
Point-of-interest (POI) recommendation has attracted great attention in the field of recommender systems over the past decade. Various techniques, such as those based on matrix factorization and deep neural networks, have demonstrated outstanding performance. However, these methods are susceptible to the impact of data sparsity. Data sparsity is a significant characteristic of POI recommendation, where some POIs have limited interaction records and, in extreme cases, become cold-start POIs with no interaction history. To alleviate the influence of data sparsity on model performance, this paper introduced FAGRec, a POI-recommendation model based on the feature-aware graph. The key idea was to construct an interaction graph between POIs and their initial features. This allows the transformation of POI features into a weighted aggregation of initial features. Different POIs can share the learned representations of initial features, thereby mitigating the issue of data sparsity. Furthermore, we proposed attention-based graph neural networks and a user preference estimation method based on delayed time factors for learning representations of POIs and users, contributing to the generation of recommendations. Experimental results on two real-world datasets demonstrate the effectiveness of FAGRec in the task of POI recommendation.
Citation: Xia Liu, Liwan Wu. FAGRec: Alleviating data sparsity in POI recommendations via the feature-aware graph learning[J]. Electronic Research Archive, 2024, 32(4): 2728-2744. doi: 10.3934/era.2024123
[1] | Xinlin Cao, Huaian Diao, Jinhong Li . Some recent progress on inverse scattering problems within general polyhedral geometry. Electronic Research Archive, 2021, 29(1): 1753-1782. doi: 10.3934/era.2020090 |
[2] | Yujie Wang, Enxi Zheng, Wenyan Wang . A hybrid method for the interior inverse scattering problem. Electronic Research Archive, 2023, 31(6): 3322-3342. doi: 10.3934/era.2023168 |
[3] | Deyue Zhang, Yukun Guo . Some recent developments in the unique determinations in phaseless inverse acoustic scattering theory. Electronic Research Archive, 2021, 29(2): 2149-2165. doi: 10.3934/era.2020110 |
[4] | Yan Chang, Yukun Guo . Simultaneous recovery of an obstacle and its excitation sources from near-field scattering data. Electronic Research Archive, 2022, 30(4): 1296-1321. doi: 10.3934/era.2022068 |
[5] | Shiqi Ma . On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, 2021, 29(3): 2391-2415. doi: 10.3934/era.2020121 |
[6] | Yao Sun, Lijuan He, Bo Chen . Application of neural networks to inverse elastic scattering problems with near-field measurements. Electronic Research Archive, 2023, 31(11): 7000-7020. doi: 10.3934/era.2023355 |
[7] | Lanfang Zhang, Jijun Ao, Na Zhang . Eigenvalue properties of Sturm-Liouville problems with transmission conditions dependent on the eigenparameter. Electronic Research Archive, 2024, 32(3): 1844-1863. doi: 10.3934/era.2024084 |
[8] | Xiaoping Fang, Youjun Deng, Wing-Yan Tsui, Zaiyun Zhang . On simultaneous recovery of sources/obstacles and surrounding mediums by boundary measurements. Electronic Research Archive, 2020, 28(3): 1239-1255. doi: 10.3934/era.2020068 |
[9] | John Daugherty, Nate Kaduk, Elena Morgan, Dinh-Liem Nguyen, Peyton Snidanko, Trung Truong . On fast reconstruction of periodic structures with partial scattering data. Electronic Research Archive, 2024, 32(11): 6481-6502. doi: 10.3934/era.2024303 |
[10] | Weishi Yin, Jiawei Ge, Pinchao Meng, Fuheng Qu . A neural network method for the inverse scattering problem of impenetrable cavities. Electronic Research Archive, 2020, 28(2): 1123-1142. doi: 10.3934/era.2020062 |
Point-of-interest (POI) recommendation has attracted great attention in the field of recommender systems over the past decade. Various techniques, such as those based on matrix factorization and deep neural networks, have demonstrated outstanding performance. However, these methods are susceptible to the impact of data sparsity. Data sparsity is a significant characteristic of POI recommendation, where some POIs have limited interaction records and, in extreme cases, become cold-start POIs with no interaction history. To alleviate the influence of data sparsity on model performance, this paper introduced FAGRec, a POI-recommendation model based on the feature-aware graph. The key idea was to construct an interaction graph between POIs and their initial features. This allows the transformation of POI features into a weighted aggregation of initial features. Different POIs can share the learned representations of initial features, thereby mitigating the issue of data sparsity. Furthermore, we proposed attention-based graph neural networks and a user preference estimation method based on delayed time factors for learning representations of POIs and users, contributing to the generation of recommendations. Experimental results on two real-world datasets demonstrate the effectiveness of FAGRec in the task of POI recommendation.
Inverse scattering problems are a central topic in applied mathematics, which have many applications of practical importance in modern technologies including radar, medical imaging, nondestructive testing, remote sensing, geophysical exploration and ultrasound tomography (cf. [27]). Inverse obstacle scattering problems and inverse medium scattering problems are two main themes in inverse scattering problems. Uniqueness issue in inverse scattering problems is concerned with the unique identifiability on the shape and/or the physical material parameters of the underlying obstacle and medium by the corresponding measurements through wave probing. In this paper, we provide an overview of some recent mathematical developments on the inverse scattering problems and the inverse medium scattering problems as well as the inverse diffraction grating problems regarding on the unique identifiability issue. Those are fundamental results in the inverse scattering theory.
In this paper, we focus on the inverse obstacle and medium scattering problems for time-harmonic acoustic waves from an impenetrable or penetrable scatterer in a homogeneous background medium. Considering the scattering of a time-harmonic acoustic wave by a bounded obstacle
ui(x;k,d)=eikx⋅d,x∈Rn, | (1) |
where
{Δu+k2Vu=0inRn,u(x)=ui(x)+us(x),limr→∞rn−12(∂us∂r−ikus)=0, | (2) |
where
If
{Δu+k2u=0inRn∖¯Ω,Bu=0on∂Ω,u=ui+usinRn,limr→∞rn−12(∂us∂r−ikus)=0. | (3) |
The boundary operator relies on the physical property of the obstacle. Precisely speaking,
B(u)=∂νu+ηu=0on ∂Ω, | (4) |
where
The Sommerfeld radiation condition in (2) or (3) leads to the asymptotic expansion (cf. [27]) for
us(x;k,d)=eikrr(n−1)/2u∞(ˆx;k,d)+O(1r(n+1)/2)asr→∞ | (5) |
uniformly with respect to all directions
FP(Ω,V)=u∞(ˆx;k,d), | (6) |
and
FI(Ω,η)=u∞(ˆx;k,d), | (7) |
where
A primary issue for the inverse medium problem (6) and inverse obstacle problem (7) is the unique identifiability, which is concerned with the sufficient conditions such that the correspondence between
By now, only under a-prior geometric assumptions on the size or the shape of the scatterer, unique identifiability results by using one incident plane wave have been established. Colton and Sleeman [29] proved that the shape of the obstacle could be uniquely determined with one incident wave when the size of the scatterer satisfies some generic conditions. The sound-soft ball can be uniquely determined by a single measurement (cf. [42]). Later, Alessandrani and Rondi in [2,3], Cheng and Yamamoto in [25] and [26], Liu and Zou [46] studied the global uniqueness results with respect to a single far-field pattern for sound-soft, sound-hard obstacles within a certain polyhedral geometry. The proofs in [2,3,25,46] mainly utilize the reflection principle for the Helmholtz equation with respect to a Dirichlet or Neumann hyperplane and the path argument, where the path argument is initially developed in [46]. Such kind of methodologies cannot tackle the uniqueness results for an impedance polygonal or polyhedral obstacle with a single far-field pattern. Please refer to Section 2 for more discussions related to the Schiffer's problem on the inverse obstacle problems.
The obstacle with the impedance boundary condition can be uniquely determined by the far-field pattern with infinite incident directions and one fixed wave number (cf. [40]). The Schiffer's problem for an impedance obstacle is mathematically challenging, which concerns with the unique determination for an impedance obstacle with a single far-field pattern. Recently, two of the authors establish the "local" unique identifiability results [23,24] on the polygonal or polyhedral obstacle with the impedance boundary condition by at most two far-field patterns, where the polygonal or polyhedral obstacles satisfy some generic geometrical conditions. Furthermore, the local arguments for uniquely determining the shape and boundary parameters of the impedance obstacle by at most two far-field patterns are developed in [23,24], which are quite different from the proofs in [2,3,25,46] for tackling the uniqueness results for sound-soft and sound-hard obstacles with respect to a single far-filed pattern. A direct consequence of the uniqueness results in [23,24] is that the shape of the convex hull of a concave polygonal or polyhedral obstacle can be uniquely determined by at most two far-field patterns. To our best knowledge, the findings in [23,24] are the first results concerning the uniqueness results with finite many far-field measurements for general concave polygonal or polyhedral obstacles. The corresponding detailed discussions for Schiffer's problem with respect to the aforementioned results can be seen in Section 2.
For the inverse medium problem (6), Nachman [51], Novikov [52], and Ramm [53] established that the refractive index
Consider the direct medium scattering problem (6) with the transmission condition
u−=u+,∂νu−=∂νu+ on ∂Ω,Ω=supp(1−V), |
where
Recently, two of the authors [30] consider the unique determination of the shape of the reflective index's support
u−=u+,∂νu−=∂νu++ηu+ on ∂Ω, |
where
Very recently, the inverse problem of recovering a conductive medium body was considered in [22]. The conductive medium body arises from several applications of practical importance, including the modeling of an electromagnetic object coated with a thin layer of a highly conducting material and the magnetotellurics in geophysics. The inverse problem is concerned with the determination of the material parameters inside the body as well as on the conductive interface by the associated electromagnetic far-field measurements. Under the transverse-magnetic polarisation, two of the authors derived two novel unique identifiability results in determining a 2D piecewise conductive medium body associated with a polygonal-nest or a polygonal-cell geometry by a single active or passive far-field measurement. The detailed discussion can be found in Section 3.
Indeed, the unique identifiability [12,22,30] for the inverse medium problem (6) by a single far-field pattern relies heavily on the geometrical structures of the interior transmission eigenfunctions [21]. The study on the interior transmission eigenvalue problem has a long history and is of significant importance in scattering theory; see [27,37]. Let
{(Δ+k2)v=0inΩ,(Δ+k2(1+V))w=0inΩ,w=v,∂νw=∂νv, on∂Ω. | (8) |
If there exits a nontrivial pair
The study on scattering problems by periodic structures especially for diffraction grating has received a lot of attention (cf. [5]). The corresponding discussions are of practical applications in many areas such as radar imaging, micro-optics and nondestructive testing. The problem was first raised by Rayleigh on the scattering of plane waves from corrugated surface. In Section 4, we will give more explanations on the recent developments and the mathematical formulations towards the direct and inverse diffraction grating problems.
The rest of the paper is organized as follows. In Section 2, we discuss some recent progress on Schiffer's problem on inverse obstacle scattering problems including sound-soft, sound-hard and impedance obstacles within a certain polyhedral geometry. The study on the unique identifiability for impedance obstacles are important due to the novelty and challenge compared with the sound-soft and sound-hard cases. In Sections 3, we present some intriguing studies on the inverse medium scattering problem together with the geometrical structures of interior transmission eigenfunctions at the corner point. Section 4 is devoted to the mathematical developments of inverse diffraction grating problems. Similar to Section 2, we mainly present the unique determination for gratings with impedance boundary. In Section 5, we conclude our review paper on recent progress on inverse scattering problems and present some intriguing open problems.
In this section, we focus on the unique determination of the shape of the obstacle in (7) by finite many far-field patterns when the obstacle is a polygon or a polyhedron. As mentioned in the introduction, the aforementioned theory is related to Schiffer's problem in inverse scattering problem, which has a long and colorful history; see a recent survey paper [28] for related developments. Indeed, for sound-soft or sound-hard polyhedral scatterers, the corresponding unique identifiability results can be found in [2], [25], [44], [45] and [46], where the mathematical arguments adopt the reflection principle and path argument. In [47], the partial determination for impedance obstacle by finite many far-field patterns was also attained by Liu and Zou following the reflection principle argument. Recently, Cao, Diao, Liu and Zou in [23] and [24] developed a completely new approach on Schiffer's problem for impedance obstacles in
In the following, we first review recent progress on the geometrical structures of Laplacian eigenfunctions (cf. [23,24]). Subsections 2.2 and 2.3 are devoted to review the unique inedibility results for sound-soft, sound-hard and impedance polyhedral scatterers.
The geometric structures of Laplacian eigenfunctions and their deep relationship to the quantitive behaviours of the underlying eigenfunctions in
For
−Δu=λu in Ω. | (9) |
Recall the following two definitions in [23].
Definition 2.1. [23,Definition 1.1] For a Laplacian eigenfunction
∂νu(x)+η(x)u(x)=0,x∈Γh, | (10) |
then
Definition 2.2. [23,Definition 1.2] Let
limr→+01rm∫B(x0,r)|u(x)|dx=0 for m=0,1,…,N+1, | (11) |
where
Vani(u;x0)=N. |
If (11) holds for any
Combining with Definition 2.1 and Definition 2.2, under the mathematical setup in [23,Section 3], where we assume that
∠(Γ+h,Γ−h)=α⋅π, α∈(0,2),andΓ+h∩Γ−h=0∈Ω, | (12) |
with
Theorem 2.3. Let
N≥n, if u(0)=0 and α≠qp,p=1,…,n−1, | (13) |
where
Theorem 2.4. Let
N≥n, if α≠qp,p=1,…,n−1, | (14) |
where
Theorem 2.5. Let
N≥n, if u(0)=0 and α≠qp,p=1,…,n−1, | (15) |
where
Next, we have vanishing orders of Laplacian eigenfunctions at the corner intersected by a generalized singular (singular) line and a nodal line.
Theorem 2.6. Let
N≥n, if α≠2q+12p,p=1,…,n−1, | (16) |
where
Theorem 2.7. Let
N≥n, if α≠2q+12p,p=1,…,n−1, | (17) |
where
Theorem 2.8. Let
N≥n, if u(0)=0 and α≠qp,p=1,…,n−1, | (18) |
where
Indeed, Theorem 2.8 is a direct corollary of Theorem 2.3 by taking
If
Theorem 2.9. Let
Vani(u;0)=0, if u(0)≠0;Vani(u;0)=+∞, if u(0)=0. |
Theorem 2.10. Let
Vani(u;0)=+∞. |
Theorem 2.11. Let
Vani(u;0)=0, if u(0)≠0;Vani(u;0)=+∞, if u(0)=0. |
Theorem 2.12. Let
Vani(u;0)=+∞. |
We can know from above theorems that the eigenfunction is generically vanishing to infinity, namely
In the subsequent studies, we only present some latest studies concerning the unique identifiability for sound-soft, sound hard and impedance polyhedral scatterers, respectively.
In this subsection, we review the existing results on the unique identifiability for sound-soft or sound-hard polyhedral scatterers by finite many far-field patterns. In 1994, Liu and Nachman [43] investigated the unique determination results for the convex hull of a polyhedral obstacle by knowledge of the far-field pattern with the help of the reflection principle for solutions of the Helmholtz equation across a flat boundary. Later, Cheng and Yamamoto [25] proved that a polygonal obstacle in
Theorem 2.13. [2,Theorem 2.2] Let us fix
Theorem 2.13 can be proved by following a reflection argument discussed in [25]. Instead of examining the boundary behavior of the nodal set of
By introducing the concept of Dirichlet/Neumann set and Dirichlet/Neumann hyperplane, Liu and Zou [46] in 2006 initiated a nowadays well-known path argument to establish the uniqueness for both the sound-soft and sound-hard cases. Following the same definition of a polyhedral scatterer in [2], Liu and Zou derived the unique identifiability result as follows.
Theorem 2.14. [46,Theorem 2] The polyhedral scatterer
Very recently, Cao, Diao, Liu and Zou [23] established a completely novel approach in dealing with the Schiffer's problem for sound-soft, sound-hard and impedance obstacles. The uniqueness results can be regarded as direct applications of the new spectral findings by utilizing the critical connection between the intersecing angles of the nodal/generalized singular lines and the vanishing order of the Laplacian eigenfunctions in
The existing studies on the unique identifiability for sound-hard obstacles with finitely many measurements depend heavily on the geometric setup of the scatterer. Cheng and Yamamoto in [25] presented the unique determination results for sound-hard polygons in
Considering the scattering problem (3) with
Theorem 2.15. [46,Theorem 1] Let
We would like to remark that the scatterer
Later, Liu, Petrini, Rondi and Xiao [44] also established an optimal stability estimate for the determination of sound-hard polyhedral scatterers in
Very recently, as we mentioned earlier in Subsection 2.2, the unique determination for sound-hard obstacles in a certain polygonal setup without any further technical restrictions was derived in [23] for the two-dimiensional case and [24] for the three-dimensional case, respectively. Compared with the existing literatures, the results in [23] and [24] hold for the scatterers of more general material properties and can be obtained by at most two far-field patterns. To avoid repetation, we also refer to Subsection 2.3 for a brief introduction on the novel findings of Laplacian eigenfunctions as well as the Schiffer's problem for sound-hard obstacles, which are presented in a unified representation on the boundary condition in (21).
As discussed in Subsection 2.2, there is a widespread consensus that the unique identifiability for a sound-soft obstacle can be derived by a single incident wave and for a sound-hard obstacle, the unique determination can be obtained by a single incident plane wave with some fixed
For the scatterer
∂Ω=∂ΩD∪U∪∂ΩI, |
where
u=0on∂ΩD,∂νu+iλu=0on∂ΩI, | (19) |
where
Theorem 2.16. [47,Theorem 2.1] For any fixed
And for the scatterer of mixed sound-soft, sound-hard and impedance type, the following boundary conditions are satisfied
u=0 on∂ΩI,∂νu=0 on∂ΩN,∂νu+iλu=0 on∂ΩI. | (20) |
In (20),
Theorem 2.17. [47,Theorem 2.2] For any fixed
In particular, if
Theorem 2.18. [47,Theorem 2.4] For any fixed
As a direct application of the geometric structures of Laplacian eigenfunctions in Subsection 2.1, we have the unique identifiability results for Schiffer's problem for a certain type of admissible complex polygonal obstacle by at most two far-field patterns. Furthermore, the constant impedance boundary parameter
Definition 2.19 [23,Definition 8.1] Suppose that
Bu=∂νu+ηu=0on∂Ω. | (21) |
Definition 2.20.
(Ω,η)=N⋃ℓ=1(Ωℓ,ηℓ),withη=N⋃ℓ=1ηℓχ∂Ωℓ∩∂Ω, | (22) |
where each
Now the unique determination results on Schiffer's problem with respect to an admissible complex irrational polygonal obstacle or an admissible complex rational polygonal obstacle of degree
Theorem 2.21. Let
u∞(ˆx,dj)=˜u∞(ˆx,dj), ˆx∈S1,j=1,2, | (23) |
then one has that
(∂Ω∖∂¯˜Ω)∪(∂˜Ω∖∂¯Ω) |
cannot have a corner on
The precise proof of this theorem can be found in [23,Theorem 8.3]. The similar unique identifiability also holds for the convex hull of an admissible complex irrational obstacle and the impedance boundary parameter
Corollary 1. Let
u∞(ˆx,dj)=˜u∞(ˆx,dj), ˆx∈S1,j=1,2, | (24) |
then one has that
CH(Ω)=CH(˜Ω):=Σ, | (25) |
and
η=˜η on ∂Ω∩∂˜Ω∩∂Σ. | (26) |
The uniqueness of the impedance boundary parameter
For an admissible complex rational obstacle of degree
Theorem 2.22. Let
L(u2⋅∇u1−u1⋅∇u2)(xc)≠0, | (27) |
where
(∂Ω∖∂¯˜Ω)∪(∂˜Ω∖∂¯Ω) |
cannot have a corner on
Theorem 2.22 is based on the fact that the rational degree of an admissible complex rational polygonal obstacle is at least 2, which is a direct conclusion of theorem 2.3 to theorem 2.12.
Remark 1. In (32), for a function
L(f)(xc):=limr→+01|Ωr(xc)|∫Ωr(xc)f(x) dx, | (28) |
if the limit exists, where
Similar to Corollary 1, the unique determination of the convex hull of an admissible complex rational obstacle can also be derived, which is omitted.
The new approach developed in the proofs of Theorem 2.21 and Theorem 2.22 can uniformly tackle the unique determination for sound-soft, sound-hard and impedance obstacles by at most two far-field measurements and is completely local, which enables us to determine an impedance obstacle as well as its surface impedance by at most two far-field patterns.
It is the first time in the literature to present a systematic study of the intriguing connections between the vanishing orders of Laplacian eigenfunctions and the intersecting angles of their nodal/generalized singular lines. The unique identifiability for the impedance or generalized impedance cases in Theorem 2.21 and 2.22 has been an open problem for a long time. Therefore, these results should be truly original and of significant interest in the spectral theory of Laplacian eigenfunctions and also Schiffer's problem for inverse obstacle scattering problems.
As an extension of [23], two of the authors [24] further investigated in
The definition for an admissible polyhedral obstacle is quite different from that of two-dimensional case due to the more complicated geometric formulation. We refer to [24,Definition 6.1] for the rigorous statements.
Similar to the two-dimensional case, we separately have the unique determination for an admissible complex irraitonal polyhedral obstacle and an admissible complex rational polyhedral obstacle of degree
Theorem 2.23. Consider a fixed
u∞(ˆx;k,dj)=˜u∞(ˆx;k,dj), forj=1,2and allˆx∈S2, | (29) |
then
η=˜ηon∂Ω∩∂˜Ω. | (30) |
Theorem 2.24. Consider a fixed
u∞j(ˆx;k,dj)=˜u∞j(ˆx;k,dj), ˆx∈S2,j=1,2, | (31) |
L(u2⋅∇u1−u1⋅∇u2)(xc)≠0 and L(˜u2⋅∇˜u1−˜u1⋅∇˜u2)(xc)≠0 | (32) |
for all vertices
The main proofs of Theorem 2.23 and Theorem 2.24 are similar to the corresponding two-dimensional case, one can also refer [24,Section 6] for detailed analyses.
We would like to emphasize that the unique identifiability results as well as the corresponding argument in both
As mentioned in the introduction, there are quite a lot of studies concerning the unique identifiability on the inverse medium scattering problems with the penetrable scatterers in the inhomogeneous medium, which shall be focused on the case that the support of the medium parameter is of polygonal or polyhedral geometry in this section. Before that, we first review some recent progress on geometrical structures of transmission eigenfunctions to (35), which have important applications in unique determinations by a single far-field pattern in the inverse medium problems.
The interior transmission eigenvalue problem was first introduced by A. Kirsch [37] in 1986. The theoretical studies on interior transmission eigenvalue problems are of significant interest in the inverse medium scattering problems. An important application is to the invisibility phenomenon. The first quantitive result on the intrinsic properties of transmission eigenfunctions was studied by Blåsten and Liu in [11]. They rigorously investigated the vanishing properties of the interior transmission eigenfunctions at a corner whose angle is less than
The mathematical argument in [11] is indirect which connects the vanishing property of the interior transmission eigenfunctions with the stability of a certain wave scattering problem with respect to variation of the wave field at the corner point. The main results in [11] can be summarized by
Theorem 3.1. [11,Theorem 3.2] Let
vj(x)=∫Sn−1eikξ⋅xgj(ξ)dσ(ξ), ξ∈Sn−1,x∈Rn. | (33) |
satisfying one of the following two assumptions
(a) the kernel
(b) the Herglotz waves
‖v−vj‖L2(Ω)≤e−j,‖gj‖L2(Sn−1)≤C(lnj)β, | (34) |
where the constants
then
limρ→01m(B(xc,r)∩Ω)∫B(xc,r)∩Ω|v(x)|dx=0, |
where
Blåsten [8] utilized an energy identity from the enclosure method and constructed a new type of planar complex geometrical optics solution whose logarithm is a branch of the square root to reveal that the transmission eigenfunction
Theorem 3.2. [8,Theorem 4.2] Let
In [13], Blåsten and Liu further extended their results on geometric structures of transmission eigenfunctions at corners intersected by line segments to the corners with curvature. Roughly speaking, they established a relationship among the value of transmission eigenfunctions, the diameter of the domain and the underlying refractive index, which yields that the interior transmission eigenfunctions must be nearly vanishing at a high-curvature point on the boundary. These new findings significantly relaxed the dependence on the geometry of the scatterer (smallness assumption) but focus on local structures, which are more practicle and interesting. The main theoretical results on the vanishing properties of transmission eigenfunctions at high-curvature point can be seen in [13,Section 3] and one can also refer to [13,Section 4] for the uniqueness results for the inverse scattering problem associated with the high curvature geometry of the underlying obstacle.
Consider the following interior transmission eigenvalue problem with a conductive boundary condition for
{Δw+k2(1+V)w=0 in Ω,Δv+k2v=0 in Ω,w=v, ∂νv+ηv=∂νw on ∂Ω, | (35) |
where
The geometric properties studied in [11] are significantly generalized in a recent paper [30] concerning the geometric structures of conductive transmission eigenfunctions to (35). Roughly speaking, the results are extended in the following three aspects. First, the conductive transmission eigenfunctions include the interior transmission eigenfunctions as a special case. The geometric structures established for the conductive transmission eigenfunctions in [30] include the results in [11] as a special case. Second, the vanishing property of the conductive transmission eigenfunctions is established for any corner as long as the corner singularity is not degenerate. Third, the regularity requirements on the interior transmission eigenfunctions in [11] are significantly relaxed for the conductive transmission eigenfunctions. Furthermore, geometrical structures have practical and important applications in the inverse medium problems. In the following, we review the intriguing discoveries regarding the geometric properties of conductive transmission eigenfunctions in
In order to present main results on the geometric properties of the conductive transmission eigenfunctions, which shall play a critical role in the unique identifiability for the conductive scatterer, we first introduce the following notations. Denote
Then the main theorems in [30] concerning the vanishing properties of conductive transmission eigenfunctions can be summarized as follows.
Theorem 3.3. [30,Theorem 2.1] Let
Γ±h(xc):=∂Wxc(θW)∩Bh(xc),ΣΛh(xc):=Sh(xc)∖Sh/2(xc),Sh/2(xc):=Ω∩Bh/2(xc)=Ω∩Wxc(θW), | (36) |
such that
(a) the transmission eigenfunction
‖v−vj‖H1(Sh)≤j−1−Υ,‖gj‖L2(S1)≤Cjϱ, | (37) |
for some constants
(b) the function
η(xc)≠0, | (38) |
(c) the open angle of the open sector
θW≠π, | (39) |
then one has
limρ→+01m(Bρ(xc)∩Ω)∫Bρ(xc)∩Ω|v(x)|dx=0, | (40) |
where
The proof of this theorem is based on microlocal analysis combining with the specific complex geometrical optics solutions introduced in [8].
If stronger regularity conditions can be fulfilled by the conductive transmission eigenfunction
Theorem 3.4. [30,Theorem 2.2] Let
(a) the function
η(xc)≠0, | (41) |
(b) the open angle of the open sector
θW≠π, |
then we have
Consider the following medium scattering system for
{Δu−+k2(1+V)u−=0 in Ω,Δu++k2u+=0 in Rn∖Ω,u+=u−,∂νu+=∂νu− on ∂Ω,u+=ui+us in Rn∖Ω,limr→∞r(n−1)/2(∂rus−ikus)=0, r=|x|, | (42) |
where
The Sommerfeld radiation condition in (42) implies that the asymptotic expansion (5) still holds with the far-field pattern
Consider the scattering problem (42) with a piecewise constant refractive index
V:=∑ℓVℓχΣℓ,¯Ω=⋃ℓ¯Σℓ,ℓ∈N, | (43) |
where
Definition 3.5. [12,Definition 2.1] An admissible cell
Based on Definition 3.5, in the following two definitions, the polyheral cell and nest geometry are defined, respectively.
Definition 3.6. [12,Definition 2.2] For
V(x)=∞∑ℓ=1VℓχΣℓ(x), |
with
Definition 3.7. [12,Definition 2.4] For
Σℓ⋑Σℓ+1. |
A bounded potential
V(x)=∞∑ℓ=1VℓχUℓ(x) |
where
Blåsten and Liu also introduced a more general case that the potential
Definition 3.8. [12,Definition 2.4] A potential
Blåsten and Liu presented the main uniqueness results for the refractive index in a certain medium structure based on the assumption that for each
Theorem 3.9. [12,Theorem 2.6] Let
Theorem 3.10. [12,Theorem 2.7] Let
For more general mediums with potentials of
Theorem 3.11. [12,Theorem 2.5] Let
u∞(ˆx;ui)=˜u∞(ˆx;ui) |
for the far-field patterns arising from
Theorems 3.9, 3.10 and 3.11 were established by investigating the singular behaviors of the transmission eigenfunctions at the corner point, which can also be proved by using the geometrical structures of transmission eigenfunctions at the corner (cf. [8,11]).
In the following, we are concerned with the time-harmonic electromagnetic wave scattering from a conductive medium body. The conductive medium body arises in several applications of practical importance, including the modeling of an electromagnetic object coated with a thin layer of a highly conducting material and the magnetotellurics in geophysics. Indeed, the following conductive scattering problem (44) can be derived by the transverse-magnetic (TM) polarisation from the time-harmonic Maxwell system
{Δu−+k2qu−=0 in Ω,Δu++k2u+=0 in R2∖Ω,u+=u−,∂νu++ηu+=∂νu− on ∂Ω,u+=ui+us in R2∖Ω,limr→∞r1/2(∂rus−ikus)=0, r=|x|, | (44) |
where
Theorem 3.3 and Theorem 3.4 can be applied directly to establish the uniquely determination of the shape of an admissible conductive scatterer by a single far-field pattern.
Definition 3.12. Let
(a)
(b) Following the notations in Theorem 3.3, if
(c) The total wave field
limρ→+01m(B(x,ρ))∫B(x,ρ)|u(x)|dx≠0. | (45) |
Theorem 3.13. [30,Theorem 4.1] Consider the conductive scattering problem (44) associated with two conductive scatterers
u∞1(ˆx;ui)=u∞2(ˆx;ui) | (46) |
for all
Ω1ΔΩ2:=(Ω1∖Ω2)∪(Ω2∖Ω1) | (47) |
cannot possess a corner. Hence, if
Ω1=Ω2. | (48) |
If conductive parameter
Theorem 3.14. [30,Theorem 4.2] Consider the conductive scattering problem (44) associated with the admissible conductive scatters
u∞1(ˆx;ui)=u∞2(ˆx;ui) | (49) |
for all
It is clear that Theorem 3.14 is established with a-prior knowledge that the medium parameter
Similar to the geometric setup proposed from Definition 3.5 to Definition 3.7, following the rigorous definitions for the polygonal-nest geometry in [22,Definition 2.3] and the polygonal-cell geometry in [22,Definition 2.4], an admissible conductive medium of polygonal-nest or polygonal-cell structure was established in [22,Definition 4.1]. Indeed, the admissibility condition in [22,Definition 4.1] indicates that the total field
Before presenting the unique identification results, similar to (43), we are supposed to introduce some necessary notions first, which shall be utilized in the subsequent discussions. For a polygonal-nest conductive medium body
(Ω;q,η)=N⋃ℓ=1(Uℓ;qℓ,ηℓ) | (50) |
and
Ω=N⋃ℓ=1Uℓ, q=N∑ℓ=1qℓχUℓ, η=N∑ℓ=1ηℓχ∂Σℓ, | (51) |
where
For a polygonal-cell conductive medium body
(Ω;q,η)=N⋃ℓ=1(Σℓ;qℓ,η∗) | (52) |
and
Ω=N⋃ℓ=1Σℓ, q=N∑ℓ=1qℓχΣℓ, η=N∑ℓ=1η∗χ∂Σℓ, | (53) |
where each
Fig. 1 (b) presents a typical polygonal-cell partition of
Definition 3.15. Let
Theorem 3.16. [22,Theorem 4.1] Consider the conductive scattering problem (44) associated with two admissible polygonal-nest or polygonal-cell conductive medium bodies
u∞1(ˆx;ui)=u∞2(ˆx;ui) | (54) |
for all
Ω1ΔΩ2:=(Ω1∖Ω2)∪(Ω2∖Ω1) | (55) |
cannot possess a corner. Furthermore, if
∂Ω1=∂Ω2. | (56) |
It is easy to see from Theorem 3.16 that for a polygonal-nest medium body, the local uniqueness results readily imply the global uniqueness results. The corresponding proof for Theorem 3.16 is based on the geometrical structures of conductive transmission eigenfunctions established in Theorem 3.4.
Next, the simultaneous unique determination for the piecewise constant medium parameter as well as the conductive surface parameter associated with an admissible polygonal-nest or polygonal-cell conductive medium body can be achieved as the following two theorems. However, a-prior knowledge on the cell structure of an admissible polygonal-cell conductive medium body is assumed to be known in advance. The proof of Theorems 3.17 and 3.18 can be obtained by utilizing certain microlocal analysis and the complex geometrical optics solution introduced in [8].
Theorem 3.17. [22,Theorem 4.2] Considering the conductive scattering problem (44) associated with two admissible polygonal-cell conductive medium bodies
qj=N∑ℓ=1qℓ,jχΣℓ,j, ηj=N∑ℓ=1η∗jχ∂Σℓ,j. | (57) |
Let
u∞1(ˆx;ui)=u∞2(ˆx;ui) | (58) |
for all
Theorem 3.18. Considering the conductive scattering problem (44) associated with two admissible polygonal-nest conductive medium bodies
qj=Nj∑ℓ=1qℓ,jχUℓ,j, ηj=Nj∑ℓ=1ηℓ,jχ∂Σℓ,j. | (59) |
Let
u∞1(ˆx;ui)=u∞2(ˆx;ui) | (60) |
for all
The study on scattering theory by periodic structures has received a lot of attention in recent years. It arises from the first work studied by Rayleigh on the scattering by plane waves from corrugated surfaces. In particular, if the corrugations are exact sinusoids, then the sinusoidally corrugated surface provides a model of a reflection grating. There are quite many applications on grating problems in spectroscopy and oceanography. For example, it can be utilized to study the structure of the ocean surface by measuring sound scattering from below or the scattering of light or radar from above. We refer to [5] and [55] for more historical discussions.
In this section, we focus on the existing studies on the uniqueness issue for inverse diffraction grating problems. It is known that a general periodic grating structure can be uniquely determined by one incident wave if the wave number
In the subsequent discussions, we just present some of the aforementioned results to illustrate the development on inverse diffraction grating problem. Consider the direct diffraction grating problem associated to Helmholtz system as:
Δu+k2u=0in Ωf;B(u)|Λf=0on Λf, | (61) |
with the generalized impedance boundary condition
B(u)=∂νu+ηu=0, | (62) |
where
In (61), for a periodic Lipschitz function
Ωf:={x∈R2;x2>f(x1),x1∈R} |
is filled with a material whose refraction index (or wave number)
Λf={(x1,x2)∈R2;x2=f(x1)}. | (63) |
The corresponding incident wave is defined as
ui(x;k,d)=eikd⋅x,d=(sinθ,−cosθ)⊤,θ∈(−π2,π2), |
which propogates to
In order to derive the uniqueness results for the inverse grating problem associated with (61), the total wave field
u(x1+2π,x2)=e2iαπ⋅u(x1,x2), |
and the scattered field
us(x;k,d)=+∞∑n=−∞uneiξn(θ)⋅xforx2>maxx1∈[0,2π]f(x1), | (64) |
where
ξn(θ)=(αn(θ),βn(θ))⊤,αn(θ)=n+ksinθ,βn(θ)={√k2−α2n(θ), if |αn(θ)|≤ki√α2n(θ)−k2, if |αn(θ)|>k. | (65) |
The existence and uniqueness of the
Introduce a measurement boundary as
Γb:={(x1,b)∈R2;0≤x1≤2π,b>maxx1∈[0,2π]|f(x1)|}. |
The inverse diffraction grating problem is to determine
F(Λf,η)=u(x;k,d),x∈Γb, |
where
The pioneer work of Kirsch in [39] determined the unknown grating structure by the knowledge of incident waves and the scattered wave fields with the help of associated expansion of incident wave as
ui(x)=ui(x1,x2)=∑n∈Zψneiαnx1+iβn(b−x2),x2≤b, | (66) |
where
Theorem 4.1. [39,Theorem 3.1] Assume that
All the literatures mentioned at the beginning of this subsections are concerning the unique recovery results on the inverse diffraction grating problem with the sound-soft (namely
First, we are supposed to give the precise definition of admissible polygonal gratings associated with the inverse diffraction grating problem.
Definition 4.2. [23,Definition 8.1] Let
Definition 4.3. Let
Similar to Theorem 2.21 and Theorem 2.22 for the study on admissible irrational and rational complex polygonal obstacles, we have the unique determination results for rational and irrational polygonal gratings, respectively. In particular, for the rational case, we are concerned with the admissible rational polygonal grating of degree
Theorem 4.4. [23,Theorem 8.13] Let
dℓ=(sinθℓ,−cosθℓ)⊤,θℓ∈(−π2,π2). |
Let
Γb:={(x1,b)∈R2;0≤x1≤2π,b>max{maxx1∈[0,2π]|f(x1)|,maxx1∈[0,2π]|˜f(x1)|}}, |
If it holds that
u(x;k,dℓ)=˜u(x;k,dℓ),ℓ=1,2,x=(x1,b)∈Γb, | (67) |
then it cannot be true that there exists a corner point of
Combining with the fact that
As the result in Theorem 2.22 for the inverse obstacle problem, the unique determination of an admissible rational polygonal grating of degree
In this paper, we present a review on some recent progress on inverse scattering problems including inverse obstacle problems, inverse medium scattering problems and inverse diffraction grating problems.
For Schiffer' problem in inverse obstacle scattering in Section 2, there has been a colorful and long history for the relevant study. We investigate the unique identification of sound-soft, sound-hard and impedance obstacles with certain polyhedral geometry, respectively. The existing literatures are mainly concerned with the uniqueness discussions for sound-soft and sound-hard obstacles, where the reflection principle is utilized for establishing the corresponding results. It has been an open problem for a long time on the study about impedance case. In [47], Liu and Zou investigated the unique determination for certain partially coated structures of the obstacle following reflection principle. And very recently, in [23] and [24], by applying the geometric structures of Laplacian eigenfunctions in Subsection 2.1, the unique identifiability for a more general structure of impedance obstacles were considered in
For inverse medium scattering problems in Section 3, we first review thegeometrical structure of interior transmission eigenfunctions and the more generalized case with the conductive boundary condition. Those theoretical novel findings play an important role in the corresponding applications for the unique identifiability of the inverse medium scattering problems within the polyhedral geometry by a single far-field measurement under certain assumptions. Recently, some simultaneous recovery results regarding the scatterer as well as the conductive surface parameter and the medium parameter were obtained in [22]. The study therein are more generalized with respect to the medium structure and the regularity assumptions.
In the study on the inverse diffraction grating problem in Section 4, we first formulate the mathematical setup with the generalized impedance boundary which is more practical. By introducing the existing uniqueness results which are mainly concerning gratings with sound-soft or sound-hard boundary, we emphasize the developments on impedance case therein. The corresponding argument is similar to the unique identifiability for impedance obstacles by making use of the spectral properties of Laplacian eigenfunctions.
At the end of our paper, we propose some interesting open problems as follows.
● Establish the global unique identifiability result for an impedance polyhedral obstacle by a single far-field measurement by relaxing the generic condition in Definition 2.19. A more challenging problem is to establish the unique identification for an impedance obstacle of a non-polyhedral geometry by a single far-field measurement.
● Develop a uniform approach to tackle the unique identification result for the Schiffer's problem associated with an impedance polyhedral obstacle in
● Establish the global unique identifiability result for a medium polyhedral scatterer by a single far-field measurement by relaxing the generic condition in Definitions 3.8 and 3.12. A more challenging problem is to establish the unique identification for a medium scatterer within more general geometries by a single far-field measurement.
● Generalize the geometrical structures of interior transmission eigenfunctions at the corner to a more general interior transmission eigenvalue problem with respect to the anisotropic metric which can be formulated as
{Δgu+k2(1+V)u=0inΩ,Δv+k2v=0inΩ,u=v,∂νgu=∂νgvon∂Ω, |
where
The authors would like to thank Professor Hongyu Liu of City University of Hong Kong for proposing this project and providing a lot of stimulating discussions and helps during the writing of the paper.
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1. | Huaian Diao, Xinlin Cao, Hongyu Liu, On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications, 2021, 46, 0360-5302, 630, 10.1080/03605302.2020.1857397 | |
2. | Xinlin Cao, Huaian Diao, Hongyu Liu, Jun Zou, Two single-measurement uniqueness results for inverse scattering problems within polyhedral geometries, 2022, 16, 1930-8337, 1501, 10.3934/ipi.2022023 | |
3. | Huaian Diao, Hongyu Liu, Long Zhang, Jun Zou, Unique continuation from a generalized impedance edge-corner for Maxwell’s system and applications to inverse problems, 2021, 37, 0266-5611, 035004, 10.1088/1361-6420/abdb42 | |
4. | Inna Kal’chuk, Yurii Kharkevych, Approximation Properties of the Generalized Abel-Poisson Integrals on the Weyl-Nagy Classes, 2022, 11, 2075-1680, 161, 10.3390/axioms11040161 | |
5. | Yujia Chang, Yi Jiang, Rongliang Chen, A parallel domain decomposition algorithm for fluid-structure interaction simulations of the left ventricle with patient-specific shape, 2022, 30, 2688-1594, 3377, 10.3934/era.2022172 | |
6. | Huaian Diao, Hongyu Liu, 2023, Chapter 8, 978-3-031-34614-9, 199, 10.1007/978-3-031-34615-6_8 |