arXiv subjects

Hongyu Liu

Publications and source records attributed to Hongyu Liu.

At least 199 records · Page 11Linked to original sources

arXiv2022

Local geometric properties of conductive transmission eigenfunctions and applications

The purpose of the paper is twofold. First, we show that partial-data transmission eigenfunctions associated with a conductive boundary condition vanish locally around a polyhedral or conic corner in $\mathbb{R}^n$, $n=2,3$. Second, we apply the spectral property to the geometrical inverse scattering problem of determining the shape as well as its boundary impedance parameter of a conductive scatterer, independent of its medium content, by a single far-field measurement. We establish several new unique recovery results. The results extend the relevant ones in [30] in two directions: first, we consider a more general geometric setup where both polyhedral and conic corners are investigated, whereas in [30] only polyhedral corners are concerned; second, we significantly relax the regularity assumptions in [30] which is particularly useful for the geometrical inverse problem mentioned above. We develop novel technical strategies to achieve these new results.

math.AP↗

arXiv2022

A novel quantitative inverse scattering scheme using interior resonant modes

This paper is devoted to a novel quantitative imaging scheme of identifying impenetrable obstacles in time-harmonic acoustic scattering from the associated far-field data. The proposed method consists of two phases. In the first phase, we determine the interior eigenvalues of the underlying unknown obstacle from the far-field data via the indicating behaviour of the linear sampling method. Then we further determine the associated interior eigenfunctions by solving a constrained optimization problem, again only involving the far-field data. In the second phase, we propose a novel iteration scheme of Newton's type to identify the boundary surface of the obstacle. By using the interior eigenfunctions determined in the first phase, we can avoid computing any direct scattering problem at each Newton's iteration. The proposed method is particularly valuable for recovering a sound-hard obstacle, where the Newton's formula involves the geometric quantities of the unknown boundary surface in a natural way. We provide rigorous theoretical justifications of the proposed method. Numerical experiments in both 2D and 3D are conducted, which confirm the promising features of the proposed imaging scheme. In particular, it can produce quantitative reconstructions of high accuracy in a very efficient manner.

math.NA↗

arXiv2022

Modal approximation for time-domain elastic scattering from metamaterial quasiparticles

This paper aims at quantitatively understanding the elastic wave scattering due to negative metamaterial structures under wide-band signals in the time domain. Specifically, we establish the modal expansion for the time-dependent field scattered by metamaterial quasiparticles in elastodynamics. By Fourier transform, we first analyze the modal expansion in the time-harmonic regime. With the presence of quasiparticles, we validate such an expansion in the static regime via quantitatively analyzing the spectral properties of the Neumann-Poincaré operator associated with the elastostatic system. We then approximate the incident field with a finite number of modes and apply perturbation theory to obtain such an expansion in the perturbative regime. In addition, we give polariton resonances as simple poles for the elastic system with non-zero frequency. Finally, we show that the low-frequency part of the scattered field in the time domain can be well approximated by using the resonant modal expansion with sharp error estimates.

math.AP↗

arXiv2022

Visibility, invisibility and unique recovery of inverse electromagnetic problems with conical singularities

In this paper, we study time-harmonic electromagnetic scattering in two scenarios, where the anomalous scatterer is either a pair of electromagnetic sources or an inhomogeneous medium, both with compact supports. We are mainly concerned with the geometrical inverse scattering problem of recovering the support of the scatterer, independent of its physical contents, by a single far-field measurement. It is assumed that the support of the scatterer (locally) possesses a conical singularity. We establish a local characterisation of the scatterer when invisibility/transparency occurs, showing that its characteristic parameters must vanish locally around the conical point. Using this characterisation, we establish several local and global uniqueness results for the aforementioned inverse scattering problems, showing that visibility must imply unique recovery. In the process, we also establish the local vanishing property of the electromagnetic transmission eigenfunctions around a conical point under the Hölder regularity or a regularity condition in terms of Herglotz approximation.

math.AP↗

arXiv2022

Elastodynamical resonances and cloaking of negative material structures beyond quasistatic approximation

Given the flexibility of choosing negative elastic parameters, we construct material structures that can induce two resonance phenomena, referred to as the elastodynamical resonances. They mimic the emerging plasmon/polariton resonance and anomalous localized resonance in optics for subwavelength particles. However, we study the peculiar resonance phenomena for linear elasticity beyond the subwavelength regime. It is shown that the resonance behaviours possess distinct characters, with some similar to the subwavelength resonances, but some sharply different due to the frequency effect. It is particularly noted that we construct a core-shell material structure that can induce anomalous localized resonance as well as cloaking phenomena beyond the quasi-static limit. The study is boiled down to analyzing the so-called elastic Neumann-Poincáre (N-P) operator in the frequency regime. We provide an in-depth analysis of the spectral properties of the N-P operator beyond the quasi-static approximation,and these results are of independent interest to the spectral theory of layer potential operators.

math-ph↗

arXiv2022

Stable determination by a single measurement, scattering bound and regularity of transmission eigenfunction

In this paper, we study an inverse problem of determining the cross section of an infinitely long cylindrical-like material structure from the transverse electromagnetic scattering measurement. We establish a sharp logarithmic stability result in determining a polygonal scatterer by a single far-field measurement. The argument in establishing the stability result is localized around a corner and can be as well used to produce two highly intriguing implications for invisibility and transmission resonance in the wave scattering theory. In fact, we show that if a generic medium scatterer possesses an admissible corner on its support, then there exists a positive lower bound of the $L^2$-norm of the associated far-field pattern. For the transmission resonance, we discover a quantitative connection between the regularity of the transmission eigenfunction at a corner and its analytic or Fourier extension.

math.AP↗

arXiv2022

Boundary localization of transmission eigenfunctions in spherically stratified media

Consider the transmission eigenvalue problem for $u \in H^1(Ω)$ and $v\in H^1(Ω)$ associated with $(Ω; σ, \mathbf{n}^2)$, where $Ω$ is a ball in $\mathbb{R}^N$, $N=2,3$. If $σ$ and $\mathbf{n}$ are both radially symmetric, namely they are functions of the radial parameter $r$ only, we show that there exists a sequence of transmission eigenfunctions $\{u_m, v_m\}_{m\in\mathbb{N}}$ associated with $k_m\rightarrow+\infty$ as $m\rightarrow+\infty$ such that the $L^2$-energies of $v_m$'s are concentrated around $\partialΩ$. If $σ$ and $\mathbf{n}$ are both constant, we show the existence of transmission eigenfunctions $\{u_j, v_j\}_{j\in\mathbb{N}}$ such that both $u_j$ and $v_j$ are localized around $\partialΩ$. Our results extend the recent studies in [15,16]. Through numerics, we also discuss the effects of the medium parameters, namely $σ$ and $\mathbf{n}$, on the geometric patterns of the transmission eigenfunctions.

math.AP↗

arXiv2022

Shape reconstructions by using plasmon resonances

We study the shape reconstruction of an inclusion from the {faraway} measurement of the associated electric field. This is an inverse problem of practical importance in biomedical imaging and is known to be notoriously ill-posed. By incorporating Drude's model of the permittivity parameter, we propose a novel reconstruction scheme by using the plasmon resonance with a significantly enhanced resonant field. We conduct a delicate sensitivity analysis to establish a sharp relationship between the sensitivity of the reconstruction and the plasmon resonance. It is shown that when plasmon resonance occurs, the sensitivity functional blows up and hence ensures a more robust and effective construction. Then we combine the Tikhonov regularization with the Laplace approximation to solve the inverse problem, which is an organic hybridization of the deterministic and stochastic methods and can quickly calculate the minimizer while capture the uncertainty of the solution. We conduct extensive numerical experiments to illustrate the promising features of the proposed reconstruction scheme.

math.NA↗

arXiv2022

On corners scattering stably and stable shape determination by a single far-field pattern

In this paper, we establish two sharp quantitative results for the direct and inverse time-harmonic acoustic wave scattering. The first one is concerned with the recovery of the support of an inhomogeneous medium, independent of its contents, by a single far-field measurement. For this challenging inverse scattering problem, we establish a sharp stability estimate of logarithmic type when the medium support is a polyhedral domain in $\mathbb{R}^n$, $n=2,3$. The second one is concerned with the stability for corner scattering. More precisely if an inhomogeneous scatterer, whose support has a corner, is probed by an incident plane-wave, we show that the energy of the scattered far-field possesses a positive lower bound depending only on the geometry of the corner and bounds on the refractive index of the medium there. This implies the impossibility of approximate invisibility cloaking by a device containing a corner and made of isotropic material. Our results sharply quantify the qualitative corner scattering results in the literature, and the corresponding proofs involve much more subtle analysis and technical arguments. As a significant byproduct of this study, we establish a quantitative Rellich's theorem that continues smallness of the wave field from the far-field up to the interior of the inhomogeneity. The result is of significant mathematical interest for its own sake and is surprisingly not yet known in the literature.

math.AP↗

arXiv2021

Plasmon resonances of nanorods in transverse electromagnetic scattering

Plasmon resonance is the resonant oscillation of conduction electrons at the interface between negative and positive permittivity material stimulated by incident light, which forms the fundamental basis of many cutting-edge industrial applications. We are concerned with the quantitative theoretical understanding of this peculiar resonance phenomenon. It is known that the occurrence of plasmon resonance as well as its quantitative behaviours critically depend on the geometry of the material structure, the corresponding material parameters and the operating wave frequency, which are delicately coupled together. In this paper, we study the plasmon resonance for a 2D nanorod structure, which presents an anisotropic geometry and arises in the transverse electromagnetic scattering. We present delicate spectral and asymptotic analysis to establish the accurate resonant conditions as well as sharply characterize the quantitative behaviours of the resonant field.

math.AP↗

arXiv2021

Large-eddy simulations of marine boundary-layer clouds associated with cold air outbreak during the ACTIVATE campaign. Part II: aerosol-meteorology-cloud interaction

Aerosol effects on micro-/macro-physical properties of marine stratocumulus clouds over the Western North Atlantic Ocean (WNAO) are investigated using in-situ measurements and large-eddy simulations (LES) for two cold air outbreak (CAO) cases (February 28 and March 1, 2020) during the Aerosol Cloud meTeorology Interactions oVer the western ATlantic Experiment (ACTIVATE). The LES is able to reproduce the vertical profiles of liquid water content (LWC), effective radius r_eff and the cloud droplet number concentration Nc from fast cloud droplet probe (FCDP) in-situ measurements for both cases. Furthermore, we show that aerosols affect cloud properties (Nc, r_eff, and LWC) via the prescribed bulk hygroscopicity of aerosols and aerosol size distributions characteristics. Nc, r_eff, and liquid water path (LWP) are positively correlated to the bulk hygroscopicity of aerosols and aerosol number concentration (Na) while cloud fractional cover (CFC) is insensitive to the bulk hygroscopicity of aerosols and aerosol size distributions for the two cases. The changes to aerosol size distribution (number concentration, width, and the geometrical diameter) allow us to disentangle aerosol effects on cloud properties from the meteorological effects. We also use the LES results to evaluate cloud properties from two reanalysis products, ERA5 and MERRA-2. Comparing to LES, the ERA5 reanalysis is able to capture the time evolution of LWP and total cloud coverage within the study domain during both CAO cases while MERRA-2 underestimates them.

physics.ao-ph↗

arXiv2021

Two unique Identifiability results for inverse scattering problems within polyhedral geometries

We consider the unique determinations of impenetrable obstacles or diffraction grating profiles in $\mathbb{R}^3$ by a single far-field measurement within polyhedral geometries. We are particularly interested in the case that the scattering objects are of impedance type. We derive two new unique identifiability results for the inverse scattering problem in the aforementioned two challenging setups. The main technical idea is to exploit certain quantitative geometric properties of the Laplacian eigenfunctions which were initiated in our recent works [8,9]. In this paper, we derive novel geometric properties that generalize and extend the related results in [9], which further enable us to establish the new unique identifiability results. It is pointed out that in addition to the shape of the obstacle or the grating profile, we can simultaneously recover the boundary impedance parameters.

math.AP↗

arXiv2021

Stable determination of an elastic medium scatterer by a single far-field measurement and beyond

We are concerned with the time-harmonic elastic scattering due to an inhomogeneous elastic material inclusion located inside a uniformly homogeneous isotropic medium. We establish a sharp stability estimate of logarithmic type in determining the support of the elastic scatterer, independent of its material content, by a single far-field measurement when the support is a convex polyhedral domain in $\mathbb{R}^n$, $n=2,3$. Our argument in establishing the stability result is localized around a corner of the medium scatterer. This enables us to further establish a byproduct result by proving that if a generic medium scatterer, not necessary to be a polyhedral shape, possesses a corner, then there exists a positive lower bound of the scattered far-field patterns. The latter result indicates that if an elastic material object possesses a corner on its support, then it scatters every incident wave stably and invisibility phenomenon does not occur.

math.AP↗

arXiv2021

Simultaneous recoveries for semilinear parabolic systems

In this paper, we study inverse boundary problems associated with semilinear parabolic systems in several scenarios where both the nonlinearities and the initial data can be unknown. We establish several simultaneous recovery results showing that the passive or active boundary Dirichlet-to-Neumann operators can uniquely recover both of the unknowns, even stably in a certain case. It turns out that the nonlinearities play a critical role in deriving these recovery results. If the nonlinear term belongs to a general $C^1$ class but fulfilling a certain growth condition, the recovery results are established by the control approach via Carleman estimates. If the nonlinear term belongs to an analytic class, the recovery results are established through successive linearization in combination with special CGO (Complex Geometrical Optics) solutions for the parabolic system.

math.AP↗

arXiv2021

Large-eddy simulations of marine boundary-layer clouds associated with cold air outbreaks during the ACTIVATE campaign-part 1: Case setup and sensitivities to large-scale forcings

Large-eddy simulation (LES) is able to capture key boundary-layer (BL) turbulence and cloud processes. Yet, large-scale forcing and surface turbulent fluxes of sensible and latent heat are often poorly prescribed for LES simulations. We derive these quantities from measurements and reanalysis obtained for two cold air outbreak (CAO) events during Phase I of the Aerosol Cloud meTeorology Interactions oVer the western ATlantic Experiment (ACTIVATE) in February-March 2020. We study the two contrasting CAO cases by performing LES and test the sensitivity of BL structure and clouds to large-scale forcings and turbulent heat fluxes. Profiles of atmospheric state and large-scale divergence and surface turbulent heat fluxes obtained from the reanalysis data ERA5 agree reasonably well with those derived from ACTIVATE field measurements for both cases at the sampling time and location. Therefore, we adopt the time evolving heat fluxes, wind and advective tendencies profiles from ERA5 reanalysis data to drive the LES. We find that large-scale thermodynamic advective tendencies and wind relaxations are important for the LES to capture the evolving observed BL meteorological states characterized by the hourly ERA5 reanalysis data and validated by the observations. We show that the divergence (or vertical velocity) is important in regulating the BL growth driven by surface heat fluxes in LES simulations. The evolution of liquid water path is largely affected by the evolution of surface heat fluxes. The liquid water path simulated in LES agrees reasonably well with the ACTIVATE measurements.This study paves the path to investigate aerosol-cloud-meteorology interactions using LES informed and evaluated by ACTIVATE field measurements.

physics.ao-ph↗

arXiv2021

Localized sensitivity analysis at high-curvature boundary points of reconstructing inclusions in transmission problems

In this paper, we are concerned with the recovery of the geometric shapes of inhomogeneous inclusions from the associated far field data in electrostatics and acoustic scattering. We present a local resolution analysis and show that the local shape around a boundary point with a high magnitude of mean curvature can be reconstructed more easily and stably. In proving this, we develop a novel mathematical scheme by analyzing the generalized polarisation tensors (GPTs) and the scattering coefficients (SCs) coming from the associated scattered fields, which in turn boils down to the analysis of the layer potential operators that sit inside the GPTs and SCs via the microlocal analysis. In a delicate and subtle manner, we decompose the reconstruction process into several steps, where all but one steps depend on the global geometry, and one particular step depends on the mean curvature at a given boundary point. Then by a sensitivity analysis with respect to local perturbations of the curvature of the boundary surface, we establish the local resolution effects. Our study opens up a new field of mathematical analysis on wave super-resolution imaging.

math.AP↗

arXiv2021

Gradient estimates for electric fields with multi-scale inclusions in the quasi-static regime

In this paper, we are concerned with the gradient estimate of the electric field due to two nearly touching dielectric inclusions, which is a central topic in the theory of composite materials. We derive accurate quantitative characterisations of the gradient fields in the transverse electromagnetic case within the quasi-static regime, which clearly indicate the optimal blowup rate or non-blowup of the gradient fields in different scenarios. There are mainly two novelties of our study. First, the sizes of the two material inclusions may be of different scales. Second, we consider our study in the quasi-static regime, whereas most of the existing studies are concerned with the static case.

math.AP↗

arXiv2021

On a local geometric property of the generalized elastic transmission eigenfunctions and application

Consider the nonlinear and completely continuous scattering map \[ \mathcal{S}\big((Ω; λ, μ, V), \mathbf{u}^i\big)=\mathbf{u}_t^\infty(\hat{\mathbf{x}}), \quad \hat{\mathbf{x}}\in\mathbb{S}^{n-1}, \] which sends an inhomogeneous elastic scatterer $(Ω; λ, μ, V)$ to its far-field pattern $\mathbf{u}_t^\infty$ due to an incident wave field $\mathbf{u}^i$ via the Lamé system. Here, $(λ, μ, V)$ signifies the medium configuration of an elastic scatterer that is compactly supported in $Ω$. In this paper, we are concerned with the intrinsic geometric structure of the kernel space of $\mathcal{S}$, which is of fundamental importance to the theory of inverse scattering and invisibility cloaking for elastic waves and has received considerable attention recently. It turns out that the study is contained in analysing the geometric properties of a certain non-selfadjoint and non-elliptic transmission eigenvalue problem. We propose a generalized elastic transmission eigenvalue problem and prove that the transmission eigenfunctions vanish locally around a corner of $\partialΩ$ under generic regularity criteria. The regularity criteria are characerized by the Hölder continuity or a certain Fourier extension property of the transmission eigenfunctions. As an interesting and significant application, we apply the local geometric property to derive several novel unique identifiability results for a longstanding inverse elastic problem by a single far-field measurement.

math.AP↗