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Hongyu Liu

Publications and source records attributed to Hongyu Liu.

At least 163 records · Page 9Linked to original sources
arXiv2023

Inverse problems for nonlinear progressive waves

We propose and study several inverse problems associated with the nonlinear progressive waves that arise in infrasonic inversions. The nonlinear progressive equation (NPE) is of a quasilinear form $\partial_t^2 u=Δf(x, u)$ with $f(x, u)=c_1(x) u+c_2(x) u^n$, $n\geq 2$, and can be derived from the hyperbolic system of conservation laws associated with the Euler equations. We establish unique identifiability results in determining $f(x, u)$ as well as the associated initial data by the boundary measurement. Our analysis relies on high-order linearisation and construction of proper Gaussian beam solutions for the underlying wave equations. In addition to its theoretical interest, we connect our study to applications of practical importance in infrasound waveform inversion.

math.AP↗
arXiv2023

Determining a stationary mean field game system from full/partial boundary measurement

In this paper, we propose and study the utilization of the Dirichlet-to-Neumann (DN) map to uniquely identify the discount functions $r, k$ and cost function $F$ in a stationary mean field game (MFG) system. This study features several technical novelties that make it highly intriguing and challenging. Firstly, it involves a coupling of two nonlinear elliptic partial differential equations. Secondly, the simultaneous recovery of multiple parameters poses a significant implementation challenge. Thirdly, there is the probability measure constraint of the coupled equations to consider. Finally, the limited information available from partial boundary measurements adds another layer of complexity to the problem. Considering these challenges and problems, we present an enhanced higher-order linearization method to tackle the inverse problem related to the MFG system. Our proposed approach involves linearizing around a pair of zero solutions and fulfilling the probability measurement constraint by adjusting the positive input at the boundary. It is worth emphasizing that this technique is not only applicable for uniquely identifying multiple parameters using full-boundary measurements but also highly effective for utilizing partial-boundary measurements.

math.OC↗
arXiv2023

Strong uniqueness principle for fractional polyharmonic operators and applications to inverse problems

In this work, we are concerned with inverse problems involving poly-fractional operators, where the poly-fractional operator is of the form \[P( (-Δ_g)^s)u := \sum_{i=1}^M α_i(-Δ_{g_i})^{s_i}u\] for $s=(s_1,\dots,s_M)$, $0<s_1<\cdots<s_M<\infty$, $s_M\in\mathbb{R}_+\backslash\mathbb{Z}$, $g=(g_1,\dots,g_M)$. There are three major contributions in this work that are new to the literature. First, we propose equations involving such poly-fractional operators $P$, which have not been previously considered in the general setting. Such equations arise naturally from the superposition of multiple stochastic processes with different scales, including classical random walks and Lévy flights. Secondly, we give novel results for the unique continuation properties for fractional polyharmonic $u$, in the sense that $u$ satisfies $\tilde{P}((-Δ_{\tilde{g}})^{\tilde{s}})=0$ in a bounded Lipschitz domain $Ω$ for some $\tilde{P}$. With these results in hand, we consider the inverse problems for $P$, and proved the uniqueness in recovering the potential, the source function in the semilinear case, and the coefficients associated to the non-isotropy of the fractional operator.

math.AP↗
arXiv2023

Lipschitz stability for determination of states and inverse source problem for the mean field game equations

In a bounded domain $Ω\subset \mathbb{R}^d$ over time interval $(0,T)$, we consider mean field game equations whose principal coefficients depend on the time and state variables with a general Hamiltonian. We attach the non-zero Robin boundary condition. We first prove the Lipschitz stability in $Ω\times (\varepsilon, T-\varepsilon)$ with given $\varepsilon>0$ for the determination of the solutions by Dirichlet data on arbitrarily chosen subboundary of $\partialΩ$. Next we prove the Lipschitz stability for an inverse problem of determining spatially varying factors of source terms and a coefficient by extra boundary data and spatial data at an intermediate time.

math.AP↗
arXiv2023

General Physical Properties of Fermi blazars

We study the general physical properties of Fermi blazars using the Fermi fourth source catalog data (4FGL-DR2). The quasi-simultaneous multiwavelength data of Fermi blazar are fitted by using the one-zone leptonic model to obtain some physical parameters, such as jet power, magnetic field and Doppler factor. We study the distributions of the derived physical parameter as a function of black hole mass and accretion disk luminosity. The main results are as follows. (1) For a standard thin accretion disk, the jet kinetic power of most FSRQs can be explained by the BP mechanism. However, the jet kinetic power of most BL Lacs can not be explained by both the BZ mechanism or the BP mechanism. The BL Lacs may have ADAFs surrounding their massive black holes. (2) After excluding the redshift, there is a moderately strong correlation between the jet kinetic power and jet radiation power and the accretion disk luminosity for Fermi blazars. These results confirm a close connection between jet and accretion. The jet kinetic power is slightly larger than the accretion disk luminosity for Fermi blazars. (3) There is a significant correlation between jet kinetic power and gamma-ray luminosity and radio luminosity for Fermi blazars, which suggests that gamma-ray luminosity and radio luminosity can be used to indicate the jet kinetic power.

astro-ph.HE↗
arXiv2023

On inverse problems for several coupled PDE systems arising in mathematical biology

In this paper, we propose and study several inverse problems of identifying/determining unknown coefficients for a class of coupled PDE systems by measuring the average flux data on part of the underlying boundary. In these coupled systems, we mainly consider the non-negative solutions of the coupled equations, which are consistent with realistic settings in biology and ecology. There are several salient features of our inverse problem study: the drastic reduction of the measurement/observation data due to averaging effects, the nonlinear coupling of multiple equations, and the non-negative constraints on the solutions, which pose significant challenges to the inverse problems. We develop a new and effective scheme to tackle the inverse problems and achieve unique identifiability results by properly controlling the injection of different source terms to obtain multiple sets of mean flux data. The approach relies on certain monotonicity properties which are related to the intrinsic structures of the coupled PDE system. We also connect our study to biological applications of practical interest.

math.AP↗
arXiv2023

XFormer: Fast and Accurate Monocular 3D Body Capture

We present XFormer, a novel human mesh and motion capture method that achieves real-time performance on consumer CPUs given only monocular images as input. The proposed network architecture contains two branches: a keypoint branch that estimates 3D human mesh vertices given 2D keypoints, and an image branch that makes predictions directly from the RGB image features. At the core of our method is a cross-modal transformer block that allows information to flow across these two branches by modeling the attention between 2D keypoint coordinates and image spatial features. Our architecture is smartly designed, which enables us to train on various types of datasets including images with 2D/3D annotations, images with 3D pseudo labels, and motion capture datasets that do not have associated images. This effectively improves the accuracy and generalization ability of our system. Built on a lightweight backbone (MobileNetV3), our method runs blazing fast (over 30fps on a single CPU core) and still yields competitive accuracy. Furthermore, with an HRNet backbone, XFormer delivers state-of-the-art performance on Huamn3.6 and 3DPW datasets.

cs.CV↗
arXiv2023

Inverse problem for a random Schrödinger equation with unknown source and potential

In this paper, we study an inverse scattering problem associated with the time-harmonic Schrödinger equation where both the potential and the source terms are unknown. The source term is assumed to be a generalised Gaussian random distribution of the microlocally isotropic type, whereas the potential function is assumed to be deterministic. The well-posedness of the forward scattering problem is first established in a proper sense. It is then proved that the rough strength of the random source can be uniquely recovered, independent of the unknown potential, by a single realisation of the passive scattering measurement. In addition to the use of a single sample of the passive measurement for two unknowns, another significant feature of our result is that there is no geometric restriction on the supports of the source and the potential: they can be separated, or overlapped, or one containing the other.

math.AP↗
arXiv2023

Stable determination of an impedance obstacle by a single far-field measurement

We establish sharp stability estimates of logarithmic type in determining an impedance obstacle in $\mathbb{R}^2$. The obstacle is of general polygonal shape and the impedance parameter can be variable. We establish the stability results by using a single far-field pattern, which constitutes a longstanding problem in the inverse scattering theory. This is the first stability result in the literature in determining an impedance obstacle by a single far-field measurement. If the obstacle is of a generally polygonal shape, the stability in determining the obstacle is established in terms of a modified Hausdorff distance and is independent of the boundary impedance parameter. If the obstacle is further known to be convex, the stability in simultaneously determining the obstacle and the boundary impedance is established in terms of the classical Hausdorff distance. There are several technical novelties and development in the mathematical strategy developed for establishing the aforementioned stability results. First, the stability analysis is conducted around a corner point in a micro-local manner. Second, our stability estimates establish explicit relationships among the geometric configurations of the obstacle and the vanishing order of the wave field at the corner point. Third, we develop novel error propagation techniques to tackle singularities of the wave field at a corner as well as to tackle the impedance boundary condition.

math.AP↗
arXiv2023

Spectral properties of an acoustic-elastic transmission eigenvalue problem with applications

We are concerned with a coupled-physics spectral problem arising in the coupled propagation of acoustic and elastic waves, which is referred to as the acoustic-elastic transmission eigenvalue problem. There are two major contributions in this work which are new to the literature. First, under a mild condition on the medium parameters, we prove the existence of an acoustic-elastic transmission eigenvalue. Second, we establish a geometric rigidity result of the transmission eigenfunctions by showing that they tend to localize on the boundary of the underlying domain. Moreover, we also consider interesting implications of the obtained results to the effective construction of metamaterials by using bubbly elastic structures and to the inverse problem associated with the fluid-structure interaction.

math.AP↗
arXiv2023

Unique continuation for a mean field game system

For a mean field game system, we prove the unique continuation which asserts that if Cauchy data are zero on arbitrarily chosen lateral subboundary, then the solution identically vanishes.

math.AP↗
arXiv2023

Surface concentration of transmission eigenfunctions

The transmission eigenvalue problem is a type of non-elliptic and non-selfadjoint spectral problem that arises in the wave scattering theory when invisibility/transparency occurs. The transmission eigenfunctions are the interior resonant modes inside the scattering medium. We are concerned with the geometric rigidity of the transmission eigenfunctions and show that they concentrate on the boundary surface of the underlying domain in two senses. This substantiates the recent numerical discovery in [10] on such an intriguing spectral phenomenon of the transmission resonance. Our argument is based on generalized Weyl's law and certain novel ergodic properties of the coupled boundary layer-potential operators which are employed to analyze the generalized transmission eigenfunctions.

math.AP↗
arXiv2023

Inverse problems for mean field games

The theory of mean field games studies the limiting behaviors of large systems where the agents interact with each other in a certain symmetric way. The running and terminal costs are critical for the agents to decide the strategies. However, in practice they are often partially known or totally unknown for the agents, while the total cost is known at the end of the game. To address this challenging issue, we propose and study several inverse problems for mean field games. When the Lagrangian is a kinetic energy, we first establish unique identifiability results, showing that one can recover either the running cost or the terminal cost from knowledge of the total cost. If the running cost is limited to the time-independent class, we can further prove that one can simultaneously recover both the running and the terminal costs. Finally, we extend the results to the setup with general Lagrangians.

math.OC↗
arXiv2023

Determining a parabolic system by boundary observation of its non-negative solutions with biological applications

In this paper, we consider the inverse problem of determining some coefficients within a coupled nonlinear parabolic system, through boundary observation of its non-negative solutions. In the physical setup, the non-negative solutions represent certain probability densities in different contexts. We innovate the successive linearisation method by further developing a high-order variation scheme which can both ensure the positivity of the solutions and effectively tackle the nonlinear inverse problem. This enables us to establish several novel unique identifiability results for the inverse problem in a rather general setup. For a theoretical perspective, our study addresses an important topic in PDE analysis on how to characterise the function spaces generated by the products of non-positive solutions of parabolic PDEs. As a typical and practically interesting application, we apply our general results to inverse problems for ecological population models, where the positive solutions signify the population densities.

math.AP↗
arXiv2023

Lipschitz stability for determination of states and inverse source problem for the mean field game equations

We consider solutions satisfying the zero Neumann boundary condition and a linearized mean field game equation in $Ω\times (0,T)$ whose principal coefficients depend on the time and spatial variables with general Hamiltonian, where $Ω$ is a bounded domain in $\Bbb R^d$ and $(0,T)$ is the time interval. We first prove the Lipschitz stability in $Ω\times (\varepsilon, T-\varepsilon)$ with given $\varepsilon>0$ for the determination of the solutions by Dirichlet data on arbitrarily chosen subboundary of $\partialΩ$. Next we prove the Lipschitz stability for an inverse problem of determining spatially varying factors of source terms and a coefficient by extra boundary data and spatial data at intermediate time.

math.AP↗
arXiv2023

Stability in determination of states for the mean field game equations

We consider solutions satisfying the Neumann zero boundary condition and a linearized mean field game system in $Ω\times (0,T)$, where $Ω$ is a bounded domain in $\mathbb{R}^d$ and $(0,T)$ is the time interval. We prove two kinds of stability results in determining the solutions. The first is Hölder stability in time interval $(ε, T)$ with arbitrarily fixed $ε>0$ by data of solutions in $Ω\times \{T\}$. The second is the Lipschitz stability in $Ω\times (ε, T-ε)$ by data of solutions in arbitrarily given subdomain of $Ω$ over $(0,T)$.

math.AP↗
arXiv2023

Hölder Stability and Uniqueness for The Mean Field Games System via Carleman Estimates

We are concerned with the mathematical study of the Mean Field Games system (MFGS). In the conventional setup, the MFGS is a system of two coupled nonlinear parabolic PDEs of the second order in a backward-forward manner, namely one terminal and one initial conditions are prescribed respectively for the value function and the population density. In this paper, we show that uniqueness of solutions to the MFGS can be guaranteed if, among all four possible terminal and initial conditions, eitheir only two terminal or only two initial conditions are given. In both cases Hölder stability estimates are proven. This means that he accuracies of the solutions are estimated in terms of the given data. Moreover, these estimates readily imply uniqueness of corresponding problems for the MFGS. The main mathematical apparatus to establish those results is two new Carleman estimates, which may find application in other contexts associated with coupled parabolic PDEs.

math.AP↗
arXiv2023

Delving StyleGAN Inversion for Image Editing: A Foundation Latent Space Viewpoint

GAN inversion and editing via StyleGAN maps an input image into the embedding spaces ($\mathcal{W}$, $\mathcal{W^+}$, and $\mathcal{F}$) to simultaneously maintain image fidelity and meaningful manipulation. From latent space $\mathcal{W}$ to extended latent space $\mathcal{W^+}$ to feature space $\mathcal{F}$ in StyleGAN, the editability of GAN inversion decreases while its reconstruction quality increases. Recent GAN inversion methods typically explore $\mathcal{W^+}$ and $\mathcal{F}$ rather than $\mathcal{W}$ to improve reconstruction fidelity while maintaining editability. As $\mathcal{W^+}$ and $\mathcal{F}$ are derived from $\mathcal{W}$ that is essentially the foundation latent space of StyleGAN, these GAN inversion methods focusing on $\mathcal{W^+}$ and $\mathcal{F}$ spaces could be improved by stepping back to $\mathcal{W}$. In this work, we propose to first obtain the precise latent code in foundation latent space $\mathcal{W}$. We introduce contrastive learning to align $\mathcal{W}$ and the image space for precise latent code discovery. %The obtaining process is by using contrastive learning to align $\mathcal{W}$ and the image space. Then, we leverage a cross-attention encoder to transform the obtained latent code in $\mathcal{W}$ into $\mathcal{W^+}$ and $\mathcal{F}$, accordingly. Our experiments show that our exploration of the foundation latent space $\mathcal{W}$ improves the representation ability of latent codes in $\mathcal{W^+}$ and features in $\mathcal{F}$, which yields state-of-the-art reconstruction fidelity and editability results on the standard benchmarks. Project page: https://kumapowerliu.github.io/CLCAE.

cs.CV↗