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Some Recent Advances in Nonlinear Inverse Scattering in 2D: Theory and Numerics

Written By

Valery Serov, Markus Harju and Georgios Fotopoulos

Submitted: October 13th, 2015 Reviewed: January 12th, 2016 Published: July 6th, 2016

DOI: 10.5772/62233

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We survey our recently published results concerning scattering problems for the nonlinear Schrödinger equation


  • Inverse scattering
  • Schrödinger equation
  • Born approximation
  • numerical solution
  • linear inverse problem

1. Introduction

We deal with the generalized nonlinear Schrödinger equation:


where E denotes the electromagnetic field in two-dimensional case, ∆ is the two-dimensional Laplacian and h describes in a general form the nonlinear contribution to the index of refraction. Considering harmonic time dependence E(xt) = e− iωtu(x) with frequency ω > 0, we obtain the steady-state nonlinear equation with fixed energy:


where k2 = ω and fixed, and u denotes the complex amplitude of the field. Concerning the nonlinearity h(xs), we pose the basic assumptions.

  1. hxscραx,αLσp2,0sρ,

  2. hxs1-hxs2cρβxs1-s2,βLσp2,0s1,s2ρ,

where cρ and cρ are constants and


Here, Lσp2 denotes the weighted Lebesgue space with the norm.


The main practical example (it can be considered as the motivation of this research) of such type equations (1) is the equation of the form:


with real number k, complex valued function q1(x) ∈ L2 and real-valued function q2(x) ∈ L2, and parameter r ≥ 0. A particular nonlinearity in (3) of cubic type (r = 0) can be met in the context of a Kerr-like nonlinear dielectric film, while the case when r > 0 corresponds to the saturation model (see [14]).

We consider the inverse scattering problems for (1). For these purposes, we are interested of the scattering solutions to (1), i.e. solutions of the form


where θ𝕊1, the unit sphere in 2, u0(xkθ) = eik(x,θ) is the incident wave and usc(xkθ) is the scattered wave. The scattered wave must satisfy the Sommerfeld radiation condition at infinity:


for fixed k > 0 and uniformly in θ𝕊1. In that case, these solutions are the unique solutions of the Lippmann-Schwinger equation.


where Gk+is the outgoing fundamental solution of the operator − ∆ − k2, i.e. the kernel of the integral operator (−∆ − k2 − i0)− 1. It is equal to


where H01 denotes the Hankel function of order zero and first kind.

The following main results concerning the direct scattering problem were proved in [5].

Under the basic assumptions and (2) for h, it is proved that for any ρ > 1 there is k0 > 0 such that for any k ≥ k0 in the ball Bρ=uL2:uρ, there is a unique scattering solution (or the solutions of the form in (4) to (5) which satisfies the condition:


uniformly in θ𝕊1. What is more, the solution is obtained as the limit




for j = 0, 1, … with u0 as above. Let the function h have the same properties as above, but now with


Then for fixed k ≥ k0, the solution u(xkθ) admits the representation


The function A(kθ′, θ) is called the scattering amplitude and it is defined as


where θ' =x| x |𝕊1 is the direction of observation. This function A gives us the scattering data for inverse problem. More precisely, the inverse problem that is considered here is to extract some information about the function h by the knowledge of the scattering amplitude A for different sets of scattering data. There are four well-known inverse scattering data sets: (i) the full (scattering) data:


(ii) the backscattering data:

D B = A k θ θ : k > 0 , θ = - θ 𝕊 1 , E17000

(iii) the fixed angle data:


and (iv) the fixed energy data:

D E = A k θ θ : k = k 0 > 0 fixed , θ , θ 𝕊 1 . E46000

We use the following notations for the direct and inverse Fourier transforms:


where (ξ, x) denotes the inner product in 2, i.e. (ξx) = ξ1x1 + ξ2x2. By C > 0, we denote a generic constant that may change from one step to another. By Ht2=W2t2,t we denote the standard L2 based Sobolev space. A weighted Sobolev space Wp,σ12 is defined here by

W p , σ 1 2 = f L σ p 2 : f L σ p 2 . E46000

The following notation for the characteristic function is used:



2. Inverse scattering problems

The direct scattering theory described above can also be reversed. The inverse scattering theory treats the function h as unknown and attempts to recover it from the knowledge of the scattering amplitude A for different data. Usually, the model in (1) is probed with one or more incident plane waves u0 and the resulting scattered waves are measured at a distance. This gives rise to several different scattering data sets which can be used to recover the unknowns.

The inverse backscattering problem for (1) was treated in [6]. Also for (1), the recovery of unknown function h is possible from the full scattering data. In addition to the two-dimensional studies mentioned above, certain particular nonlinear cases of (1) have been investigated in other dimensions too. In one-space dimension, we refer to [7] and the references therein. In higher dimensions n ≥ 3 we are only aware of [8,9]. Similar problems with formally more general equation but with bounded h are considered in [10] and [11].

Our point of view is that the nonlinearity may contain local singularities in the space coordinate x, and therefore we work in the frame of weighted Lebesgue spaces. These local singularities can be recovered from the scattering amplitude using the method of Born approximation. As a unifying result, we obtain mathematically more general results that have far wider applicability in physical experiments.

Let us set


In the subsections that follow we consider the inverse problems of recovering information about h0 from the knowledge of full data D, backscattering data DB, fixed angle data DA and fixed energy data DE.

2.1. Full scattering data

The inverse problem with full data D was investigated in [5]. Here we summarize the main results without proofs.

Theorem 1 (Saito’s formula). Under the basic assumptions and (7) for the function h we have,


where the limit is valid in the sense of distributions for 4/3 < p ≤ 2 and pointwise (even uniformly) for 2 < p ≤ ∞.

Corollary 1 (Uniqueness). Let σ be as in (7). Consider the scattering problems for two sets of potentials h and h̃. If the scattering amplitudes coincide for some sequence kj → ∞ and for all θ′, θ𝕊1, then


holds in the sense of distributions for 4/3 < p ≤ ∞.

Corollary 2 (Representation formula). Let σ be as in (7). Then the representation


holds in the sense of distributions for 4/3 < p ≤ ∞.

Remark 1. In addition to providing the above results, Saito’s formula might be applied numerically too. It can be written as a convolution equation:


where the function f can be computed from the full scattering data in principle. A numerical inversion of this equation would yield a full recovery of h0 but this is an open problem as far as we know. What is more, unlike the representation formula above it holds pointwise in the important case of bounded (p = ∞) nonlinearities.

We assume that the function x ↦ h(xs) is real-valued and recall that


For reasons of purely technical nature we define the scattering solutions u(xkθ) for negative k as


These are the unique solutions of the integral equation:


provided that h is real-valued. This allows us to extend A to negative k ≤ − k0 by


We also put A(kθ′, θ) = 0 for |k| ≤ k0. Splitting


we have that




where F denotes the Fourier transform (8). Using the basic assumptions for the function h and (6), we can easily obtain that

| A R ( k, θ ,θ )| 2 |h( y,|u| )u( y,k,θ ) h 0 (y) u 0 | dyC 2 ( α( y )+β( y ) )| u sc ( y )|dy C | | u sc || ( | |α|| 1 + | |β|| 1 )0,|k|. E34000

We have used here the fact that the basic assumptions for the function h guarantee that the functions α and β both belong to L1(2).

Hence, for k large, we have approximately that


These considerations and real valuedness of h suggest and justify the following definition:

We define the inverse Born approximation qB via the equality


which is understood in the sense of tempered distributions. In order to recover main singularities of h0 from qB, we must study their difference and show that it is locally less singular than h0. To this end, we have the following main result from [5].

Theorem 2. Let σ be as in (7). Then

qB-h0Hloct2, E37000

where t < 3 − 4/p if 4/3 < p ≤ 3/2 and t < 1 − 1/p if 3/2 < p ≤ ∞.

Remark 2. Theorem 2 means that, for 4/3 < p < ∞, the main singularities of h0 can be recovered from the inverse scattering Born approximation qB with full data D. In the case of p = ∞, we have no singularities but may have finite jumps. Under such circumstances, we record the following special case.

Corollary 3. If a piecewise smooth compactly supported function h0 contains a jump over a smooth curve, then the curve and the height function of the jump are uniquely determined by the full scattering data. Especially, for the function h0 being the characteristic function of a smooth bounded domain, this domain is uniquely determined by the full scattering data.

Concluding, in this part of the work, the uniqueness of the direct problem for the nonlinearities h satisfying the appropriate properties was proved. These properties allow local singularities and do not require compact support, but rather some sufficient decay at infinity. Under similar properties, we were also able to establish the asymptotic behaviour of scattering solutions, which gives us the scattering data so we can investigate the inverse scattering problems. Note that both results were proved without assuming smallness of the norm of the nonlinearity as is necessary in dimensions three and higher.

What can we regard as the main result of this section is the Saito’s formula since it implies a uniqueness result and a representation formula for our unknown function h0. In addition, we managed to extract more information about the nonlinearity by applying the method of Born approximation. More precisely, the main singularities (or jumps over smooth curve) of h0 can be recovered from the Born approximation qB which corresponds to the full scattering data D.

2.2. Backscattering and fixed angle data

In this section, we consider backscattering data DB and fixed angle scattering data DA following [12]. Using (10) we introduce the inverse backscattering and inverse fixed angle scattering Born approximations qBb and qBθ0 as follows:




where θ0 is fixed.

Furthermore, we assume in addition that the nonlinearity h possesses the Taylor expansion:




uniformly in s ∈ (0, s0), s0 > 0 and with η1,η2LσpR2, where σ as in (7). From this we obtain the expansion:


where g2x=12shx1,g1x=h0x+g2x and ηLσp2 with the same σ as above.

Again, the main result for the recovery of main singularities is formulated as the following theorem.

Theorem 3. Let σ be as in (7) with 2 < p ≤ ∞. Suppose in addition


and 2 < s′ < p ≤ ∞, where s′ is the Hölder conjugate of s. Then


for any t < 1 − 1/p if 1 < s ≤ 4/3 and for any t < min{1 − 1/p, 4/s − 2} if 4/3 < s < 2.

Let us sketch the main ideas in the proof of Theorem 3. Using the definition, we may expand the difference in several terms as

q B θ 0 - h 0 = q 1 θ 0 + q θ 0 + q 2 θ 0 + q R θ 0 . E111000

In straightforward manner one sees that qθ0C2 and q2θ0,qRθ0Ht2 with t < 3 − 4/p if 4/3 < p ≤ 3/2 and t < 1 − 1/p if 3/2 < p ≤ ∞. The first nonlinear term q1θ0 cannot be analyzed directly from its definition. Instead, we first proved the representation


where F4-1 denotes the four-dimensional inverse Fourier transform. This formula might have independent interest too, but primarily it allows us to prove the following regularity: the term q1θ0 belongs to the space

  1. C2L2 if 1 < s < 4/3;

  2. H12 if s = 4/3;

  3. Ht2,t<4/s-2 if 4/3 < s < 2.

If we combine all these steps, we obtain Theorem 3 for fixed angle scattering.

The inverse backscattering Born approximation is treated similarly. Namely, we write

q B b - h 0 = q 1 b + q b + q 2 b + q R b . E46000

The latter four terms have exactly the same regularity results as their counterparts in fixed angle scattering. For the first nonlinear term, the representation is now


The additional assumption h 0WS1(2) in Theorem 3 implies that h0L1s2.This explains why we restrict s′ < p. By Sobolev embedding, we obtain


with some positive  < min{1/p, 1 − 2/p}. Hence, h0 is locally more singular than either of these differences. That’s why both Born approximations recover the main singularities of h0. On the other hand, we may perform the comparison also in the scale of Sobolev spaces. Indeed, if h0Hcompr2 with some 0 < r < 1, then


for any t < 2r if 0 < r ≤ 1/3 and for any t < (1 + r)/2 if 1/3 < r < 1. In both cases this t can be made bigger than r. It means that we can reconstruct all singularities from Sobolev space Hr, 0 < r < 1 from data DB and DA by the method of Born approximation.

Corollary 4. If a piecewise smooth compactly supported function h0 contains a jump over a smooth curve, then the curve and the height function of the jump are uniquely determined by backscattering and fixed angle scattering data. Especially, for the function h0 being the characteristic function of a smooth bounded domain, this domain is uniquely determined by backscattering and fixed angle scattering data.

Concluding, in this section we proved that all singularities and jumps (in the absence of a uniqueness theorem) of the nonlinearity h can be recovered from the inverse scattering Born approximation corresponding to fixed angle scattering and backscattering data DA and DB, respectively. No assumptions about the smallness of the norm of nonlinearity h were used as it were in previous publications even in the linear case.

2.3. Fixed energy data

The two-dimensional fixed energy problemx1 was a long-standing open problem. In the case of linear Schrödinger operator, the first uniqueness and reconstruction algorithm was proved by Nachman [13] via ¯-methods for potentials of conductivity type. Sun and Uhlmann [14] proved uniqueness for potentials satisfying nearness conditions to each other. The question of global uniqueness for the linear Schrödinger equation with fixed energy was settled only in 2008 by Bukhgeim [15] for compactly supported potentials from Lp, p > 2. This result has recently been improved and extended to related inverse problems (see for example [16] and [17]). Note that Grinevich and Novikov [18] proved that in two dimensions there are nonzero real potentials of the Schwartz class with zero scattering amplitude at fixed positive energy. Thus, the compactness of the supports of the potentials is very natural condition in our considerations.

The results of this section are proved in [19] and they slightly generalize the linear case to a special type of nonlinearity. It turned out that (as we can see in this section) inverse fixed energy scattering problem is much more difficult than the others.

In fixed energy scattering problem, instead of the scattering solutions (4) we need the so-called complex geometrical optics solutions. Complex geometrical optics (CGO) solutions or exponentially growing solutions of the form:


with R decaying at infinity for |z| large for the homogeneous Schrödinger equation


are obtained as follows. Substituting (14) into (15) yields


It means that using the Faddeev Green’s function


as the fundamental solution of the differential operator


we see that the function R solves the integral equation

Rxz=-2gzx-yhy,|eiyz1+Ryz|1+Ryzdy. E16

It remains to establish unique solvability of this equation. To this end we again use iterations in the sense of next theorem.

We assume that h is compactly supported in Ω2 and

  1. |h(xs)| ≤ α(x) with some α ∈ L2(Ω) and s ≥ 0

  2. |h(x, |ei(x,z)(1 + R1)|) − h(x, |ei(x,z)(1 + R2)|)| ≤ β(x)|R1 − R2| with some β ∈ L2(Ω) and for any R1,R2L2 such that ‖Rj < 1 − δ, j = 1, 2 for some fixed δ ∈ (0, 1) and for any z2

  3. α2, ‖β2 > 0

Theorem 4. Under the above conditions for h, there exists a constant C0 > 0 such that for all |z| ≥ C0 the equation (16) has a unique solution in L2 and this solution can be obtained as limjRj in L2, where


with R0 = 0. Moreover, the following estimates hold


The proof of Theorem 4 is based on the fact that for any γ < 1 there is constant cγ > 0 such that


for any f ∈ L2(Ω), see [20].

Turning now to the inverse fixed energy scattering problem, we define the scattering transform by


and Th(ξ) = 0 for ξ<2C0 . Here z=12ξ-iJξ,


and e0=eixz=e12x1ξ2-x2ξ1. What is more, we have the uniform limit Th(ξ)=limjTh,j(ξ), where


We point out that the scattering transform is somehow an auxiliary object (see DE in Introduction). But it is connected to the scattering amplitude as follows. It is well known that the scattering amplitude at fixed energy uniquely determines the Dirichlet-to-Neumann map Λh-k02 which in turn uniquely determines the scattering transform (see the details, for example, in [19] and [20]). Recall that Λhf = ∂νu, where u satisfies the Dirichlet problem:


Here, Ω is a domain with boundary ∂Ω and outward unit normal vector ν.

Next, we define the inverse fixed energy Born approximation by


In contrast to the preceding inverse problems, we now set the unknown function to be


In linear case h0 is the actual potential appearing in the Schrödinger equation, but otherwise the connection to physical scatterers is not known to us.

We assume that the nonlinearity h admits the Taylor expansion


where |∂sh(xe0(1 + s))|s = 0| ≤ β1(x) and O(β1(x)s2) with β1(x) ∈ L2(Ω) and with small s in the neighbourhood of zero and where O is uniform in x ∈ Ω and such s.

Suppose in addition that the nonlinearity h satisfies the asymptotic expansions


where |αj|,|αj̃|αx/2 . Then we have the following main result concerning the recovery of singularities of h0(x) defined by (18).

Theorem 5. Under the foregoing conditions for the potential function h, it is true that


for any t < 1 modulo C2 - functions.

Remark 3. The embedding theorem for Sobolev spaces says that the difference qBf-h0 belongs to Lq(R2) for any q < ∞ modulo C(R2) functions. It means that all singularities from LlocpR2,p< of unknown function h0 can be obtained exactly by the Born approximation which corresponds to the inverse scattering problem with fixed positive energy.

We note that under fixed energy data we have some additional assumptions on h. This limits the applicability of the main result to, for example, saturation type nonlinearities. In particular, cubic nonlinearity is excluded from these considerations. Moreover, the unknown function h0 has no direct connection to original scatterers in nonlinear cases.


3. Numerical examples

Here we discuss the numerical computation of the Born approximation first proposed in [21] for backscattering and fixed angle data. We assume that the scatterer h0(x) is supported in the rectangle R0=-1,1×-1,12. We consider the following examples:

Example 1: h(x,| u |)=χΩ(x (linear)

Example 2: h(x,| u |)=χΩ(x)| u || u |21+| u |2 (saturation)

Example 3: h(x,| u |)=0.75χΩ1(x)| u |+χΩ2(x)| u |21+| u |2 (linear, saturation)

Example 4: h(x,| u |)=0.75χΩ2(x)| u |21+| u |2+ χΩ1(x (saturation, linear)

Here Ω, Ω1, Ω2 are the shifted ellipses rotated in angle Θ theta (counter-clockwise) detailed in Table 1 and shown in Figure 1.

Semi axis Θ Centre
Ω 1/2, 1/5 17π/18 (−0.3, − 0.4)
Ω1 1/2, 1/4 π/3 (0.5, 0)
Ω2 1/3, 1/5 2π/3 (−0.5, 0)

Table 1.

Geometries of the ellipses Ω, Ω1 and Ω2

Figure 1.

Geometries of the ellipses Ω, Ω1 and Ω2.

Consider the Born approximation for full data (11) in the form


To discretize the unknown function f we divide the rectangle R0 into N = n × n disjoint subrectangles rj of equal size, i.e.


Then, we represent f on R0 in piecewise constant form:


where fj’s are the unknown values on rj’s. Substitution into (19) yields


Evaluating this at some points k=ks,θ=θt and θ = θp leads us to


If we denote M = N1N2N3, we may form one linearized index l = l(stp), l = 1, …, M and write the latter equation as


where Ejl=rjeiksθt-θp,ydy are easily evaluated and gl=Aksθtθp needs to be computed. The computation of gl is carried out using numerical integration, see [21] for details. In matrix form (20) is clearly Ef = g.

For backscattering data and fixed angle data the system (20) is modified accordingly. We note that the system Ef = g does not depend on the scatterer but only on data type and measurement setup.

The fixed energy case differs from the first three data sets. In fixed energy inversion we approximate the scattering transform as


We choose M = m × m points ξ uniformly from the rectangle [−ss] × [−ss]. The function Th,1(ξ) is evaluated by numerical integration, see [19,21] for details. Then the inverse Born approximation (17) is computed similarly to (19).

We use the following parameter values:


We use N = 2500 and N1 = 12 for each data set. For full data we use N2 = N3 = 6. For backscattering and fixed angle data, we use N2 = 24. For fixed energy scattering we use m = 40 and s = 6.

In all four cases we obtain the linear system Ef = g whose coefficient matrix E is of size M × N. The data g is corrupted with zero mean Gaussian noise with standard deviation σ = 0.01 max |g|.

The size M as well as the ranks r(E) and (approximate) condition numbers log10κ(E) measuring the ill-posedness of the linear system EF = g are shown in Table 2.

Data M r(E) log10κ(E)
Full 1728 360 63
Backscattering 288 281 15
Fixed angle 288 241 17
Fixed energy 1600 268 15

Table 2.

Matrix sizes, ranks and condition numbers

As the linear system is rank-deficient and ill-posed, we use regularization method to solve it. More precisely, we apply the total variation method (TV) which is defined as


where the matrix L implements the differences between neighbouring elements in horizontal and vertical directions (for details, see [21]). As in [21] we formulate this minimization problem as a quadratic problem in standard form for more efficient solution. As the regularization parameter we use δ = 2 ⋅ 10− 3 for DE and δ = 10− 3 otherwise.

All computations are carried out in a UNIX system with 512 GB of memory and 40 logical CPU cores, each running at 2.8 GHz. The software platform is MATLAB R2015b. We have used 12 workers in parallel in computing the right-hand side g. We note that a desktop PC with 8 logical cores running at 3.4 GHz and 16 GB of memory is also capable of our computations, but with 7 workers it is considerably slower in computing g.

The computational times to both form and solve the linear system are shown in Table 3. We point out that the right-hand side g contains synthetic data and actual physical measurements may take longer or shorter time.

Example 1 Example 3
Data E g TV g TV
Full 1.3s 47s 70s 3m20s 63s
Backscattering 0.4s 9s 59s 37s 45s
Fixed angle 0.3s 9s 66s 33s 58s
Fixed energy 0.1s 4m11s 78s - -

Table 3.

Computational times

The contour plots of scatterers h0(x) and their TV reconstructions via Born approximation from full data, backscattering data and fixed angle data (with θ0 = (1, 0)) are shown in Figures 25 for all examples. For fixed energy scattering we only consider the linear Example 1, since otherwise we do not have direct comparison to a scatterer. The TV reconstruction is shown in Figure 6. In each figure solid white line indicates the true geometry of the scatterer.

We see that the location of the scatterer is located quite nicely in all cases. The shape of the scatterer is best seen from full data and backscattering data. By computing the Born approximation from full angle data, we close an open problem from [5].

Figure 2.

Scatterer h0(x) and its TV reconstruction via Born approximation, Example 1. (a) scatterer; (b) full data; (c) backscattering data; (d) fixed angle data.

Figure 3.

Scatterer h0(x) and its TV reconstruction via Born approximation, Example 2. (a) scatterer; (b) full data; (c) backscattering data; (d) fixed angle data

Figure 4.

Scatterer h0(x) and its TV reconstruction via Born approximation, Example 3. (a) scatterer; (b) full data; (c) backscattering data; (d) fixed angle data.

Figure 5.

Scatterer h0(x) and its TV reconstruction via Born approximation, Example 4. (a) scatterer; (b) full data; (c) backscattering data; (d) fixed angle data.

Figure 6.

Scatterer h0(x) and its TV reconstruction via Born approximation, Example 1. (a) scatterer; (b) fixed energy.



This work was supported by the Academy of Finland (application number 250215, Finnish Programme for Centres of Excellence in Research 2012–2017).


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Written By

Valery Serov, Markus Harju and Georgios Fotopoulos

Submitted: October 13th, 2015 Reviewed: January 12th, 2016 Published: July 6th, 2016