Computational inversion using complex geometric optics solutions and nonlinear Fourier transforms

Transcription

1 Computational inversion using complex geometric optics solutions and nonlinear Fourier transforms Samuli Siltanen Department of Mathematics and Statistics University of Helsinki, Finland 17th Intensive Course on Complex Analysis, its Generalizations and Applications University of Aveiro, Portugal, March 26 28, 2012

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3 This is a joint work with David Isaacson, Rensselaer Polytechnic Institute, USA Kim Knudsen, Technical University of Denmark Matti Lassas, University of Helsinki, Finland Jon Newell, Rensselaer Polytechnic Institute, USA Jennifer Mueller, Colorado State University, USA Andreas Stahel, Bern University of Applied Sciences, Switzerland

4 Outline Electrical impedance tomography Regularization of nonlinear inverse problems in general The D-bar method for EIT with infinite-precision data Regularization of EIT using non-linear low-pass filtering The Novikov-Veselov equation

5 Outline Electrical impedance tomography Regularization of nonlinear inverse problems in general The D-bar method for EIT with infinite-precision data Regularization of EIT using non-linear low-pass filtering The Novikov-Veselov equation

6 Electrical impedance tomography (EIT) is an emerging medical imaging technique Feed electric currents through electrodes. Measure the resulting voltages. Repeat the measurement for several current patterns. Reconstruct distribution of electric conductivity inside the patient. Different tissues have different conductivities, so EIT gives an image of the patient s inner structure. EIT is a harmless and painless imaging method suitable for long-term monitoring.

7 The most promising use of EIT is detection of breast cancer in combination with mammography ACT4 and mammography devices Radiolucent electrodes Cancerous tissue is up to four times more conductive than healthy breast tissue [Jossinet 1998]. The above experiment by David Isaacson s team measures 3D X-ray mammograms and EIT data at the same time.

8 Which of these three breasts have cancer?

9 Spectral EIT can detect cancerous tissue [Kim, Isaacson, Xia, Kao, Newell & Saulnier 2007]

10 This talk concentrates on applications of EIT to chest imaging Applications: monitoring cardiac activity, lung function, and pulmonary perfusion. Also, electrocardiography (ECG) can be enhanced using knowledge about conductivity distribution.

11 The mathematical model of EIT is the inverse conductivity problem introduced by Calderón Let Ω R 2 be the unit disc and let conductivity σ : Ω R satisfy 0 < M 1 σ(z) M. Ω Applying voltage f at the boundary Ω leads to the elliptic PDE { σ u = 0 in Ω, u Ω = f. Boundary measurements are modelled by the Dirichlet-to-Neumann map Λ σ : f σ u n Ω. Calderón s problem is to recover σ from the knowledge of Λ σ. It is a nonlinear and ill-posed inverse problem.

12 Why is Calderón s problem nonlinear? Define a quadratic form P σ for functions f : Ω R by P σ (f ) = σ u 2 dz, (1) where u is the solution of the Dirichlet problem { σ u = 0 in Ω, u Ω = f. Ω Now the map σ P σ is nonlinear because u depends on σ in (1). Physically, P σ (f ) is the power needed for maintaining the voltage potential f on the boundary Ω. Integrate by parts in (1): P σ (f ) = Ω f (σ u n ) ds = Ω Thus the map σ Λ σ cannot be linear in σ. f (Λ σ f ) ds.

13 We illustrate the ill-posedness of Calderón s problem using a simulated example σ 1 σ 2

14 We apply the voltage distribution f (θ) = cos θ at the boundary of the two different phantoms σ 1 u 1 θ σ 2 u 2

15 The measurement is the distribution of current through the boundary σ 1 u 1 θ σ 1 u 1 n θ σ 2 u 2 σ 2 u 2 n θ

16 The measurements are very similar, although the conductivities are quite different σ 1 σ 1 u 1 n σ 2 u 2 n σ 2

17 Let us apply the more oscillatory distribution f (θ) = cos 2θ of voltage at the boundary σ 1 u 1 σ 2 u 2

18 The measurement is again the distribution of current through the boundary σ 1 u 1 σ 1 u 1 n θ σ 2 u 2 σ 2 u 2 n θ

19 The current distribution measurements are again very similar σ 1 σ 1 u 1 n σ 2 u 2 n σ 2

20 EIT is an ill-posed problem: big differences in conductivity cause only small effect in data σ 1 cos θ cos 4θ cos 2θ cos 5θ σ 2 cos 3θ cos 6θ

21 EIT is an ill-posed problem: noise in data causes serious difficulties in interpreting the data σ 1 cos θ cos 4θ cos 2θ cos 5θ σ 2 cos 3θ cos 6θ

22 Many different types of reconstruction methods have been suggested for EIT in the literature Linearization: Barber, Bikowski, Brown, Calderón, Cheney, Isaacson, Mueller, Newell Iterative regularization: Dobson, Gehre, Hua, Jin, Kaipio, Kindermann, Kluth, Leitão, Lechleiter, Lipponen, Maass, Neubauer, Rieder, Rondi, Santosa, Seppänen, Tompkins, Webster, Woo Bayesian inversion: Fox, Kaipio, Kolehmainen, Nicholls, Pikkarainen, Ronkanen, Somersalo, Vauhkonen, Voutilainen Resistor network methods: Borcea, Druskin, Mamonov, Vasquez Layer stripping: Cheney, Isaacson, Isaacson, Somersalo D-bar methods: Astala, Bikowski, Bowerman, Delbary, Hansen, Isaacson, Kao, Knudsen, Lassas, Mueller, Murphy, Nachman, Newell, Päivärinta, Perämäki, Saulnier, S, Tamasan Teichmüller space methods: Kolehmainen, Lassas, Ola, S Methods for partial information: Alessandrini, Ammari, Bilotta, Brühl, Eckel, Erhard, Gebauer, Hanke, Harrach, Hyvönen, Ide, Ikehata, Isozaki, Kang, Kim, Kress, Kwon, Lechleiter, Lim, Morassi, Nakamura, Nakata, Potthast, Rossetand, Seo, Sheen, S, Turco, Uhlmann, Wang, and others

23 History of CGO-based methods for real 2D EIT Infinite-precision data Practical data 1980 Calderón 2008 Bikowski-Mueller 1987 Sylvester-Uhlmann (d 3) 2008 Boverman-Isaacson-Kao-Saulnier-Newell 1988 Nachman 2010 Bikowski-Knudsen-Mueller 1988 R. G. Novikov 2011 Delbary-Hansen-Knudsen 1996 Nachman (σ C 2 (Ω)) 2000 S-Mueller-Isaacson 1997 Liu 2003 Mueller-S 2004 Isaacson-Mueller-Newell-S 2006 Isaacson-Mueller-Newell-S 2007 Murphy-Mueller 2008 Knudsen-Lassas-Mueller-S 2009 Knudsen-Lassas-Mueller-S 2012 S-Tamminen 1997 Brown-Uhlmann (σ C 1 (Ω)) 2001 Knudsen-Tamasan 2001 Barceló-Barceló-Ruiz 2003 Knudsen 2000 Francini 2012 Hamilton-Herrera-Mueller-Herrmann 2010 Beretta-Francini 2003 Astala-Päivärinta (σ L (Ω)) 2009 Astala-Mueller-Päivärinta-S 2005 Astala-Lassas-Päivärinta 2011 Astala-Mueller-Päivärinta Barceló-Faraco-Ruiz Perämäki-S 2008 Clop-Faraco-Ruiz

24 Outline Electrical impedance tomography Regularization of nonlinear inverse problems in general The D-bar method for EIT with infinite-precision data Regularization of EIT using non-linear low-pass filtering The Novikov-Veselov equation

25 The forward map F : X D(F ) Y of an ill-posed problem does not have a continuous inverse Model space X Data space Y x F yδ y δ D(F ) F (D(F ))

26 Regularization means constructing a continuous map Γ α : Y X that inverts F approximately Model space X Data space Y x F yδ y δ Γ α D(F ) F (D(F ))

27 A regularization strategy needs to be constructed so that the assumptions below are satisfied A family Γ α : Y X of continuous mappings parameterized by 0 < α < is a regularization strategy for F if for each fixed x D(F ). lim Γ α(y) x X = 0 α 0 Further, a regularization strategy with a choice α = α(δ) of regularization parameter is called admissible if α(δ) 0 as δ 0, and for any fixed x D(F ) the following holds: sup y δ { Γα(δ) (y δ ) x X : y δ y Y δ } 0 as δ 0.

28 Two main methods for nonlinear regularization: In the Tikhonov approach, one writes a penalty functional such as Φ(x) = F (x) y δ 2 Y + α x 2 X, and Γ α (y δ ) is defined by Φ(Γ α (y δ )) = min x X {Φ(x)}. Pro: The same code applies to many problems. Con: Prone to get stuck in local minima. See [Bissantz, Burger, Engl, Hanke, Hofmann, Hohage, Justen, Kaltenbacher, Kindermann, Lechleiter, Lu, Mathé, Morozov, Munk, Neubauer, Pereverzev, Pöschl, Pricop, Ramlau, Ramm, Resmerita, Rieder, Scherzer, Seidman, Teschke, Vogel, Yagola] In the problem-specific approach, one constructs a regularization strategy tailored for a given inverse problem. Pro: Can deal efficiently with a specific nonlinearity. Con: Each code applies to only one problem.

29 In practice, efficient regularized algorithms are needed for linear and nonlinear inverse problems Assume given a forward map F and noisy data y δ. An efficient regularized inversion algorithm should compute a numerical approximation to Γ α(δ) (y δ ) quickly and accurately, where Γ α is a regularization strategy with an admissible choice of regularization parameter. The Tikhonov approach provides efficient regularized inversion algorithms only for linear and almost linear forward maps F. Electrical impedance tomography is the only strongly nonlinear inverse problem with efficient regularized inversion algorithms, based on the problem-specific approach. Inclusions: Ikehata 2002, Ikehata-S 2004, Lechleiter Global coefficient recovery: Knudsen-Lassas-Mueller-S 2009.

30 Outline Electrical impedance tomography Regularization of nonlinear inverse problems in general The D-bar method for EIT with infinite-precision data Regularization of EIT using non-linear low-pass filtering The Novikov-Veselov equation

31 Nachman s 1996 uniqueness proof in 2D uses complex geometric optics (CGO) solutions Define a potential q by setting q(z) 0 for z outside Ω and q(z) = σ(z) σ(z) for z Ω. Then q C 0 (Ω). We look for solutions of the Schrödinger equation ( + q)ψ(, k) = 0 in R 2 parametrized by k C \ 0 and satisfying the asymptotic condition e ikz ψ(z, k) 1 W 1, p (R 2 ), where p > 2 and ikz = i(k 1 + ik 2 )(x + iy).

32 The CGO solutions are constructed using a generalized Lippmann-Schwinger equation Define µ(z, k) = e ikz ψ(z, k). Then ( + q)ψ = 0 implies ( 4ik z + q)µ(, k) = 0, (2) where the D-bar operator is defined by z = 1 2 ( x + i y ). A solution of (2) satisfying µ(z, k) 1 W 1, p (R 2 ) can be constructed using the Lippmann-Schwinger type equation µ = 1 g k (qµ), where g k satisfies ( 4ik z )g k = δ and is defined by g k (z) = 1 4π 2 R2 e iz ξ ξ 2 + 2k(ξ 1 + iξ 2 ) dξ 1dξ 2.

33 The Faddeev fundamental solution g 1 (z) has a logarithmic singularity at z = 0 Real part of g 1 (z) Imaginary part of g 1 (z) It is enough to know g 1 (z) because of the relation g k (z) = g 1 (kz).

34 One of the breakthroughs in Nachman s 1996 article is showing uniqueness of µ A solution of ( 4ik z + q)µ(, k) = 0 satisfying µ(z, k) 1 W 1, p (R 2 ) can be constructed using the formula µ 1 = [I + g k (q )] 1 (g k q), provided that the inverse operator exists. Now q L p (R 2 ) with 1 < p < 2 and 1/ p = 1/p 1/2, and q : W 1, p (R 2 ) L p (R 2 ) is bounded, g k : L p (R 2 ) W 1, p (R 2 ) is compact. Thus I + g k (q ) : W 1, p (R 2 ) W 1, p (R 2 ) is Fredholm of index zero, and Nachman proved injectivity for all k 0.

35 The conductivity σ can be recovered from the functions µ(z, k) at k = 0 Recall that with the asymptotics Substituting k = 0 gives ( 4ik z + q)µ(, k) = 0 µ(z, k) 1 W 1, p (R 2 ). ( + σ σ )µ(, 0) = 0, (3) and setting µ(z, 0) = σ(z) gives the unique solution of (3) satisfying µ(z, 0) 1 W 1, p (R 2 ).

36 The crucial intermediate object in the proof is the non-physical scattering transform t(k) We denote z = x + iy C or z = (x, y) R 2 whenever needed. The scattering transform t : C C is defined by t(k) := e ikz q(z)ψ(z, k) dxdy. (4) R 2 Sometimes (4) is called the nonlinear Fourier transform of q. This is because asymptotically ψ(z, k) e ikz as z, and substituting e ikz in place of ψ(z, k) into (4) results in e i(kz+kz) q(z)dxdy R 2 = e i( 2k 1,2k 2 ) (x,y) q(z)dxdy R 2 = q( 2k 1, 2k 2 ).

37 Alessandrini s equation gives a way to write t in terms of Λ σ and traces of the CGO solutions The following boundary integral equation is a Fredholm equation of the second kind and uniquely solvable in the space H 1/2 ( Ω): ψ(, k) Ω = e ikz Ω S k (Λ σ Λ 1 )ψ(, k). Here S k is the single-layer operator with Faddeev Green s function: (S k φ)(z) := G k (z ζ)φ(ζ) ds(ζ), Ω where G k (z) := e ikz g k (z) satisfies G k = δ. The scattering transform can be evaluated by t(k) = e ikz (Λ σ Λ 1 )ψ(, k) ds. Ω

38 The difference between the usual Green s function and Faddeev Green s function is exponential Usual: G 0 (z) = 1 log z 2π Faddeev: G 1(z) = e iz g 1 (z)

39 The functions µ can be recovered from the scattering transform t using a D-bar equation It is natural to ask whether µ(z, k) depends analytically on the parameter k. If it does, the D-bar operator k = 1 2 ( + i k 1 will give zero when applied to µ(z, k). ) k 2 It turns out that the k-differential of g k is a rank-one operator, and differentiating µ = 1 g k (qµ) yields k µ(z, k) = 1 4πk t(k)e k(z)µ(z, k). Thus the dependence of µ(z, k) on k is not analytic. The D-bar equation was discovered by Beals and Coifman in the 1980 s.

40 These are the steps of Nachman s 1996 proof: Solve boundary integral equation ψ(, k) Ω = e ikz S k (Λ σ Λ 1 )ψ for every complex number k C \ 0. Fredholm equation of 2nd kind, ill-posedness shows up here. Evaluate the scattering transform: t(k) = e ikz (Λ σ Λ 1 )ψ(, k) ds. Ω Simple integration. Fix z Ω. Solve D-bar equation k µ(z, k) = t(k) 4πk e i(kz+kz) µ(z, k) with µ(z, ) 1 L r L (C). Reconstruct: σ(z) = (µ(z, 0)) 2. Well-posed problem, can be analyzed by scattering theory. Trivial step.

41 Outline Electrical impedance tomography Regularization of nonlinear inverse problems in general The D-bar method for EIT with infinite-precision data Regularization of EIT using non-linear low-pass filtering The Novikov-Veselov equation

42 Infinite-precision data: Solve boundary integral equation ψ(, k) Ω = e ikz S k (Λ σ Λ 1 )ψ for every complex number k C \ 0. Practical data: Solve boundary integral equation ψ δ (, k) Ω = e ikz S k (Λ δ σ Λ 1 )ψ δ for all 0 < k < R = 1 10 log δ. Evaluate the scattering transform: t(k) = e ikz (Λ σ Λ 1 )ψ(, k) ds. Ω Fix z Ω. Solve D-bar equation k µ(z, k) = t(k) 4πk e i(kz+kz) µ(z, k) with µ(z, ) 1 L r L (C). For k R set t δ R (k) = 0. For k < R t δ R(k) = e ikz (Λ δ σ Λ 1 )ψ δ (, k) ds. Ω Fix z Ω. Solve D-bar equation k µδ R(z, k) = tδ R (k) 4πk e i(kz+kz) µ δ R (z, k) with µ δ R (z, ) 1 Lr L (C). Reconstruct: σ(z) = (µ(z, 0)) 2. Set Γ 1/R(δ) (Λ δ σ) := (µ δ R (z, 0))2.

43 We define spaces for our regularization strategy Consider F : X D(F ) Y with X = L (Ω). Let M > 0 and 0 < ρ < 1. Now D(F ) consists of functions σ : Ω R satisfying σ C 2 (Ω) M, σ(z) M 1, and σ(z) 1 for ρ < z < 1. Y comprises bounded linear operators A : H 1/2 ( Ω) H 1/2 ( Ω) satisfying A(1) = 0 and A(f ) ds = 0. Ω

44 Main result: nonlinear low-pass filtering yields a regularization strategy with convergence speed Theorem (Knudsen, Lassas, Mueller & S 2009) There exists a constant 0 < δ 0 < 1, depending only on M and ρ, with the following properties. Let σ D(F ) be arbitrary and assume given noisy data Λ δ σ satisfying Then Γ α with the choice R(δ) = 1 10 Λ δ σ Λ σ Y δ < δ 0. log δ, α(δ) = 1 R(δ), is well-defined, admissible and satisfies the estimate Γ α(δ) (Λ δ σ) σ L (Ω) C( log δ) 1/14.

45 The proof of the main theorem is divided into several steps. First a D-bar estimate: Lemma 1. Let 4/3 < r 0 < 2 and suppose that φ 1, φ 2 L r (R 2 ) for all r r 0. Let µ 1, µ 2 be the solutions of µ j (z, k) = φ j (k ) (2π) 2 R 2 (k k ) µ j(z, k ) dk 1dk 2, j = 1, 2. Then for fixed z Ω we have µ 1 (z, ) µ 2 (z, ) C β (R 2 ) C φ 1 φ 2 L r 0 L r 0(R 2 ), where β < 2/r 0 1 and 1/r 0 = 1 1/r 0. Proof. Combination of well-known results.

46 These results follow from careful analysis of Faddeev Green s function Lemma 2. Let φ 0 H 1/2 ( Ω) with Ω φ 0 ds = 0. Then S k φ 0 H 1/2 ( Ω) Ce2 k (1 + k ) φ 0 H 1/2 ( Ω). Lemma 3. For k C we have the estimate [I + S k (Λ σ Λ 1 )] 1 L(H 1/2 ( Ω)) Ce2 k (1 + k ), where C depends only on M and ρ.

47 Combining previous results, a perturbation argument, and delicate L p analysis shows Lemma 4. There exists δ 0 > 0, depending only on M and ρ, such that the equation ψ δ (, k) Ω = e ikz S k (Λ δ σ Λ 1 )ψ δ (, k) Ω is solvable in H 1/2 ( Ω) for all 0 < δ < δ 0 and k < R with R = R(δ) = 1 log δ. 10 Furthermore, for p > 1 we have the estimate t(k) t δ R (k) k L p ( k <R) where C is independent of p and R and δ. Cδ 1/10 ( 1 10 log δ ) 2/p,

48 Sketch of the proof that nonlinear low-pass filtering gives a regularization strategy for EIT We need to show the following: (i) lim α 0 Γ α (Λ σ ) σ X = 0, (ii) α(δ) 0 as δ 0, { (iii) sup Γα(δ) (Λ δ σ) σ X : Λ δ σ Λ σ Y δ } 0 as δ 0. Λ δ σ Claim (i) follows from Lemma 1 and careful choices of Lebesgue exponents in Nachman s original proof. Claim (ii) holds by the definition α(δ) = 1 R(δ) = 10(log δ) 1.

49 Proof of claim (iii) To prove that the worst-case reconstruction error { sup Γα(δ) (Λ δ σ) σ X : Λ δ σ Λ σ Y δ } Λ δ σ tends to zero as δ 0 we combine Nachman s results with Lemmas 1 and 4 to estimate µ(z, ) µ R (z, ) C β (R 2 ) C t(k) t δ R (k) k L p L p (C) C t(k) t δ R (k) k + C t(k) L p L p ( k <R) k L p ( k >R) C (( 110 ) log δ)10/7 δ 1/10 + R(δ) 1/7 + R(δ) 1/14 ( C ( log δ) 10/7 δ 1/10 + ( log δ) 1/7 + ( log δ) 1/14).

50 We still need to define the regularization strategy on all of the data space Y, not only near F (D(F )) The previous results show the claim only for operators δ 0 -close to the range F (D(F )) Y. The structure of the set F (D(F )) is not understood at the moment. However, the proof can be extended to the whole data space Y using spectral-theoretic arguments.

51 Let us analyze how the regularization works using a simulated heart-and-lungs phantom

52 Numerical solution of traces of CGO solutions from the boundary integral equation Define Fourier basis functions ϕ n (θ) = 1 2π e inθ. Electrodes Theory We invert the linear operator appearing in the equation ψ δ (, k) Ω = [I +S k (Λ δ σ Λ 1 )] 1 e ikz Ω as a matrix in span ( {ϕ n } N n= N). The single-layer operator (S k φ)(z) = G k (z w)φ(w) ds(w) Ω uses Faddeev s Green s function.

53 This is how the actual scattering transform looks like in the disc k < 10, computed by knowing σ Real part of t(k) Imaginary part

54 Scattering transform in the disc k < 10, here computed from noisy measurement Λ δ σ Real part of t(k) Imaginary part

55 Numerical solution of the D-bar equation is based on the periodization approach of G. Vainikko The generalization of Vainikko s method for the D-bar equation is described in [Knudsen, Mueller & S 2004]. The D-bar equation k µδ R = 1 4πk tδ R(k)e z (k)µ δ R R Q together with the asymptotics µ δ R(z, ) 1 L r L (C) 2R can be combined in a generalized Lippmann-Schwinger equation: µ δ R(z, k) = 1 1 t δ R (k ) 4π 2 C (k k )k e z(k )µ δ R (z, k ) dk 1dk 2.

56 The D-bar equation is not complex-linear, so real and imaginary parts must be written separately The grid points are numbered with one index as shown. Any function φ : Q C is represented by a vector of its values at the grid points, organized as follows: Reφ(z 1 ) Reφ(z 2 ). Reφ(z 64 ) Imφ(z 1 ) Imφ(z 2 ). Imφ(z 64 ) R Q

57 This is the real-linear operation given to GMRES 1 πk 1 πk 1 πk 1 πk 1 πk 1 πk 1 πk 1 1 πk 0 πk 1 πk 1 πk 1 πk 1 πk 1 πk 1 πk FFT Element-wise multiplication T R φ T R φ T R φ T R φ T R φ T R φ T R φ T R φ FFT T R φ T R φ T R φ T R φ T R φ T R φ T R φ T R φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ IFFT φ 1 πk (T Rφ)

58 Regularized reconstructions from simulated data with noise amplitude δ = Λ δ σ Λ σ Y δ 10 6 δ 10 5 δ 10 4 δ 10 3 δ 10 2 The percentages are the relative square norm errors in the reconstructions.

59 The observed radii are better (=larger) than those given by the theoretical formula R(δ) = 1 10 log δ Practical radii R R(δ) = 1 10 log δ δ

60 The method works for real data as well, including laboratory phantoms and in vivo human data [Isaacson, Mueller, Newell & S 2006]

61 Unknown boundary shape can be estimated from EIT data using Teichmüller space methods [Kolehmainen, Lassas, Ola & S 2012]

62 Outline Electrical impedance tomography Regularization of nonlinear inverse problems in general The D-bar method for EIT with infinite-precision data Regularization of EIT using non-linear low-pass filtering The Novikov-Veselov equation

63 Report on waves, J. Scott Russell (1845), British Association for the Advancement of Science

64 John Scott Russell ( ) Heriot-Watt University 1995

65 Korteweg and de Vries formulated in 1895 an equation for waves in shallow water u τ + u xxx + 6uu x = 0, x R, τ 0 Assumptions: wave height is small compared to the depth, which in turn is small compared to the length of the wave. The KdV equation is a nonlinear, dispersive wave equation. It allows solitary wave solutions observed by Russell and studied by Boussinesq (1871) and lord Rayleigh (1876).

66 Gardner, Greene, Kruskal and Miura (1967) found a striking connection between the KdV equation and Schrödinger scattering (λ n, c n, R(k)) (λ n, c n e 4k3 n τ, R(k)e 8ik3τ ) q 0 (x) KdV The inverse scattering step is due to 1946 Borg 1949 Levinson 1951 Gelfand-Levitan 1952 Marchenko 1953 Krein q τ (x)

67 Nonlinear collision of two KdV solitons t x Source: Nonlinear Dynamical Systems Group at San Diego State University

68 Novikov-Veselov equation is the most natural 2D generalization of the KdV equation Korteveg-de Vries equation, dimension (1+1): q + 3 q q + 6q x 3 x = 0. Kadomtsev-Petviashvili equation, dimension (2+1): ( ) q + 3 q q + 6q = ± 2 q x x 3 x y 2. Novikov-Veselov equation, dimension (2+1): q + 3 z + 3 z 3 z (qv) 3 z (qv) = 0, z q = z v. Here z = x + iy and z = 1 2 ( x + i y ).

69 This is another comparison between the KdV and Novikov-Veselov equations Korteveg-de Vries equation: ( ) q = 2 q x x 6q q 2 x. Novikov-Veselov equation: ( ) q = x 2 q x q 2 y 2 +6Re[( q x )Sq] + 6Re[q x (Sq)]

70 The inverse scattering method is one way to solve the Novikov-Veselov equation t 0 (k) T Q exp(iτ(k 3 + k 3 )) t τ (k) T Q q 0 (z) nonlinear NV evolution q IS τ (z) qτ NV (z),

71 The direct and inverse nonlinear Fourier transforms T and Q are defined as follows: The direct transform q τ T q τ is given by (T q τ )(k) = e ikz q τ (z)ψ τ (z, k)dz, R 2 where ( + q τ )ψ τ (, k) = 0 and ψ τ (z, k) e ikz as z. The inverse transform t τ Qt τ is given by (Qt τ )(z) = i π 2 t τ (k) z e ikz ψ τ (z, k)dk, k C where ψ τ (z, k) = e ikz µ τ (z, k) and µ τ satisfies the D-bar equation k µ τ (z, k) = t τ (k) 4πk e i(kz+kz) µ τ (z, k), µ τ (z, ) 1.

72 What is known about the inverse scattering diagram of the Novikov-Veselov equation? 1984 Novikov & Veselov: New (2+1)-dimensional periodic generalization of KdV, proof based on algebraic geometry Boiti, Leon, Manna & Pempinelli: Formal evolution of scattering data for the non-periodic NV equation, formula for the inverse transform (assuming no exceptional points) Tsai: Formal derivation of non-periodic NV equation and a hierarchy of related equations (assuming no exceptional points) Nachman: Conductivity-type q 0 have no exceptional points Lassas, Mueller & S: Inverse scattering evolution q IS τ well-defined for conductivity-type initial data q Lassas, Mueller, S & Stahel: Evolution qτ IS preserves conductivity-type, numerical evidence for qτ IS = qτ NV Perry: The identity q IS τ = q NV τ holds for a large class of q 0.

73 Let s look at an example. Here is a smooth and rotationally symmetric conductivity function σ(z) y x z

74 This is the initial potential q 0 (z) = σ 1/2 (z) σ 1/2 (z) y x z

75 This is the initial scattering transform t 0 (k) k 2 k 1 k

76 D. Isaacson, J. Mueller, J. Newell, and S. Siltanen, Imaging cardiac activity by the D-bar method for electrical impedance tomography, Physiological Measurement, 27 (2006), pp. S43 S50. D. Isaacson, J. L. Mueller, J. C. Newell, and S. Siltanen, Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography, IEEE Transactions on Medical Imaging, 23 (2004), pp K. Knudsen, M. Lassas, J. Mueller, and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2009), pp M. Lassas, J. L. Mueller, and S. Siltanen, Mapping properties of the nonlinear Fourier transform in dimension two, Communications in Partial Differential Equations, 32 (2007), pp M. Lassas, J. L. Mueller, S. Siltanen, and A. Stahel, The Novikov-Veselov Equation and the Inverse Scattering Method, Part I: Analysis, ArXiv e-prints, (2011). J. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM Journal on Scientific Computing, 24 (2003), pp A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), pp S. Siltanen, J. Mueller, and D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2-D inverse conductivity problem, Inverse Problems, 16 (2000), pp