arXiv:2201.03494v1 [math.NA] 10 Jan 2022

High-frequency limit of the inverse scattering problem: asymptotic convergence frominverse Helmholtz to inverse Liouville˚

Shi Chen: , Zhiyan Ding; , Qin Li§ , and Leonardo Zepeda-Nunez¶

Abstract. We investigate the asymptotic relation between the inverse problems relying on the Helmholtz equationand the radiative transfer equation (RTE) as physical models, in the high-frequency limit. In particular,we evaluate the asymptotic convergence of a generalized version of inverse scattering problem based onthe Helmholtz equation, to the inverse scattering problem of the Liouville equation (a simplified versionof RTE). The two inverse problems are connected through the Wigner transform that translates the wave-type description on the physical space to the kinetic-type description on the phase space, and the Husimitransform that models data localized both in location and direction. The finding suggests that imping-ing tightly concentrated monochromatic beams can indeed provide stable reconstruction of the medium,asymptotically in the high-frequency regime. This fact stands in contrast with the unstable reconstructionfor the classical inverse scattering problem when the probing signals are plane-waves.

Key words. Inverse Scattering, Wigner Transform, Husimi Transform, High-frequency Limit

AMS subject classifications. 65N21, 78A46, 81S30

1. Introduction. The wave-particle duality of light has been one of the greatest enigmas in thenatural sciences, dating back to Euclid’s treatise in light, Catoptrics (280 B.C.) and spanning morethan two millennia. In a nutshell, light can be either described as an electromagnetic (EM) wavegoverned by the Maxwell’s equations, or as a stream of particles, called photons, governed by theradiative transport equation (RTE).

Although the advent of quantum mechanics at the onset of the last century partially solvedthe riddle, due to computational considerations, light continues to be modeled either as a particleor as a wave depending on the target application. Among those applications, inverse problems areperhaps the ones that have gained the most attention in the last decades, which in return havefueled many breakthroughs in telecommunications [52, 53], radar [16], biomedical imaging [44, 4]and, more recently, in chip manufacturing [29]. In this context, inverse problems can be roughlydescribed as reconstructing unknown parameters within a domain of interest by data comprised ofobservations on its boundary.

Unfortunately, the properties of the inverse problems are highly dependent on the specificmodeling of the underlying physical phenomena, even though, in principle, they share the samemicroscopic description. In particular, the stability of the inverse problem, i.e., how sensitive is thereconstruction of the unknown parameter to perturbations in the data, is surprisingly disparate [35,

˚Submitted to the editors DATE.Funding: The work of Q. L. is supported in part by the UW-Madison Data Initiative, Vilas Young Investigation

Award, National Science Foundation under the grant DMS-1750488 and the Office of Naval Research under the grantONR-N00014-21-1-2140. The work of L. Z.-N. is supported in part by the National Science Foundation under the grantDMS-2012292. In addition, Q. L. and L. Z.-N. are supported by the NSF TRIPODS award 1740707. The views expressedin the article do not necessarily represent the views of the any funding agencies. The authors are grateful for the support.

:Department of Mathematics, University of Wisconsin-Madison, Madison WI 53706 ([email protected], https://simonchenthu.github.io/.

;Department of Mathematics, University of Wisconsin-Madison, Madison WI 53706 ([email protected], https://people.math.wisc.edu/„zding49/).

§Department of Mathematics, University of Wisconsin-Madison, Madison WI 53706 ([email protected], https://people.math.wisc.edu/„qinli/).

¶Department of Mathematics, University of Wisconsin-Madison, Madison WI 53706 ([email protected], https://people.math.wisc.edu/„lzepeda/). Now at Google Research.

1

arX

iv:2

201.

0349

4v2

[m

ath.

NA

] 1

1 Ju

n 20

22

2 S. CHEN, Z. DING, Q. LI, AND L. ZEPEDA-NUNEZ

13], thus creating an important gap between the wave and particle descriptions, which we seekto bridge in this paper. We point out that understanding this gap is not only of theoreticalimportance, it would also play an important role in designing new reconstruction algorithms withimproved stability applicable to a broader set of wave-based inverse problems, which are ubiquitousin science [48, 42, 38] and engineering [39, 2, 18].

For simplicity, we consider a time-harmonic wave-like description governed by the Helmholtzequation, which can be derived from the time-harmonic Maxwell’s equations after some simpli-fications. Alternatively, the Helmholtz equation can also be obtained by computing the Fouriertransform of the constant-density acoustic wave equation at frequency k, and is given by1

(1.1)`

∆` k2n˘

upxq “ 0 ,

where u is the wave field, and npxq is the refractive index of the medium. We point out that evenif this is a simplified model, it retains the core difficulty of more complex physics.

We also consider a particle-like description governed by the Liouville equation, which is asimplified RTE, given by:

(1.2) v ¨∇xf ´∇xn ¨∇vf “ 0 ,

where fpx, vq is the distribution of photon particles, and n is still the refractive index. The Liouvilleequation describes the trajectories of photons via its characteristics: 9x “ v and 9v “ ´∇xn. Forsimplicity we neglect the photon interactions which are usually encoded by the collision operator.

Following the wave and photon descriptions, we define the forward problem as calculating eitherthe wave-field, or the photon distribution from the refractive index by solving either the Helmholtzor the Liouville equations. The wave-particle duality, when translated to mathematical language,corresponds to the fact that the solutions obtained by the Helmholtz and Liouville equations areasymptotically close when k Ñ8, see [3].

For the sake of conciseness, we consider a simplified inverse problem consisting of reconstructingan unknown environment within a domain of interest by probing it with tightly concentratedmonochromatic beams originated from the the boundary of the domain, in which the response ofthe unknown medium to the impinging beam is measured at its boundary. This measurement isperformed by a measurement operator that is model-specific and it will play an important role inwhat follows. For simplicity, we consider the full aperture regime, i.e., we can probe the mediumfrom any direction, and we sample its impulse response in all possible directions. When the beamis modeled as a wave, i.e., using the Helmholtz equation as a forward model, this process can beconsidered as a generalized version of the inverse scattering wave problem (which we, for the sake ofclarity, just refer to as the generalized Helmholtz scattering problem. When the beam is modeled asa flux of photons, i.e., using the Liouville equation as a forward model, this process is often referredto as the optical tomography problem, but we will refer to as the Liouville scattering problem in thismanuscript.

Although the two different formulations seek to solve the same underlying physical problem,our understanding of the two inverse problems seems to suggest different stability properties. Thetraditional inverse scattering problem, using either near-field or far-field data is ill-posed: smallperturbations in the measurements usually lead to large deviations in the reconstructions [19, 27].Thus, sophisticated algorithms [32, 22, 21, 40, 15, 8, 5] have been designed to artificially stabilizethe process by appropriately restricting the class of possible unknown environments, usually in theform of band-limited environments. Conversely, the inverse Liouville equation is well-conditioned:a small perturbation is reflected by a small error in the reconstruction [37].

1The domain of definition, source, and boundary conditions will be specified in Section 2.

INVERSE SCATTERING IN THE HIGH-FREQUENCY LIMIT 3

Thus the observation that the stability for both problems is different seems to be at odds withthe fact that the Liouville equation and the Helmholtz equation are asymptotically close in thehigh-frequency regime. Fortunately, as what we will see, this somewhat contradictory propertystems from the inability of traditional formulations of the inverse problems to agree in the high-frequency limit. When the measurement operators are accordingly adjusted, we show that thenew formulations, which we call the generalized inverse scattering, are equivalent in the limit ask Ñ 8, producing a stable inverse problem. The convergence from the Helmholtz equation tothe Liouville equation is conducted through the Wigner transform [24, 43, 3], and the convergenceof the measuring operators is achieved through the Husimi transform [7]. Both convergences areobtained asymptotically in the k Ñ8 limit. This convergence allows us to conclude the following:

The inverse Liouville scattering problem is asymptotically equivalent to the generalized inverseHelmholtz scattering problem in the high-frequency regime.

The current manuscript is dedicated to formulating the statement above in a mathematicallyprecise manner, while providing extensive numerical evidence supporting the statement.

On the mathematical level, the current paper carries the following important features:‚ The result connects the two seemingly distinct inverse problems, and suggests that in the

high-frequency regime, probing an unknown object with a single frequency is already enoughfor its reconstruction, with properly prepared data in the generalized inverse scatteringsetting. This partially answers the stability question regarding the inverse scattering.

‚ The result can be viewed as the counterpart of the asymptotic multiscale study conductedin the forward setting. In particular, semi-classical limit is a theory that connects quantummechanical and the classical mechanical description: the proposed formulation for the in-verse scattering problem can be regarded as taking the (semi-)classical limit in the inversesetting, and thus the work carries conceptual merits. This is in line with [35, 13]. Seealso [31] for a different setting.

These mathematical understandings also naturally bring numerical and practical benefits. Thenew inverse wave scattering formulation coupled with PDE-constrained optimization seems to beempirically less prone to cycle-skipping, i.e., convergence to spurious local minima [51], than itsstandard counterparts [50, 8], thus potentially opening the way to more robust algorithmic pipelinesfor inverse problems.

We point out that even though this current study is motivated by the wave-particle duality oflight, the current results are also applicable to other oscillatory phenomena, see [13] for a discussionon inverse Schrodinger problem in the classical limit.

Organization. In Section 2, we briefly review the Helmholtz equation and present the corre-sponding inverse problem that fits the particular experimental setup that allows passing the systemto the k Ñ8 limit. In Section 3, we discuss the limiting Liouville equation and the inverse Liouvillescattering problem, by conducting the Wigner and Husimi transforms. The connections betweenthe two inverse problems will thus be immediate. Finally, we present our numerical evidencesthat justify the convergence in Section 4 and we showcase the stability of the inverse problem inSection 5.

2. Experimental setup and inverse problem formulation. Suppose we use tightly concentratedmonochromatic beams, or laser beams, to probe the medium. Each beam impinges in the area ofinterest, thus producing a scattered field which is then measured by directional receivers2 placedon a manifold around the domain of interest. The data, which is used to reconstruct the optical

2Experimentally, this is often achieved by placing a collimator before the receiver, and changing the orientationof the collimator.

4 S. CHEN, Z. DING, Q. LI, AND L. ZEPEDA-NUNEZ

properties of the medium, is the intensity captured by each receiver for each incoming beam. Thus,the data is indexed by the position and direction of the impinging beam, and the location anddirection of the receivers.

𝑣𝑣𝑠𝑠𝑥𝑥𝑠𝑠𝑣𝑣𝑟𝑟

𝑥𝑥𝑟𝑟

Figure 1: Illustration of the setup.

In this section, we set up the experiment and provide the mathematical formulation, using boththe wave and the particle forms for the forward model. This prepares us to link the two problemsin Section 3.

2.1. Helmholtz equation and inverse wave scattering problem. The Helmholtz equation isa model equation for time-harmonic wave propagation. After some approximations, both theconstant-density acoustic wave equation and the Maxwell equations for the EM waves, can berecast, through the Fourier transform in t, into the Helmholtz equation. It writes as:

(2.1) ∆uk ` k2npxquk “ Skpxq .

In the equation, uk is the wave-field, with the superscript, k ą 0 represents the wave number (thatcarries the frequency information, and thus in the paper we use the two words interchangeably).npxq is a complex-valued refractive index having non-negative imaginary part, Impnpxqq ě 0, re-flecting the heterogeneity of the medium. We assume npxq is the constant one in all Rd except in aconvex bounded open set Ω Ă Rd, meaning supppn´ 1q Ă Ω. In order to streamline the notation,we let Ω “ B1, the ball with radius 1 centered around the origin. The right-hand side Skpxq is thesource term, which is wave-number dependent.

The classical setup for the scattering problem is to probe the medium with an incident wave-fieldui,k that triggers the response from the medium. Noting that the total field, which satisfies (2.1),is the sum of the incident and the scattered wave-fields, we can write:

uk “ ui,k ` us,k ,

and derive the equation for the scattered wave-field us,k. Suppose the incident wave is designed sothat it absorbs all the external source information:

(2.2) ∆ui,k ` k2ui,k “ Skpxq ,

then by simply subtracting it from (2.1), we have the equation for us,k:

(2.3)

∆us,k ` k2npxqus,k “ k2p1´ npxqqui,k x P Rd ,Bus,k

Br´ ikus,k “ Opr´pd`1q2q as r “ |x| Ñ 8 .

In this equation, we can view the incident wave ui,k impinging in the perturbation n ´ 1 as thesource term for us,k. Clearly, this source term k2p1 ´ npxqqui,k is zero outside B1, the support ofn´ 1. The Sommerfeld radiation condition is imposed at infinity to ensure the uniqueness for us,k.

INVERSE SCATTERING IN THE HIGH-FREQUENCY LIMIT 5

When d “ 3, a typical approach is to set Spxq “ δy, a point Dirac delta, then the solution ui,k

to (2.2) becomes the fundamental solution to the homogeneous Helmholtz equation in R3

Φpx; yq “ ´1

4π

exppik|x´ y|q

|x´ y|, x, y P R3, x ‰ y ,

for any given y. We can clearly observe that the function is radially symmetric centered in ythus it is often termed a spherical wave. If |y| " |x|, i.e., y is far away from the origin, we havethe far-field regime, in which the fundamental solution is approximately a plain wave: Φpx; yq «

´ eik|y|

4π|y| expp´iky ¨ xq.In this case, however, instead of using the Dirac delta, we handcraft a specially designed source

term, which will be crucial for the re-scaling proposed in this article. In particular, we choose SkHpxqto be the following:

(2.4) SkHpx;xs, vsq “ ´k3`d2 Svspkpx´ xsqq x P Rd ,

where the subscript H stands for Helmholtz, and

(2.5) Svspxq “ Cpσ, dq exp

ˆ

´σ2 |x|2

2` ivs ¨ x

˙

.

Here Cpσ, dq is the normalization constant Cpσ, dq “?

2´

σ?π

¯d`12

.

Physically this source term can be understood as the source generating a tight beam being shoneonto the medium from the location xs in the direction of vs. The profile of this tight beam, or“laser beam”, is a Gaussian centered around the light-up location xs and the width of the Gaussianis characterized by pkσq´1. With σ fixed, as k Ñ8, the beam is more and more concentrated.

Following the explanation above we incorporate the source term in (2.4) into (2.1)-(2.2), toprobe the medium from the positions, xs, in the direction of vs, that are physically pertinent. Inparticular, we let pxs, vsq P Γ´ where

Γ˘ “ tpx, vq P BB1 ˆ Sd´1 : ˘v ¨ νpxq ą 0u .

In which, νpxq denotes the outer-normal direction at x P BB1. This means the laser beams shinefrom the boundary of B1 in the direction v pointing inward the interior of the domain.

From (2.4) we can observe that as k Ñ 8, the laser beam becomes increasingly concentrated.In particular, in the k Ñ 8 limit, the incident wave ui,k becomes a ray, propagating through astraight line3.

As usual in inverse problems (in particular, in non-intrusive experimental setups), we takemeasurements of uk near the boundary BB1. To take such measurement we design a family of testfunctions of the form:

(2.6) φkvpxq “ kd4χ´?

kx¯

e´ikv¨x ,

where χ : Rd Ñ R is a smooth radially symmetric function that vanishes as |x| Ñ 8.We define the measurement of uk as its Husimi transform

(2.7) Hkukpx, vq “

ˆ

k

2π

˙d ˇˇ

ˇuk ˚ φkv

ˇ

ˇ

ˇ

2for px, vq P Γ`.

3The incoming ray propagates in a straight line due to the assumption that the background is constant. Otherwise,the ray would bend if a smooth non-constant background is considered.

6 S. CHEN, Z. DING, Q. LI, AND L. ZEPEDA-NUNEZ

The measurement then consists of the intensity of the field that convolves with the test function.This measurement is conducted only on the boundary, and only in the directions pointing outsidethe domain.

This measurement has a clear physical interpretation: it measures the intensity of the wave-fieldat location x propagating in direction v, using χ as the impulse response of the receiver, or testfunction.

One typical choice for the family of test functions is to set χ as a Gaussian (normalized in L2

norm)

(2.8) χpxq “

ˆ

1

π

˙d4

exp

ˆ

´|x|2

2

˙

.

It is straightforward to see that as k Ñ 8, the test function φkv concentrates around zero due tothe

?k scaling. As such, the measurement uk ˚ φkv at a location xs only takes value of uk in a very

small neighborhood around xs.

Remark 2.1. We note that the choice of χ in (2.8) is not essential. We use this specific form tomake the calculation explicit, as it will be shown in Proposition 3.4. Other forms of χ would alsowork well as long as the corresponding Gk “ W krφk0s converges to a Dirac delta when k Ñ 8, asit will be explained in Remark 3.5.

Forward Map: now we have all the elements to define the forward map. For any pxs, vsq P Γ´,we shine laser beam into B1 according to the format in (2.4), then the solution to the Helmholtzequation (2.1), uk is tested by φkvpxq and evaluated on Γ`:

(2.9) Λkn : SkHpx;xs, vsq Ñ Hkukpxr, vrq|Γ` .

As a consequence, the dataset generated by this forward map is the collection of:

(2.10) Dkrns “!´

SkHpx;xs, vsq,ΛknrS

kHspxr, vrq

¯

: pxs, vsq P Γ´, pxr, vrq P Γ`

)

.

We now formulate the generalized inverse scattering problem as

(2.11) to reconstruct n using the information in Dkrns.

2.1.1. Traditional inverse scattering problem. Given that we use a non-standard formulationof the inverse scattering problem, we will stress a couple of similarities and differences between thegeneralized and classical inverse scattering problems.

In particular, the form of the forward map introduced in our setting differs from the classicalone, where the incident wave is typically a plane wave, meaning ui,kpx; vsq “ exppikvs ¨ xq, wherevs P Sd´1, see [30].

So the forward map is given by the far field map, rΛkn:

rΛkn : ui,kpx; vsq Ñ u8,kpx; vsq ,

where u8,k : Sd´1 Ñ C is defined as

u8,kpx; vsq “ limrÑ8

rus,kprx; vsq expp´ikrq|xPSk´1 , @x P Sd´1 ,

with us,k being the solution of (2.3), where we leverage that ui,kpx; vsq satisfies (2.2) with S “ 0.Therefore in this setting, the data set induced by the forward map is defined as:

rDkrns “!´

ui,kpx; vsq, rΛkn

”

ui,kı

pxq¯

: vs P Sd´1, x P Sd´1)

.

INVERSE SCATTERING IN THE HIGH-FREQUENCY LIMIT 7

The well-posedness and stability of the inverse scattering problem in this context has been studiedin [27, Theorem 1.2].

The differences from the classical inverse scattering formulation is two fold: i) we use a richer setof probing functions, instead of using incident waves that are directionally localized (as plane waves)or whose sources are localized (as Green’s functions), we use tight beams that combine these twoproperties, and ii) instead of measuring the scattered wave-field on a manifold around the domainof interest, we multiply it with a set of directional filters localized on the same manifold, and wecompute its intensity. We should emphasize that this difference is significant. Take the plane-waveas the probing wave, as an example, it is only the direction of the incoming wave that can betuned, and this composes 2 dimensions of degrees of freedom in 3D with vs P Sd´1. The way oursource term is designed automatically carries 4 dimensions of degrees of freedom with pxs, vsq P Γ´.Similarly, the way data gets taken also expands the degrees of freedom the measuring operator canaccess. It is a widely accepted fact that more data leads to more stable reconstruction. This willbe indeed demonstrated in the later sections.

Remark 2.2. We note that even though the conventional inverse scattering problem has beenshown to be ill-conditioned, a couple of strategies have been introduced in the literature to stabilizethe problem. The most prominent strategy is to add the phase information (microlocally) [6, 45, 17].At the first look, the Husimi data (2.7) also extracts the phase information, by integrating thescattered wave with an oscillatory test function (2.6) that is localized in position and direction. Invery simple cases, we can even show that the two sets of information is equivalent. For example,suppose the wave field is of the simple form of ukpxq “ Apxqeikp¨v with p P Sd´1 and Apxq ě 0, forall x P Rd. Then in the limit k Ñ8, we can fully recover ukpxq, both the amplitude and the phase,on the boundary BB1 using the Husimi data (2.7)

limkÑ8

Hkukpx, vq “ |Apxq|2δpv ´ pq , @px, vq P BB1 ˆ Sd´1 .

However, in general cases, we are not aware of results that translate Husimi data to the phase data.Indeed, according to [25, 26, 1], this might be a very complicated phase retrieval problem that isbeyond the scope of the current paper.

Remark 2.3. Another strategy to stabilize the inverse scattering problem is to transform theHelmholtz equation back to the time-domain, and solve the inverse acoustic wave problem, witheither full or partial data available for all time T ě 0. In various settings [28, 46, 54], it isproved that the time-domain data is sufficient to reconstruct the medium. The wave equation andHelmholtz equation are Fourier transform of each other in time. Roughly speaking, the temporaldata collected on the boundary translates to the boundary information for all frequency k. Assuch, the temporal data has wide-band information instead of being monochromatic, and thus isexpected to be more stable. In our setting, though we require k " 1, we still use monochromaticinformation, and thus the data does not directly translate.

We should note, however, that though the time-domain data is expected to be more informativein theory, in practice, however, especially within the optimization-based reconstruction algorithmframework, the typical `2 misfit loss function results in an extremely non-linear problem that oftenleads to cycle-skipping, and convergence to spurious, non-physical, local minima. The numericalproblem is usually attenuated by using the time/frequency duality and localizing the frequencycontent of the data, which is then processed in a hierarchical fashion [15, 40]. These are beyondthe focus of the paper.

2.2. High-frequency limit and inverse Liouville scattering problem. The Liouville equation isa well studied classical model for describing particle propagation. Any system with a large number

8 S. CHEN, Z. DING, Q. LI, AND L. ZEPEDA-NUNEZ

of identical particles can be described by the Liouville equation, or its variants, which is oftenwritten as:

(2.12) v ¨∇xf `1

2∇xn ¨∇vf “ SLpx, vq ,

where fpx, vq characterizes the number of particles on the phase space px, vq. Following the char-acteristics, we see that the particles follow Newton’s second law:

9x “ v , 9v “1

2∇xn .

As usual in classical mechanics, we can define the Hamiltonian for each particle to be

Hpxptq, vptqq “ 2|vptq|2 ´ npxptqq ,

which is preserved along the characteristics of the particles, i.e., dHdt “ 0.

We use (2.12) to describe photon propagation, and use the same setup as that in Section 2.1.The source term SLpx, vq on the right-hand side of (2.12) describes how laser beams are shone intothe medium, and takes the form of:

(2.13) SLpx, v;xs, vsq “ φpx´ xsqψpv ´ vsq , with pxs, vsq P Γ´ ,

where both φ : Rd Ñ R and ψ : Rd Ñ R are radially symmetric smooth functions that concentrateat the origin. By setting pxs, vsq P Γ´, we have the laser beam shining from the boundary BB1

inward to the domain. The concentration of the beam is determined by φ and ψ in physical- andvelocity-space respectively.

Similar to the previous section, we take the measurements of the light intensity at the boundarypointing outside of the domain. To do so, we set the test function ζpx, vq and the measurementswould be its convolution with the solution to (2.12):

(2.14) Lfpx, vq “ f ˚ ζpx, vq .

The physical setup is clear. Imaging ζ a blob centers around px, vq “ p0, 0q, then Lfpxr, vrqessentially represents a measuring equipment that takes in light intensity concentrated aroundpxr, vrq with the concentration determined by the size of the blob. The specific format of ζ will bespecified in Section 3.Forward Map: we define the forward map in a similar fashion as in Section 2.1. For any pxs, vsq PΓ´, we solve (2.12) with SL defined in (2.13), and test the solution on ζpx, vq evaluated on Γ`:

Λn : SLpx, v;xs, vsq Ñ Lfpxr, vrq|Γ` .

As a consequence, the dataset generated by this forward map is the collection of:

(2.15) Drns “ tpSLpx, v;xs, vsq,ΛnrSLspxr, vrqq : pxs, vsq P Γ´, pxr, vrq P Γ`u .

While the forward problem is to compute and construct this Drns for any given n, the inverseproblem amounts to inferring n using the information in Drns.

3. Relation between the two problems in the high-frequency regime. In this section wediscuss the connection between the forward maps for the wave- and particle-like descriptions intro-duced in the section above. We start introducing the Wigner transform, and we use it to presentthe equivalence of the two descriptions for the forward maps in the high-frequency regime. Thenwe introduce the Husimi transform to take the limit of the measuring operator, and this is used toshow the equivalence of the two inverse problems. Finally, we briefly introduce the stability of theinverse Liouville problem.

INVERSE SCATTERING IN THE HIGH-FREQUENCY LIMIT 9

3.1. High-frequency limit of the forward problem. We first present their connection in theforward setting. We discuss the derivation of the Liouville equation as the limiting equation forthe Helmholtz. This process is typically called taking the “classical”-limit, to reflect the passagefrom quantum mechanics to classical mechanics by linking the Schrodinger equation to the Liouvilleequation in the small ~ regime.

Among the multiple techniques to derive the classical limit we utilize the Wigner transform [24,43, 3, 14]. Compared to other techniques, such as WKB expansion [23] and Gaussian beam ex-pansion [47, 33, 41] , Wigner transform presents the equation on the phase space, and avoids theemerging singularities during the evolution. Let uk1 and uk2 be two functions, then the correspondingWigner transform is defined as

(3.1) W kruk1, uk2spx, vq “

1

p2πqd

ż

Rdeiv¨yuk1

´

x´y

2k

¯

uk2

´

x`y

2k

¯

dy .

Here uk2 is the complex conjugate of uk2. We furthermore abbreviate W kruk1, uk2s to be W kruks.

The Wigner transform W kruks is defined on the phase space, is always real-valued, and themoments in v of W kruks carry interesting physical meanings. In particular, the first momentrecovers the energy density Ek:

(3.2) Ekpxq “ż

RdW krukspx, vqdv “

ˇ

ˇ

ˇukpxq

ˇ

ˇ

ˇ

2,

and its second moment expresses the energy flux Fk:

(3.3) Fkpxq “

ż

RdvW krukspx, vqdv “

1

kIm

´

ukpxq∇xukpxq

¯

.

Most importantly, if uk solves the Helmholtz equation (2.1), one can show that W kruks solvesan equation in the form of the radiative transfer equation, and in the k Ñ8 limit, this degeneratesto the Liouville equation (2.12). In what follows we seek to make this statement more precise bydefining the functional space and an appropriate metric.

Let λ ą 0, we define Xλ a space that contains all scalar real valued functions defined on thephase-space R3 ˆ R3:

(3.4) Xλ “

"

φpx, yq

ˇ

ˇ

ˇ

ˇ

ż

R3

supxPR3

p1` |x| ` |ξ|q1`λ|φpx, ξq|dξ ă 8

*

,

with associated norm given by

φXλ “

ż

R3

supxPR3

p1` |x| ` |ξ|q1`λ|φpx, ξq|dξ ,

where φpx, ξq “ 1p2πqd

ş

Rd φpx, yqe´iξ¨ydy is the Fourier transform in velocity-space. Now we cite a

result from [7, Theorem 3.11, 3.12].

Theorem 3.1. Let npxq be a C2pRd;R`q function that satisfies certain conditions (see Remark 3.2).Let uk be the solution to (2.1) with radiation conditions, where the source term SkH is defined in (2.4).Then the Wigner transform of uk, denoted by fkpx, vq “W krukspx, vq solves

(3.5) v ¨∇xfk `

1

2Lknrfks “ ´

1

kIm

´

W kruk, Sks¯

, px, vq P R2d ,

10 S. CHEN, Z. DING, Q. LI, AND L. ZEPEDA-NUNEZ

with the operator Lkn defined as

(3.6) Lknrfks :“i

p2πqd

ż

R2d

δkrnspx, yqfkpx, pqeiypv´pq dy dp .

Here δkrnspx, yq “ k“

n`

x` y2k

˘

´ n`

x´ y2k

˘‰

. Furthermore, when k Ñ8, fk converges in weak-‹sense to fpx, vq in pXλq

‹, the solution to the Liouville equation (2.12) with the radiation conditionlim|x|Ñ8 fpx, vq “ 0 for all x ¨ v ă 0, and the source SLpx, vq is:

(3.7) SLpx, vq “ p2πqdπ

2δpx´ xsq|Svspvq|

2δ`

|v|2 “ npxsq˘

.

Here Svs denotes the Fourier transform, and the delta function δ`

|v|2 “ npxsq˘

P D1pRdq means

xδ`

|v|2 “ npxsq˘

, gy “

ż

|v|2“npxsqgpvqdSv, @g P SpRdq .

Suppose Sv takes the form of (2.4), we can explicitly calculate its Fourier transform:

|Svspvq|2 “ Cpσ, dq2

1

p2πqdσ2de´

|v´vs|2

σ2 .

Remark 3.2. The formal derivation of the limit is shown in Appendix A. To prove it rigorously,we refer to [7, Theorem 3.11, 3.12] and [11]. The conditions for a rigorous proof are rather com-plicated to obtain. However, we mention that if n is radially symmetric, i.e., npxq “ np|x|q, thestatement of the theorem holds true rigorously.

Theorem 3.1 suggests that the wave model and the particle model are asymptotically equivalentin the high-frequency regime. According to (3.7), the source term concentrates at pxs, vsq, the sourcelocation and the source velocity, when k Ñ8. The concentration on x is already achieved by takingto limit as k Ñ8, but the concentration profile in v still needs to be tuned by σ. Smaller σ resultsin a more concentrated source in this limiting regime. Let σ Ñ 0, we have the source term SL

turning into:

(3.8)p2πqd

π

2δpx´ xsq|Svspvq|

2δp|v| “ 1q “ δpx´ xsq

ˆ

1

σ?π

˙d´1

e´|v´vs|

2

σ2 δp|v| “ 1q

Ñ δpx´ xsqδpv ´ vsq ,

where we used npxsq “ 1, given that xs is out of the domain interest B1.In this specific limit, we have the explicit solution to the Liouville equation (2.12):

(3.9) fpx, vq “ δpxps;pxs,vsqq,vps;pxs,vsqqq , k Ñ8 ,

where pxps; pxs, vsqq, vps; pxs, vsqqq are the location and velocity of a particle at time s that startsoff at pxs, vsq, meaning pxp0; pxs, vsqq, vp0; pxs, vsqqq “ pxs, vsq and

(3.10)

$

’

&

’

%

dxps; pxs, vsqq

ds“ vps; pxs, vsqq ,

dvps; pxs, vsqq

ds“

1

2∇xnpxps; pxs, vsqqq .

The formulation in (3.9) means in this limit, with k Ñ 8 and σ ! 1, the wave becomes a curvedray that follows the trajectory of the particle that is governed by Newton’s laws. As a consequence,recall the definition of energy and energy flux in (3.2)-(3.3):

limσÑ0

limkÑ8

Ekpxq “ 1są0δxps;pxs,vsqq, limσÑ0

limkÑ8

Fkpxq “ 1są0δxps;pxs,vsqqvps; pxs, vsqq ,

suggesting that Ek and Fk respectively show approximately the location and velocity of the trajec-tory.

INVERSE SCATTERING IN THE HIGH-FREQUENCY LIMIT 11

3.2. High-frequency limit of the inverse problem. In the prequel we linked the two forwardproblems. We now proceed to connect the two inverse problems, by evaluating the convergence ofthe measurements. To do so, we first introduce Lemma 3.3 from [20, Section 2.5] that connects theHusimi and Wigner transforms.

Lemma 3.3. Assume u P L2pRd;Rq , and let Hku be the Husimi transform defined in (2.7) withφkv being the test function (defined in (2.6)). Denote fk “ W krus, and Gk “ W krφk0s, the Wignertransform of uk and φk0 respectively. Here φk0 “ φkv“0. Then

(3.11) Hkupx, vq “ fk ˚Gkpx, vq , @px, vq P R2d .

Proof. This theorem is a directly result of the Moyal identity

(3.12) pW krh1s,Wkrh2sqL2pR2dq “

ˆ

k

2π

˙d

|ph1, h2qL2pRdq|2 , @h1, h2 P L

2pRd;Rq ,

and the fact that

(3.13) W krφkvpx´ ¨qspy, pq “W krφk0spx´ y, v ´ pq .

Using (2.7), we have

Hkupx, vq “

ˆ

k

2π

˙d ˇˇ

ˇu ˚ φkv

ˇ

ˇ

ˇ

2

“

ˆ

k

2π

˙d ˇˇ

ˇ

ˇ

´

up¨q, φkvpx´ ¨q¯

L2pRdq

ˇ

ˇ

ˇ

ˇ

2

“

´

W krus,W krφkvpx´ ¨qs¯

L2pR2dq

“

´

W krus,W krφk0spx´ ¨, v ´ ¨q¯

L2pR2dq

“ fk ˚Gk ,

where we use (3.12) in the third equality, (3.13) in the fourth equality, and the definitions of fk

and Gk in the last equality.

This lemma connects the measurement of uk with the measurement on the phase space. Testinguk using the test function φk0 is translated to testing fk using the test function Gk. This allowsus to pass to the limit on the phase space. Combining with Theorem 3.1, we have the followingproposition:

Proposition 3.4. Let the assumption in Theorem 3.1 hold true. Denote fk “ W kruks, with uk

solving the Helmholtz equation (2.1) with the source term SH defined in (2.4), and denote f thesolution to the Liouville equation (2.12) with source term SL defined in (3.7). If χ takes the formof (2.8), so that Gk takes the form of:

(3.14) Gkpx, vq “

ˆ

k

π

˙d

exp`

´k`

|x|2 ` |v|2˘˘

,

as k Ñ8, we have:fk ˚Gkpx, vq Ñ fpx, vq

weak-‹ in pXλq‹.

12 S. CHEN, Z. DING, Q. LI, AND L. ZEPEDA-NUNEZ

Proof. Given the form of Gk in (3.14), for any φ P Xλ, as k Ñ8:

Gk ˚ φpx, vq ÝÑ φpx, vq in Xλ .

Thus,

limkÑ8

ż

R3ˆR3

´

fk ˚Gkpx, vq¯

φpx, vqdx dv “ limkÑ8

ż

R3ˆR3

fkpx, vq´

Gk ˚ φpx, vq¯

dx dv

“ limkÑ8

ż

R3ˆR3

fkpx, vqφpx, vqdx dv

“

ż

R3ˆR3

fpx, vqφpx, vqdx dv ,

where we use fkpXλq˚ being bounded in the second equality, and fk Ñ f in the weak-‹ sense inthe last equality.

Remark 3.5. We note that the statement of the proposition indeed uses the explicit form ofχ as defined in (2.8), but the use only lies in the fact that Gk ˚ φpx, vq ÝÑ φpx, vq in the highfrequency limit. Other forms of χ works equally well as long as this Gk serves as a delta measurewhen k Ñ8.

Theorem 3.6. Let the assumptions in Theorem 3.1 and Lemma 3.3 hold true, then:

limkÑ8

Hkukpx, vq “ limkÑ8

fk ˚Gkpx, vq ÝÝÝÝÑweak´‹

fpx, vq,

in pXλq˚. Furthermore, if Hkuk and f are continuous, then each element in Dkrns has a limit in

Drns. More specifically:

(3.15) pSkHpx;xs, vsq ,ΛknrS

kHspxr, vrqq Ñ pSLpx, v;xs, vsq ,ΛnrSLspxr, vrqq

where SL takes the form of (3.7), and ΛnrSLspxr, vrq “ fpxr, vrq. In particular, if σ Ñ 0,

(3.16) ΛnrSLspxr, vrq “ f ˚ δp~0,~0q|Γ` “ fpxr, vrq|Γ` “ δpx´ xrsqδpv ´ vrsq ,

with pxrs , vrsq being the outgoing location and velocity when the photon particle leaves the domain,namely:

(3.17) xrs “ xpS; pxs, vsqq, vrs “ vpS; pxs, vsqq ,

where S “ supsě0 ts|xps; pxs, vsqq P B1u and txps; pxs, vsq, vps; pxs, vsqqu solves (3.10).

This theorem naturally links the two inverse problems. In the k Ñ 8 limit, the two datasets(2.10),(2.15) are asymptotically close with ζ “ δ

p~0,~0qpx, vq in (2.14). In the limit of k Ñ 8 and

σ Ñ 0, the dataset (2.10) is asymptotically approximately equivalent to

(3.18) D8rns “ tppxs, vsq, pxr, vrqq : pxs, vsq P Γ´, pxr, vrq from (3.17)u .

3.3. Stability of Liouville inverse problem. In this section, we consider the stability of Liouvilleinverse problem. In particular, we focus on the stability of (3.18). We will show that when n isclose enough to 1, D8n almost contains the information of the X-ray transforms of npxq and ∇xnpxq,while the inverse of X-ray transform is a well-posed inverse problem.

INVERSE SCATTERING IN THE HIGH-FREQUENCY LIMIT 13

We first introduce the X-ray transform. Define

TSd´1 “

!

px, vqˇ

ˇ

ˇx P Rd, v P Sd´1, xv, xy “ 0

)

.

Assuming that npxq is continuous, we introduce the X-ray transform P , which maps npxq,∇xnpxqinto functions Pn P CpTSd´1,Rq and P p∇xnq P CpTS

d´1,Rdq, such that

Pnpv, xq “

ż 8

´8

nptv ` xqdt, P p∇xnqpv, xq “

ż 8

´8

∇xnptv ` xqdt.

To connect D8n with X-ray transform, we define a projection map P : BB1 ˆ Sd´1 Ñ Rd ˆ Sd´1

Pppx, vqq “ px´ xx, vy v, vq

that projects x to the plane with normal vector v. We also define in-out map L : Γ´ Ñ Γ`corresponding to (3.17):

Lppxs, vsqq “ pxr, vrq .

Remark 3.7. We remark that the in-out map may not be well-defined for arbitrarily given n.Suppose npxq ě c0 for all x P Rd and some c0 ą 0, then according to the conservation of Hamiltonian

Hpx, vq “1

2|v|2 ´

1

2npxq “

1

2´

1

2“ 0 ,

the velocity of the particle satisfies

|vpsq| “a

npxpsqq ě?c0 ą 0 ,

for all time s ě 0. This by no means suggests the non-trapping property, but it at least ensuresthat the potential is not a sink. In the general case, we do assume that n is non-trapping, sothat any incoming particle can eventually be expelled out of the domain again, making the map Lwell-defined. Such non-trapping condition is closely related to geodesic X-ray transforms, and welist references [45, 17, 34] for interested readers.

We note that Pppx, vqq P TSd´1 for any px, vq P BB1ˆSd´1, and P|Γ´ : Γ´ Ñ RdˆSd´1,P|Γ` :Γ` Ñ Rd ˆ Sd´1 are invertible. Now, we are ready to introduce the following approximationtheorem [37, Theorem 4.1]:

Theorem 3.8. Assume

∇npxqL8 ď ∆, HnpxqF L8 ď ∆

for some ∆ ą 0, then for any pv, xq P TSd´1, we haveˇ

ˇ

ˇpPnpv, xq, P p∇xnqpv, xqq ´ P|Γ` ˝ L ˝

`

P|Γ´˘´1

pv, xqˇ

ˇ

ˇď C∆2 ,

where C ą 0 is a constant only depends on d.

According to Theorem 3.8, if n is almost a constant (close enough to 1), then we can use the dataset to recover X-ray transform of n,∇npxq. Thus, we can separate (3.18) into two inverse problems

D8rns ùñ pPnpv, xq, P p∇xnqpv, xqq ùñ npxq ,

where the first one can be approximately calculated if n is almost constant 1 and the second one isthe inverse of X-ray transform that is well-posed according to [36, Theorem 5.1].

14 S. CHEN, Z. DING, Q. LI, AND L. ZEPEDA-NUNEZ

4. Numerical experiments. In this section we provide numerical evidence showcasing the the-ory developed above. In particular, we would like to demonstrate that as k increases, the mea-surement taken on the solution to the Helmholtz equation through the Husimi transform convergesto the pointwise evaluation of the solution to the Liouville equation, and that the data becomesmore and more sensitive to the perturbation in media, making the inverse problem more and morestable.

We first summarize the numerical setup and unify the notations, and then present a class ofnumerical results.

4.1. Numerical setup. We set up our experiment in a two dimensional domain that takes theform of:

(4.1) ∆uk ` k2npxquk “ ´k52Svspkpx´ xsqq, x P R2 .

The Sommerfeld radiation condition is imposed at infinity as well. The source term is given by

(4.2) Svspxq “?

2

ˆ

σ?π

˙32

exp

ˆ

´σ2 |x|2

2` ivs ¨ x

˙

,

for pxs, vsq P Γ´. We denote the solution to (4.1) by ukxs,vs whenever the source center and theincident direction are relevant for the discussion. The Husimi transform defined in (2.7) takes theform

(4.3) Hkukpxr, vrq “

ˆ

k

2π

˙d ˇˇ

ˇuk ˚ φkvrpxrq

ˇ

ˇ

ˇ

2,

with pxr, vrq P Γ` . We let the refractive index npxq set to be npxq “ 1` qpxq with the support ofthe heterogeneity qpxq Ă Bprq. The measurement is taken on BBpRq with R ą r. See Figure 2 foran illustration of the configuration.

Computationally we set the domain D “ r´L2, L2s2, with L significantly bigger than R, andchoose the spatial mesh size h “ 1N with N being an even integer. For simplicity of representation,we use the angles θs and θr to denote the center of the sources and the center of the receivers,respectively, and the angles θi and θo are used to denote the incident and outgoing direction of thesources and receivers, respectively, so that:

(4.4)xs “ pR cos θs, R sin θsq ,

vs “ p´ cospθs ` θiq,´ sinpθs ` θiqq ,

and

(4.5)xr “ pR cospθs ` θrq, R sinpθs ` θrqq

vr “ pcospθs ` θr ` θoq, sinpθs ` θr ` θoqq .

The angles θi and θo take values in r0, 2πq, whereas the angles θi and θo take values in p´π2 ,

π2 q.

An illustration of the angles can be found in Figure 2. Since the mapping between pθs, θi, θs, θoq

and the corresponding pxs, vs, xr, vrq is one-to-one, we present the quantities uk and Hkuk on the θcoordinate system whenever there is no confusion.

The angles are discretized with step size ∆θ and the angular grids are denoted by θjs , θjr “ j∆θ

for all j “ 0, ¨ ¨ ¨ , 2π∆θ ´ 1, and θji , θjo “ ´

π2 ` j∆θ for all j “ 1, ¨ ¨ ¨ , π∆θ ´ 1.

INVERSE SCATTERING IN THE HIGH-FREQUENCY LIMIT 15

𝐵𝐵(𝑅𝑅)

𝐵𝐵(𝑟𝑟)

D

𝜃𝜃𝑠𝑠

𝜃𝜃𝑖𝑖

𝑥𝑥𝜃𝜃𝑟𝑟

𝜃𝜃𝑜𝑜

Figure 2: (left) illustration of the setup for numerical experiments, (right) sketch of the definitionof the angles on the circle BBpRq used to parameterize the data.

To compare the Husimi transform of the solutions, we further define two quantities. The firstquantity is the Husimi transform integrated in the outgoing direction

(4.6) Mko pxs, vs, xrq :“

ż

S`xrHkukxs,vspxr, vrqdvr “

ż π2

´π2Hkukθs,θipθr, θoqdθo ,

where S˘xr “ tv P S1 : ˘νpxrq ¨ v ą 0u and νpxq is the unit outer normal vector at x P BΩ. Similarly,

we also define the Husimi transform integrated along the outgoing boundary

(4.7) Mkr pxs, vi, vrq :“

ż

BΩ`vr

Hkukxs,vipxr, vrqdxr “

ż

p´π2`θor,π2`θorqHkukθs,θipθr, θor ´ θrqdθr ,

where we denote θor “ θo ` θr P r0, 2πq, and define BΩ˘vr “ tx P BΩ : ˘νpxq ¨ vr ą 0u.To solve the Helmholtz equation (4.1), we use the truncated kernel method [49], and solve

for the Lippmann-Schwinger equation to obtain the scattered field us,k. This allows us to pushfor high-frequency without suffering from the numerical pollution that Finite Differences or FiniteElements methods often have. The scattered field is then combined with the incident field ui,k toyield uk.

4.2. Numerical examples. In the first example, we set L “ 1, R “ 0.3 and r “ 0.25. For themedium, we set the heterogeneity to be the radially symmetric smooth function

(4.8) qpxq “

#

A exp´

´ 11´|x|2r2

¯

, |x| ă r ,

0 , otherwise .

Clearly, the support of qpxq is contained in Bprq; see Figure 3. For the source term, we fix σ “ 2´5

in the following experiments. Noting that the medium npxq is radially symmetric, one can studythe scattered data for a fixed source location. We choose θs “ π4; see Figure 3. For discretization,we choose spatial step size h “ 1p2kq in the truncated kernel solver, and ∆θ “ π30 for the angulargrids.

We first show the solution’s behavior as k increases in Figure 4. As k increases, the solutionconverges to a narrow beam that follows the characteristic equation (3.10).

We compute the Husimi transform Hkuk for different k and we compare them with the trajec-tories of the Liouville equation. The results are shown in Figure 5, where we can observe that for

16 S. CHEN, Z. DING, Q. LI, AND L. ZEPEDA-NUNEZ

-0.4 -0.2 0 0.2 0.4

x

-0.4

-0.2

0

0.2

0.4

y

0.5

0.6

0.7

0.8

0.9

1

-0.4 -0.2 0 0.2 0.4

x

-0.4

-0.2

0

0.2

0.4

y

0

1

2

3

4

5

6

105

Figure 3: The left plot illustrates the medium npxq “ 1 ` qpxq in (4.8) with A “ ´0.5. The rightplot shows the amplitude of source |Svspkpx´ xsqq| with k “ 211, σ “ 2´5 and θs “ π4.

-0.4 -0.2 0 0.2 0.4x

-0.4

-0.2

0

0.2

0.4

y

-3

-2

-1

0

1

2

3

-0.4 -0.2 0 0.2 0.4x

-0.4

-0.2

0

0.2

0.4

y

-4

-3

-2

-1

0

1

2

3

4

-0.4 -0.2 0 0.2 0.4x

-0.4

-0.2

0

0.2

0.4

y

-6

-4

-2

0

2

4

6

-0.4 -0.2 0 0.2 0.4x

-0.4

-0.2

0

0.2

0.4

y

-6

-4

-2

0

2

4

6

-0.4 -0.2 0 0.2 0.4x

-0.4

-0.2

0

0.2

0.4

y

-6

-4

-2

0

2

4

6

-0.4 -0.2 0 0.2 0.4x

-0.4

-0.2

0

0.2

0.4

y

-6

-4

-2

0

2

4

6

Figure 4: The real part of uk for k “ 29 (left), k “ 210 (middle) and k “ 211 (right). The blue linesshow the Liouville trajectory that solves (3.10). The medium (4.8) has amplitude A “ ´0.5. Theincident direction θi “ 0 (upper) and θi “ ´π6 (lower).

a fixed θi, Hkuk converges to a delta function on the θr-θo plane, as k increases. This agrees with

the statement in Theorem 3.1, especially equation (3.16).We then compare the integrated Husimi transform defined in (4.6) and (4.7). In Figure 6 and

Figure 7, we demonstrate the convergence of Mko and Mk

r as k increases.As k increases, the outgoing data becomes more and more sparse, and fewer and fewer detectors

can receive outgoing light, leading to the sparser matrix presentation of Λkn (see definition in (2.9).This is shown in Figure 8 for different k.

INVERSE SCATTERING IN THE HIGH-FREQUENCY LIMIT 17

100 150 200 250

r

-60

-30

0

30

60

o

20 40 60 80 100 120 140

100 150 200 250

r

-60

-30

0

30

60

o

100 200 300 400 500

100 150 200 250

r

-60

-30

0

30

60

o

500 1000 1500

50 100 150

r

-30

0

30

60

90

o

100 200 300 400 500

50 100 150

r

-30

0

30

60

90

o

500 1000 1500

50 100 150

r

-30

0

30

60

90

o

1000 2000 3000 4000 5000

Figure 5: The Husimi transform Hkuk for k “ 29 (left), k “ 210 (middle) and k “ 211 (right).The upper row shows the results with θi “ 0, and the lower row shows the results with θi “ ´π6.The red crosses show the outgoing position and direction (3.17) of the Liouville trajectory. Themedium (4.8) has amplitude A “ ´0.5.

0 50 100 150 200 250 300 350

r

-90

-60

-30

0

30

60

90

i

20 40 60 80 100 120

0 50 100 150 200 250 300 350

r

-90

-60

-30

0

30

60

90

i

100 200 300 400 500 600 700 800

Figure 6: The averaged Husimi transform Mko for k “ 29 (left) and k “ 211 (right). The red lines

show the outgoing position (3.17) of the Liouville trajectory. The medium (4.8) has amplitudeA “ ´0.5.

Finally we compare the change of Λkn as n differs, for different k. Let n0pxq “ 1 as thebackground media whose corresponding map is denoted Λk0, and by adjusting A we design a sequenceof npxq. We measure how the Frobenius norm Λkn ´ Λk0F changes with respect to n´ n0L8 fordifferent k. As can be seen in Figure 9, as k increases, the slope of Λk ´ Λk0F as n´ n0L8 Ñ 0increases. This confirms that bigger k sees more sensitivity of the data when n changes, hence the

18 S. CHEN, Z. DING, Q. LI, AND L. ZEPEDA-NUNEZ

0 100 200 300

or

-90

-60

-30

0

30

60

90

i

100 200 300 400 500

0 100 200 300

or

-90

-60

-30

0

30

60

90

i

500 1000 1500 2000 2500 3000

Figure 7: The averaged Husimi transform Mkr for k “ 29 (left) and k “ 211 (right). The red lines

show the outgoing direction (3.17) of the Liouville trajectory. The medium (4.8) has amplitudeA “ ´0.5.

0 500 1000 1500

0

200

400

600

800

1000

1200

1400

1600

Figure 8: Sparsity of the matrix Λkn for k “ 24 (left) and k “ 211 (right). Rows represent differentpθr, θoq, and columns represent different pθs, θiq. Elements that are larger than half of the maximalelement in Λkn are shown. For k “ 24, we use larger computational domain r´8, 8s2, and the stepsize is h “ 2´8.

reconstruction is expected to be better for higher k.

5. Inversion Algorithm. The inverse problem that we study in this article has a different setupfrom the conventional one. While the conventional setup has either the concentration in the in-coming direction, or in the incoming source location, our experimental setup requires concentrationin both direction and source location. Naturally we expect a better stability in the reconstructionprocess, compared to the traditional formulation. In this section we showcase such stability.

Numerically the reconstruction process is formulated as a PDE-constrained minimization prob-lem, where we seek to minimize the misfit between the data and the forward model:

(5.1) minn

›

›

›D ´Dkrns

›

›

›

2

L2pΓ´ˆΓ`q,

INVERSE SCATTERING IN THE HIGH-FREQUENCY LIMIT 19

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4104

Figure 9: The dependence of Λk ´ Λk0F on the medium perturbation n ´ n0L8 . Differentn´ n0L8 is obtained by tuning the amplitude A in the medium (4.8).

or equivalently, in the discretized form:

(5.2) minn

J rns, where J rns :“1

2nrcvnsrc

nrcvÿ

i“1

nsrcÿ

j“1

ˇ

ˇ

ˇ

ˇ

Di,j ´

´

Dkrns¯i,j

ˇ

ˇ

ˇ

ˇ

2

.

In particular, nrcv and nsrc stand for the number of receivers and sources, and each point pDkrnsqi,j

is the intensity squared of the impulse response generated by illuminating the medium n with atight beam given by (2.4) originated at xis with direction vis, which is then filtered using (2.6)centered at xjr with direction vjr . See definition in (4.3), with pxr, vrq replaced by pxjr , v

jr q, and uk

solving (4.1) with pxs, vsq replaced by pxis, visq.

We employ quasi-newton methods for finding a local minimum4, thus we need to efficientlycompute the gradient of the misfit function. In order to provide a fully self-contained exposition webriefly summarize below how to compute the gradient for only one data point using the adjoint-statemethods. From there the computation for the full gradient can be easily deduced.

We can readily compute the application of the gradient to a perturbation δn by using the chainrule, which results in

(5.3) ∇J rnsδn “ˆ

k

2π

˙d´

D ´Hkukpxr, vrq

¯

Real´

2puk ˚ φkvrpxrqqpφkvrpxrq ˚ F rnsδnq

¯

,

where F rns is linearized forward wave-propagation operator, given by the Born approximation ofthe scattered wave-field [9]. Thus the gradient can be easily computed by applying the adjoint ofthe Born approximation to the residual times the filter function, i.e.,

∇J rns “ 2

ˆ

k

2π

˙d

Real´

F rns˚´

`

D ´Hkukpxr, vrq˘

puk ˚ φkvrpxrqqpφkvrpxr ´ xqq¯¯

.

Fortunately, the application of the adjoint of the Born approximation operator is well studied: itcan be performed by solving the adjoint equation followed by a multiplication by the solution ofthe forward wave problem5. In this case the adjoint equation is the same Helmholtz equation, but

4Given that the problem is very non-linear, there is not guarantees that we can find the global minimum.5We redirect the interest readers to [8] for a modern self-contained presentation.

20 S. CHEN, Z. DING, Q. LI, AND L. ZEPEDA-NUNEZ

with adjoint Sommerfeld radiation conditions, i.e., we solve

(5.4)∆v ` k2npxqv “

´

D ´Hkukpxr, vrq

¯

puk ˚ φkvrpxrqqpφkvrpxr ´ xqq x P Rd ,

Bv

Br` ikv “ Opr´pd`1q2q as r “ |x| Ñ 8 .

Thus, using (5.4), we can easily compute the application of the adjoint of the Born approximation

(5.5) F rns˚´

`

D ´Hkukpxr, vrq˘

puk ˚ φkvrpxrqqpφkvrpxr ´ xqq¯

“ ukv.

where v solves (5.4).We point out that in (5.5), the source for the adjoint is conjugated, thus following (2.6), we can

see that it means that the pφkvrpx´ xrqq is pointing towards the interior of the domain in direction´vr.

We solve (5.2) using L-BFGS [10, 55], a quasi-Newton method in Matlab. We consider theinitial perturbation equal to zero. We set a first order optimality tolerance of 10´5 and let thealgorithm run for a maximum of 300 iterations or until the tolerance is achieved.

To avoid the inverse crime [19], the data is generated by solving the Lippmann-Schwingerequation discretized by the truncated kernel method [49] as in Section 4, and the inversion isperformed with an 4th-order finite difference scheme for both (4.1) and (5.4). To generate thedata, we set the computational domain to be K “ r´1, 1s2 with NLS “ 2562 “ 65536 grid pointsso that there are at least 12 points per wavelength for the largest k “ 26. In the inversion, wediscretize the same domain K with NFD “ 1632 “ 26569 grid points so that there are at least 8points per wavelength for k “ 26. We enclose the domain K with perfect matching layer (PML) toavoid reflection. We choose the thickness of PML to be 2.5 times wavelength.

The measurement is taken on BBpRq with R “ 0.4 in all the examples. To generate the probingray, we set σ “ 2´2 in (4.2). We compute the data with the source position and incident direction

xi1s “ pR cos θi1s , R sin θi1s q

vi1,i2s “ p´ cospθi1s ` θi2i q,´ sinpθi1s ` θ

i2i qq

where θi1s “ π` i1π48 for all i1 “ 0, . . . , 95 and θi2i “ ´

π2 ` i2

π49 for all i2 “ 1, . . . , 48, and the receiver

position

xj1r “ pR cos θj1r , R sin θj1r q

vj1,j2r “ pcospθj1r ` θj2o q, sinpθ

j1r ` θ

j2o qq

where θj1r “ j1π48 for all j1 “ 0, . . . , 95 and θj2o “ ´π

2 ` j2π49 for all j2 “ 1, . . . , 48.

In all the examples, the scattered data is perturbed with the noise in the form

(5.6) rDi,j “ Di,j ` 0.05εDi,j

|Di,j |

where ε is symmetric Bernoulli random variable that takes the values ˘1.All the experiments are reported on a server with 64-core Intel Xeon CPU and 256 Gigabytes

RAM. The code accompanying this manuscript are publicly available [12].In order to illustrate the reconstruction using Husimi data, we choose three examples of in-

creasing complexity. The exact contrast function qpxq’s are shown in Figure 10.

INVERSE SCATTERING IN THE HIGH-FREQUENCY LIMIT 21

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4y

0

0.1

0.2

0.3

0.4

0.5

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

0

0.1

0.2

0.3

0.4

0.5

Figure 10: The contrast function qpxq for our three examples: a bump function (left), a delocalizedfunction (middle) and the Shepp-Logan phantom (right).

In the first example, we consider a single bump in the form (4.8) with A “ 0.5 and r “ 0.2,which is shown in Figure 10 (left). We run the minimization loop as described above using k “ 24

and k “ 26, and the resulting reconstruction are shown in Figure 11. From Figure 11 we can clearlysee that as k becomes larger, the reconstruction becomes closer to the true medium. The solutiontime for k “ 26 is 15787.1 seconds.

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

0

0.1

0.2

0.3

0.4

0.5

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

0

0.1

0.2

0.3

0.4

0.5

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.1

-0.05

0

0.05

0.1

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.1

-0.05

0

0.05

0.1

Figure 11: Recovering a single bump contrast function. The upper row shows the estimated contrastfunction and the lower row shows the reconstruction error at k “ 26 (left) and k “ 24 (right).

In the second example, we consider a delocalized medium. The delocalized contrast function

22 S. CHEN, Z. DING, Q. LI, AND L. ZEPEDA-NUNEZ

qpxq is obtained by convolving a pointwise independent Gaussian random field with a Gaussianmollifier. The main difference with the single bump example is that the refractive index, can besmaller than the background one, thus allowing for more complex ray paths as shown in Figure 10(center). We repeat the same experiments, whose results are shown in Figure 12. The solutiontime required for k “ 26 is 13185.3 seconds.

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.2

-0.1

0

0.1

0.2

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.2

-0.1

0

0.1

0.2

Figure 12: Recovering a delocalized contrast function. The upper row shows the estimated contrastfunction and the lower row shows the reconstruction error at k “ 26 (left) and k “ 24 (right).

Finally, for the third example, we consider the more challenging, and more practical, problemof recovering the Shepp-Logan phantom, depicted in Figure 10 (right). In this case we have verysharp transitions of the refractive index, which will generate a strong reflection, compared tothe refraction-dominated media considered before. In addition, the interior of the still acts as aresonant cavity, thus creating a large amount of interior reflections, which are exacerbated as thefrequency increases. We perform the same experiments as above, whose results are depicted in inFigure 13. The solution time required for k “ 26 is 14640.4 seconds. In this case, the reconstructionis qualitative worse than before. We can still see the shape of the phantom, but with a largeamount of artifacts. These artifacts are common to the three examples, but are somewhat morenotorious for the Shepp-Logan phantom. Indeed, these artifacts can be in part explained by thelarge difference in the dispersion relation between the forward and backwards discretizations. TheLippmann-Schwinger discretization used for the forward problem is known to be highly accurate ifthe media is smooth. In the cases before, the data generated by the Lippmann-Schwinger solver isclose to the analytical solution, and the artifacts seems to come mostly for the phase errors in the

INVERSE SCATTERING IN THE HIGH-FREQUENCY LIMIT 23

Finite-Difference discretization. However, in this case the phantom is discontinuous thus creatinglarge phase errors in the solution of the equation, and therefore the forward map, which in returnproduce more notorious artifacts.

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 13: Recovering the Shepp-Logan phantom. The estimated contrast functions are shown fork “ 26 (left) and k “ 24 (right).

To avoid inverse crime, we have used two different solvers for computing the equation. Thetwo solvers produce relatively large phase errors that propagate in the reconstruction. The recon-struction can be significantly improved if we use the same PDE solvers in generating data andreconstructing the media. In Figure 14, we show the reconstructions of the same single bumpmedium as in Figure 11 but with the 4th-order finite difference for both data generation and in-version. It can be seen that the artifacts in the estimated medium is much smaller for larger k andthe reconstructed medium achieves a relative L2 error of 0.0389 for k “ 26. Better reconstructioncan also be seen in Figure 15 for the reconstructed delocalized medium, whose relative L2 error is0.0341 for k “ 26. In Figure 16, we show the reconstruction for the Shepp-Logan phantom. Wecan observe that as the frequency increased the reconstruction becomes better, although due tocomputational limitations induced by the current implementation we were unable to test with ahigher frequency, however, we would expect to obtain even a better reconstruction.

Lastly, we compare the conventional inverse scattering problem and our new inverse problem

using the Husimi data. We choose the incident wave ui,k “ eiωθ¨x with θ P S1 in (2.2), and measurethe scattered far field data us,k. Again we cast the problem as a nonlinear least square problem,and solve it using L-BFGS. We consider the initial perturbation equal to zero, and set a first orderoptimality tolerance of 10´5.

For simplicity, we use 4th-order finite difference for both data generation and inversion. Thesetup of the computational domain and the discretization are the same as in the previous examples.

The far field measurement is taken on the boundary BBp rRq with rR “ 1. We compute the datawith 180 incident directions θ that are equally distributed on S1 and 180 receivers that are equallydistributed on BBp rRq. We add 5% noise to the scattered data in the form of (5.6).

Finally, we test the robustness of the new formulation with respect to the non-convexity ofthe loss function. The ill-posedness of the inverse scattering problem is often manifested as a verynon-convex loss function with a myriad of local minima. As a consequence, any PDE constrainedoptimization-based reconstruction has a higher chance of converging to a non-physical minimum,a process that is often called cycle-skipping [50]. For comparing the new formulation and thetraditional one we also run the classical full-wave form inversion in frequency domain, using data

24 S. CHEN, Z. DING, Q. LI, AND L. ZEPEDA-NUNEZ

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

0

0.1

0.2

0.3

0.4

0.5

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

0

0.1

0.2

0.3

0.4

0.5

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.15

-0.1

-0.05

0

0.05

0.1

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.15

-0.1

-0.05

0

0.05

0.1

Figure 14: Recovering a single bump contrast function with 4th-order finite difference solver forboth data and inversion. The upper row shows the estimated contrast function and the lower rowshows the reconstruction error at k “ 26 (left) and k “ 24 (right).

at a single frequency, using the delocalized media in Figure 10. As discussed in Section 2.1.1, inthe classical formulation one probes the medium with plane waves, and the measurement operatorsamples the wavefield directly on the boundary of the domain of interest. Numerically, we minimizethe `2 misfit of the wavefield at the boundary, using the same L-BFGS solver as before. Initialguess is zero. We repeat the experiments for two different wave numbers that are used in the newformulation as well. The results are shown in Figures 17, 18, and 19, respectively. In the plotswe can observe that at low-frequencies we recover a smoothed version of the medium, but as thefrequency increases we encounter cycle-skipping, i.e., the algorithm converges to a spurious medium.This is an stark contrast with the inversion results of the new formulation shown in Figures 14, 15,and 16, where at low-frequency the reconstruction does not perform as well, but it is more stableat high-frequencies, providing an accurate reconstruction.

In summary the numerical experiments seems to indicate that the new inverse formulation isfar more robust to cycle skipping than its traditional counterpart.

6. Conclusions. To reconstruct an unknown medium, the generalized Helmholtz inverse scat-tering problem uses data pairs consisting of the impinging and scattered wave fields, while Liouvilleinverse scattering problems uses data pairs consisting of incoming and outgoing wave location anddirection. The former is regarded ill-posed in the high-frequency regime, while the latter is well-posed. This is intuitively contradicting to the fact that Liouville equation is the asymptotic limitof the Helmholtz equation.

INVERSE SCATTERING IN THE HIGH-FREQUENCY LIMIT 25

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4y

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.1

0

0.1

0.2

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.1

0

0.1

0.2

Figure 15: Recovering a delocalized contrast function with 4th-order finite difference solver for bothdata and inversion. The upper row shows the estimated contrast function and the lower row showsthe reconstruction error at k “ 26 (left) and k “ 24 (right).

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

0

0.1

0.2

0.3

0.4

0.5

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

0

0.1

0.2

0.3

0.4

0.5

Figure 16: Recovering the Shepp-Logan phantom with 4th-order finite difference solver for bothdata and inversion. The estimated contrast functions are shown for k “ 26 (left) and k “ 24 (right).

26 S. CHEN, Z. DING, Q. LI, AND L. ZEPEDA-NUNEZ

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.2

0

0.2

0.4

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.2

0

0.2

0.4

Figure 17: Recovering a single bump contrast function by plane waves. The estimated contrastfunction at k “ 26 (left) and k “ 24 (right) are shown. 4th-order finite difference solver is used forboth data and inversion.

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 18: Recovering a delocalized contrast function by plane wave. The estimated contrastfunction at k “ 26 (left) and k “ 24 (right) are shown. 4th-order finite difference solver is used forboth data and inversion.

We investigate this issue in this paper. In particular, we develop a new formulation for studyingthe Helmholtz inverse scattering problem with a new data collection process, and we show that thisnew formulation, in the high-frequency limit, becomes the Liouville inverse scattering problem, andthus inherits the well-posedness nature. This discovery bares the conceptual merit of providing themathematical description of the wave-particle duality for light propagation in the inverse setting.In addition, this discovery also suggests a more stable numerical reconstruction process for studyingthe Helmholtz inverse scattering problem, which we showcase using several numerical experiments.

REFERENCES

[1] R. Alaifari, I. Daubechies, P. Grohs, and R. Yin, Stable phase retrieval in infinite dimensions, Foundationsof Computational Mathematics, 19 (2019), pp. 869–900.

[2] D. Atkinson and N. D. Aparicio, An inverse problem method for crack detection in viscoelastic materialsunder anti-plane strain, Int. J. Eng. Sci., 35 (1997), pp. 841 – 849.

INVERSE SCATTERING IN THE HIGH-FREQUENCY LIMIT 27

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.05

0

0.05

0.1

0.15

-0.6 -0.4 -0.2 0 0.2 0.4

x

-0.6

-0.4

-0.2

0

0.2

0.4

y

-0.05

0

0.05

0.1

0.15

Figure 19: Recovering the Shepp-Logan phantom by plane wave. The estimated contrast functionat k “ 26 (left) and k “ 24 (right) are shown. 4th-order finite difference solver is used for bothdata and inversion.

[3] G. Bal, G. Papanicolaou, and L. Ryzhik, Radiative transport limit for the random Schrodinger equation,Nonlinearity, 15 (2002), p. 513.

[4] G. Bal and G. Uhlmann, Inverse diffusion theory of photoacoustics, Inverse Problems, 26 (2010), p. 085010.[5] G. Bao, P. Li, J. Lin, and F. Triki, Inverse scattering problems with multi-frequencies, Inverse Problems, 31

(2015), p. 093001.[6] C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control, and stabiliza-

tion of waves from the boundary, SIAM journal on control and optimization, 30 (1992), pp. 1024–1065.[7] J.-D. Benamou, F. Castella, T. Katsaounis, and B. Perthame, High frequency limit of the Helmholtz

equations, Rev. Mat. Iberoam., 18 (2002), pp. 187–209.[8] C. Borges, A. Gillman, and L. Greengard, High resolution inverse scattering in two dimensions using

recursive linearization, SIAM J. Imaging Sci., 10 (2017), pp. 641–664.[9] M. Born, Quantenmechanik der stoßvorgange, Zeitschrift fur Physik, 38 (1926), pp. 803–827.

[10] R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, A limited memory algorithm for bound constrained optimization,§SIAM J. Sci. Comput., 16 (1995), pp. 1190–1208.

[11] F. Castella, B. Perthame, and O. Runborg, High frequency limit of the Helmholtz equation. II. Source ona general smooth manifold, Comm. Partial Differential Equations, (2002).

[12] S. Chen, Z. Ding, Q. Li, and L. Zepeda-Nunez, Inverse scattering with Husimi data, 12 2021, https://github.com/Forgotten/inverse scattering/tree/husimi.

[13] S. Chen and Q. Li, Semiclassical limit of an inverse problem for the Schrodinger equation, Res. Math. Sci., 8(2021), pp. 1–18.

[14] S. Chen, Q. Li, and X. Yang, Classical limit for the varying-mass Schrodinger equation with random inho-mogeneities, J. Comput. Phys., 438 (2021), p. 110365.

[15] Y. Chen, Inverse scattering via Heisenberg’s uncertainty principle, Inverse Problems, 13 (1997), p. 253.[16] M. Cheney, A mathematical tutorial on synthetic aperture radar, SIAM Rev., 43 (2001), pp. 301–312.[17] E. Chung, J. Qian, G. Uhlmann, and H. Zhao, A new phase space method for recovering index of refraction

from travel times, Inverse Problems, 23 (2007), p. 309.[18] A. Colli, D. Prati, M. Fraquelli, S. Segato, P. P. Vescovi, F. Colombo, C. Balduini,

S. Della Valle, and G. Casazza, The use of a pocket-sized ultrasound device improves physical ex-amination: Results of an in-and outpatient cohort study, PLOS ONE, 10 (2015), pp. 1–10.

[19] D. L. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93, Springer, 2019.[20] M. Combescure and D. Robert, Coherent States and Applications in Mathematical Physics, Springer, 2012.[21] M. de Buhan and M. Darbas, Numerical resolution of an electromagnetic inverse medium problem at fixed

frequency, Comput. Math. Appl., 74 (2017), pp. 3111 – 3128.[22] M. de Buhan and M. Kray, A new approach to solve the inverse scattering problem for waves: combining the

TRAC and the adaptive inversion methods, Inverse Problems, 29 (2013), p. 085009.[23] B. Engquist and O. Runborg, Computational high frequency wave propagation, Acta Numer., 12 (2003),

p. 181–266.

28 S. CHEN, Z. DING, Q. LI, AND L. ZEPEDA-NUNEZ

[24] P. Gerard, P. A. Markowich, N. J. Mauser, and F. Poupaud, Homogenization limits and Wigner trans-forms, Comm. Pure Appl. Math., 50 (1997), pp. 323–379.

[25] P. Grohs, S. Koppensteiner, and M. Rathmair, Phase retrieval: Uniqueness and stability, SIAM Review,62 (2020), pp. 301–350.

[26] P. Grohs and M. Rathmair, Stable gabor phase retrieval and spectral clustering, Communications on Pureand Applied Mathematics, 72 (2019), pp. 981–1043.

[27] P. Hahner and T. Hohage, New stability estimates for the inverse acoustic inhomogeneous medium problemand applications, SIAM J. Math. Anal., 33 (2001), pp. 670–685.

[28] V. Isakov, Inverse Problems for Partial Differential Equations, vol. 127 of Applied Mathematical Sciences,Springer, Cham, third ed., 2017.

[29] M. C. KING, Chapter 2 - principles of optical lithography, in VLSI Electronics: Microstructure Science, N. G.Einspruch, ed., vol. 1 of VLSI Electronics Microstructure Science, Elsevier, 1981, pp. 41–81.

[30] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, vol. 120, Springer, 01 2011.[31] R.-Y. Lai, Q. Li, and G. Uhlmann, Inverse problems for the stationary transport equation in the diffusion

scaling, SIAM J. Appl. Math., 79 (2019), pp. 2340–2358.[32] Y. E. Li and L. Demanet, Full-waveform inversion with extrapolated low-frequency data, Geophysics, 81

(2016), pp. R339–R348.[33] J. Lu and X. Yang, Frozen gaussian approximation for high frequency wave propagation, Commun. Math. Sci.,

9 (2010), pp. 663–683.[34] F. Monard, P. Stefanov, and G. Uhlmann, The geodesic ray transform on riemannian surfaces with con-

jugate points, Communications in Mathematical Physics, 337 (2015).[35] S. Nagayasu, G. Uhlmann, and J.-N. Wang, Increasing stability in an inverse problem for the acoustic

equation, Inverse Problems, 29 (2013), p. 025012.[36] F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001.[37] R. G. Novikov, Small angle scattering and X-ray transform in classical mechanics, Ark. Mat., 37 (1999),

pp. 141–169.[38] R. D. Oldham, The constitution of the interior of the Earth, as revealed by earthquakes, Quarterly Journal of

the Geological Society, 62 (1906), pp. 456–475.[39] J. R. Pettit, A. E. Walker, and M. J. S. Lowe, Improved detection of rough defects for ultrasonic non-

destructive evaluation inspections based on finite element modeling of elastic wave scattering, IEEE T.Ultrason. Ferr., 62 (2015), pp. 1797–1808.

[40] R. G. Pratt, Seismic waveform inversion in the frequency domain; part 1: Theory and verification in a physicalscale model, Geophysics, 64 (1999), pp. 888–901.

[41] J. Qian and L. Ying, Fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beams for thewave equation, Multiscale Model. Simul., 8 (2010).

[42] N. Rawlinson, S. Pozgay, and S. Fishwick, Seismic tomography: a window into deep Earth, Phys. EarthPlanet. Int., 178 (2010), pp. 101–135.

[43] L. Ryzhik, G. Papanicolaou, and J. B. Keller, Transport equations for elastic and other waves in randommedia, Wave motion, 24 (1996), pp. 327–370.

[44] H. Schomberg, An improved approach to reconstructive ultrasound tomography, J. of Phys. D: Appl. Phys., 11(1978), pp. L181–L185.

[45] P. Stefanov, G. Uhlmann, A. Vasy, and H. Zhou, Travel time tomography, Acta Mathematica Sinica,English Series, 35 (2019), pp. 1085–1114.

[46] Z. Sun, On continuous dependence for an inverse initial boundary value problem for the wave equation, Journalof Mathematical Analysis and Applications, 150 (1990), pp. 188–204.

[47] N. M. Tanushev, J. Qian, and J. V. Ralston, Mountain waves and gaussian beams, Multiscale Model.Simul., 6 (2007), pp. 688–709.

[48] A. Tarantola, Inversion of seismic reflection data in the acoustic approximation, Geophysics, 49 (1984),pp. 1259–1266.

[49] F. Vico, L. Greengard, and M. Ferrando, Fast convolution with free-space Green’s functions, J. Comput.Phys., 323 (2016), pp. 191–203.

[50] J. Virieux, A. Asnaashari, R. Brossier, L. Metivier, A. Ribodetti, and W. Zhou, 6. An introductionto full waveform inversion, Society of Exploration Geophysicists, 2017, pp. R1–1–R1–40.

[51] J. Virieux and S. Operto, An overview of full-waveform inversion in exploration geophysics, Geophysics, 74(2009), pp. WCC1–WCC26.

[52] A. D. Wheelon, Electromagnetic Scintillation, vol. 1, Cambridge University Press, 2001.[53] A. D. Wheelon, Electromagnetic Scintillation, vol. 2, Cambridge University Press, 2003.[54] O. Yu. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave

equations, Communications in Partial Differential Equations, 26 (2001), pp. 1409–1425.

INVERSE SCATTERING IN THE HIGH-FREQUENCY LIMIT 29

[55] C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, Algorithm 778: L-bfgs-b: Fortran subroutines for large-scalebound-constrained optimization, ACM Trans. Math. Softw., 23 (1997), p. 550–560.

Appendix A. Formal derivation of Theorem 3.1.We start from the equation

(A.1) ikαkuk `∆uk ` k2npxquk “ ´Skpxq “ ´kd`32 Spkpx´ xsqq , x P Rd ,

and assume that αk Ñ α ě 0 in the limit k Ñ 8. We denote the density matrix of uk satisfy-ing (A.1) by

(A.2) gkpx, yq “ uk´

x´y

2k

¯

uk´

x`y

2k

¯

,

and the Fourier transform of a generic u by

(A.3) pupvq “ FyÑvupyq “1

p2πqd

ż

Rde´iyvupyqdy .

The inverse Fourier transform is then

(A.4) F´1vÑxupvq “

ż

Rdeixvupvqdv .

Now we compute the equation satisfied by the Wigner transform. The first step is to compute thederivatives of gk

(A.5) ∇y ¨∇xgkpx, yq “ ´

1

2k

”

∆uk´

x´y

2k

¯

uk´

x`y

2k

¯

´ uk´

x´y

2k

¯

∆uk´

x`y

2k

¯ı

,

and thus we have

(A.6)

αkgk ` i∇y ¨∇xgkpx, yq `

ik

2

”

n´

x`y

2k

¯

´ n´

x´y

2k

¯ı

gkpx, yq “

“ σkpx, yq

:“i

2k

”

Sk´

x´y

2k

¯

uk´

x`y

2k

¯

´ Sk´

x`y

2k

¯

uk´

x´y

2k

¯ı

.

Therefore, after a Fourier transform, we obtain the following transport equation on the Wignertransform fk

(A.7) αkfkpx, vq ` v ¨∇xfkpx, vq ` Zkpx, vq ˚v f

kpx, vq “ Qkpx, vq ,

where the last term denotes the convolution in v

Zkpx, vq ˚v fkpx, vq “

ż

RdZkpx, v ´ pqfkpx, pqdp

and the quantities Zk, Qk arising in this equation are given by

(A.8)

Zkpx, vq “1

p2πqdik

2F´1yÑv

”

n´

x`y

2k

¯

´ n´

x´y

2k

¯ı

,

Qkpx, vq “1

p2πqdF´1yÑvσ

kpx, yq .

30 S. CHEN, Z. DING, Q. LI, AND L. ZEPEDA-NUNEZ

From this equation we can compute the formally compute the limits. For Zk we have that

(A.9) Zkpx, vqkÑ8ÝÝÝÑ

1

p2πqdi

2pF´1

yÑvyq ¨∇xnpxq “ ´1

2∇xnpxq ¨∇vδpvq .

The limit of the source term Qk is slightly more involved. First, we define the complex valuedfunction

(A.10) wkpyq “1

kd´12

uk´

xs `y

k

¯

,

which after a change of variable can be rewritten as

(A.11) ukpxq “ kd´12 wkpkpx´ xsqq ,

where function wk satisfies the rescaled Helmholtz equation

(A.12) iαk

kwk `∆wk ` n

´

xs `y

k

¯

wk “ ´Spyq .

In the high-frequency limit, wk converges towards a solution w of

(A.13) ∆w ` npxsqw “ ´Spyq .

The second step is to compute the Fourier transform of w. To do so, we add an absorption termto the equation above, resulting in

(A.14) iβw `∆w ` npxsqw “ ´Spyq .

where β ą 0. This new term, is used as a broadening factor, which helps to smooth the Fouriertransform. We perform a Fourier transform on both sides, which leads to

(A.15) pwpvq “´Spvq

npxsq ´ |v|2 ` iβ“ SpvqGpv;βq .

where Gpv;βq denotes the Fourier transform of the outgoing Green’s function that vanishes atinfinity

(A.16) Gpv;βq ” ´1

npxsq ´ |v|2 ` iβ“ ´

npxsq ´ |v|2

pnpxsq ´ |v|2q2 ` β2`

iβ

pnpxsq ´ |v|2q2 ` β2, β ą 0 .

As usual, we take the limit β Ñ 0`. The first term converges weakly to the principal value

(A.17) ´npxsq ´ |v|

2

pnpxsq ´ |v|2q2 ` β2

βÑ0`ÝÝÝÝÑ ´P.V.

ˆ

1

npxsq ´ |v|2

˙

.

The second term converges to a delta function on the sphere t|v|2 “ npxsqu as β Ñ 0`

(A.18)iβ

pnpxsq ´ |v|2q2 ` β2

βÑ0`ÝÝÝÝÑ

iπ

2δp|v|2 “ npxsqq .

In summary, we obtain the Fourier transform of the outgoing solution to (A.13)

(A.19) pwpvq “ limβÑ0`

SpvqGpv;βq “ Spvq

„

iπ

2δp|v|2 “ npxsqq ´ P.V.

ˆ

1

npxsq ´ |v|2

˙

.

INVERSE SCATTERING IN THE HIGH-FREQUENCY LIMIT 31

Now we are ready to compute Qk. We take two test functions φpxq and ψpyq

(A.20)

ż

R2d

σkpx, yqφpxqψpyq dx dy

“i

2k

ż

R2d

”

Sk´

x´y

2k

¯

uk´

x`y

2k

¯

´ Sk´

x`y

2k

¯

uk´

x´y

2k

¯ı

φpxqψpyqdxdy

“ikd

2

ż

Rd

„

S´

k´

x´y

2k´ xs

¯¯

wk´

k´

x`y

2k´ xs

¯¯

´ S´

k´

x`y

2k´ xs

¯¯

wk´

k´

x´y

2k´ xs

¯¯

φpxqψpyqdxdy

“i

2

ż

R2d

”

Spzqwkpz ` yqφ´z

k`

y

2k` xs

¯

´ Spzqwkpz ´ yqφ´z

k´

y

2k` xs

¯ı

ψpyqdzdy

kÑ8ÝÝÝÑ

i

2φpxsq

ż

R2d

“

Spzqwpz ` yq ´ Spzqwpz ´ yq‰

ψpyqdzdy .

In other words, we have formally obtained that

(A.21) σkpx, yqkÑ8ÝÝÝÑ

i

2δpx´ xsq

ż

Rd

“

Spzqwpz ` yq ´ Spzqwpz ´ yq‰

dz ,

which after a Fourier transform gives

(A.22)

Qkpx, vq “1

p2πqdF´1yÑvσ

kpx, yq

kÑ8ÝÝÝÑ

1

p2πqdi

2δpx´ xsqF´1

yÑv

"ż

Rd

“

Spzqwpz ` yq ´ Spzqwpz ´ yq‰

dz

*

“i

2δpx´ xsqp2πq

d”

Spvqwpvq ´ Spvqwpvqı

“ p2πqdδpx´ xsqIm”

Spvqwpvqı

.

We finally obtain

(A.23) Qkpx, vqkÑ8ÝÝÝÑ p2πqd

π

2δpx´ xsq|Spvq|

2δp|v|2 “ npxsqq .

by substituting (A.19) in (A.22).