Recovering point sources in unknown environment with differential data

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1 Recovering point sources in unknown environment with differential data Yimin Zhong University of Texas at Austin April 29, 2013

2 Introduction Introduction The field u satisfies u(x) + k 2 (1 + n(x))u(x) = f(x) in Ω (1) u = 0 on Ω (2) k is the given frequency, n(x) is refractive index, supported in compact domain Ω and f(x) is source function as superposition of point sources. f 1 = λ 1δ(x x 1) f 3 = λ 3δ(x x 3) Ω source function f(x) = m j=1 λjδ(x xj) u = 0 on Ω differential data: u on Ω n f 2 = λ 2δ(x x 2) Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

3 Forward source problem Introduction Problem (Forward source problem) We assume source f(x) is the superposition of some point sources, find a function u H 1 (Ω) such that B(u, v) = f, v (3) where sesquilinear form B : H 1 (Ω) H 1 (Ω) C is defined as: B(u, v) = u, v L 2 (Ω) k 2 (1 + n)u, v L 2 (Ω) Λγu, γv L 2 ( Ω) (4) where γ : H 1 (Ω) H 1/2 ( Ω) is the trace operator, and Λ : H 1/2 ( Ω) H 1/2 ( Ω) is the Dirichlet to Neumann operator. Here we assume 0 is not an eigenvalue of operator + k 2 (1 + n(x)) and we can obtain well-posedness from the forward problem by utilizing Resiz-Fredholm theory. Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

4 Inverse source problem Introduction Problem (Inverse source problem) Let s assume the unknown source function f is the superposition of several point sources and u be the solution to the forward source problem for some given frequency k K, provided the Cauchy data on the boundary Ω, finding the source f. Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

5 Inverse source problem Introduction Problem (Inverse source problem) Let s assume the unknown source function f is the superposition of several point sources and u be the solution to the forward source problem for some given frequency k K, provided the Cauchy data on the boundary Ω, finding the source f. Multi-frequency Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

6 Inverse source problem Introduction Problem (Inverse source problem) Let s assume the unknown source function f is the superposition of several point sources and u be the solution to the forward source problem for some given frequency k K, provided the Cauchy data on the boundary Ω, finding the source f. Multi-frequency Eller M and Valdivia N, IOP, 2009 Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

7 Inverse source problem Introduction Problem (Inverse source problem) Let s assume the unknown source function f is the superposition of several point sources and u be the solution to the forward source problem for some given frequency k K, provided the Cauchy data on the boundary Ω, finding the source f. Multi-frequency Eller M and Valdivia N, IOP, 2009 Bao G, Lin J and Triki F, C.R.Acad.Sci, 2011 Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

8 Inverse source problem Introduction Problem (Inverse source problem) Let s assume the unknown source function f is the superposition of several point sources and u be the solution to the forward source problem for some given frequency k K, provided the Cauchy data on the boundary Ω, finding the source f. Multi-frequency Eller M and Valdivia N, IOP, 2009 Bao G, Lin J and Triki F, C.R.Acad.Sci, 2011 Unknown heterogeneous media Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

9 Inverse source problem Introduction Problem (Inverse source problem) Let s assume the unknown source function f is the superposition of several point sources and u be the solution to the forward source problem for some given frequency k K, provided the Cauchy data on the boundary Ω, finding the source f. Multi-frequency Eller M and Valdivia N, IOP, 2009 Bao G, Lin J and Triki F, C.R.Acad.Sci, 2011 Unknown heterogeneous media Stenfanov P and Uhlmann G, IOP, 2009 Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

10 Inverse source problem Introduction Problem (Inverse source problem) Let s assume the unknown source function f is the superposition of several point sources and u be the solution to the forward source problem for some given frequency k K, provided the Cauchy data on the boundary Ω, finding the source f. Multi-frequency Eller M and Valdivia N, IOP, 2009 Bao G, Lin J and Triki F, C.R.Acad.Sci, 2011 Unknown heterogeneous media Stenfanov P and Uhlmann G, IOP, 2009 Point sources Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

11 Inverse source problem Introduction Problem (Inverse source problem) Let s assume the unknown source function f is the superposition of several point sources and u be the solution to the forward source problem for some given frequency k K, provided the Cauchy data on the boundary Ω, finding the source f. Multi-frequency Eller M and Valdivia N, IOP, 2009 Bao G, Lin J and Triki F, C.R.Acad.Sci, 2011 Unknown heterogeneous media Stenfanov P and Uhlmann G, IOP, 2009 Point sources Alves C, Kress R and Serranho P, IOP, 2009 Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

12 Scheme Our approach Our Objective To recover both the refractive index n(x) and source function f(x) using differential data To study stability of recovering f(x) with respect to perturbation of n(x) We are going to formulate our scheme as two steps: Reconstruct refractive index n(x) by adding test sources and measuring differential data Reconstruct source function f(x) from the reconstructed ˆn(x) Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

13 Recovering refractive index Our approach The solution field u satisfies: u + k 2 (1 + n)u = f (5) If we put some test sources g in our domain Ω R n, then the modified field ũ should satisfy: Subtract (5) from (6), taking ω = ũ u: ũ + k 2 (1 + n)ũ = f + g (6) ω + k 2 (1 + n)ω = g (7) with data as Λγω = Λγũ Λγu on the boundary of the domain Ω. Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

14 Recovering refractive index Our approach If we choose g carefully, we can modify (7) as: ˆω + k 2 (1 + n)ˆω = 0 in Ω (8) ˆω = h on Ω (9) where h is the modified boundary condition corresponding to the test source function, multi-frequency k K, where K is an admissible set, and measurements are ˆω = Λγ ˆω n on the boundary. Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

15 Recovering refractive index Our approach Theorem (Uniqueness) If Dirichlet to Neumann map Λ : h(x) h(x) associated with refractive index n(x) is n provided for each h(x) C 1 (R n ), then n(x) can be recovered uniquely. Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

16 Recovering refractive index Our approach Theorem (Uniqueness) If Dirichlet to Neumann map Λ : h(x) h(x) associated with refractive index n(x) is n provided for each h(x) C 1 (R n ), then n(x) can be recovered uniquely. Unfortunately this problem is extremely ill-posed. Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

17 Recovering refractive index Our approach Theorem (Uniqueness) If Dirichlet to Neumann map Λ : h(x) h(x) associated with refractive index n(x) is n provided for each h(x) C 1 (R n ), then n(x) can be recovered uniquely. Unfortunately this problem is extremely ill-posed. In practice we use finite number of test source functions {g i} I i=1. Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

18 Recovering refractive index Our approach Theorem (Uniqueness) If Dirichlet to Neumann map Λ : h(x) h(x) associated with refractive index n(x) is n provided for each h(x) C 1 (R n ), then n(x) can be recovered uniquely. Unfortunately this problem is extremely ill-posed. In practice we use finite number of test source functions {g i} I i=1. Qustion Can we get a good recovery of n(x) by using a few test source functions only? Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

19 Stability Our approach Theorem (Stability estimation,nagayasu-uhlmann-wang,iop,2013) Suppose n 1(x), n 2(x) are two refractive indices associated with Dirichlet to Neumann maps Λ 1, Λ 2 respectively. And assume that for s > n 2 + 1, n l(x) H s M, supp(n 1 n 2) Ω R n,there exists a constant C(n, s, Ω), C 1(n, s, Ω), such that if k 2 1 C 1 M, Λ1 Λ2 e 1, n 1 n 2 H s C k 2 exp(ck2 ) Λ 1 Λ 2 + C(k 2 log( Λ 1 Λ 2 )) n 2s We can see there is an exponential instability here. And we can see there is an increasing stability by increasing frequencies k. Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

20 Algorithm Our approach We formulate the approach as the following minimization problem: where, J(n) = k K d j Ω min J(n) n Λγ ˆω j Λh j 2 dσ + α n 2 dx Ω where Λh j is the differential data for test source g j = e ikx d j, α is regularization parameter. Algorithm 1 pseudocode for recovering n(x) 1: select initial n = n 0(x) 2: while J(n) ɛ do 3: update n by using Newton s method and linear search 4: for j = 1 to I do 5: solve ˆω j of Helmholtz equation (8) associated with the updated n for each d j 6: end for 7: compute J(n) 8: end while Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

21 Numerical results Recovering refractive index: Smooth Ex.1 Our approach For n(x) is chosen as: n(x) = { cos(2π x /R) if x if x > 0.4 (10) Figure : Ex.1 Exact index Figure : Ex.1 Reconstruction Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

22 Numerical results Recovering refractive index: Smooth Ex.1 Our approach Error of the reconstruction in relative L 2 norm is 2.7%, and in relative L norm is 6.1%. Figure : Ex.1 Cross section on diagonal Figure : Ex.1 Relative error Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

23 Numerical results Recovering refractive index: Smooth Ex.2 Our approach For n(x) is chosen as: n(x) = { cos(2π x /R) if x if x > 0.45 (11) Figure : Ex.2 Exact index Figure : Ex.2 Reconstruction Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

$Numerical results Recovering refractive index: Smooth Ex.2 Our approach Error of the reconstruction in relative L 2 norm is 1.5%, and in relative L norm is 2.$

24 Numerical results Recovering refractive index: Smooth Ex.2 Our approach Error of the reconstruction in relative L 2 norm is 1.5%, and in relative L norm is 2.6%. Figure : Ex.2 Cross section on diagonal Figure : Ex.2 Relative error Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

25 Our approach Numerical results Recovering refractive index: Smooth but bumpy Ex.3 n(x) is chosen to have two bumps on diagonal with the same shape as the previous examples. Figure : Ex.3 Exact index Figure : Ex.3 Reconstruction Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

$Our approach Numerical results Recovering refractive index: Smooth but bumpy Ex.3 Error of the reconstruction in relative L 2 norm is 3.$

26 Our approach Numerical results Recovering refractive index: Smooth but bumpy Ex.3 Error of the reconstruction in relative L 2 norm is 3.94%, and in relative L norm is 7.32%. Figure : Ex.3 Cross section on diagonal Figure : Ex.3 Relative error Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

27 Our approach Numerical results Recovering refractive index: Smooth but bumpy Ex.4 n(x) is chosen to have 4 bumps on 4 corners with the same shape as the previous examples. Figure : Ex.4 Exact index Figure : Ex.4 Reconstruction Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

$Our approach Numerical results Recovering refractive index: Smooth but bumpy Ex.4 Error of the reconstruction in relative L 2 norm is 2.$

28 Our approach Numerical results Recovering refractive index: Smooth but bumpy Ex.4 Error of the reconstruction in relative L 2 norm is 2.0%, and in relative L norm is 6.0%. Figure : Ex.4 Cross section on diagonal Figure : Ex.4 Relative error Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

29 Numerical results Recovering refractive index: Non-smooth Ex.5 Our approach n(x) is chosen to be as 0.12 if x y n(x) = 0.8 if 0.5 x 0.75, 0.25 y else cases (12) Figure : Ex.5 Exact index Figure : Ex.5 Reconstruction Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

$Numerical results Recovering refractive index: Non-smooth Ex.5 Our approach Error of the reconstruction in relative L 2 norm is 1.91%, and in relative L norm is 6.$

30 Numerical results Recovering refractive index: Non-smooth Ex.5 Our approach Error of the reconstruction in relative L 2 norm is 1.91%, and in relative L norm is 6.81%. Figure : Ex.5 Cross section on diagonal Figure : Ex.5 Relative error Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

31 Recovering source Our approach After we have reconstructed the refractive index ˆn(x), we then shall use the reconstructed refractive index ˆn to recover the source. Measurement is the differential data Λγu on Ω. u + k 2 (1 + ˆn)u = f in Ω (13) u = 0 on Ω (14) Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

32 Stability Our approach Theorem (Hölder Stability for recovering location in R 3 ) Suppose u l, for l = 1, 2 be the solutions to the problem associated with sources f l = s j=1 P jδ(x l x l j) and refractive index n l (x) respectively. Assume the intensity is bounded below, and the locations of point sources are not too close. The there exists a permutation π of {1, 2,..., s} such that max S(x 1 j) S(x 2 π(j)) C j ( ) 1 Ω diam(ω) 2s 1 s n 1 n 2 L 2 ( Ω) (15) ρη s 1 where ρ = min j( P 1 j, P 2 j ), η = min l min k j dist(x l k, x l j) and C = C(k, Ω, {x l j}) is a constant. S is a projection operator to a manifold. Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

33 Stability Our approach Theorem (Hölder Stability for recovering location in R 3 ) Suppose u l, for l = 1, 2 be the solutions to the problem associated with sources f l = s j=1 P jδ(x l x l j) and refractive index n l (x) respectively. Assume the intensity is bounded below, and the locations of point sources are not too close. The there exists a permutation π of {1, 2,..., s} such that max S(x 1 j) S(x 2 π(j)) C j ( ) 1 Ω diam(ω) 2s 1 s n 1 n 2 L 2 ( Ω) (15) ρη s 1 where ρ = min j( P 1 j, P 2 j ), η = min l min k j dist(x l k, x l j) and C = C(k, Ω, {x l j}) is a constant. S is a projection operator to a manifold. Question How about the stability of intensities? Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

34 Stability Our approach Theorem (Hölder Stability for recovering location in R 3 ) Suppose u l, for l = 1, 2 be the solutions to the problem associated with sources f l = s j=1 P jδ(x l x l j) and refractive index n l (x) respectively. Assume the intensity is bounded below, and the locations of point sources are not too close. The there exists a permutation π of {1, 2,..., s} such that max S(x 1 j) S(x 2 π(j)) C j ( ) 1 Ω diam(ω) 2s 1 s n 1 n 2 L 2 ( Ω) (15) ρη s 1 where ρ = min j( P 1 j, P 2 j ), η = min l min k j dist(x l k, x l j) and C = C(k, Ω, {x l j}) is a constant. S is a projection operator to a manifold. Question How about the stability of intensities? Question How about 2D? Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

35 Algorithm Our approach We again formulate our inverse problem as a minimization problem: min K(f) f where f(x) is superposition of point sources and, K(f) = Λγu h 2 dσ Ω Algorithm 2 pseudocode for recovering source f(x) 1: select initial f = s j=1 Pjδ(x xj) 2: while K(f) ɛ do 3: for j = 1 to s do 4: update x j and P j by using Newton s method and linear search 5: end for 6: solve Helmholtz equation (13) associated with the updated x j and P j and compute K(f) 7: end while Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

36 Numerical results Recovering source function Numerical simulation To be convenient in computing, we didn t choose δ(x) as our source function, we choose another function D(x) to approximate δ(x). ( ) (x x0) 2 (y y 0) 2 D(x 0, y 0, h 0) = h 0 exp (16) ε 2 And here we choose ε as 1 2 of the source. of the mesh size. h0 is the intensity, (x0, y0) is the location Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

37 Numerical simulation Numerical results Recovering source function Ex 1. one point source Consider there is only one point source in the field with the following smooth numerical reconstructed refractive index. Figure : Ex.1 Numerical refractive index Figure : Ex.1 Numerical refractive index Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

Numerical simulation Numerical results Recovering source function Ex 1. one point source The absolute error on location is (2.556 10 5, 4.

38 Numerical simulation Numerical results Recovering source function Ex 1. one point source The absolute error on location is ( , ), relative error on intensity is Figure : Ex.1 Exact source location Figure : Ex.1 Numerical source location Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

39 Numerical simulation Numerical results Recovering source function Ex 2. one point source Consider there is only one point source with following bumpy refractive index. Figure : Ex.2 Numerical refractive index Figure : Ex.2 Numerical refractive index Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

40 Numerical simulation Numerical results Recovering source function Ex 2. one point source The absolute error on location is ( , ), relative error on intensity is Figure : Ex.2 Exact source location Figure : Ex.2 Numerical source location Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

41 Numerical simulation Numerical results Recovering source function Ex 3. two point sources Consider there are two point sources with different intensities associated with following 4-cornered bumpy refractive index. Figure : Ex.3 Numerical refractive index Figure : Ex.3 Numerical refractive index Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

42 Numerical simulation Numerical results Recovering source function Ex 3. two point sources The absolute error on location is ( , ), relative error on intensity is Figure : Ex.3 Exact source location Figure : Ex.3 Numerical source location Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

43 Numerical simulation Numerical results Recovering source function Ex 4. two point sources Consider there are two points sources with different intensities associated with following non-smooth refractive index. Figure : Ex.4 Numerical refractive index Figure : Ex.4 Numerical refractive index Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

Numerical simulation Numerical results Recovering source function Ex 4. two point sources The absolute error on location is (1.474 10 5, 8.

44 Numerical simulation Numerical results Recovering source function Ex 4. two point sources The absolute error on location is ( , ), relative error on intensity is Figure : Ex.4 Exact source location Figure : Ex.4 Numerical source location Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

45 Summary Numerical simulation Add test sources and measure differential data Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

46 Summary Numerical simulation Add test sources and measure differential data Reconstruction for both refractive index n(x) and source f(x) Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

47 Summary Numerical simulation Add test sources and measure differential data Reconstruction for both refractive index n(x) and source f(x) Stability of recovering source with respect to perturbation of n(x) Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

48 Numerical simulation Thanks. Y.Zhong (UT Austin) Recovering point source in unknown environment April 29, / 34

Increasing stability in an inverse problem for the acoustic equation

Increasing stability in an inverse problem for the acoustic equation Sei Nagayasu Gunther Uhlmann Jenn-Nan Wang Abstract In this work we study the inverse boundary value problem of determining the refractive