Open access peer-reviewed chapter

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

By Valery Serov, Markus Harju and Georgios Fotopoulos

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

DOI: 10.5772/62233

Downloaded: 1530


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 Edenotes the electromagnetic field in two-dimensional case, ∆ is the two-dimensional Laplacian and hdescribes 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 = ωand fixed, and udenotes 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 cρand cρare constants and


Here, Lσp2denotes 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 H01denotes 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 hhave 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|𝕊1is the direction of observation. This function Agives us the scattering data for inverse problem. More precisely, the inverse problem that is considered here is to extract some information about the function hby the knowledge of the scattering amplitude Afor different sets of scattering data. There are four well-known inverse scattering data sets: (i) the full (scattering) data:


(ii) the backscattering data:


(iii) the fixed angle data:


and (iv) the fixed energy data:


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,twe denote the standard L2 based Sobolev space. A weighted Sobolev space Wp,σ12is defined here by


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 has unknown and attempts to recover it from the knowledge of the scattering amplitude Afor 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 his 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 hare 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 DAand fixed energy data DE.

2.1. Full scattering data

The inverse problem with full data Dwas 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 hwe 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 σbe as in (7). Consider the scattering problems for two sets of potentials hand 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 σbe 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 fcan 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 kas


These are the unique solutions of the integral equation:


provided that his real-valued. This allows us to extend Ato negative k ≤ − k0 by


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


we have that




where Fdenotes the Fourier transform (8). Using the basic assumptions for the function hand (6), we can easily obtain that

|AR(k,θ,θ)|2|h(y,|u|)u(y,k,θ)h0(y)u0| dyC2(α(y)+β(y))|usc(y)|dyC||usc||(||α||1+||β||1)0,|k|.E34000

We have used here the fact that the basic assumptions for the function hguarantee that the functions αand βboth belong to L1(2).

Hence, for klarge, we have approximately that


These considerations and real valuedness of hsuggest and justify the following definition:

We define the inverse Born approximation qBvia 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 σbe as in (7). Then


where t < 3 − 4/pif 4/3 < p ≤ 3/2 and t < 1 − 1/pif 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 qBwith 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 hsatisfying 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 qBwhich corresponds to the full scattering data D.

2.2. Backscattering and fixed angle data

In this section, we consider backscattering data DBand fixed angle scattering data DAfollowing [12]. Using (10) we introduce the inverse backscattering and inverse fixed angle scattering Born approximations qBband qBθ0as follows:




where θ0 is fixed.

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




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


where g2x=12shx1,g1x=h0x+g2xand ηLσp2with the same σas above.

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

Theorem 3.Let σbe 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/pif 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


In straightforward manner one sees that qθ0C2and q2θ0,qRθ0Ht2with t < 3 − 4/pif 4/3 < p ≤ 3/2 and t < 1 − 1/pif 3/2 < p ≤ ∞. The first nonlinear term q1θ0cannot be analyzed directly from its definition. Instead, we first proved the representation


where F4-1denotes 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 q1θ0belongs to the space

  1. C2L2if 1 < s < 4/3;

  2. H12if s = 4/3;

  3. Ht2,t<4/s-2if 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


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 h0WS1(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 h0Hcompr2with some 0 < r < 1, then


for any t < 2rif 0 < r ≤ 1/3 and for any t < (1 + r)/2 if 1/3 < r < 1. In both cases this tcan be made bigger than r. It means that we can reconstruct all singularities from Sobolev space Hr, 0 < r < 1 from data DBand DAby 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 hcan be recovered from the inverse scattering Born approximation corresponding to fixed angle scattering and backscattering data DAand DB, respectively. No assumptions about the smallness of the norm of nonlinearity hwere 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 Rdecaying 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 Rsolves the integral equation


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 his compactly supported in Ω2and

  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,R2L2such 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 L2and this solution can be obtained as limjRjin 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 DEin 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-k02which in turn uniquely determines the scattering transform (see the details, for example, in [19] and [20]). Recall that Λhf = ∂νu, where usatisfies 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 hadmits 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 sin the neighbourhood of zero and where Ois uniform in x ∈ Ω and such s.

Suppose in addition that the nonlinearity hsatisfies 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-h0belongs 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 axisΘCentre
Ω1/2, 1/517π/18(−0.3, − 0.4)
Ω11/2, 1/4π/3(0.5, 0)
Ω21/3, 1/52π/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 fwe divide the rectangle R0 into N = n × ndisjoint subrectangles rjof equal size, i.e.


Then, we represent fon 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=θtand θ = θpleads us to


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


where Ejl=rjeiksθt-θp,ydyare easily evaluated and gl=Aksθtθpneeds to be computed. The computation of glis 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 = gdoes 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 × mpoints ξuniformly 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 = gwhose coefficient matrix Eis of size M × N. The data gis corrupted with zero mean Gaussian noise with standard deviation σ = 0.01 max |g|.

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

Fixed angle28824117
Fixed energy160026815

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 Limplements 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 DEand 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 gcontains synthetic data and actual physical measurements may take longer or shorter time.

Example 1Example 3
Fixed angle0.3s9s66s33s58s
Fixed energy0.1s4m11s78s--

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.

Scattererh0(x) and its TV reconstruction via Born approximation, Example 1. (a) scatterer; (b) full data; (c) backscattering data; (d) fixed angle data.

Figure 3.

Scattererh0(x) and its TV reconstruction via Born approximation, Example 2. (a) scatterer; (b) full data; (c) backscattering data; (d) fixed angle data

Figure 4.

Scattererh0(x) and its TV reconstruction via Born approximation, Example 3. (a) scatterer; (b) full data; (c) backscattering data; (d) fixed angle data.

Figure 5.

Scattererh0(x) and its TV reconstruction via Born approximation, Example 4. (a) scatterer; (b) full data; (c) backscattering data; (d) fixed angle data.

Figure 6.

Scattererh0(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).

© 2016 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Valery Serov, Markus Harju and Georgios Fotopoulos (July 6th 2016). Some Recent Advances in Nonlinear Inverse Scattering in 2D: Theory and Numerics, Applied Linear Algebra in Action, Vasilios N. Katsikis, IntechOpen, DOI: 10.5772/62233. Available from:

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