 1 LS = T ∗ c 1 α 1 1 + T ∗ c 2 α 1 α 1 i ∈ ℓ 2 ( − N i , N i ) 2 , Then, for i = 1 , 2 � � − 1 1 − ( 2 N 1 + 1 )( 2 N 2 + 1 ) ∥ γ 1 − γ 0 ∥ 2 ∥ α 1 i − α 0 i ∥ 2 2 ≤ 2 | c 1 − c 2 | Uncertainty principles Roland Griesmaier

Stability of data completion by least squares Suppose that γ 0 , γ 1 ∈ L 2 ( S 1 ) , c ∈ R 2 , N ∈ N and Ω ⊂ S 1 such that Theorem: ( 2 N + 1 ) | Ω | < 1 2 π and let γ 0 LS = β 0 + T ∗ c α 0 , α 0 ∈ ℓ 2 ( − N , N ) and β 0 ∈ L 2 ( Ω ) γ 1 LS = β 1 + T ∗ c α 1 , α 1 ∈ ℓ 2 ( − N , N ) and β 1 ∈ L 2 ( Ω ) Then � � − 1 1 − ( 2 N + 1 ) | Ω | ∥ α 1 − α 0 ∥ 2 ∥ γ 1 − γ 0 ∥ 2 2 ≤ 2 2 π and � � − 1 1 − ( 2 N + 1 ) | Ω | ∥ β 1 − β 0 ∥ 2 ∥ γ 1 − γ 0 ∥ 2 2 ≤ 2 2 π Uncertainty principles Roland Griesmaier

ℓ 1 corollaries of the uncertainty principle Uncertainty principles Roland Griesmaier